Properties

Label 2.0.7.1-86.4-a2
Base field \(\Q(\sqrt{-7}) \)
Conductor \((5a-9)\)
Conductor norm \( 86 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{-7}) \)

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

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

Weierstrass equation

\({y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a-111\right){x}-110a-352\)
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([1,0]),K([-111,-1]),K([-352,-110])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,1])),Pol(Vecrev([1,0])),Pol(Vecrev([-111,-1])),Pol(Vecrev([-352,-110]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,1],K![1,0],K![-111,-1],K![-352,-110]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((5a-9)\) = \((-a+1)\cdot(2a+5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 86 \) = \(2\cdot43\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16648913a+40384587)\) = \((-a+1)^{45}\cdot(2a+5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1512927999819776 \) = \(2^{45}\cdot43\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2687211387898023049}{1512927999819776} a + \frac{1649707309527639061}{756463999909888} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.716038413420136 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 0.541274163165409 \)
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\((j)_{-}\)
\((-a+1)\) \(2\) \(1\) \(I_{45}\) Non-split multiplicative \(1\) \(1\) \(45\) \(45\)
\((2a+5)\) \(43\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 86.4-a consists of curves linked by isogenies of degrees dividing 9.

Base change

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