Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-100.1-a4 Conductor $$(10)$$ Conductor norm $$100$$ CM no base-change yes: 20.a1,320.a1 Q-curve yes Torsion order $$2$$ Rank $$1$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 2)

gp: K = nfinit(a^2 + 2);

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

Weierstrass equation

$$y^2 = x^{3} + x^{2} - 41 x - 116$$
sage: E = EllipticCurve(K, [0, 1, 0, -41, -116])

gp: E = ellinit([0, 1, 0, -41, -116],K)

magma: E := ChangeRing(EllipticCurve([0, 1, 0, -41, -116]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(10)$$ = $$\left(a\right)^{2} \cdot \left(5\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$100$$ = $$2^{2} \cdot 25$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(2000)$$ = $$\left(a\right)^{8} \cdot \left(5\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$4000000$$ = $$2^{8} \cdot 25^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{488095744}{125}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: $$1$$

sage: E.rank()

magma: Rank(E);

Generator: $\left(-6 : 5 a : 1\right)$

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

Height: 0.2756246093394419

sage: [P.height() for P in gens]

magma: [Height(P):P in gens];

Regulator: 0.275624609339

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

Torsion subgroup

Structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-4 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$1$$ $$IV^*$$ Additive $$-1$$ $$2$$ $$8$$ $$0$$
$$\left(5\right)$$ $$25$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

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 100.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base-change of elliptic curves 20.a1, 320.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.