# Properties

 Label 2.2.8.1-4802.1-z10 Base field $$\Q(\sqrt{2})$$ Conductor norm $$4802$$ CM no Base change yes Q-curve yes Torsion order $$4$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([-2, 0, 1]));

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

## Weierstrass equation

$${y}^2+{x}{y}={x}^{3}+{x}^{2}-133795{x}+18781197$$
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([0,0]),K([-133795,0]),K([18781197,0])])

gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([0,0]),Polrev([-133795,0]),Polrev([18781197,0])], K);

magma: E := EllipticCurve([K![1,0],K![1,0],K![0,0],K![-133795,0],K![18781197,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(49a)$$ = $$(a)\cdot(-2a+1)^{2}\cdot(2a+1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$4802$$ = $$2\cdot7^{2}\cdot7^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(2951578112)$$ = $$(a)^{18}\cdot(-2a+1)^{8}\cdot(2a+1)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$8711813351237484544$$ = $$2^{18}\cdot7^{8}\cdot7^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{2251439055699625}{25088}$$ 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: $$\Z/2\Z\oplus\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-224 a - 106 : 112 a + 53 : 1\right)$ $\left(\frac{843}{4} : -\frac{843}{8} : 1\right)$ 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: $$1.0039583008428659672203286732196336592$$ Tamagawa product: $$32$$  =  $$2\cdot2^{2}\cdot2^{2}$$ Torsion order: $$4$$ Leading coefficient: $$0.70990572255451447651953290981238027339$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$I_{18}$$ Non-split multiplicative $$1$$ $$1$$ $$18$$ $$18$$
$$(-2a+1)$$ $$7$$ $$4$$ $$I_{2}^{*}$$ Additive $$-1$$ $$2$$ $$8$$ $$2$$
$$(2a+1)$$ $$7$$ $$4$$ $$I_{2}^{*}$$ Additive $$-1$$ $$2$$ $$8$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 6, 9 and 18.
Its isogeny class 4802.1-z consists of curves linked by isogenies of degrees dividing 36.

## Base change

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

Base field Curve
$$\Q$$ 98.a1
$$\Q$$ 3136.e1