# Properties

 Base field 3.3.733.1 Label 3.3.733.1-8.2-a1 Conductor $$(8,-a)$$ Conductor norm $$8$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 3.3.733.1

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 7 x + 8$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 7*x + 8)

gp: K = nfinit(a^3 - a^2 - 7*a + 8);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);

## Weierstrass equation

$$y^2 + \left(a^{2} - 4\right) y = x^{3} + \left(a - 1\right) x^{2} + \left(39 a^{2} + 56 a - 128\right) x - 1199 a^{2} - 1822 a + 3800$$
sage: E = EllipticCurve(K, [0, a - 1, a^2 - 4, 39*a^2 + 56*a - 128, -1199*a^2 - 1822*a + 3800])

gp: E = ellinit([0, a - 1, a^2 - 4, 39*a^2 + 56*a - 128, -1199*a^2 - 1822*a + 3800],K)

magma: E := ChangeRing(EllipticCurve([0, a - 1, a^2 - 4, 39*a^2 + 56*a - 128, -1199*a^2 - 1822*a + 3800]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(8,-a)$$ = $$\left(a^{2} - 6\right)^{3}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$8$$ = $$2^{3}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(2048,a + 1544,a^{2} + 1984)$$ = $$\left(a^{2} - 6\right)^{11}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$2048$$ = $$2^{11}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-919922 a^{2} + 3416322 a - 2769394$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

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

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## 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$$(j)_{-}$$
$$\left(a^{2} - 6\right)$$ $$2$$ $$1$$ $$II^*$$ Additive $$-1$$ $$3$$ $$11$$ $$0$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 8.2-a consists of this curve only.

## Base change

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