# Properties

 Base field 3.3.733.1 Label 3.3.733.1-8.1-c3 Conductor $$(2,2)$$ Conductor norm $$8$$ CM no base-change no Q-curve no Torsion order $$10$$ 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

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

gp: E = ellinit([1, -a^2 + a + 4, a, a^2 - 3*a - 3, -a^2 + a + 1],K)

magma: E := ChangeRing(EllipticCurve([1, -a^2 + a + 4, a, a^2 - 3*a - 3, -a^2 + a + 1]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(2,2)$$ = $$\left(a^{2} + a - 3\right) \cdot \left(a^{2} - 6\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$8$$ = $$2 \cdot 4$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1024,4 a + 32,4 a^{2} + 768)$$ = $$\left(a^{2} + a - 3\right)^{2} \cdot \left(a^{2} - 6\right)^{10}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$16384$$ = $$2^{10} \cdot 4^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{229488321}{1024} a^{2} - \frac{40784951}{1024} a + \frac{1560076799}{1024}$$ 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: $$\Z/10\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-a + 2 : -a + 1 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(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$$(j)_{-}$$
$$\left(a^{2} - 6\right)$$ $$2$$ $$10$$ $$I_{10}$$ Split multiplicative $$-1$$ $$1$$ $$10$$ $$10$$
$$\left(a^{2} + a - 3\right)$$ $$4$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## 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
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 8.1-c consists of curves linked by isogenies of degrees dividing 10.

## Base change

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