Properties

Label 2.2.13.1-27.2-a7
Base field \(\Q(\sqrt{13}) \)
Conductor \((-3 a + 3)\)
Conductor norm \( 27 \)
CM no
Base change no
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{13}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 3 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 3)
 
gp: K = nfinit(a^2 - a - 3);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -1, 1]);
 

Weierstrass equation

\(y^2+\left(a+1\right)xy+\left(a+1\right)y=x^{3}-ax^{2}+\left(57a-194\right)x-767a+1595\)
sage: E = EllipticCurve(K, [a + 1, -a, a + 1, 57*a - 194, -767*a + 1595])
 
gp: E = ellinit([a + 1, -a, a + 1, 57*a - 194, -767*a + 1595],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, -a, a + 1, 57*a - 194, -767*a + 1595]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-3 a + 3)\) = \( \left(-a\right) \cdot \left(-a + 1\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \( 3^{3} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((-5347215 a + 46786491)\) = \( \left(-a\right)^{24} \cdot \left(-a + 1\right)^{8} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1853020188851841 \) = \( 3^{32} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1794398270320625}{282429536481} a + \frac{2024459751037750}{282429536481} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-7 a + 12 : -22 a + 58 : 1\right)$
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: \( 2.34957712739991 \)
Tamagawa product: \( 48 \)  =  \(( 2^{3} \cdot 3 )\cdot2\)
Torsion order: \(6\)
Leading coefficient: \(0.868873929076694\)
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\((j)_{-}\)
\( \left(-a\right) \) \(3\) \(24\) \(I_{24}\) Split multiplicative \(-1\) \(1\) \(24\) \(24\)
\( \left(-a + 1\right) \) \(3\) \(2\) \(I_{2}^*\) Additive \(-1\) \(2\) \(8\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 27.2-a consists of curves linked by isogenies of degrees dividing 24.

Base change

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