# Properties

 Label 2.0.4.1-8100.2-a1 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$8100$$ CM no Base change no Q-curve yes Torsion order $$6$$ Rank $$1$$

# Related objects

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

$${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-21i+3\right){x}-20i+35$$
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([3,-21]),K([35,-20])])

gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([0,0]),Polrev([3,-21]),Polrev([35,-20])], K);

magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![3,-21],K![35,-20]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(90)$$ = $$(i+1)^{2}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$8100$$ = $$2^{2}\cdot5\cdot5\cdot9^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-75600i+259200)$$ = $$(i+1)^{8}\cdot(-i-2)^{2}\cdot(2i+1)^{6}\cdot(3)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$72900000000$$ = $$2^{8}\cdot5^{2}\cdot5^{6}\cdot9^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{42660324}{15625} i + \frac{45166032}{15625}$$ 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: $$1$$ Generator $\left(4 i + 8 : -21 i - 22 : 1\right)$ Height $$0.40074192602945562281038178849545286386$$ Torsion structure: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(4 i - 2 : -i + 18 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.40074192602945562281038178849545286386$$ Period: $$1.6358717644015549446037367326795824484$$ Tamagawa product: $$72$$  =  $$3\cdot2\cdot( 2 \cdot 3 )\cdot2$$ Torsion order: $$6$$ Leading coefficient: $$2.6222496064139319502147628395855432672$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$3$$ $$IV^{*}$$ Additive $$-1$$ $$2$$ $$8$$ $$0$$
$$(-i-2)$$ $$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(2i+1)$$ $$5$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$(3)$$ $$9$$ $$2$$ $$III$$ Additive $$1$$ $$2$$ $$3$$ $$0$$

## 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.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 8100.2-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

This elliptic curve is a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.