# Properties

 Label 2.0.4.1-72.1-a1 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$72$$ CM no Base change yes Q-curve yes Torsion order $$8$$ Rank $$0$$

# Learn more

Show commands: Magma / PariGP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))

gp: K = nfinit(Polrev([1, 0, 1]));

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

## Weierstrass equation

$${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}-i{x}^{2}+\left(-i-4\right){x}+22i$$
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([1,1]),K([-4,-1]),K([0,22])])

gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([1,1]),Polrev([-4,-1]),Polrev([0,22])], K);

magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,1],K![-4,-1],K![0,22]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(6i+6)$$ = $$(i+1)^{3}\cdot(3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$72$$ = $$2^{3}\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(209952)$$ = $$(i+1)^{10}\cdot(3)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$44079842304$$ = $$2^{10}\cdot9^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{207646}{6561}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-11 i : -22 i - 33 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$1.8176735089657194402812466671253839988$$ Tamagawa product: $$16$$  =  $$2\cdot2^{3}$$ Torsion order: $$8$$ Leading coefficient: $$0.45441837724142986007031166678134599969$$ Analytic order of Ш: $$1$$ (rounded)

## 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$2$$ $$III^{*}$$ Additive $$1$$ $$3$$ $$10$$ $$0$$
$$(3)$$ $$9$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 72.1-a consists of curves linked by isogenies of degrees dividing 8.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 24.a6
$$\Q$$ 48.a6