# Properties

 Base field $$\Q(\sqrt{33})$$ Label 2.2.33.1-196.1-a6 Conductor $$(14)$$ Conductor norm $$196$$ CM no base-change yes: 1694.e1,126.b1 Q-curve yes Torsion order $$6$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{33})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 8$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 8)

gp: K = nfinit(a^2 - a - 8);

## Weierstrass equation

$$y^2 + x y + \left(a + 1\right) y = x^{3} + \left(a + 1\right) x^{2} + \left(-1004831 a - 2383742\right) x + 927667860 a + 2200689139$$
magma: E := ChangeRing(EllipticCurve([1, a + 1, a + 1, -1004831*a - 2383742, 927667860*a + 2200689139]),K);

sage: E = EllipticCurve(K, [1, a + 1, a + 1, -1004831*a - 2383742, 927667860*a + 2200689139])

gp: E = ellinit([1, a + 1, a + 1, -1004831*a - 2383742, 927667860*a + 2200689139],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(14)$$ = $$\left(-a - 2\right) \cdot \left(-a + 3\right) \cdot \left(7\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$196$$ = $$2^{2} \cdot 49$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(25088)$$ = $$\left(-a - 2\right)^{9} \cdot \left(-a + 3\right)^{9} \cdot \left(7\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$629407744$$ = $$2^{18} \cdot 49^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{2251439055699625}{25088}$$ 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: $$\Z/6\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(199 a + 473 : -2732 a - 6481 : 1\right)$ 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$$(j)_{-}$$
$$\left(-a - 2\right)$$ $$2$$ $$9$$ $$I_{9}$$ Split multiplicative $$-1$$ $$1$$ $$9$$ $$9$$
$$\left(-a + 3\right)$$ $$2$$ $$9$$ $$I_{9}$$ Split multiplicative $$-1$$ $$1$$ $$9$$ $$9$$
$$\left(7\right)$$ $$49$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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, 6, 9 and 18.
Its isogeny class 196.1-a consists of curves linked by isogenies of degrees dividing 18.

## Base change

This curve is the base-change of elliptic curves 1694.e1, 126.b1, defined over $$\Q$$, so it is also a $$\Q$$-curve.