Properties

Label 2.0.31.1-14.3-a4
Base field \(\Q(\sqrt{-31}) \)
Conductor norm \( 14 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{-31}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 8 \); class number \(3\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -1, 1]))
 
gp: K = nfinit(Polrev([8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((14,a+9)\) = \((2,a+1)\cdot(7,a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 14 \) = \(2\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a+5)\) = \((2,a+1)^{2}\cdot(7,a+2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 28 \) = \(2^{2}\cdot7\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{17387}{28} a + \frac{19846}{7} \)
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: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(0 : -1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 10.099537487736980767950029351068013831 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\)
Torsion order: \(3\)
Leading coefficient: \( 0.80619132480699370705893434103738946096 \)
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\((j)_{-}\)
\((2,a+1)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((7,a+2)\) \(7\) \(1\) \(I_{1}\) 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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 9 and 27.
Its isogeny class 14.3-a consists of curves linked by isogenies of degrees dividing 27.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.