Properties

Label 2.2.17.1-32.4-a11
Base field \(\Q(\sqrt{17}) \)
Conductor \((2a-8)\)
Conductor norm \( 32 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

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

\({y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(23a-87\right){x}-129a+354\)
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([0,1]),K([-87,23]),K([354,-129])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,1])),Pol(Vecrev([-87,23])),Pol(Vecrev([354,-129]))], K);
 
magma: E := EllipticCurve([K![0,1],K![0,1],K![0,1],K![-87,23],K![354,-129]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a-8)\) = \((-a+2)^{4}\cdot(-a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 32 \) = \(2^{4}\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((28a+176)\) = \((-a+2)^{13}\cdot(-a-1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 32768 \) = \(2^{13}\cdot2^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{54503407609}{4} a + \frac{42555672073}{2} \)
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: \(0\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-2 a + 7 : -7 a + 14 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 23.3146986046129 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.41366124965088 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(2\) \(I_5^{*}\) Additive \(-1\) \(4\) \(13\) \(1\)
\((-a-1)\) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 32.4-a consists of curves linked by isogenies of degrees dividing 24.

Base change

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