# Properties

 Base field 3.3.316.1 Label 3.3.316.1-32.5-a3 Conductor $$(16,-a^{2} - a + 4)$$ Conductor norm $$32$$ CM no base-change no Q-curve no Torsion order $$4$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field 3.3.316.1

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 4 x + 2$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 4*x + 2)

gp: K = nfinit(a^3 - a^2 - 4*a + 2);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y + \left(a^{2} + a - 2\right) y = x^{3} - a x^{2} + \left(623 a^{2} - 451 a - 2866\right) x - 13706 a^{2} + 8249 a + 60042$$
sage: E = EllipticCurve(K, [a + 1, -a, a^2 + a - 2, 623*a^2 - 451*a - 2866, -13706*a^2 + 8249*a + 60042])

gp: E = ellinit([a + 1, -a, a^2 + a - 2, 623*a^2 - 451*a - 2866, -13706*a^2 + 8249*a + 60042],K)

magma: E := ChangeRing(EllipticCurve([a + 1, -a, a^2 + a - 2, 623*a^2 - 451*a - 2866, -13706*a^2 + 8249*a + 60042]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(16,-a^{2} - a + 4)$$ = $$\left(-a\right) \cdot \left(-a + 1\right)^{4}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$32$$ = $$2^{5}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1024,2 a + 778,a^{2} + a + 108)$$ = $$\left(-a\right) \cdot \left(-a + 1\right)^{10}$$ 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)); $$j$$ = $$\frac{118715864416425}{2} a^{2} + \frac{159426274721303}{2} a - 50669979335827$$ 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: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-\frac{3}{2} a^{2} + 4 a + \frac{23}{2} : \frac{143}{4} a^{2} - \frac{99}{4} a - \frac{325}{2} : 1\right)$ 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$$(j)_{-}$$
$$\left(-a\right)$$ $$2$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(-a + 1\right)$$ $$2$$ $$4$$ $$I_{2}^*$$ Additive $$1$$ $$4$$ $$10$$ $$0$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 32.5-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

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