Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-225.2-a1 Conductor $$(15)$$ Conductor norm $$225$$ CM no base-change no Q-curve yes Torsion order $$8$$ Rank $$0$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{-1})$$

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

magma: R<x> := PolynomialRing(Rationals()); K<i> := NumberField(R![1, 0, 1]);

sage: x = polygen(QQ); K.<i> = NumberField(x^2 + 1)

gp (2.8): K = nfinit(i^2 + 1);

Weierstrass equation

$$y^2 + x y + y = x^{3} + x^{2} + \left(-105 i + 395\right) x - 2982 i - 1054$$
magma: E := ChangeRing(EllipticCurve([1, 1, 1, -105*i + 395, -2982*i - 1054]),K);

sage: E = EllipticCurve(K, [1, 1, 1, -105*i + 395, -2982*i - 1054])

gp (2.8): E = ellinit([1, 1, 1, -105*i + 395, -2982*i - 1054],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(15)$$ = $$\left(3\right) \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$225$$ = $$5^{2} \cdot 9$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-19305000 i + 22036875)$$ = $$\left(3\right) \cdot \left(-i - 2\right)^{16} \cdot \left(2 i + 1\right)^{4}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$858306884765625$$ = $$5^{20} \cdot 9$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{117751185817608007}{457763671875} i - \frac{2360548126387992}{152587890625}$$ 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: $$\Z/8\Z$$ 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] $\left(11 i : -13 i - 3 : 1\right)$ 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$$(j)_{-}$$
$$\left(-i - 2\right)$$ $$5$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$
$$\left(2 i + 1\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8 and 16.
Its isogeny class 225.2-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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