Properties

Label 2.0.7.1-6272.5-d4
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 6272 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
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

\({y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+282\right){x}-1458a+588\)
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([282,1]),K([588,-1458])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([282,1]),Polrev([588,-1458])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![282,1],K![588,-1458]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-56a+56)\) = \((a)^{3}\cdot(-a+1)^{4}\cdot(-2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 6272 \) = \(2^{3}\cdot2^{4}\cdot7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-34420736)\) = \((a)^{11}\cdot(-a+1)^{11}\cdot(-2a+1)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1184787066781696 \) = \(2^{11}\cdot2^{11}\cdot7^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3543122}{49} \)
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 $\left(8 a - 5 : 4 a + 6 : 1\right)$
Height \(1.7823824021101342879977891732597127016\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(7 a - 3 : 0 : 1\right)$
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: \( 1.7823824021101342879977891732597127016 \)
Period: \( 0.60434055726777155990521678828441348423 \)
Tamagawa product: \( 8 \)  =  \(1\cdot2\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 3.2570437581212591893847535091229747616 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(II^{*}\) Additive \(-1\) \(3\) \(11\) \(0\)
\((-a+1)\) \(2\) \(2\) \(I_{3}^{*}\) Additive \(1\) \(4\) \(11\) \(0\)
\((-2a+1)\) \(7\) \(4\) \(I_{4}^{*}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 6272.5-d consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve.

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