Properties

Label 2.2.217.1-28.1-f1
Base field \(\Q(\sqrt{217}) \)
Conductor norm \( 28 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{217}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 54 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-54, -1, 1]))
 
gp: K = nfinit(Polrev([-54, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-54, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-8796257a+69186625\right){x}-110141681624a+866314983587\)
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([0,1]),K([69186625,-8796257]),K([866314983587,-110141681624])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([0,1]),Polrev([69186625,-8796257]),Polrev([866314983587,-110141681624])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,1],K![0,1],K![69186625,-8796257],K![866314983587,-110141681624]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-996a+7834)\) = \((-a+8)\cdot(-a-7)\cdot(-498a+3917)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28 \) = \(2\cdot2\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2809856)\) = \((-a+8)^{13}\cdot(-a-7)^{13}\cdot(-498a+3917)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 7895290740736 \) = \(2^{13}\cdot2^{13}\cdot7^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{529475129}{2809856} \)
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 $\left(-411 a + 3230 : 110254 a - 867201 : 1\right)$
Height \(0.030816291130993076019681928927298566342\)
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: \( 0.030816291130993076019681928927298566342 \)
Period: \( 2.3493301061268639564132105536629099409 \)
Tamagawa product: \( 1014 \)  =  \(13\cdot13\cdot( 2 \cdot 3 )\)
Torsion order: \(1\)
Leading coefficient: \( 9.9669549715627850650792823659342955084 \)
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\((j)_{-}\)
\((-a+8)\) \(2\) \(13\) \(I_{13}\) Split multiplicative \(-1\) \(1\) \(13\) \(13\)
\((-a-7)\) \(2\) \(13\) \(I_{13}\) Split multiplicative \(-1\) \(1\) \(13\) \(13\)
\((-498a+3917)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 28.1-f consists of this curve only.

Base change

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

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