Properties

Label 2.2.17.1-676.5-i7
Base field \(\Q(\sqrt{17}) \)
Conductor \((20a-42)\)
Conductor norm \( 676 \)
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: 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

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-956a-1508\right){x}+22281a+34805\)
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([1,0]),K([-1508,-956]),K([34805,22281])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([-1508,-956])),Pol(Vecrev([34805,22281]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,-1],K![1,0],K![-1508,-956],K![34805,22281]]);
 

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: \((189376a+1096000)\) = \((-a+2)^{6}\cdot(-a-1)^{12}\cdot(2a+1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1265319018496 \) = \(2^{6}\cdot2^{12}\cdot13^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{203862548967}{4096} a + \frac{130566616997}{1024} \)
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 $\left(6 a + 11 : -20 a - 33 : 1\right)$
Height \(0.329942588567746\)
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: $\left(-\frac{71}{4} a - \frac{105}{4} : \frac{71}{8} a + \frac{101}{8} : 1\right)$ $\left(\frac{27}{4} a + 12 : -\frac{27}{8} a - \frac{13}{2} : 1\right)$
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: \( 0.329942588567746 \)
Period: \( 6.56584108801035 \)
Tamagawa product: \( 48 \)  =  \(( 2 \cdot 3 )\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.15250318775630 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((-a-1)\) \(2\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\((2a+1)\) \(13\) \(4\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 676.5-i 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.