Properties

Label 2.0.7.1-42592.6-d2
Base field \(\Q(\sqrt{-7}) \)
Conductor norm \( 42592 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
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

\({y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-17a+12\right){x}+52a-49\)
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([0,1]),K([12,-17]),K([-49,52])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,1]),Polrev([0,1]),Polrev([12,-17]),Polrev([-49,52])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,1],K![0,1],K![12,-17],K![-49,52]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-66a+220)\) = \((a)^{4}\cdot(-a+1)\cdot(-2a+3)^{2}\cdot(2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 42592 \) = \(2^{4}\cdot2\cdot11^{2}\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1115136a+1239040)\) = \((a)^{15}\cdot(-a+1)^{10}\cdot(-2a+3)^{2}\cdot(2a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5403974828032 \) = \(2^{15}\cdot2^{10}\cdot11^{2}\cdot11^{3}\)
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 $\left(-a - 6 : 17 a - 5 : 1\right)$
Height \(0.034494406318325220733215264193173553181\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.034494406318325220733215264193173553181 \)
Period: \( 1.1978587692432013483088884265270675319 \)
Tamagawa product: \( 120 \)  =  \(2^{2}\cdot( 2 \cdot 5 )\cdot1\cdot3\)
Torsion order: \(1\)
Leading coefficient: \( 7.4962922343508609157975574092484598716 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{7}^{*}\) Additive \(-1\) \(4\) \(15\) \(3\)
\((-a+1)\) \(2\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)
\((-2a+3)\) \(11\) \(1\) \(II\) Additive \(-1\) \(2\) \(2\) \(0\)
\((2a+1)\) \(11\) \(3\) \(I_{3}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 42592.6-d consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

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