# Properties

 Label 3.3.1620.1-25.1-a3 Base field 3.3.1620.1 Conductor norm $$25$$ CM no Base change yes Q-curve yes Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field3.3.1620.1

Generator $$a$$, with minimal polynomial $$x^{3} - 12 x - 14$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -12, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-14, -12, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -12, 0, 1]);

## Weierstrass equation

$${y}^2+\left(a^{2}-a-7\right){y}={x}^{3}+\left(a^{2}-a-8\right){x}^{2}+\left(2053627a^{2}-2856402a-20670534\right){x}+2802623882a^{2}-3898187220a-28209473416$$
sage: E = EllipticCurve([K([0,0,0]),K([-8,-1,1]),K([-7,-1,1]),K([-20670534,-2856402,2053627]),K([-28209473416,-3898187220,2802623882])])

gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([-8,-1,1])),Pol(Vecrev([-7,-1,1])),Pol(Vecrev([-20670534,-2856402,2053627])),Pol(Vecrev([-28209473416,-3898187220,2802623882]))], K);

magma: E := EllipticCurve([K![0,0,0],K![-8,-1,1],K![-7,-1,1],K![-20670534,-2856402,2053627],K![-28209473416,-3898187220,2802623882]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^2-3a-3)$$ = $$(a+3)\cdot(2a^2-3a-19)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$25$$ = $$5\cdot5$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(5)$$ = $$(a+3)^{2}\cdot(2a^2-3a-19)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$125$$ = $$5^{2}\cdot5$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{884736}{5}$$ 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 $\left(-\frac{82199}{100} a^{2} + \frac{57163}{50} a + \frac{206816}{25} : \frac{4500237}{125} a^{2} - \frac{25037727}{500} a - \frac{181187153}{500} : 1\right)$ Height $$1.58568958279732$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(351 a^{2} - \frac{977}{2} a - 3533 : -\frac{1}{2} a^{2} + \frac{1}{2} a + \frac{7}{2} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.58568958279732$$ Period: $$64.7693507143025$$ Tamagawa product: $$2$$  =  $$2\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$3.82755524972352$$ 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$$(j)_{-}$$
$$(a+3)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(2a^2-3a-19)$$ $$5$$ $$1$$ $$I_{1}$$ 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
$$2$$ 2B
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 25.1-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following elliptic curve:

Base field Curve
$$\Q$$ 405.d1