# Properties

 Label 3.3.1620.1-25.1-b2 Base field 3.3.1620.1 Conductor $$(a^2-3a-3)$$ Conductor norm $$25$$ CM no Base change yes: 405.a1 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

$${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+9\right){x}^{2}+\left(20a^{2}-28a-200\right){x}+108a^{2}-151a-1090$$
sage: E = EllipticCurve([K([0,0,0]),K([9,2,-1]),K([1,1,0]),K([-200,-28,20]),K([-1090,-151,108])])

gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([9,2,-1])),Pol(Vecrev([1,1,0])),Pol(Vecrev([-200,-28,20])),Pol(Vecrev([-1090,-151,108]))], K);

magma: E := EllipticCurve([K![0,0,0],K![9,2,-1],K![1,1,0],K![-200,-28,20],K![-1090,-151,108]]);

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: $$\frac{36864}{5}$$ 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 $\left(-4 a^{2} + 5 a + 42 : -23 a^{2} + 32 a + 229 : 1\right)$ Height $$0.272277013504516$$ 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: $\left(a^{2} - \frac{3}{2} a - 10 : -\frac{1}{2} a - \frac{1}{2} : 1\right)$ 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: $$0.272277013504516$$ Period: $$147.106329377527$$ Tamagawa product: $$2$$  =  $$2\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$1.49271222347893$$ 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$$(j)_{-}$$
$$(a+3)$$ $$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(2a^2-3a-19)$$ $$5$$ $$1$$ $$I_{1}$$ Non-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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 25.1-b consists of curves linked by isogenies of degree 2.

## Base change

This curve is the base change of 405.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.