# Properties

 Base field $$\Q(\sqrt{-3})$$ Label 2.0.3.1-15876.1-c2 Conductor $$(54 a - 144)$$ Conductor norm $$15876$$ 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 $$\Q(\sqrt{-3})$$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)
gp (2.8): K = nfinit(a^2 - a + 1);

## Weierstrass equation

$$y^2 + a x y + a y = x^{3} + \left(-a + 1\right) x^{2} + \left(-44 a - 31\right) x + 377 a - 123$$
magma: E := ChangeRing(EllipticCurve([a, -a + 1, a, -44*a - 31, 377*a - 123]),K);
sage: E = EllipticCurve(K, [a, -a + 1, a, -44*a - 31, 377*a - 123])
gp (2.8): E = ellinit([a, -a + 1, a, -44*a - 31, 377*a - 123],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(54 a - 144)$$ = $$\left(2\right) \cdot \left(-2 a + 1\right)^{4} \cdot \left(-3 a + 1\right)^{2}$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$15876$$ = $$3^{4} \cdot 4 \cdot 7^{2}$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(5983632 a - 39675096)$$ = $$\left(2\right)^{3} \cdot \left(-2 a + 1\right)^{12} \cdot \left(-3 a + 1\right)^{9}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$1372515920491968$$ = $$3^{12} \cdot 4^{3} \cdot 7^{9}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{1192725}{1372} a - \frac{2098143}{2744}$$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): 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()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: Trivial magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1]

## 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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$3$$ $$1$$ $$II^*$$ Additive $$1$$ $$4$$ $$12$$ $$0$$
$$\left(-3 a + 1\right)$$ $$7$$ $$2$$ $$I_{3}^*$$ Additive $$-1$$ $$2$$ $$9$$ $$3$$
$$\left(2\right)$$ $$4$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 15876.1-c consists of curves linked by isogenies of degrees dividing 9.

## Base change

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