Elliptic curves over number fields.

EXAMPLES:

sage: k.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([i,2])
sage: E.j_invariant()
-23328/365*i + 864/365
sage: E.simon_two_descent()
(1, 1, [(2*i : -2*i + 2 : 1)])
sage: P = E([2*i,-2*i+2])
sage: P+P
(15/32*i + 3/4 : 139/256*i + 339/256 : 1)
class sage.schemes.elliptic_curves.ell_number_field.EllipticCurve_number_field(x, y=None)

Elliptic curve over a number field.

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: EllipticCurve([i, i - 1, i + 1, 24*i + 15, 14*i + 35])
Elliptic Curve defined by y^2 + i*x*y + (i+1)*y = x^3 + (i-1)*x^2 + (24*i+15)*x + (14*i+35) over Number Field in i with defining polynomial x^2 + 1
__init__(x, y=None)

Allow some ways to create an elliptic curve over a number field in addition to the generic ones.

INPUT:

  • x, y – see examples.

EXAMPLES:

A curve from the database of curves over \QQ, but over a larger field:

sage: K.<i>=NumberField(x^2+1) sage: EllipticCurve(K,‘389a1’) Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-2)*x over Number Field in i with defining polynomial x^2 + 1

Making the field of definition explicitly larger:

sage: EllipticCurve(K,[0,-1,1,0,0])
Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in i with defining polynomial x^2 + 1
_get_local_data(P, proof)

Internal function to create data for this elliptic curve at the prime P.

This function handles the caching of local data. It is called by local_data() which is the user interface and which parses the input parameters P and proof.

INPUT:

  • P – either None or a prime ideal of the base field of self.
  • proof – whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve(K,[0,1,0,-160,308])
sage: p = K.ideal(i+1)
sage: E._get_local_data(p, False)
Local data at Fractional ideal (i + 1):
Reduction type: good
Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-10)*x + (-10) over Number Field in i with defining polynomial x^2 + 1
Minimal discriminant valuation: 0
Conductor exponent: 0
Kodaira Symbol: I0
Tamagawa Number: 1

Verify that we cache based on the proof value:

sage: E._get_local_data(p, False) is E._get_local_data(p, True)
False
_tidy_model()

Transforms the elliptic curve to a model in which a_1, a_2, a_3 are reduced modulo 2, 3, 2 respectively.

Note

This only works on integral models, i.e. it requires that a_1, a_2 and a_3 lie in the ring of integers of the base field.

EXAMPLES:

sage: K.<a>=NumberField(x^2-38)
sage: E=EllipticCurve([a, -5*a + 19, -39*a + 237, 368258520200522046806318224*a - 2270097978636731786720858047, 8456608930180227786550494643437985949781*a - 52130038506835491453281450568107193773505])
sage: E.ainvs()
[a,
-5*a + 19,
-39*a + 237,
368258520200522046806318224*a - 2270097978636731786720858047,
8456608930180227786550494643437985949781*a - 52130038506835491453281450568107193773505]
sage: E._tidy_model().ainvs()
[a,
a + 1,
a + 1,
368258520200522046806318444*a - 2270097978636731786720859345,
8456608930173478039472018047583706316424*a - 52130038506793883217874390501829588391299]
sage: EllipticCurve([101,202,303,404,505])._tidy_model().ainvs()
[1, 1, 0, -2509254, 1528863051]
sage: EllipticCurve([-101,-202,-303,-404,-505])._tidy_model().ainvs()
[1, -1, 0, -1823195, 947995262]
_torsion_bound(number_of_places=20)

An upper bound on the order of the torsion subgroup.

INPUT:

  • number_of_places (positive integer, default = 20) – the number of places that will be used to find the bound.

OUTPUT:

(integer) An upper bound on the torsion order.

ALGORITHM:

An upper bound on the order of the torsion.group of the elliptic curve is obtained by counting points modulo several primes of good reduction. Note that the upper bound returned by this function is a multiple of the order of the torsion group, and in general will be greater than the order.

