# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-3.1-a2 Conductor $$(3,a)$$ Conductor norm $$3$$ CM no base-change no Q-curve no Torsion order $$11$$ Rank not available

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Show commands for: Magma / SageMath / Pari/GP

## Base field 4.4.19821.1

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

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

## Weierstrass equation

$$y^2 + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\right) x y + \left(a^{2} - 4\right) y = x^{3} + a x^{2} + \left(-\frac{7}{3} a^{3} + \frac{2}{3} a^{2} + 22 a + 9\right) x - \frac{10}{3} a^{3} + \frac{5}{3} a^{2} + 32 a + 7$$
magma: E := ChangeRing(EllipticCurve([1/3*a^3 + 1/3*a^2 - 2*a, a, a^2 - 4, -7/3*a^3 + 2/3*a^2 + 22*a + 9, -10/3*a^3 + 5/3*a^2 + 32*a + 7]),K);

sage: E = EllipticCurve(K, [1/3*a^3 + 1/3*a^2 - 2*a, a, a^2 - 4, -7/3*a^3 + 2/3*a^2 + 22*a + 9, -10/3*a^3 + 5/3*a^2 + 32*a + 7])

gp: E = ellinit([1/3*a^3 + 1/3*a^2 - 2*a, a, a^2 - 4, -7/3*a^3 + 2/3*a^2 + 22*a + 9, -10/3*a^3 + 5/3*a^2 + 32*a + 7],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(3,a)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$3$$ = $$3$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(729,243 a,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 37 a + 697,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 116 a + 71)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{11}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$177147$$ = $$3^{11}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{73343}{729} a^{3} + \frac{24131}{729} a^{2} + \frac{615892}{729} a + \frac{18719}{27}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

magma: Rank(E);

sage: E.rank()

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/11\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-a : \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 3 a + 1 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ $$3$$ $$11$$ $$I_{11}$$ Split multiplicative $$-1$$ $$1$$ $$11$$ $$11$$

## Galois Representations

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

prime Image of Galois Representation
$$11$$ 11B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 11.
Its isogeny class 3.1-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.