# Properties

 Base field $$\Q(\sqrt{-3})$$ Label 2.0.3.1-3249.1-a4 Conductor $$(63 a - 15)$$ Conductor norm $$3249$$ CM no base-change no Q-curve no Torsion order $$2$$ Rank $$0$$

# 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

$$y^2 + \left(a + 1\right) x y + \left(a + 1\right) y = x^{3} + \left(-600 a + 426\right) x - 2310 a + 5603$$
magma: E := ChangeRing(EllipticCurve([a + 1, 0, a + 1, -600*a + 426, -2310*a + 5603]),K);

sage: E = EllipticCurve(K, [a + 1, 0, a + 1, -600*a + 426, -2310*a + 5603])

gp (2.8): E = ellinit([a + 1, 0, a + 1, -600*a + 426, -2310*a + 5603],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(63 a - 15)$$ = $$\left(-2 a + 1\right)^{2} \cdot \left(-5 a + 3\right)^{2}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$3249$$ = $$3^{2} \cdot 19^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-20940606 a + 8145279)$$ = $$\left(-2 a + 1\right)^{9} \cdot \left(-5 a + 3\right)^{8}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$334287471336003$$ = $$3^{9} \cdot 19^{8}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{363527109}{361} a + \frac{287391186}{361}$$ 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: $$0$$
magma: Rank(E);

sage: E.rank()

magma: Generators(E); // includes torsion

sage: E.gens()

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/2\Z$$ 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] $\left(-\frac{51}{4} a - 2 : \frac{53}{4} a - \frac{47}{8} : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$3$$ $$2$$ $$III^*$$ Additive $$1$$ $$2$$ $$9$$ $$0$$
$$\left(-5 a + 3\right)$$ $$19$$ $$4$$ $$I_{2}^*$$ Additive $$-1$$ $$2$$ $$8$$ $$2$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 3249.1-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

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