EXAMPLES:

sage: CDB=CremonaDatabase() 
sage: [E._torsion_bound() for E in CDB.iter([14])]
[6, 6, 6, 6, 6, 6]
sage: [E.torsion_order() for E in CDB.iter([14])] 
[6, 6, 2, 6, 2, 6]    
conductor()

Returns the conductor of this elliptic curve as a fractional ideal of the base field.

OUTPUT:

(fractional ideal) The conductor of the curve.

EXAMPLES:

sage: K.<i>=NumberField(x^2+1)
sage: EllipticCurve([i, i - 1, i + 1, 24*i + 15, 14*i + 35]).conductor()
Fractional ideal (21*i - 3)
sage: K.<a>=NumberField(x^2-x+3)
sage: EllipticCurve([1 + a , -1 + a , 1 + a , -11 + a , 5 -9*a  ]).conductor()
Fractional ideal (-6*a)

A not so well known curve with everywhere good reduction:

sage: K.<a>=NumberField(x^2-38)
sage: E=EllipticCurve([0,0,0, 21796814856932765568243810*a - 134364590724198567128296995, 121774567239345229314269094644186997594*a - 750668847495706904791115375024037711300])
sage: E.conductor()
Fractional ideal (1)

An example which used to fail (see trac #5307):

sage: K.<w>=NumberField(x^2+x+6)
sage: E=EllipticCurve([w,-1,0,-w-6,0])
sage: E.conductor()
Fractional ideal (86304, w + 5898)
global_integral_model()

Return a model of self which is integral at all primes.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
sage: P1,P2 = K.primes_above(5)
sage: E.global_integral_model()
Elliptic Curve defined by y^2 + (-i)*x*y + (-25*i)*y = x^3 + 5*i*x^2 + 125*i*x + 3125*i over Number Field in i with defining polynomial x^2 + 1
global_minimal_model(proof=None)

Returns a model of self that is integral, minimal at all primes.

Note

This is only implemented for class number 1. In general, such a model may or may not exist.

INPUT:

  • proof – whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.

OUTPUT:

A global integral and minimal model.

EXAMPLES:

sage: K.<a> = NumberField(x^2-38)
sage: E = EllipticCurve([0,0,0, 21796814856932765568243810*a - 134364590724198567128296995, 121774567239345229314269094644186997594*a - 750668847495706904791115375024037711300])
sage: E.global_minimal_model()
Elliptic Curve defined by y^2 + a*x*y + (a+1)*y = x^3 + (a+1)*x^2 + (368258520200522046806318444*a-2270097978636731786720859345)*x + (8456608930173478039472018047583706316424*a-52130038506793883217874390501829588391299) over Number Field in a with defining polynomial x^2 - 38
has_additive_reduction(P)

Return True if this elliptic curve has (bad) additive reduction at the prime P.

INPUT:

  • P – a prime ideal of the base field of self, or a field element generating such an ideal.

OUTPUT:

(bool) True if the curve has additive reduction at P, else False.

EXAMPLES:

sage: E=EllipticCurve('27a1')
sage: [(p,E.has_additive_reduction(p)) for p in prime_range(15)]
[(2, False), (3, True), (5, False), (7, False), (11, False), (13, False)]

sage: K.<a>=NumberField(x^3-2)
sage: P17a, P17b = [P for P,e in K.factor(17)]
sage: E = EllipticCurve([0,0,0,0,2*a+1])
sage: [(p,E.has_additive_reduction(p)) for p in [P17a,P17b]]           
[(Fractional ideal (4*a^2 - 2*a + 1), False),
(Fractional ideal (2*a + 1), True)]
has_bad_reduction(P)

Return True if this elliptic curve has bad reduction at the prime P.

INPUT:

  • P – a prime ideal of the base field of self, or a field element generating such an ideal.

OUTPUT:

(bool) True if the curve has bad reduction at P, else False.

Note

This requires determining a local integral minimal model; we do not just check that the discriminant of the current model has valuation zero.

