# Properties

 Base field 3.3.316.1 Label 3.3.316.1-4.3-a1 Conductor $$(4,a + 1)$$ Conductor norm $$4$$ CM no base-change no Q-curve no Torsion order $$2$$ 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

$$y^2 + \left(a^{2} + a - 2\right) x y + \left(a + 1\right) y = x^{3} + \left(a^{2} + a - 3\right) x^{2} + \left(-2770 a^{2} + 1478 a + 11792\right) x + 1679357 a^{2} - 888891 a - 7135787$$
sage: E = EllipticCurve(K, [a^2 + a - 2, a^2 + a - 3, a + 1, -2770*a^2 + 1478*a + 11792, 1679357*a^2 - 888891*a - 7135787])

gp: E = ellinit([a^2 + a - 2, a^2 + a - 3, a + 1, -2770*a^2 + 1478*a + 11792, 1679357*a^2 - 888891*a - 7135787],K)

magma: E := ChangeRing(EllipticCurve([a^2 + a - 2, a^2 + a - 3, a + 1, -2770*a^2 + 1478*a + 11792, 1679357*a^2 - 888891*a - 7135787]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(4,a + 1)$$ = $$\left(-a + 1\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$4$$ = $$2^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(16,a + 5,a^{2} + 7)$$ = $$\left(-a + 1\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$16$$ = $$2^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$342336 a^{2} + 460416 a - 292544$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-41 a^{2} + \frac{41}{2} a + \frac{345}{2} : -\frac{99}{4} a^{2} + \frac{63}{4} a + \frac{221}{2} : 1\right)$ 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$$(j)_{-}$$
$$\left(-a + 1\right)$$ $$2$$ $$1$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$

## 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.
Its isogeny class 4.3-a consists of curves linked by isogenies of degree 2.

## Base change

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