# Properties

 Label 6.6.371293.1-53.5-a1 Base field $$\Q(\zeta_{13})^+$$ Conductor norm $$53$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\zeta_{13})^+$$

Generator $$a$$, with minimal polynomial $$x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))

gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);

## Weierstrass equation

$${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+4a^{2}-a-3\right){x}^{2}+\left(-5a^{5}-3a^{4}+21a^{3}+12a^{2}-14a-4\right){x}-4a^{5}-2a^{4}+15a^{3}+9a^{2}-7a-2$$
sage: E = EllipticCurve([K([-1,1,1,0,0,0]),K([-3,-1,4,4,-1,-1]),K([-1,3,1,-4,0,1]),K([-4,-14,12,21,-3,-5]),K([-2,-7,9,15,-2,-4])])

gp: E = ellinit([Polrev([-1,1,1,0,0,0]),Polrev([-3,-1,4,4,-1,-1]),Polrev([-1,3,1,-4,0,1]),Polrev([-4,-14,12,21,-3,-5]),Polrev([-2,-7,9,15,-2,-4])], K);

magma: E := EllipticCurve([K![-1,1,1,0,0,0],K![-3,-1,4,4,-1,-1],K![-1,3,1,-4,0,1],K![-4,-14,12,21,-3,-5],K![-2,-7,9,15,-2,-4]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-a^5+a^4+3a^3-3a^2+2)$$ = $$(-a^5+a^4+3a^3-3a^2+2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$53$$ = $$53$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(a^5-5a^3-a^2+5a)$$ = $$(-a^5+a^4+3a^3-3a^2+2)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-53$$ = $$-53$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{167274219}{53} a^{5} - \frac{608694462}{53} a^{4} + \frac{282958399}{53} a^{3} + \frac{924381730}{53} a^{2} - \frac{673826397}{53} a - \frac{212698186}{53}$$ 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 $\left(a^{4} - a^{3} - 3 a^{2} + a + 2 : -a^{5} + a^{4} + 3 a^{3} - 2 a^{2} - 2 a + 1 : 1\right)$ Height $$0.0038375737672750227010403851891003939277$$ 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: $$0.0038375737672750227010403851891003939277$$ Period: $$60162.441045792087048848477885082004309$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$2.27340$$ 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$$(j)_{-}$$
$$(-a^5+a^4+3a^3-3a^2+2)$$ $$53$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 53.5-a consists of curves linked by isogenies of degree 3.

## Base change

This elliptic curve is not a $$\Q$$-curve.

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