# 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 $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## 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.