Properties

Label 2.0.3.1-124.1-a3
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-12a+2)\)
Conductor norm \( 124 \)
CM no
Base change no
Q-curve no
Torsion order \( 5 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-3}) \)

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

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

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-15a+5\right){x}-7a+21\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,1]),K([5,-15]),K([21,-7])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,1])),Pol(Vecrev([5,-15])),Pol(Vecrev([21,-7]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![0,1],K![5,-15],K![21,-7]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-12a+2)\) = \((2)\cdot(-6a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 124 \) = \(4\cdot31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-139168a-52032)\) = \((2)^{5}\cdot(-6a+1)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 29316250624 \) = \(4^{5}\cdot31^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{511363962461}{916132832} a + \frac{1018073036305}{916132832} \)
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: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(4 a - 6 : -7 a - 7 : 1\right)$
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: \( 1.84215893028954 \)
Tamagawa product: \( 5 \)  =  \(1\cdot5\)
Torsion order: \(5\)
Leading coefficient: \( 0.425428381717095 \)
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\((j)_{-}\)
\((2)\) \(4\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((-6a+1)\) \(31\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 124.1-a consists of curves linked by isogenies of degrees dividing 25.

Base change

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