# Properties

 Label 2.2.8.1-4802.1-z2 Base field $$\Q(\sqrt{2})$$ Conductor $$(49a)$$ Conductor norm $$4802$$ CM no Base change yes: 98.a5,3136.e5 Q-curve yes Torsion order $$2$$ 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

$${y}^2+{x}{y}={x}^{3}+{x}^{2}-25{x}-111$$
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([0,0]),K([-25,0]),K([-111,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-25,0])),Pol(Vecrev([-111,0]))], K);

magma: E := EllipticCurve([K![1,0],K![1,0],K![0,0],K![-25,0],K![-111,0]]);

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: $$(-3294172)$$ = $$(a)^{4}\cdot(-2a+1)^{7}\cdot(2a+1)^{7}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$10851569165584$$ = $$2^{4}\cdot7^{7}\cdot7^{7}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{15625}{28}$$ 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: $\left(6 : -3 : 1\right)$ 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: $$1.00395830084287$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$0.709905722554514$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$(-2a+1)$$ $$7$$ $$2$$ $$I_{1}^{*}$$ Additive $$-1$$ $$2$$ $$7$$ $$1$$
$$(2a+1)$$ $$7$$ $$2$$ $$I_{1}^{*}$$ Additive $$-1$$ $$2$$ $$7$$ $$1$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6, 9, 12, 18 and 36.
Its isogeny class 4802.1-z consists of curves linked by isogenies of degrees dividing 36.

## Base change

This curve is the base change of 98.a5, 3136.e5, defined over $$\Q$$, so it is also a $$\Q$$-curve.