Properties

Label 2.0.3.1-102675.1-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 102675 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
 
gp: K = nfinit(Polrev([1, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(770a+3631\right){x}+210053a-149676\)
sage: E = EllipticCurve([K([1,1]),K([1,-1]),K([0,0]),K([3631,770]),K([-149676,210053])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([0,0]),Polrev([3631,770]),Polrev([-149676,210053])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,-1],K![0,0],K![3631,770],K![-149676,210053]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((365a-235)\) = \((-2a+1)\cdot(5)\cdot(-7a+4)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 102675 \) = \(3\cdot25\cdot37^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5966275530600a+6616926718515)\) = \((-2a+1)^{32}\cdot(5)\cdot(-7a+4)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 118858570873683396049224225 \) = \(3^{32}\cdot25\cdot37^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{147281603041}{215233605} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-\frac{10903242414425132819223}{227291244218697273316} a - \frac{38714620905543369416}{56822811054674318329} : \frac{53890811987965329919389259622836}{428335502252173016267165118733} a - \frac{302446325895585888318816549361351}{3426684018017384130137320949864} : 1\right)$
Height \(24.145269411621517677362867527445990175\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{369}{4} a - 53 : -\frac{263}{4} a + \frac{581}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 24.145269411621517677362867527445990175 \)
Period: \( 0.091886774367793679127325410583210268342 \)
Tamagawa product: \( 4 \)  =  \(2\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 5.1237086412940690662013739803697711034 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{32}\) Non-split multiplicative \(1\) \(1\) \(32\) \(32\)
\((5)\) \(25\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-7a+4)\) \(37\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 102675.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.