EXAMPLES:

sage: E=EllipticCurve('14a1')
sage: [(p,E.has_bad_reduction(p)) for p in prime_range(15)]
[(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]

sage: K.<a>=NumberField(x^3-2)
sage: P17a, P17b = [P for P,e in K.factor(17)]
sage: E = EllipticCurve([0,0,0,0,2*a+1])
sage: [(p,E.has_bad_reduction(p)) for p in [P17a,P17b]]           
[(Fractional ideal (4*a^2 - 2*a + 1), False),
(Fractional ideal (2*a + 1), True)]
has_good_reduction(P)

Return True if this elliptic curve has good reduction at the prime P.

INPUT:

  • P – a prime ideal of the base field of self, or a field element generating such an ideal.

OUTPUT:

(bool) – True if the curve has good reduction at P, else False.

Note

This requires determining a local integral minimal model; we do not just check that the discriminant of the current model has valuation zero.

EXAMPLES:

sage: E=EllipticCurve('14a1')
sage: [(p,E.has_good_reduction(p)) for p in prime_range(15)]
[(2, False), (3, True), (5, True), (7, False), (11, True), (13, True)]

sage: K.<a>=NumberField(x^3-2)
sage: P17a, P17b = [P for P,e in K.factor(17)]
sage: E = EllipticCurve([0,0,0,0,2*a+1])
sage: [(p,E.has_good_reduction(p)) for p in [P17a,P17b]]           
[(Fractional ideal (4*a^2 - 2*a + 1), True),
(Fractional ideal (2*a + 1), False)]
has_multiplicative_reduction(P)

Return True if this elliptic curve has (bad) multiplicative reduction at the prime P.

Note

See also has_split_multiplicative_reduction() and has_nonsplit_multiplicative_reduction().

INPUT:

  • P – a prime ideal of the base field of self, or a field

    element generating such an ideal.

OUTPUT:

(bool) True if the curve has multiplicative reduction at P, else False.

EXAMPLES:

sage: E=EllipticCurve('14a1')
sage: [(p,E.has_multiplicative_reduction(p)) for p in prime_range(15)]
[(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]

sage: K.<a>=NumberField(x^3-2)
sage: P17a, P17b = [P for P,e in K.factor(17)]
sage: E = EllipticCurve([0,0,0,0,2*a+1])
sage: [(p,E.has_multiplicative_reduction(p)) for p in [P17a,P17b]]           
[(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)]
has_nonsplit_multiplicative_reduction(P)

Return True if this elliptic curve has (bad) non-split multiplicative reduction at the prime P.

INPUT:

  • P – a prime ideal of the base field of self, or a field element generating such an ideal.

OUTPUT:

(bool) True if the curve has non-split multiplicative reduction at P, else False.

EXAMPLES:

sage: E=EllipticCurve('14a1')
sage: [(p,E.has_nonsplit_multiplicative_reduction(p)) for p in prime_range(15)]
[(2, True), (3, False), (5, False), (7, False), (11, False), (13, False)]

sage: K.<a>=NumberField(x^3-2)
sage: P17a, P17b = [P for P,e in K.factor(17)]
sage: E = EllipticCurve([0,0,0,0,2*a+1])
sage: [(p,E.has_nonsplit_multiplicative_reduction(p)) for p in [P17a,P17b]]           
[(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)]
has_split_multiplicative_reduction(P)

Return True if this elliptic curve has (bad) split multiplicative reduction at the prime P.

INPUT:

  • P – a prime ideal of the base field of self, or a field element generating such an ideal.

OUTPUT:

(bool) True if the curve has split multiplicative reduction at P, else False.

