# Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-4050.2-c2 Conductor $$(45 i + 45)$$ Conductor norm $$4050$$ CM no base-change yes: 720.j8,90.c8 Q-curve yes Torsion order $$4$$ Rank $$0$$

# Learn more about

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(i^2 + 1);

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

## Weierstrass equation

$$y^2 + x y + y = x^{3} - x^{2} + 13 x - 61$$
sage: E = EllipticCurve(K, [1, -1, 1, 13, -61])

gp: E = ellinit([1, -1, 1, 13, -61],K)

magma: E := ChangeRing(EllipticCurve([1, -1, 1, 13, -61]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(45 i + 45)$$ = $$\left(i + 1\right) \cdot \left(3\right)^{2} \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$4050$$ = $$2 \cdot 5^{2} \cdot 9^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1574640)$$ = $$\left(i + 1\right)^{8} \cdot \left(3\right)^{9} \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$2479491129600$$ = $$2^{8} \cdot 5^{2} \cdot 9^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{357911}{2160}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$0$$

sage: E.rank()

magma: Rank(E);

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(9 : -32 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(i + 1\right)$$ $$2$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$\left(-i - 2\right)$$ $$5$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(2 i + 1\right)$$ $$5$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(3\right)$$ $$9$$ $$4$$ $$I_{3}^*$$ Additive $$1$$ $$2$$ $$9$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 4050.2-c consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base-change of elliptic curves 720.j8, 90.c8, defined over $$\Q$$, so it is also a $$\Q$$-curve.