# Properties

 Label 2.0.4.1-21025.5-a4 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$21025$$ CM no Base change no Q-curve yes Torsion order $$2$$ 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

$${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}+\left(-45i+43\right){x}-76i-164$$
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([43,-45]),K([-164,-76])])

gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([43,-45]),Polrev([-164,-76])], K);

magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![43,-45],K![-164,-76]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(145)$$ = $$(-i-2)\cdot(2i+1)\cdot(-2i+5)\cdot(2i+5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$21025$$ = $$5\cdot5\cdot29\cdot29$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-144855i+207640)$$ = $$(-i-2)^{4}\cdot(2i+1)\cdot(-2i+5)\cdot(2i+5)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$64097340625$$ = $$5^{4}\cdot5\cdot29\cdot29^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{43875392736951}{442050625} i + \frac{12075778344168}{442050625}$$ 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 $\left(-i - 4 : 22 i : 1\right)$ Height $$0.58445762986456711207078421002164796832$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(4 i + \frac{5}{4} : -\frac{9}{8} i + 2 : 1\right)$ 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: $$0.58445762986456711207078421002164796832$$ Period: $$1.3320355406399877122231578510461998792$$ Tamagawa product: $$4$$  =  $$2\cdot1\cdot1\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.5570366699556289628436820961048402430$$ 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$$(j)_{-}$$
$$(-i-2)$$ $$5$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$(2i+1)$$ $$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(-2i+5)$$ $$29$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(2i+5)$$ $$29$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$

## 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 and 4.
Its isogeny class 21025.5-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

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

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