# Properties

 Base field $$\Q(\sqrt{33})$$ Label 2.2.33.1-300.1-g8 Conductor $$(-20 a + 70)$$ Conductor norm $$300$$ CM no base-change yes: 3630.w1,90.c1 Q-curve yes Torsion order $$6$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(a^2 - a - 8);

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

## Weierstrass equation

$$y^2 + x y + \left(a + 1\right) y = x^{3} + \left(-a - 1\right) x^{2} + \left(1962736 a - 6618897\right) x - 2535640706 a + 8550893793$$
sage: E = EllipticCurve(K, [1, -a - 1, a + 1, 1962736*a - 6618897, -2535640706*a + 8550893793])

gp: E = ellinit([1, -a - 1, a + 1, 1962736*a - 6618897, -2535640706*a + 8550893793],K)

magma: E := ChangeRing(EllipticCurve([1, -a - 1, a + 1, 1962736*a - 6618897, -2535640706*a + 8550893793]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-20 a + 70)$$ = $$\left(-a - 2\right) \cdot \left(-a + 3\right) \cdot \left(-2 a + 7\right) \cdot \left(5\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$300$$ = $$2^{2} \cdot 3 \cdot 25$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(81000)$$ = $$\left(-a - 2\right)^{3} \cdot \left(-a + 3\right)^{3} \cdot \left(-2 a + 7\right)^{8} \cdot \left(5\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$6561000000$$ = $$2^{6} \cdot 3^{8} \cdot 25^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{16778985534208729}{81000}$$ 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: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-391 a + 1321 : -4035 a + 13604 : 1\right)$ 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$$(j)_{-}$$
$$\left(-a - 2\right)$$ $$2$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(-a + 3\right)$$ $$2$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(-2 a + 7\right)$$ $$3$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$\left(5\right)$$ $$25$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## 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, 4, 6 and 12.
Its isogeny class 300.1-g consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base-change of elliptic curves 3630.w1, 90.c1, defined over $$\Q$$, so it is also a $$\Q$$-curve.