Properties

 Base field $$\Q(\sqrt{5})$$ Label 2.2.5.1-1156.1-b3 Conductor $$(34)$$ Conductor norm $$1156$$ CM no base-change yes: 850.e3,34.a3 Q-curve yes Torsion order $$6$$ Rank $$0$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

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

Generator $$\phi$$, with minimal polynomial $$x^{2} - x - 1$$; class number $$1$$.

sage: x = polygen(QQ); K.<phi> = NumberField(x^2 - x - 1)

gp: K = nfinit(phi^2 - phi - 1);

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

Weierstrass equation

$$y^2+xy=x^{3}-43x+105$$
sage: E = EllipticCurve(K, [1, 0, 0, -43, 105])

gp: E = ellinit([1, 0, 0, -43, 105],K)

magma: E := ChangeRing(EllipticCurve([1, 0, 0, -43, 105]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(34)$$ = $$\left(2\right) \cdot \left(17\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$1156$$ = $$4 \cdot 289$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(2312)$$ = $$\left(2\right)^{3} \cdot \left(17\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$5345344$$ = $$4^{3} \cdot 289^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{8805624625}{2312}$$ 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: $$0$$

sage: E.rank()

magma: Rank(E);

sage: gens = E.gens(); gens

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

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

Torsion subgroup

Structure: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(8 : 13 : 1\right)$ 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$$(j)_{-}$$
$$\left(2\right)$$ $$4$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(17\right)$$ $$289$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 1156.1-b consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base-change of elliptic curves 850.e3, 34.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.