Properties

Label 2.2.28.1-648.1-m4
Base field \(\Q(\sqrt{7}) \)
Conductor \((18a+54)\)
Conductor norm \( 648 \)
CM no
Base change yes: 7056.q3,72.a3
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 \( x^{2} - 7 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-7, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2631a-6945\right){x}-114640a+303321\)
sage: E = EllipticCurve([K([1,1]),K([1,1]),K([0,0]),K([-6945,2631]),K([303321,-114640])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-6945,2631])),Pol(Vecrev([303321,-114640]))], K);
 
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,0],K![-6945,2631],K![303321,-114640]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((18a+54)\) = \((a+3)^{3}\cdot(-a+2)^{2}\cdot(-a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 648 \) = \(2^{3}\cdot3^{2}\cdot3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((944784)\) = \((a+3)^{8}\cdot(-a+2)^{10}\cdot(-a-2)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 892616806656 \) = \(2^{8}\cdot3^{10}\cdot3^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1556068}{81} \)
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 $\left(-44 a + 114 : -620 a + 1645 : 1\right)$
Height \(0.322644888696544\)
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: $\left(25 a - 69 : 22 a - 53 : 1\right)$ $\left(-11 a + 27 : -8 a + 25 : 1\right)$
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: \( 0.322644888696544 \)
Period: \( 7.57801135666878 \)
Tamagawa product: \( 64 \)  =  \(2^{2}\cdot2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.69650257072685 \)
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\((j)_{-}\)
\((a+3)\) \(2\) \(4\) \(I_1^{*}\) Additive \(-1\) \(3\) \(8\) \(0\)
\((-a+2)\) \(3\) \(4\) \(I_4^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)
\((-a-2)\) \(3\) \(4\) \(I_4^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)

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 648.1-m consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of 7056.q3, 72.a3, defined over \(\Q\), so it is also a \(\Q\)-curve.