Properties

Label 2.0.3.1-77841.3-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor \((315a-216)\)
Conductor norm \( 77841 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}+\left(-243a+249\right){x}-108a-1251\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,0]),K([249,-243]),K([-1251,-108])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([249,-243])),Pol(Vecrev([-1251,-108]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![1,0],K![249,-243],K![-1251,-108]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((315a-216)\) = \((-2a+1)^{4}\cdot(6a-5)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 77841 \) = \(3^{4}\cdot31^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-52044768a+245865213)\) = \((-2a+1)^{10}\cdot(6a-5)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 50362362869853609 \) = \(3^{10}\cdot31^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2138112}{961} a + \frac{5738496}{961} \)
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 $\left(-3 a + 18 : 45 a - 100 : 1\right)$
Height \(0.614294300077049\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.614294300077049 \)
Period: \( 0.517271645642963 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2^{2}\)
Torsion order: \(1\)
Leading coefficient: \( 2.93531364896580 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(1\) \(IV^{*}\) Additive \(-1\) \(4\) \(10\) \(0\)
\((6a-5)\) \(31\) \(4\) \(I_2^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 77841.3-a consists of this curve only.

Base change

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