Properties

 Base field $$\Q(\sqrt{2}, \sqrt{5})$$ Label 4.4.1600.1-400.1-a5 Conductor $$(10,2 a^{2} - 6)$$ Conductor norm $$400$$ CM no base-change yes: 320.f3,1600.c3,20.a3,100.a3 Q-curve yes Torsion order $$12$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{2}, \sqrt{5})$$

Generator $$a$$, with minimal polynomial $$x^{4} - 6 x^{2} + 4$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -6, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 6*x^2 + 4)

gp: K = nfinit(a^4 - 6*a^2 + 4);

Weierstrass equation

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

sage: E = EllipticCurve(K, [0, 1, 0, -1, 0])

gp: E = ellinit([0, 1, 0, -1, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(10,2 a^{2} - 6)$$ = $$\left(\frac{1}{2} a^{3} - 2 a\right)^{2} \cdot \left(-a^{2} + 3\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$400$$ = $$4^{2} \cdot 25$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(80,40 a^{2} - 80,40 a^{3} - 160 a,80 a)$$ = $$\left(\frac{1}{2} a^{3} - 2 a\right)^{8} \cdot \left(-a^{2} + 3\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$40960000$$ = $$4^{8} \cdot 25^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{16384}{5}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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 not available.

magma: Rank(E);

sage: E.rank()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

Torsion subgroup

Structure: $$\Z/2\Z\times\Z/6\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-1 : 1 : 1\right)$,$\left(\frac{1}{2} a^{2} - 2 : 0 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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$$(j)_{-}$$
$$\left(\frac{1}{2} a^{3} - 2 a\right)$$ $$4$$ $$3$$ $$IV^*$$ Additive $$1$$ $$2$$ $$8$$ $$0$$
$$\left(-a^{2} + 3\right)$$ $$25$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 400.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 320.f3, 1600.c3, 20.a3, 100.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.