EXAMPLES:

sage: E=EllipticCurve('14a1')
sage: [(p,E.has_split_multiplicative_reduction(p)) for p in prime_range(15)]
[(2, False), (3, False), (5, False), (7, True), (11, False), (13, False)]

sage: K.<a>=NumberField(x^3-2)
sage: P17a, P17b = [P for P,e in K.factor(17)]
sage: E = EllipticCurve([0,0,0,0,2*a+1])
sage: [(p,E.has_split_multiplicative_reduction(p)) for p in [P17a,P17b]]           
[(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)]
integral_model()

Return a model of self which is integral at all primes.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
sage: P1,P2 = K.primes_above(5)
sage: E.global_integral_model()
Elliptic Curve defined by y^2 + (-i)*x*y + (-25*i)*y = x^3 + 5*i*x^2 + 125*i*x + 3125*i over Number Field in i with defining polynomial x^2 + 1
is_global_integral_model()

Return true iff self is integral at all primes.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
sage: P1,P2 = K.primes_above(5)
sage: Emin = E.global_integral_model()
sage: Emin.is_global_integral_model()
True
is_local_integral_model(*P)

Tests if self is integral at the prime ideal P, or at all the primes if P is a list or tuple.

INPUT:

  • *P – a prime ideal, or a list or tuple of primes.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: P1,P2 = K.primes_above(5)
sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
sage: E.is_local_integral_model(P1,P2)
False
sage: Emin = E.local_integral_model(P1,P2)
sage: Emin.is_local_integral_model(P1,P2)
True
kodaira_symbol(P, proof=None)

Returns the Kodaira Symbol of this elliptic curve at the prime P.

INPUT:

  • P – either None or a prime ideal of the base field of self.
  • proof – whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.

OUTPUT:

(string) The Kodaira Symbol of the curve at P.

EXAMPLES:

sage: K.<a>=NumberField(x^2-5)
sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
sage: bad_primes = E.discriminant().support(); bad_primes
[Fractional ideal (-a),
Fractional ideal (7/2*a - 81/2),
Fractional ideal (a + 52),
Fractional ideal (2)]
sage: [E.kodaira_symbol(P) for P in bad_primes]
[I0, I1, I1, II]
sage: K.<a> = QuadraticField(-11)
sage: E = EllipticCurve('11a1').change_ring(K)
sage: [E.kodaira_symbol(P) for P in K(11).support()]
[I10]
local_data(P=None, proof=None)

Local data for this elliptic curve at the prime P.

INPUT:

  • P – either None or a prime ideal of the base field of self.
  • proof – whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.

OUTPUT:

If P is specified, returns the EllipticCurveLocalData object associated to the prime P for this curve. Otherwise, returns a list of such objects, one for each prime P in the support of the discriminant of this model.

Note

The model is not required to be integral on input.

For principal P, a generator is used as a uniformizer, and integrality or minimality at other primes is not affected. For non-principal P, the minimal model returned will preserve integrality at other primes, but not minimality.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([1 + i, 0, 1, 0, 0])
sage: E.local_data()
[Local data at Fractional ideal (2*i + 1):
Reduction type: bad non-split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1
Minimal discriminant valuation: 1
Conductor exponent: 1
Kodaira Symbol: I1
Tamagawa Number: 1, 
Local data at Fractional ideal (-3*i - 2):
Reduction type: bad split multiplicative
Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1
Minimal discriminant valuation: 2
Conductor exponent: 1
Kodaira Symbol: I2
Tamagawa Number: 2]
sage: E.local_data(K.ideal(3))
Local data at Fractional ideal (3):
Reduction type: good
Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1
Minimal discriminant valuation: 0
Conductor exponent: 0
Kodaira Symbol: I0
Tamagawa Number: 1

An example raised in #3897:

sage: E = EllipticCurve([1,1])
sage: E.local_data(3)
Local data at Principal ideal (3) of Integer Ring:
Reduction type: good
Local minimal model: Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
Minimal discriminant valuation: 0
Conductor exponent: 0
Kodaira Symbol: I0
Tamagawa Number: 1
local_information(P=None, proof=None)
code{local_information} has been renamed code{local_data} and is being deprecated.
local_integral_model(*P)

Return a model of self which is integral at the prime ideal P.

Note

The integrality at other primes is not affected, even if P is non-principal.

INPUT:

  • *P – a prime ideal, or a list or tuple of primes.

EXAMPLES:

sage: K.<i> = NumberField(x^2+1)
sage: P1,P2 = K.primes_above(5)
sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
sage: E.local_integral_model((P1,P2))
Elliptic Curve defined by y^2 + (-i)*x*y + (-25*i)*y = x^3 + 5*i*x^2 + 125*i*x + 3125*i over Number Field in i with defining polynomial x^2 + 1
local_minimal_model(P, proof=None)

Returns a model which is integral at all primes and minimal at P.

