Properties

Label 2.0.3.1-100156.6-a2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((364a-210)\)
Conductor norm \( 100156 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-273a-39\right){x}+413a+488\)
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([1,0]),K([-39,-273]),K([488,413])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([-39,-273])),Pol(Vecrev([488,413]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,-1],K![1,0],K![-39,-273],K![488,413]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((364a-210)\) = \((2)\cdot(-3a+1)\cdot(3a-2)^{2}\cdot(9a-8)\)
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: \((335180160a-1918715456)\) = \((2)^{6}\cdot(-3a+1)\cdot(3a-2)^{10}\cdot(9a-8)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3150699387214360576 \) = \(4^{6}\cdot7\cdot7^{10}\cdot73^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{28238503095}{87139808} a + \frac{299121167277}{174279616} \)
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: \(0\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.394815323647074 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 0.911786933551310 \)
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\((j)_{-}\)
\((2)\) \(4\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-3a+1)\) \(7\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((3a-2)\) \(7\) \(1\) \(II^{*}\) Additive \(-1\) \(2\) \(10\) \(0\)
\((9a-8)\) \(73\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 100156.6-a consists of curves linked by isogenies of degree 3.

Base change

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