Properties

Label 2.2.85.1-3.1-b2
Base field \(\Q(\sqrt{85}) \)
Conductor \((3,a)\)
Conductor norm \( 3 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{85}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 21 \); class number \(2\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 21)
 
gp: K = nfinit(a^2 - a - 21);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21, -1, 1]);
 

Weierstrass equation

\(y^2+\left(a+1\right)xy+\left(a+1\right)y=x^{3}+\left(-4a-19\right)x-29a-121\)
sage: E = EllipticCurve(K, [a + 1, 0, a + 1, -4*a - 19, -29*a - 121])
 
gp: E = ellinit([a + 1, 0, a + 1, -4*a - 19, -29*a - 121],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, 0, a + 1, -4*a - 19, -29*a - 121]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3,a)\) = \( \left(3, a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 3 \) = \( 3 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((194 a - 615)\) = \( \left(3, a\right)^{12} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 531441 \) = \( 3^{12} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \)
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 $\left(-\frac{113}{49} a + \frac{232}{49} : \frac{1982}{343} a - \frac{3718}{343} : 1\right)$
Height \(2.71500852777962\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a - 2 : a + 14 : 1\right)$
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: \( 2.71500852777962 \)
Period: \( 6.29513781937821 \)
Tamagawa product: \( 12 \)
Torsion order: \(6\)
Leading coefficient: \(1.23587833378182\)
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\((j)_{-}\)
\( \left(3, a\right) \) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)

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.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 3.1-b consists of curves linked by isogenies of degrees dividing 6.

Base change

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