Properties

Label 2.2.13.1-81.1-a4
Base field \(\Q(\sqrt{13}) \)
Conductor norm \( 81 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((9)\) = \((-a)^{2}\cdot(-a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 81 \) = \(3^{2}\cdot3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2187a-8748)\) = \((-a)^{7}\cdot(-a+1)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 43046721 \) = \(3^{7}\cdot3^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{24125}{27} a - \frac{8500}{9} \)
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: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 4 : -6 a + 4 : 1\right)$
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: \( 4.2982869860845826158904549165029528337 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 1.1921303173067333268356614256236372911 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(4\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((-a+1)\) \(3\) \(4\) \(I_{3}^{*}\) Additive \(-1\) \(2\) \(9\) \(3\)

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

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 81.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

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

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