# Properties

 Base field 3.3.733.1 Label 3.3.733.1-8.1-b3 Conductor $$(2,2)$$ Conductor norm $$8$$ CM no base-change no Q-curve no Torsion order $$6$$ 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

$$y^2 + x y + \left(a^{2} - 4\right) y = x^{3} + \left(-a + 1\right) x^{2} + \left(398163 a^{2} + 83660 a - 2735554\right) x - 225568877 a^{2} - 41410904 a + 1534686372$$
sage: E = EllipticCurve(K, [1, -a + 1, a^2 - 4, 398163*a^2 + 83660*a - 2735554, -225568877*a^2 - 41410904*a + 1534686372])

gp: E = ellinit([1, -a + 1, a^2 - 4, 398163*a^2 + 83660*a - 2735554, -225568877*a^2 - 41410904*a + 1534686372],K)

magma: E := ChangeRing(EllipticCurve([1, -a + 1, a^2 - 4, 398163*a^2 + 83660*a - 2735554, -225568877*a^2 - 41410904*a + 1534686372]),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)); $$\mathfrak{D}$$ = $$(64,64 a,8 a^{2} + 56 a + 8)$$ = $$\left(a^{2} + a - 3\right)^{6} \cdot \left(a^{2} - 6\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$32768$$ = $$2^{3} \cdot 4^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$15216 a^{2} - \frac{319193}{64} a - 83228$$ 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: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-180 a^{2} - 145 a + 1507 : -3221 a^{2} + 2007 a + 15370 : 1\right)$ 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$$(j)_{-}$$
$$\left(a^{2} - 6\right)$$ $$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$\left(a^{2} + a - 3\right)$$ $$4$$ $$6$$ $$I_{6}$$ 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
$$2$$ 2B
$$3$$ 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 8.1-b consists of curves linked by isogenies of degrees dividing 6.

## Base change

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