Properties

 Base field $$\Q(\sqrt{-11})$$ Label 2.0.11.1-1600.2-c1 Conductor $$(40)$$ Conductor norm $$1600$$ CM no base-change yes: 4840.f4,40.a4 Q-curve yes Torsion order $$4$$ Rank $$0$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{-11})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x + 3$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 3)
gp (2.8): K = nfinit(a^2 - a + 3);

Weierstrass equation

$$y^2 = x^{3} + 13 x - 34$$
magma: E := ChangeRing(EllipticCurve([0, 0, 0, 13, -34]),K);
sage: E = EllipticCurve(K, [0, 0, 0, 13, -34])
gp (2.8): E = ellinit([0, 0, 0, 13, -34],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(40)$$ = $$\left(2\right)^{3} \cdot \left(-a - 1\right) \cdot \left(a - 2\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$1600$$ = $$4^{3} \cdot 5^{2}$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(640000)$$ = $$\left(2\right)^{10} \cdot \left(-a - 1\right)^{4} \cdot \left(a - 2\right)^{4}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$409600000000$$ = $$4^{10} \cdot 5^{8}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{237276}{625}$$ 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: $$0$$
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: $$\Z/4\Z$$ 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] $\left(7 : -20 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

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$$(j)_{-}$$
$$\left(-a - 1\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a - 2\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2\right)$$ $$4$$ $$2$$ $$III^*$$ Additive $$-1$$ $$3$$ $$10$$ $$0$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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 and 4.
Its isogeny class 1600.2-c consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 4840.f4, 40.a4, defined over $$\Q$$, so it is also a $$\Q$$-curve.