INPUT:

  • P – either None or a prime ideal of the base field of self.
  • proof – whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.

OUTPUT:

A model of the curve which is minimal (and integral) at P.

Note

The model is not required to be integral on input.

For principal P, a generator is used as a uniformizer, and integrality or minimality at other primes is not affected. For non-principal P, the minimal model returned will preserve integrality at other primes, but not minimality.

EXAMPLES:

sage: K.<a>=NumberField(x^2-5)
sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
sage: P=K.ideal(a)
sage: E.local_minimal_model(P).ainvs()
[0, 1, 0, a - 33, -2*a + 64]
period_lattice(embedding)

Returns the period lattice of the elliptic curve for the given embedding of its base field.

INPUT:

  • embedding - an embedding of the base number field into \RR or \CC.

Note

The precision of the embedding is ignored: we only use the given embedding to determine which embedding into QQbar to use. Once the lattice has been initialized, periods can be computed to arbitrary precision.

EXAMPLES:

First define a field with two real embeddings:

sage: K.<a> = NumberField(x^3-2)
sage: E=EllipticCurve([0,0,0,a,2])                            
sage: embs=K.embeddings(CC); len(embs)
3

For each embedding we have a different period lattice:

sage: E.period_lattice(embs[0])
Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Field
Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I

sage: E.period_lattice(embs[1])
Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Field
Defn: a |--> -0.6299605249474365? + 1.091123635971722?*I

sage: E.period_lattice(embs[2])
Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:   Algebraic Field
Defn: a |--> 1.259921049894873?

Although the original embeddings have only the default precision, we can obtain the basis with higher precision later:

sage: L=E.period_lattice(embs[0])
sage: L.basis()
(1.86405007647981 - 0.903761485143226*I, -0.149344633143919 - 2.06619546272945*I)

sage: L.basis(prec=100)
(1.8640500764798108425920506200 - 0.90376148514322594749786960975*I, -0.14934463314391922099120107422 - 2.0661954627294548995621225062*I)
reduction(place)

Return the reduction of the elliptic curve at a place of good reduction.

INPUT:

  • place – a prime ideal in the base field of the curve

OUTPUT:

An elliptic curve over a finite field, the residue field of the place.

EXAMPLES:

sage: K.<i> = QuadraticField(-1)
sage: EK = EllipticCurve([0,0,0,i,i+3])
sage: v = K.fractional_ideal(2*i+3)
sage: EK.reduction(v)
Elliptic Curve defined by y^2  = x^3 + 5*x + 8 over Residue field of Fractional ideal (2*i + 3)
sage: EK.reduction(K.ideal(1+i))     
...  
AttributeError: The curve must have good reduction at the place.
sage: EK.reduction(K.ideal(2))  
...  
AttributeError: The ideal must be prime.
simon_two_descent(verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30)

Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points.

INPUT:

  • verbose – 0, 1, 2, or 3 (default: 0), the verbosity level
  • lim1 – (default: 5) limit on trivial points on quartics
  • lim3 – (default: 50) limit on points on ELS quartics
  • limtriv – (default: 10) limit on trivial points on elliptic curve
  • maxprob – (default: 20)
  • limbigprime – (default: 30) to distinguish between small and large prime numbers. Use probabilistic tests for large primes. If 0, don’t use probabilistic tests.

OUTPUT:

(lower, upper, list) where lower is a lower bound on the rank, upper is an upper bound (the 2-Selmer rank) and list is a list of independent points on the Weierstrass model. The length of list is equal to either lower, or lower-1, since when lower is less than upper and of different parity, the value of lower is increased by 1.

Note

For non-quadratic number fields, this code does return, but it takes a long time.

IMPLEMENTATION:

Uses Denis Simon’s GP/PARI scripts from url{http://www.math.unicaen.fr/~simon/}.

