Properties

Label 2.2.53.1-7.1-a2
Base field \(\Q(\sqrt{53}) \)
Conductor norm \( 7 \)
CM no
Base change no
Q-curve no
Torsion order \( 7 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

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

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

Weierstrass equation

\({y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-1\right){x}-a-4\)
sage: E = EllipticCurve([K([0,0]),K([1,-1]),K([1,1]),K([-1,1]),K([-4,-1])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,-1]),Polrev([1,1]),Polrev([-1,1]),Polrev([-4,-1])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,-1],K![1,1],K![-1,1],K![-4,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a-2)\) = \((-a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 7 \) = \(7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a-908)\) = \((-a-2)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -823543 \) = \(-7^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1840001024}{823543} a + \frac{7615438848}{823543} \)
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: \(1\)
Generator $\left(\frac{47}{49} a - \frac{95}{49} : -\frac{192}{343} a + \frac{92}{343} : 1\right)$
Height \(3.0127273193334040302324070995063316628\)
Torsion structure: \(\Z/7\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 : -3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 3.0127273193334040302324070995063316628 \)
Period: \( 10.922421609854650675504715088956066487 \)
Tamagawa product: \( 7 \)
Torsion order: \(7\)
Leading coefficient: \( 1.2914356858321198627617303947711573863 \)
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\((j)_{-}\)
\((-a-2)\) \(7\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 7.1-a consists of curves linked by isogenies of degree 7.

Base change

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

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