Properties

Label 2.2.53.1-36.1-a1
Base field \(\Q(\sqrt{53}) \)
Conductor \((6)\)
Conductor norm \( 36 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{53}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 13 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 13)
 
gp: K = nfinit(a^2 - a - 13);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);
 

Weierstrass equation

\(y^2+\left(a+1\right)xy+\left(a+1\right)y=x^{3}+\left(a-1\right)x^{2}+\left(3a+6\right)x+4a+11\)
sage: E = EllipticCurve(K, [a + 1, a - 1, a + 1, 3*a + 6, 4*a + 11])
 
gp: E = ellinit([a + 1, a - 1, a + 1, 3*a + 6, 4*a + 11],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, a - 1, a + 1, 3*a + 6, 4*a + 11]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((6)\) = \( \left(2\right) \cdot \left(3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 36 \) = \( 4 \cdot 9 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((18)\) = \( \left(2\right) \cdot \left(3\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 324 \) = \( 4 \cdot 9^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{20833}{18} a - \frac{43171}{9} \)
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: \( 7.10641560802924 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(1\)
Leading coefficient: \(1.95228251114532\)
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\((j)_{-}\)
\( \left(2\right) \) \(4\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(3\right) \) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 36.1-a consists of this curve only.

Base change

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