EXAMPLES:

sage: K.<a> = NumberField(x^2 + 23, 'a')
sage: E = EllipticCurve(K, '37')
sage: E == loads(dumps(E))
True
sage: E.simon_two_descent()
(2, 2, [(-1 : 0 : 1), (1/2*a - 5/2 : -1/2*a - 13/2 : 1)])
sage: K.<a> = NumberField(x^2 + 7, 'a')
sage: E = EllipticCurve(K, [0,0,0,1,a]); E
Elliptic Curve defined by y^2  = x^3 + x + a over Number Field in a with defining polynomial x^2 + 7
sage: v = E.simon_two_descent(verbose=1); v
courbe elliptique : Y^2 = x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)
A = 0
B = Mod(1, y^2 + 7)
C = Mod(y, y^2 + 7)
LS2gen = [Mod(Mod(-5, y^2 + 7)*x^2 + Mod(-3*y, y^2 + 7)*x + Mod(8, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y - 1/2, y^2 + 7)*x - 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))]
#LS2gen = 2
 Recherche de points triviaux sur la courbe 
points triviaux sur la courbe = [[1, 1, 0], [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]]
zc = Mod(Mod(-5, y^2 + 7)*x^2 + Mod(-3*y, y^2 + 7)*x + Mod(8, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
 symbole de Hilbert (Mod(2, y^2 + 7),Mod(-5, y^2 + 7)) = -1
 zc = Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y - 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
 symbole de Hilbert (Mod(-2*y + 2, y^2 + 7),Mod(1, y^2 + 7)) = 0
 sol de Legendre = [1, 0, 1]~
 zc*z1^2 = Mod(Mod(2*y - 2, y^2 + 7)*x + Mod(2*y + 10, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
 quartique : (-1/2*y + 1/2)*Y^2 = x^4 + (-3*y - 15)*x^2 + (-8*y - 16)*x + (-11/2*y - 15/2)
 reduite: Y^2 = (-1/2*y + 1/2)*x^4 - 4*x^3 + (-3*y + 3)*x^2 + (2*y - 2)*x + (1/2*y + 3/2)
 non ELS en [2, [0, 1]~, 1, 1, [1, 1]~]
zc = Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
 vient du point trivial [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]
m1 = 1
m2 = 1
#S(E/K)[2]    = 2
#E(K)/2E(K)   = 2
#III(E/K)[2]  = 1
rang(E/K)     = 1
listpointsmwr = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]]
(1, 1, [(1/2*a + 3/2 : -a - 2 : 1)])

A curve with 2-torsion:

sage: K.<a> = NumberField(x^2 + 7, 'a')
sage: E = EllipticCurve(K, '15a')
sage: v = E.simon_two_descent(); v  # long time (about 10 seconds), points can vary
(1, 3, [...])
tamagawa_exponent(P, proof=None)

Returns the Tamagawa index of this elliptic curve at the prime P.

INPUT:

  • P – either None or a prime ideal of the base field of self.
  • proof – whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.

OUTPUT:

(positive integer) The Tamagawa index of the curve at P.

EXAMPLES:

sage: K.<a>=NumberField(x^2-5)
sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
sage: [E.tamagawa_exponent(P) for P in E.discriminant().support()]
[1, 1, 1, 1]
sage: K.<a> = QuadraticField(-11)
sage: E = EllipticCurve('11a1').change_ring(K)
sage: [E.tamagawa_exponent(P) for P in K(11).support()]
[10]
tamagawa_number(P, proof=None)

Returns the Tamagawa number of this elliptic curve at the prime P.

INPUT:

  • P – either None or a prime ideal of the base field of self.
  • proof – whether to only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.

OUTPUT:

(positive integer) The Tamagawa number of the curve at P.

EXAMPLES:

sage: K.<a>=NumberField(x^2-5)
sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
sage: [E.tamagawa_number(P) for P in E.discriminant().support()]
[1, 1, 1, 1]
sage: K.<a> = QuadraticField(-11)
sage: E = EllipticCurve('11a1').change_ring(K)
sage: [E.tamagawa_number(P) for P in K(11).support()]
[10]
torsion_order()

Returns the order of the torsion subgroup of this elliptic curve.

