Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-17424.5-p1 Conductor $$(132)$$ Conductor norm $$17424$$ CM no base-change yes: 2112.r3,528.j3 Q-curve yes Torsion order $$8$$ Rank not available

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: K = nfinit(a^2 + 2);

Weierstrass equation

$$y^2 + a x y + a y = x^{3} - 47 x - 598$$
magma: E := ChangeRing(EllipticCurve([a, 0, a, -47, -598]),K);

sage: E = EllipticCurve(K, [a, 0, a, -47, -598])

gp: E = ellinit([a, 0, a, -47, -598],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(132)$$ = $$\left(a\right)^{4} \cdot \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(a + 3\right) \cdot \left(a - 3\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$17424$$ = $$2^{4} \cdot 3^{2} \cdot 11^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(151797888)$$ = $$\left(a\right)^{14} \cdot \left(-a - 1\right)^{4} \cdot \left(a - 1\right)^{4} \cdot \left(a + 3\right)^{4} \cdot \left(a - 3\right)^{4}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$23042598801260544$$ = $$2^{14} \cdot 3^{8} \cdot 11^{8}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{192100033}{2371842}$$ 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: $$\Z/2\Z\times\Z/4\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-6 : -14 a : 1\right)$,$\left(4 a - 5 : 2 a + 4 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$4$$ $$I_{6}^*$$ Additive $$1$$ $$4$$ $$14$$ $$2$$
$$\left(-a - 1\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a - 1\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a + 3\right)$$ $$11$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a - 3\right)$$ $$11$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 17424.5-p consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 2112.r3, 528.j3, defined over $$\Q$$, so it is also a $$\Q$$-curve.