Properties

Label 2.0.7.1-30492.5-f2
Base field \(\Q(\sqrt{-7}) \)
Conductor \((-132a+66)\)
Conductor norm \( 30492 \)
CM no
Base change yes: 462.a3,3234.n3
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-7}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}+{x}^{2}-16{x}-20\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([0,0]),K([-16,0]),K([-20,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-16,0])),Pol(Vecrev([-20,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![0,0],K![-16,0],K![-20,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-132a+66)\) = \((a)\cdot(-a+1)\cdot(-2a+1)\cdot(3)\cdot(-2a+3)\cdot(2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 30492 \) = \(2\cdot2\cdot7\cdot9\cdot11\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((213444)\) = \((a)^{2}\cdot(-a+1)^{2}\cdot(-2a+1)^{4}\cdot(3)^{2}\cdot(-2a+3)^{2}\cdot(2a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 45558341136 \) = \(2^{2}\cdot2^{2}\cdot7^{4}\cdot9^{2}\cdot11^{2}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{498677257}{213444} \)
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 $\left(-3 a - 5 : -6 a + 22 : 1\right)$
Height \(0.125851028964798\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-4 : 2 : 1\right)$ $\left(-\frac{5}{4} : \frac{5}{8} : 1\right)$
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: \( 0.125851028964798 \)
Period: \( 1.75387213061083 \)
Tamagawa product: \( 128 \)  =  \(2\cdot2\cdot2^{2}\cdot2\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 2.66965816642823 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-a+1)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-2a+1)\) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((3)\) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-2a+3)\) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((2a+1)\) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 30492.5-f consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of 462.a3, 3234.n3, defined over \(\Q\), so it is also a \(\Q\)-curve.