# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-9.1-a2 Conductor $$(3,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 4)$$ Conductor norm $$9$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

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Show commands for: Magma / Pari/GP / SageMath

## Base field 4.4.19821.1

Generator $$a$$, with minimal polynomial $$x^{4} - x^{3} - 8 x^{2} + 6 x + 3$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 8*x^2 + 6*x + 3)

gp: K = nfinit(a^4 - a^3 - 8*a^2 + 6*a + 3);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);

## Weierstrass equation

$$y^2 + \left(a^{2} - 4\right) x y + \left(a^{2} - 4\right) y = x^{3} + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 4 a\right) x^{2} + \left(-a^{3} + 3 a^{2} + 8 a + 4\right) x + 3 a^{3} + 8 a^{2} - 11 a - 3$$
sage: E = EllipticCurve(K, [a^2 - 4, 1/3*a^3 + 1/3*a^2 - 4*a, a^2 - 4, -a^3 + 3*a^2 + 8*a + 4, 3*a^3 + 8*a^2 - 11*a - 3])

gp: E = ellinit([a^2 - 4, 1/3*a^3 + 1/3*a^2 - 4*a, a^2 - 4, -a^3 + 3*a^2 + 8*a + 4, 3*a^3 + 8*a^2 - 11*a - 3],K)

magma: E := ChangeRing(EllipticCurve([a^2 - 4, 1/3*a^3 + 1/3*a^2 - 4*a, a^2 - 4, -a^3 + 3*a^2 + 8*a + 4, 3*a^3 + 8*a^2 - 11*a - 3]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(3,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 4)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$9$$ = $$3^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(19683,6561 a,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 4654 a + 8716,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 4976 a + 5903)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{17}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$129140163$$ = $$3^{17}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{73343}{729} a^{3} + \frac{24131}{729} a^{2} + \frac{615892}{729} a + \frac{18719}{27}$$ 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$$(j)_{-}$$
$$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ $$3$$ $$2$$ $$I_{11}^*$$ Additive $$-1$$ $$2$$ $$17$$ $$11$$

## Galois Representations

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

prime Image of Galois Representation
$$11$$ 11B.10.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 11.
Its isogeny class 9.1-a consists of curves linked by isogenies of degree 11.

## Base change

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