# Properties

 Base field $$\Q(\sqrt{33})$$ Label 2.2.33.1-12.1-a8 Conductor $$(-4 a + 14)$$ Conductor norm $$12$$ CM no base-change no Q-curve yes Torsion order $$6$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 8)

gp: K = nfinit(a^2 - a - 8);

## Weierstrass equation

$$y^2 + x y + y = x^{3} + \left(-a - 1\right) x^{2} + \left(-345 a - 819\right) x + 6409 a + 15203$$
magma: E := ChangeRing(EllipticCurve([1, -a - 1, 1, -345*a - 819, 6409*a + 15203]),K);

sage: E = EllipticCurve(K, [1, -a - 1, 1, -345*a - 819, 6409*a + 15203])

gp: E = ellinit([1, -a - 1, 1, -345*a - 819, 6409*a + 15203],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-4 a + 14)$$ = $$\left(-a - 2\right) \cdot \left(-a + 3\right) \cdot \left(-2 a + 7\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$12$$ = $$2^{2} \cdot 3$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(5832)$$ = $$\left(-a - 2\right)^{3} \cdot \left(-a + 3\right)^{3} \cdot \left(-2 a + 7\right)^{12}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$34012224$$ = $$2^{6} \cdot 3^{12}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{8934171875}{5832}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

magma: Rank(E);

sage: E.rank()

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/6\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(11 a + 25 : 59 a + 140 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-a - 2\right)$$ $$2$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(-a + 3\right)$$ $$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$\left(-2 a + 7\right)$$ $$3$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 12.1-a consists of curves linked by isogenies of degrees dividing 36.

## Base change

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