Properties

Label 6.6.371293.1-53.1-a2
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 53 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\zeta_{13})^+\)

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+3\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+3a^{2}+9a+1\right){x}^{2}+\left(a^{5}-2a^{4}-8a^{3}+6a^{2}+17a+4\right){x}+2a^{5}-a^{4}-11a^{3}+2a^{2}+15a+4\)
sage: E = EllipticCurve([K([-1,1,1,0,0,0]),K([1,9,3,-6,-1,1]),K([3,5,-4,-5,1,1]),K([4,17,6,-8,-2,1]),K([4,15,2,-11,-1,2])])
 
gp: E = ellinit([Polrev([-1,1,1,0,0,0]),Polrev([1,9,3,-6,-1,1]),Polrev([3,5,-4,-5,1,1]),Polrev([4,17,6,-8,-2,1]),Polrev([4,15,2,-11,-1,2])], K);
 
magma: E := EllipticCurve([K![-1,1,1,0,0,0],K![1,9,3,-6,-1,1],K![3,5,-4,-5,1,1],K![4,17,6,-8,-2,1],K![4,15,2,-11,-1,2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+a^3+3a^2-2a+1)\) = \((-a^4+a^3+3a^2-2a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 53 \) = \(53\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^5+a^4+a^3-7a^2-13a+7)\) = \((-a^4+a^3+3a^2-2a+1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -148877 \) = \(-53^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{15669618624}{148877} a^{5} + \frac{14755246679}{148877} a^{4} - \frac{49850712165}{148877} a^{3} - \frac{33922667128}{148877} a^{2} + \frac{28694128602}{148877} a + \frac{8142934591}{148877} \)
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 $\left(-5 a^{5} + 7 a^{4} + 23 a^{3} - 29 a^{2} - 23 a + 23 : -33 a^{5} + 38 a^{4} + 156 a^{3} - 155 a^{2} - 159 a + 123 : 1\right)$
Height \(0.0012791912557583409003467950630334646415\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.0012791912557583409003467950630334646415 \)
Period: \( 60162.441045792087048848477885082004309 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 2.27340 \)
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\((j)_{-}\)
\((-a^4+a^3+3a^2-2a+1)\) \(53\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 53.1-a consists of curves linked by isogenies of degree 3.

Base change

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

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