Properties

Label 2.2.17.1-576.6-e5
Base field \(\Q(\sqrt{17}) \)
Conductor \((15 a - 27)\)
Conductor norm \( 576 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 4)
 
gp: K = nfinit(a^2 - a - 4);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);
 

Weierstrass equation

\(y^2+\left(a+1\right)xy=x^{3}+\left(a-1\right)x^{2}+\left(-4a-21\right)x+11a+4\)
sage: E = EllipticCurve(K, [a + 1, a - 1, 0, -4*a - 21, 11*a + 4])
 
gp: E = ellinit([a + 1, a - 1, 0, -4*a - 21, 11*a + 4],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, a - 1, 0, -4*a - 21, 11*a + 4]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((15 a - 27)\) = \( \left(-a - 1\right)^{6} \cdot \left(3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 576 \) = \( 2^{6} \cdot 9 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((-297 a + 4797)\) = \( \left(-a - 1\right)^{18} \cdot \left(3\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 21233664 \) = \( 2^{18} \cdot 9^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{14326000}{9} a + \frac{36913625}{9} \)
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 $\left(\frac{71}{64} a + \frac{13}{64} : -\frac{251}{512} a - \frac{585}{512} : 1\right)$
Height \(1.95697104151459\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{7}{4} a - \frac{17}{4} : \frac{31}{8} a + \frac{45}{8} : 1\right)$ $\left(-a + 4 : -a : 1\right)$
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: \( 1.95697104151459 \)
Period: \( 7.24262255020968 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \(3.43760356448729\)
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\((j)_{-}\)
\( \left(-a - 1\right) \) \(2\) \(4\) \(I_{8}^*\) Additive \(1\) \(6\) \(18\) \(0\)
\( \left(3\right) \) \(9\) \(2\) \(I_{2}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 576.6-e consists of curves linked by isogenies of degrees dividing 16.

Base change

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