Properties

Label 2.0.7.1-5324.7-c2
Base field \(\Q(\sqrt{-7}) \)
Conductor \((44a+22)\)
Conductor norm \( 5324 \)
CM no
Base change no
Q-curve yes
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(Pol(Vecrev([2, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(48a-320\right){x}+3251a+758\)
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([1,0]),K([-320,48]),K([758,3251])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([-320,48])),Pol(Vecrev([758,3251]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,1],K![1,0],K![-320,48],K![758,3251]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((44a+22)\) = \((a)\cdot(-a+1)\cdot(-2a+3)\cdot(2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5324 \) = \(2\cdot2\cdot11\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-7949012720a+11148070736)\) = \((a)^{4}\cdot(-a+1)^{32}\cdot(-2a+3)^{2}\cdot(2a+1)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 162036931496379416576 \) = \(2^{4}\cdot2^{32}\cdot11^{2}\cdot11^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{31504873455125}{519691042816} a + \frac{50524809992625}{259845521408} \)
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 $\left(5 a - 2 : -19 a - 38 : 1\right)$
Height \(0.245619348511687\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(28 a - 7 : 150 a - 201 : 1\right)$
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: \( 0.245619348511687 \)
Period: \( 0.288454494899430 \)
Tamagawa product: \( 512 \)  =  \(2\cdot2^{5}\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.42768446016436 \)
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 \(1\) \(1\) \(4\) \(4\)
\((-a+1)\) \(2\) \(32\) \(I_{32}\) Split multiplicative \(-1\) \(1\) \(32\) \(32\)
\((-2a+3)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2a+1)\) \(11\) \(4\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 5324.7-c consists of curves linked by isogenies of degrees dividing 8.

Base change

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