Properties

Label 2.0.7.1-30492.5-c2
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 30492 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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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(Polrev([2, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-5a-4\right){x}-26a+32\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-4,-5]),K([32,-26])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-4,-5]),Polrev([32,-26])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-4,-5],K![32,-26]]);
 

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: \((457380a+60984)\) = \((a)^{4}\cdot(-a+1)^{2}\cdot(-2a+1)^{2}\cdot(3)^{2}\cdot(-2a+3)^{2}\cdot(2a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 450004838976 \) = \(2^{4}\cdot2^{2}\cdot7^{2}\cdot9^{2}\cdot11^{2}\cdot11^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{430100035}{4919376} a - \frac{302575447}{14758128} \)
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 $\left(3 a - 1 : -7 a + 11 : 1\right)$
Height \(0.14086867692112875896876823827174028120\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{1}{4} a - \frac{15}{4} : -\frac{1}{8} a + \frac{11}{8} : 1\right)$ $\left(-2 a + 1 : a - 1 : 1\right)$
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: \( 0.14086867692112875896876823827174028120 \)
Period: \( 1.4920236626445363664885568402263771191 \)
Tamagawa product: \( 128 \)  =  \(2\cdot2\cdot2\cdot2\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 2.5420910685025005657204317900938282739 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-a+1)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-2a+1)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((3)\) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-2a+3)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2a+1)\) \(11\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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-c consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is not a \(\Q\)-curve.

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