OUTPUT:

(integer) the order of the torsion subgroup of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: K.<t> = NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101)
sage: EK = E.base_extend(K)
sage: EK.torsion_order()
25
sage: E = EllipticCurve('15a1')
sage: K.<t> = NumberField(x^2 + 2*x + 10)
sage: EK = E.base_extend(K)
sage: EK.torsion_order()
16
sage: E = EllipticCurve('19a1')
sage: K.<t> = NumberField(x^9-3*x^8-4*x^7+16*x^6-3*x^5-21*x^4+5*x^3+7*x^2-7*x+1)
sage: EK = E.base_extend(K)
sage: EK.torsion_order()
9
sage: K.<i> = QuadraticField(-1)
sage: EK = EllipticCurve([0,0,0,i,i+3])
sage: EK.torsion_order()
1
torsion_points()

Returns a list of the torsion points of this elliptic curve.

OUTPUT:

(list) A sorted list of the torsion points.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: E.torsion_points()
[(0 : 1 : 0), (5 : -6 : 1), (5 : 5 : 1), (16 : -61 : 1), (16 : 60 : 1)]
sage: K.<t> = NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101)
sage: EK = E.base_extend(K)
sage: EK.torsion_points()
[(t : 1/11*t^3 + 6/11*t^2 + 19/11*t + 48/11 : 1),
(1/11*t^3 - 5/11*t^2 + 19/11*t - 40/11 : -6/11*t^3 - 3/11*t^2 - 26/11*t - 321/11 : 1),
(1/11*t^3 - 5/11*t^2 + 19/11*t - 40/11 : 6/11*t^3 + 3/11*t^2 + 26/11*t + 310/11 : 1),
(t : -1/11*t^3 - 6/11*t^2 - 19/11*t - 59/11 : 1),
(16 : 60 : 1),
(-3/55*t^3 - 7/55*t^2 - 2/55*t - 133/55 : 6/55*t^3 + 3/55*t^2 + 25/11*t + 156/55 : 1),
(14/121*t^3 - 15/121*t^2 + 90/121*t + 232/121 : 16/121*t^3 - 69/121*t^2 + 293/121*t - 46/121 : 1),
(-26/121*t^3 + 20/121*t^2 - 219/121*t - 995/121 : -15/121*t^3 - 156/121*t^2 + 232/121*t - 2887/121 : 1),
(10/121*t^3 + 49/121*t^2 + 168/121*t + 73/121 : -32/121*t^3 - 60/121*t^2 + 261/121*t + 686/121 : 1),
(5 : 5 : 1),
(-9/121*t^3 - 21/121*t^2 - 127/121*t - 377/121 : -7/121*t^3 + 24/121*t^2 + 197/121*t + 16/121 : 1),
(3/55*t^3 + 7/55*t^2 + 2/55*t + 78/55 : 7/55*t^3 - 24/55*t^2 + 9/11*t + 17/55 : 1),
(-5/121*t^3 + 36/121*t^2 - 84/121*t + 24/121 : -34/121*t^3 + 27/121*t^2 - 305/121*t - 829/121 : 1),
(5/121*t^3 - 14/121*t^2 - 158/121*t - 453/121 : 49/121*t^3 + 129/121*t^2 + 315/121*t + 86/121 : 1),
(5 : -6 : 1),
(5/121*t^3 - 14/121*t^2 - 158/121*t - 453/121 : -49/121*t^3 - 129/121*t^2 - 315/121*t - 207/121 : 1),
(-5/121*t^3 + 36/121*t^2 - 84/121*t + 24/121 : 34/121*t^3 - 27/121*t^2 + 305/121*t + 708/121 : 1),
(3/55*t^3 + 7/55*t^2 + 2/55*t + 78/55 : -7/55*t^3 + 24/55*t^2 - 9/11*t - 72/55 : 1),
(-9/121*t^3 - 21/121*t^2 - 127/121*t - 377/121 : 7/121*t^3 - 24/121*t^2 - 197/121*t - 137/121 : 1),
(16 : -61 : 1),
(10/121*t^3 + 49/121*t^2 + 168/121*t + 73/121 : 32/121*t^3 + 60/121*t^2 - 261/121*t - 807/121 : 1),
(-26/121*t^3 + 20/121*t^2 - 219/121*t - 995/121 : 15/121*t^3 + 156/121*t^2 - 232/121*t + 2766/121 : 1),
(14/121*t^3 - 15/121*t^2 + 90/121*t + 232/121 : -16/121*t^3 + 69/121*t^2 - 293/121*t - 75/121 : 1),
(-3/55*t^3 - 7/55*t^2 - 2/55*t - 133/55 : -6/55*t^3 - 3/55*t^2 - 25/11*t - 211/55 : 1),
(0 : 1 : 0)]
sage: E = EllipticCurve('15a1')
sage: K.<t> = NumberField(x^2 + 2*x + 10)
sage: EK = E.base_extend(K)
sage: EK.torsion_points()
[(t : t - 5 : 1),
(-1 : 0 : 1),
(t : -2*t + 4 : 1),
(8 : 18 : 1),
(1/2 : 5/4*t + 1/2 : 1),
(-2 : 3 : 1),
(-7 : 5*t + 8 : 1),
(3 : -2 : 1),
(-t - 2 : 2*t + 8 : 1),
(-13/4 : 9/8 : 1),
(-t - 2 : -t - 7 : 1),
(8 : -27 : 1),
(-7 : -5*t - 2 : 1),
(-2 : -2 : 1),
(1/2 : -5/4*t - 2 : 1),
(0 : 1 : 0)]
sage: K.<i> = QuadraticField(-1)
sage: EK = EllipticCurve(K,[0,0,0,0,-1])            
sage: EK.torsion_points ()              
[(-2 : -3*i : 1),
(0 : -i : 1),
(1 : 0 : 1),
(0 : i : 1),
(-2 : 3*i : 1),
(0 : 1 : 0)]
torsion_subgroup()

