# Properties

 Base field 4.4.17725.1 Label 4.4.17725.1-19.1-b1 Conductor $$(19,a + 1)$$ 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 4.4.17725.1

Generator $$a$$, with minimal polynomial $$x^{4} - 2 x^{3} - 12 x^{2} + 13 x + 41$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 - 12*x^2 + 13*x + 41)

gp: K = nfinit(a^4 - 2*a^3 - 12*a^2 + 13*a + 41);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 13, -12, -2, 1]);

## Weierstrass equation

$$y^2 + \left(a^{3} - a^{2} - 6 a\right) x y + \left(a^{3} - a^{2} - 6 a\right) y = x^{3} + \left(-a^{3} + 2 a^{2} + 7 a - 7\right) x^{2} + \left(a^{3} - 5 a^{2} - a + 33\right) x + 2 a^{2} + a - 11$$
sage: E = EllipticCurve(K, [a^3 - a^2 - 6*a, -a^3 + 2*a^2 + 7*a - 7, a^3 - a^2 - 6*a, a^3 - 5*a^2 - a + 33, 2*a^2 + a - 11])

gp: E = ellinit([a^3 - a^2 - 6*a, -a^3 + 2*a^2 + 7*a - 7, a^3 - a^2 - 6*a, a^3 - 5*a^2 - a + 33, 2*a^2 + a - 11],K)

magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 6*a, -a^3 + 2*a^2 + 7*a - 7, a^3 - a^2 - 6*a, a^3 - 5*a^2 - a + 33, 2*a^2 + a - 11]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(19,a + 1)$$ = $$\left(-a - 1\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$19$$ = $$19$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(6859,a^{2} - a + 3646,a + 2034,a^{3} - a^{2} - 7 a + 4954)$$ = $$\left(-a - 1\right)^{3}$$ 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)); $$j$$ = $$-\frac{2454829497}{6859} a^{3} - \frac{3223188207}{6859} a^{2} + \frac{18885670830}{6859} a + \frac{30593217186}{6859}$$ 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$$(j)_{-}$$
$$\left(-a - 1\right)$$ $$19$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 19.1-b consists of this curve only.

## Base change

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