Properties

Label 2.0.7.1-16128.5-n7
Base field \(\Q(\sqrt{-7}) \)
Conductor \((-96 a + 48)\)
Conductor norm \( 16128 \)
CM no
Base change yes: 336.d4,2352.l4
Q-curve yes
Torsion order \( 8 \)
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(Pol(Vecrev([2, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\(y^2=x^{3}+x^{2}-1344x-19404\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-1344,0]),K([-19404,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-1344,0])),Pol(Vecrev([-19404,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-1344,0],K![-19404,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-96 a + 48)\) = \( \left(a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(3\right) \cdot \left(-2 a + 1\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16128 \) = \( 2^{8} \cdot 7 \cdot 9 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((260112384)\) = \( \left(a\right)^{16} \cdot \left(-a + 1\right)^{16} \cdot \left(3\right)^{4} \cdot \left(-2 a + 1\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 67658452310163456 \) = \( 2^{32} \cdot 7^{4} \cdot 9^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{65597103937}{63504} \)
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 $\left(-\frac{156}{7} : \frac{180}{49} a - \frac{90}{49} : 1\right)$
Height \(2.31081181470692\)
Torsion structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-6 a - 18 : -6 a - 60 : 1\right)$ $\left(-22 : 0 : 1\right)$
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: \( 2.31081181470692 \)
Period: \( 0.342545916533007 \)
Tamagawa product: \( 256 \)  =  \(2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2}\)
Torsion order: \(8\)
Leading coefficient: \(4.78689979783825\)
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(4\) \(I_{8}^*\) Additive \(-1\) \(4\) \(16\) \(4\)
\( \left(-a + 1\right) \) \(2\) \(4\) \(I_{8}^*\) Additive \(-1\) \(4\) \(16\) \(4\)
\( \left(-2 a + 1\right) \) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) 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 and 4.
Its isogeny class 16128.5-n consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of elliptic curves 336.d4, 2352.l4, defined over \(\Q\), so it is also a \(\Q\)-curve.