Properties

Label 2.2.17.1-512.2-f2
Base field \(\Q(\sqrt{17}) \)
Conductor \((-16a-16)\)
Conductor norm \( 512 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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

\({y}^2={x}^{3}-a{x}^{2}+\left(90a-235\right){x}-633a+1626\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([-235,90]),K([1626,-633])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-235,90])),Pol(Vecrev([1626,-633]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![-235,90],K![1626,-633]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-16a-16)\) = \((-a+2)^{4}\cdot(-a-1)^{5}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 512 \) = \(2^{4}\cdot2^{5}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((4096a+8192)\) = \((-a+2)^{13}\cdot(-a-1)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 33554432 \) = \(2^{13}\cdot2^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1592342311}{2} a + \frac{4078872403}{2} \)
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 $\left(-2 a + 5 : -4 a + 8 : 1\right)$
Height \(0.235830817702805\)
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: $\left(-3 a + 6 : 0 : 1\right)$
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: \( 0.235830817702805 \)
Period: \( 12.2230737240707 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.79651091476216 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(4\) \(I_{5}^{*}\) Additive \(-1\) \(4\) \(13\) \(1\)
\((-a-1)\) \(2\) \(2\) \(I_{3}^{*}\) Additive \(1\) \(5\) \(12\) \(0\)

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 512.2-f consists of curves linked by isogenies of degrees dividing 4.

Base change

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