# Properties

 Label 2.0.4.1-5202.2-c2 Base field $$\Q(\sqrt{-1})$$ Conductor $$(51i+51)$$ Conductor norm $$5202$$ CM no Base change yes: 816.d4,102.b4 Q-curve yes Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / 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(Pol(Vecrev([1, 0, 1])));

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

## Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}+1809{x}-37790$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([1809,0]),K([-37790,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1809,0])),Pol(Vecrev([-37790,0]))], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![1809,0],K![-37790,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(51i+51)$$ = $$(i+1)\cdot(3)\cdot(i+4)\cdot(i-4)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5202$$ = $$2\cdot9\cdot17\cdot17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-1001033261568)$$ = $$(i+1)^{18}\cdot(3)^{4}\cdot(i+4)^{6}\cdot(i-4)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1002067590765467905818624$$ = $$2^{18}\cdot9^{4}\cdot17^{6}\cdot17^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{655215969476375}{1001033261568}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(\frac{71}{4} : -\frac{75}{8} : 1\right)$ 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: $$0.138622644973780$$ Tamagawa product: $$32$$  =  $$2\cdot2^{2}\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.10898115979024$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$2$$ $$I_{18}$$ Non-split multiplicative $$1$$ $$1$$ $$18$$ $$18$$
$$(3)$$ $$9$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(i+4)$$ $$17$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$(i-4)$$ $$17$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$

## 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 and 6.
Its isogeny class 5202.2-c consists of curves linked by isogenies of degrees dividing 6.

## Base change

This curve is the base change of 816.d4, 102.b4, defined over $$\Q$$, so it is also a $$\Q$$-curve.