# Properties

 Label 2.2.53.1-36.1-e1 Base field $$\Q(\sqrt{53})$$ Conductor $$(6)$$ Conductor norm $$36$$ CM no Base change no Q-curve yes Torsion order $$5$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{53})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 13$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 13)

gp: K = nfinit(a^2 - a - 13);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);

## Weierstrass equation

$$y^2+axy=x^{3}+x^{2}+\left(-21a-62\right)x+68a+215$$
sage: E = EllipticCurve(K, [a, 1, 0, -21*a - 62, 68*a + 215])

gp: E = ellinit([a, 1, 0, -21*a - 62, 68*a + 215],K)

magma: E := ChangeRing(EllipticCurve([a, 1, 0, -21*a - 62, 68*a + 215]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(6)$$ = $$\left(2\right) \cdot \left(3\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$36$$ = $$4 \cdot 9$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(18)$$ = $$\left(2\right) \cdot \left(3\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$324$$ = $$4 \cdot 9^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{3307949}{18}$$ 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: $$\Z/5\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-1 : -3 a - 11 : 1\right)$ 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: $$48.1486013196905$$ Tamagawa product: $$2$$  =  $$1\cdot2$$ Torsion order: $$5$$ Leading coefficient: $$0.529097522449063$$ 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$$(j)_{-}$$
$$\left(2\right)$$ $$4$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$\left(3\right)$$ $$9$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 36.1-e consists of curves linked by isogenies of degree 5.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.