# Properties

 Base field $$\Q(\sqrt{-3})$$ Label 2.0.3.1-61731.3-a3 Conductor $$(-285 a + 114)$$ Conductor norm $$61731$$ CM no base-change no Q-curve yes 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: K = nfinit(a^2 - a + 1);

## Weierstrass equation

$$y^2 + y = x^{3} + \left(-a - 1\right) x^{2} + \left(-6699 a - 18\right) x + 251154 a - 123514$$
magma: E := ChangeRing(EllipticCurve([0, -a - 1, 1, -6699*a - 18, 251154*a - 123514]),K);

sage: E = EllipticCurve(K, [0, -a - 1, 1, -6699*a - 18, 251154*a - 123514])

gp: E = ellinit([0, -a - 1, 1, -6699*a - 18, 251154*a - 123514],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-285 a + 114)$$ = $$\left(-2 a + 1\right)^{2} \cdot \left(-5 a + 3\right) \cdot \left(-5 a + 2\right)^{2}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$61731$$ = $$3^{2} \cdot 19^{3}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-529099522605 a + 246827970333)$$ = $$\left(-2 a + 1\right)^{6} \cdot \left(-5 a + 3\right) \cdot \left(-5 a + 2\right)^{15}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$210273790490795831239449$$ = $$3^{6} \cdot 19^{16}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{14306161739497472}{322687697779} a - \frac{12817090105540608}{322687697779}$$ 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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$3$$ $$2$$ $$I_{0}^*$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$\left(-5 a + 3\right)$$ $$19$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$\left(-5 a + 2\right)$$ $$19$$ $$4$$ $$I_{9}^*$$ Additive $$-1$$ $$2$$ $$15$$ $$9$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 61731.3-a 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 a $$\Q$$-curve.