Properties

 Base field $$\Q(\sqrt{-1})$$ Label 2.0.4.1-5929.1-c2 Conductor $$(77)$$ Conductor norm $$5929$$ CM no base-change yes: 1232.a1,77.c1 Q-curve yes Torsion order $$2$$ Rank $$1$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Base field $$\Q(\sqrt{-1})$$

Generator $$i$$, with minimal polynomial $$x^{2} + 1$$; class number $$1$$.

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

gp: K = nfinit(i^2 + 1);

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

Weierstrass equation

$$y^2 + x y = x^{3} + x^{2} - 51 x + 110$$
sage: E = EllipticCurve(K, [1, 1, 0, -51, 110])

gp: E = ellinit([1, 1, 0, -51, 110],K)

magma: E := ChangeRing(EllipticCurve([1, 1, 0, -51, 110]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(77)$$ = $$\left(7\right) \cdot \left(11\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$5929$$ = $$49 \cdot 121$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(1294139)$$ = $$\left(7\right)^{6} \cdot \left(11\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$1674795751321$$ = $$49^{6} \cdot 121$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{15124197817}{1294139}$$ 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: $$1$$

sage: E.rank()

magma: Rank(E);

Generator: $\left(-\frac{19}{2} : -\frac{49}{4} i + \frac{19}{4} : 1\right)$

sage: gens = E.gens(); gens

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

Height: 0.6755381582936599

sage: [P.height() for P in gens]

magma: [Height(P):P in gens];

Regulator: 0.675538158294

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

Torsion subgroup

Structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(\frac{11}{4} : -\frac{11}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(7\right)$$ $$49$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(11\right)$$ $$121$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 5929.1-c consists of curves linked by isogenies of degree 2.

Base change

This curve is the base-change of elliptic curves 1232.a1, 77.c1, defined over $$\Q$$, so it is also a $$\Q$$-curve.