Properties

Label 2.2.8.1-578.1-d6
Base field \(\Q(\sqrt{2}) \)
Conductor \((17a)\)
Conductor norm \( 578 \)
CM no
Base change yes: 34.a3,1088.l3
Q-curve yes
Torsion order \( 12 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}-43{x}+105\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-43,0]),K([105,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-43,0])),Pol(Vecrev([105,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-43,0],K![105,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((17a)\) = \((a)\cdot(-3a-1)\cdot(3a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 578 \) = \(2\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2312)\) = \((a)^{6}\cdot(-3a-1)^{2}\cdot(3a-1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5345344 \) = \(2^{6}\cdot17^{2}\cdot17^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{8805624625}{2312} \)
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 $\left(6 : 6 a - 3 : 1\right)$
Height \(0.937230688943121\)
Torsion structure: \(\Z/2\Z\times\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(4 a - 2 : -2 a + 1 : 1\right)$ $\left(a + 2 : 3 a - 5 : 1\right)$
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: \( 0.937230688943121 \)
Period: \( 20.2109887435620 \)
Tamagawa product: \( 24 \)  =  \(( 2 \cdot 3 )\cdot2\cdot2\)
Torsion order: \(12\)
Leading coefficient: \( 2.23237840548924 \)
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\((j)_{-}\)
\((a)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((-3a-1)\) \(17\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((3a-1)\) \(17\) \(2\) \(I_{2}\) Non-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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 578.1-d consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base change of elliptic curves 34.a3, 1088.l3, defined over \(\Q\), so it is also a \(\Q\)-curve.