Properties

Label 2.0.3.1-100156.5-c2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((350a-266)\)
Conductor norm \( 100156 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-929a+1539\right){x}-13161a+1058\)
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([1,0]),K([1539,-929]),K([1058,-13161])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([1539,-929])),Pol(Vecrev([1058,-13161]))], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![1,0],K![1539,-929],K![1058,-13161]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((350a-266)\) = \((2)\cdot(-3a+1)\cdot(3a-2)^{2}\cdot(-9a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 100156 \) = \(4\cdot7\cdot7^{2}\cdot73\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((108870129648a+46864751416)\) = \((2)^{3}\cdot(-3a+1)\cdot(3a-2)^{10}\cdot(-9a+1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 19151181617437014190528 \) = \(4^{3}\cdot7\cdot7^{10}\cdot73^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1454621964199278075}{1453413909279556} a + \frac{3568835637322210611}{2906827818559112} \)
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 $\left(50 a - 23 : -179 a + 299 : 1\right)$
Height \(1.17359093781688\)
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: $\left(\frac{137}{4} a - 3 : -\frac{131}{4} a + \frac{145}{8} : 1\right)$
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: \( 1.17359093781688 \)
Period: \( 0.189118281655471 \)
Tamagawa product: \( 24 \)  =  \(1\cdot1\cdot2^{2}\cdot( 2 \cdot 3 )\)
Torsion order: \(2\)
Leading coefficient: \( 3.07539479405303 \)
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\((j)_{-}\)
\((2)\) \(4\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((-3a+1)\) \(7\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((3a-2)\) \(7\) \(4\) \(I_{4}^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)
\((-9a+1)\) \(73\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

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
\(3\) 3B[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 100156.5-c consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.