Properties

Label 2.0.7.1-5324.6-a1
Base field \(\Q(\sqrt{-7}) \)
Conductor \((-44a+66)\)
Conductor norm \( 5324 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
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

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-45a+7\right){x}-280a+120\)
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([1,1]),K([7,-45]),K([120,-280])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([1,1])),Pol(Vecrev([7,-45])),Pol(Vecrev([120,-280]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,1],K![1,1],K![7,-45],K![120,-280]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-44a+66)\) = \((a)\cdot(-a+1)\cdot(-2a+3)^{2}\cdot(2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5324 \) = \(2\cdot2\cdot11^{2}\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-18069656a+51057160)\) = \((a)^{3}\cdot(-a+1)^{10}\cdot(-2a+3)^{8}\cdot(2a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2337273205645312 \) = \(2^{3}\cdot2^{10}\cdot11^{8}\cdot11^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{531165637}{1362944} a + \frac{64260511}{681472} \)
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 $\left(-2 a - 6 : 14 a - 10 : 1\right)$
Height \(0.804718677304910\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a : 11 a - 17 : 1\right)$
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: \( 0.804718677304910 \)
Period: \( 0.722336016257565 \)
Tamagawa product: \( 18 \)  =  \(1\cdot2\cdot3\cdot3\)
Torsion order: \(3\)
Leading coefficient: \( 1.75761729726168 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((-a+1)\) \(2\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((-2a+3)\) \(11\) \(3\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((2a+1)\) \(11\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 5324.6-a consists of curves linked by isogenies of degree 3.

Base change

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