# Properties

 Base field 3.3.1129.1 Label 3.3.1129.1-19.1-b2 Conductor $$(19,-a^{2} - a + 4)$$ Conductor norm $$19$$ 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 3.3.1129.1

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

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

gp: K = nfinit(a^3 - 7*a - 3);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -7, 0, 1]);

## Weierstrass equation

$$y^2 + \left(a + 1\right) y = x^{3} + \left(a^{2} - a - 4\right) x^{2} + \left(-2 a^{2} + 4 a + 6\right) x + a^{2} - 3 a - 3$$
sage: E = EllipticCurve(K, [0, a^2 - a - 4, a + 1, -2*a^2 + 4*a + 6, a^2 - 3*a - 3])

gp: E = ellinit([0, a^2 - a - 4, a + 1, -2*a^2 + 4*a + 6, a^2 - 3*a - 3],K)

magma: E := ChangeRing(EllipticCurve([0, a^2 - a - 4, a + 1, -2*a^2 + 4*a + 6, a^2 - 3*a - 3]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(19,-a^{2} - a + 4)$$ = $$\left(-a^{2} - a + 4\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$19$$ = $$19$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(6859,a + 2008,a^{2} - a + 5879)$$ = $$\left(-a^{2} - a + 4\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$6859$$ = $$19^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{11923591168}{6859} a^{2} + \frac{28591403008}{6859} a + \frac{14920290304}{6859}$$ 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$$(j)_{-}$$
$$\left(-a^{2} - a + 4\right)$$ $$19$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 19.1-b consists of curves linked by isogenies of degree 3.

## Base change

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