# Properties

 Base field $$\Q(\sqrt{2})$$ Label 2.2.8.1-3721.1-a1 Conductor $$(61)$$ Conductor norm $$3721$$ CM no base-change yes: 61.a1,3904.j1 Q-curve yes Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 2)
gp (2.8): K = nfinit(a^2 - 2);

## Weierstrass equation

$$y^2 + x y = x^{3} - 2 x + 1$$
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -2, 1]),K);
sage: E = EllipticCurve(K, [1, 0, 0, -2, 1])
gp (2.8): E = ellinit([1, 0, 0, -2, 1],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(61)$$ = $$\left(61\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$3721$$ = $$3721$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(61)$$ = $$\left(61\right)$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$3721$$ = $$3721$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-\frac{912673}{61}$$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$1$$
magma: Rank(E);
sage: E.rank()

Generator: $\left(1 : -1 : 1\right)$

Height: 0.07918773136204195

magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 0.079187731362

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: Trivial magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1]

## Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(61\right)$$ $$3721$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 3721.1-a consists of this curve only.

## Base change

This curve is the base-change of elliptic curves 61.a1, 3904.j1, defined over $$\Q$$, so it is also a $$\Q$$-curve.