Returns the torsion subgroup of this elliptic curve.

OUTPUT:

(EllipticCurveTorsionSubgroup) The EllipticCurveTorsionSubgroup associated to this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: K.<t>=NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101)
sage: EK=E.base_extend(K)
sage: tor = EK.torsion_subgroup()
sage: tor
Torsion Subgroup isomorphic to Multiplicative Abelian Group isomorphic to C5 x C5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in t with defining polynomial x^4 + x^3 + 11*x^2 + 41*x + 101
sage: tor.gens()
((16 : 60 : 1), (t : 1/11*t^3 + 6/11*t^2 + 19/11*t + 48/11 : 1))
sage: E = EllipticCurve('15a1')
sage: K.<t>=NumberField(x^2 + 2*x + 10)
sage: EK=E.base_extend(K)
sage: EK.torsion_subgroup()
Torsion Subgroup isomorphic to Multiplicative Abelian Group isomorphic to C4 x C4 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-10)*x + (-10) over Number Field in t with defining polynomial x^2 + 2*x + 10
sage: E = EllipticCurve('19a1')
sage: K.<t>=NumberField(x^9-3*x^8-4*x^7+16*x^6-3*x^5-21*x^4+5*x^3+7*x^2-7*x+1)
sage: EK=E.base_extend(K)
sage: EK.torsion_subgroup()
Torsion Subgroup isomorphic to Multiplicative Abelian Group isomorphic to C9 associated to the Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-9)*x + (-15) over Number Field in t with defining polynomial x^9 - 3*x^8 - 4*x^7 + 16*x^6 - 3*x^5 - 21*x^4 + 5*x^3 + 7*x^2 - 7*x + 1
sage: K.<i> = QuadraticField(-1)
sage: EK = EllipticCurve([0,0,0,i,i+3])
sage: EK.torsion_subgroup ()
Torsion Subgroup isomorphic to Trivial Abelian Group associated to the Elliptic Curve defined by y^2  = x^3 + i*x + (i+3) over Number Field in i with defining polynomial x^2 + 1

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