# Properties

 Base field $$\Q(\sqrt{69})$$ Label 2.2.69.1-92.1-a2 Conductor $$(-6 a + 26)$$ Conductor norm $$92$$ CM no base-change yes: 9522.p1,46.a1 Q-curve yes Torsion order $$2$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{69})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 17$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 17)

gp: K = nfinit(a^2 - a - 17);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, -1, 1]);

## Weierstrass equation

$$y^2 + x y = x^{3} - x^{2} - 170 x - 812$$
sage: E = EllipticCurve(K, [1, -1, 0, -170, -812])

gp: E = ellinit([1, -1, 0, -170, -812],K)

magma: E := ChangeRing(EllipticCurve([1, -1, 0, -170, -812]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-6 a + 26)$$ = $$\left(2\right) \cdot \left(-3 a + 13\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$92$$ = $$4 \cdot 23$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(16928)$$ = $$\left(2\right)^{5} \cdot \left(-3 a + 13\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$286557184$$ = $$4^{5} \cdot 23^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{545138290809}{16928}$$ 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 not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-\frac{29}{4} : \frac{29}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(-3 a + 13\right)$$ $$23$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2\right)$$ $$4$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$

## 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 92.1-a consists of curves linked by isogenies of degree 2.

## Base change

This curve is the base-change of elliptic curves 9522.p1, 46.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.