Properties

 Base field $$\Q(\sqrt{57})$$ Label 2.2.57.1-121.1-d2 Conductor $$(11)$$ Conductor norm $$121$$ CM no base-change yes: 11.a2,35739.a2 Q-curve yes Torsion order $$5$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{57})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 14$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 14)

gp: K = nfinit(a^2 - a - 14);

Weierstrass equation

$$y^2 + y = x^{3} - x^{2} - 10 x - 20$$
magma: E := ChangeRing(EllipticCurve([0, -1, 1, -10, -20]),K);

sage: E = EllipticCurve(K, [0, -1, 1, -10, -20])

gp: E = ellinit([0, -1, 1, -10, -20],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(11)$$ = $$\left(11\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$121$$ = $$121$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(161051)$$ = $$\left(11\right)^{5}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$25937424601$$ = $$121^{5}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{122023936}{161051}$$ 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: $$\Z/5\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(5 : -6 : 1\right)$ 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$$(j)_{-}$$
$$\left(11\right)$$ $$121$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$

Galois Representations

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

prime Image of Galois Representation
$$5$$ 5Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 121.1-d consists of curves linked by isogenies of degrees dividing 25.

Base change

This curve is the base-change of elliptic curves 11.a2, 35739.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.