# Properties

 Label 2.2.33.1-300.1-g6 Base field $$\Q(\sqrt{33})$$ Conductor $$(-20 a + 70)$$ Conductor norm $$300$$ CM no Base change yes: 3630.w3,90.c3 Q-curve yes Torsion order $$12$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{33})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 8$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -1, 1]))

gp: K = nfinit(Pol(Vecrev([-8, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);

## Weierstrass equation

$$y^2+xy+\left(a+1\right)y=x^{3}+\left(-a-1\right)x^{2}+\left(122736a-413897\right)x-39525706a+133291793$$
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,1]),K([-413897,122736]),K([133291793,-39525706])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,1])),Pol(Vecrev([-413897,122736])),Pol(Vecrev([133291793,-39525706]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,1],K![-413897,122736],K![133291793,-39525706]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-20 a + 70)$$ = $$\left(-a - 2\right) \cdot \left(-a + 3\right) \cdot \left(-2 a + 7\right) \cdot \left(5\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$300$$ = $$2^{2} \cdot 3 \cdot 25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(9000000)$$ = $$\left(-a - 2\right)^{6} \cdot \left(-a + 3\right)^{6} \cdot \left(-2 a + 7\right)^{4} \cdot \left(5\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$81000000000000$$ = $$2^{12} \cdot 3^{4} \cdot 25^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4102915888729}{9000000}$$ 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: $$\Z/2\Z\times\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-71 a + 241 : -435 a + 1464 : 1\right)$ $\left(-81 a + \frac{1099}{4} : 40 a - \frac{1103}{8} : 1\right)$ 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: $$5.36748913421879$$ Tamagawa product: $$864$$  =  $$( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2^{2}\cdot( 2 \cdot 3 )$$ Torsion order: $$12$$ Leading coefficient: $$5.60615956111491$$ 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$$(j)_{-}$$
$$\left(-a - 2\right)$$ $$2$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(-a + 3\right)$$ $$2$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(-2 a + 7\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(5\right)$$ $$25$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 300.1-g consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base change of elliptic curves 3630.w3, 90.c3, defined over $$\Q$$, so it is also a $$\Q$$-curve.