# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-27.2-f2 Conductor $$(9,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 1)$$ Conductor norm $$27$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

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

$$y^2 + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\right) x y + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\right) y = x^{3} + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\right) x^{2} + \left(-\frac{19}{3} a^{3} - \frac{19}{3} a^{2} + 33 a + 9\right) x - \frac{67}{3} a^{3} - \frac{124}{3} a^{2} + 59 a + 22$$
magma: E := ChangeRing(EllipticCurve([-1/3*a^3 + 2/3*a^2 + 3*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 2, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -19/3*a^3 - 19/3*a^2 + 33*a + 9, -67/3*a^3 - 124/3*a^2 + 59*a + 22]),K);

sage: E = EllipticCurve(K, [-1/3*a^3 + 2/3*a^2 + 3*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 2, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -19/3*a^3 - 19/3*a^2 + 33*a + 9, -67/3*a^3 - 124/3*a^2 + 59*a + 22])

gp: E = ellinit([-1/3*a^3 + 2/3*a^2 + 3*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 2, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -19/3*a^3 - 19/3*a^2 + 33*a + 9, -67/3*a^3 - 124/3*a^2 + 59*a + 22],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(9,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 1)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{3}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$27$$ = $$3^{3}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(243,81 a,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 37 a + 211,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 35 a + 71)$$ = $$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{9}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$19683$$ = $$3^{9}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{2659993414}{3} a^{3} + \frac{3059637059}{3} a^{2} + 7384788197 a - 8211399121$$ 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: Trivial magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]

## 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$$(j)_{-}$$
$$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ $$3$$ $$3$$ $$IV^*$$ Additive $$-1$$ $$3$$ $$9$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 27.2-f 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.