Properties

 Label 2.0.8.1-5202.5-i8 Base field $$\Q(\sqrt{-2})$$ Conductor $$(51a)$$ Conductor norm $$5202$$ CM no Base change yes: 102.c1,3264.bc1 Q-curve yes Torsion order $$4$$ Rank $$0$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{-2})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))

gp: K = nfinit(Pol(Vecrev([2, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);

Weierstrass equation

$${y}^2+{x}{y}={x}^{3}-27744{x}-1781010$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([-27744,0]),K([-1781010,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-27744,0])),Pol(Vecrev([-1781010,0]))], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-27744,0],K![-1781010,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(51a)$$ = $$(a)\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5202$$ = $$2\cdot3\cdot3\cdot17\cdot17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(5202)$$ = $$(a)^{2}\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(-2a+3)^{2}\cdot(2a+3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$27060804$$ = $$2^{2}\cdot3^{2}\cdot3^{2}\cdot17^{2}\cdot17^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{2361739090258884097}{5202}$$ 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: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-96 : -3 a + 48 : 1\right)$ 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: $$0.183754147211948$$ Tamagawa product: $$32$$  =  $$2\cdot2\cdot2\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$4.15788171407103$$ Analytic order of Ш: $$16$$ (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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(-a-1)$$ $$3$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(a-1)$$ $$3$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(-2a+3)$$ $$17$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(2a+3)$$ $$17$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

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 5202.5-i consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of 102.c1, 3264.bc1, defined over $$\Q$$, so it is also a $$\Q$$-curve.