Properties

Label 2.2.13.1-1936.1-a1
Base field \(\Q(\sqrt{13}) \)
Conductor \((44)\)
Conductor norm \( 1936 \)
CM no
Base change yes: 44.a1,7436.d1
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: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 3)
 
gp: K = nfinit(a^2 - a - 3);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -1, 1]);
 

Weierstrass equation

\(y^2=x^{3}+x^{2}-77x-289\)
sage: E = EllipticCurve(K, [0, 1, 0, -77, -289])
 
gp: E = ellinit([0, 1, 0, -77, -289],K)
 
magma: E := ChangeRing(EllipticCurve([0, 1, 0, -77, -289]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((44)\) = \( \left(2\right)^{2} \cdot \left(11\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 1936 \) = \( 4^{2} \cdot 121 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((340736)\) = \( \left(2\right)^{8} \cdot \left(11\right)^{3} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 116101021696 \) = \( 4^{8} \cdot 121^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{199794688}{1331} \)
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 $\left(14 : 22 a - 11 : 1\right)$
Height \(1.36753340625236\)
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: \( 1.36753340625236 \)
Period: \( 0.647455648107670 \)
Tamagawa product: \( 9 \)  =  \(3\cdot3\)
Torsion order: \(1\)
Leading coefficient: \(4.42026999028636\)
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\((j)_{-}\)
\( \left(2\right) \) \(4\) \(3\) \(IV^*\) Additive \(1\) \(2\) \(8\) \(0\)
\( \left(11\right) \) \(121\) \(3\) \(I_{3}\) 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.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 1936.1-a consists of curves linked by isogenies of degree 3.

Base change

This curve is the base change of elliptic curves 44.a1, 7436.d1, defined over \(\Q\), so it is also a \(\Q\)-curve.