Properties

 Base field $$\Q(\sqrt{-11})$$ Label 2.0.11.1-6400.2-k4 Conductor $$(80)$$ Conductor norm $$6400$$ CM no base-change yes: 9680.q1,80.a1 Q-curve yes Torsion order $$4$$ Rank $$1$$

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

$$y^2 = x^{3} - 107 x + 426$$
magma: E := ChangeRing(EllipticCurve([0, 0, 0, -107, 426]),K);

sage: E = EllipticCurve(K, [0, 0, 0, -107, 426])

gp (2.8): E = ellinit([0, 0, 0, -107, 426],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(80)$$ = $$\left(2\right)^{4} \cdot \left(-a - 1\right) \cdot \left(a - 2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$6400$$ = $$4^{4} \cdot 5^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(5120)$$ = $$\left(2\right)^{10} \cdot \left(-a - 1\right) \cdot \left(a - 2\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$26214400$$ = $$4^{10} \cdot 5^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{132304644}{5}$$ 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: $\left(-\frac{5}{9} a + \frac{20}{3} : -\frac{65}{27} a + \frac{26}{9} : 1\right)$

Height: 2.1780391255541485

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: 2.17803912555

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

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 6400.2-k consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 9680.q1, 80.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.