Properties

Label 2.2.13.1-256.1-d1
Base field \(\Q(\sqrt{13}) \)
Conductor norm \( 256 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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

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

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

Weierstrass equation

\({y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-1894a+4271\right){x}+34785a-79853\)
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,0]),K([4271,-1894]),K([-79853,34785])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,1]),Polrev([0,0]),Polrev([4271,-1894]),Polrev([-79853,34785])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![4271,-1894],K![-79853,34785]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((16)\) = \((2)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 256 \) = \(4^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-131072)\) = \((2)^{17}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 17179869184 \) = \(4^{17}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1250637664527933}{32} a - \frac{1629300280935823}{32} \)
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 $\left(-\frac{62}{9} a + \frac{61}{3} : \frac{2336}{27} a - \frac{1792}{9} : 1\right)$
Height \(0.28133767664042888038709673104676728432\)
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: \( 0.28133767664042888038709673104676728432 \)
Period: \( 2.9613879770120062423840255273475678148 \)
Tamagawa product: \( 4 \)
Torsion order: \(1\)
Leading coefficient: \( 1.8485939029697846680635351417838982186 \)
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\((j)_{-}\)
\((2)\) \(4\) \(4\) \(I_{9}^{*}\) Additive \(-1\) \(4\) \(17\) \(5\)

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
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5, 9, 15 and 45.
Its isogeny class 256.1-d consists of curves linked by isogenies of degrees dividing 45.

Base change

This elliptic curve is a \(\Q\)-curve.

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