Properties

Label 2.2.17.1-676.5-h1
Base field \(\Q(\sqrt{17}) \)
Conductor \((20a-42)\)
Conductor norm \( 676 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{17}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-4, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-5a-8\right){x}+5a+8\)
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([0,0]),K([-8,-5]),K([8,5])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-8,-5])),Pol(Vecrev([8,5]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,1],K![0,0],K![-8,-5],K![8,5]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((20a-42)\) = \((-a+2)\cdot(-a-1)\cdot(2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 676 \) = \(2\cdot2\cdot13^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-28a-820)\) = \((-a+2)^{2}\cdot(-a-1)^{10}\cdot(2a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 692224 \) = \(2^{2}\cdot2^{10}\cdot13^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{7851401}{1024} a - \frac{3092357}{256} \)
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 $\left(2 : -a : 1\right)$
Height \(0.0154532545293924\)
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: \( 0.0154532545293924 \)
Period: \( 21.3806647550581 \)
Tamagawa product: \( 20 \)  =  \(2\cdot( 2 \cdot 5 )\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 3.20535911003276 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-a-1)\) \(2\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)
\((2a+1)\) \(13\) \(1\) \(II\) Additive \(1\) \(2\) \(2\) \(0\)

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 676.5-h consists of this curve only.

Base change

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