# Properties

 Base field $$\Q(\sqrt{11})$$ Label 2.2.44.1-200.1-e2 Conductor $$(10 a + 30)$$ Conductor norm $$200$$ CM no base-change no Q-curve no Torsion order $$2$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{11})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 11$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 11)

gp: K = nfinit(a^2 - 11);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y = x^{3} + \left(-a + 1\right) x^{2} + \left(117 a - 387\right) x + 1500 a - 4975$$
sage: E = EllipticCurve(K, [a + 1, -a + 1, 0, 117*a - 387, 1500*a - 4975])

gp: E = ellinit([a + 1, -a + 1, 0, 117*a - 387, 1500*a - 4975],K)

magma: E := ChangeRing(EllipticCurve([a + 1, -a + 1, 0, 117*a - 387, 1500*a - 4975]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(10 a + 30)$$ = $$\left(a + 3\right)^{3} \cdot \left(a - 4\right) \cdot \left(-a - 4\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$200$$ = $$2^{3} \cdot 5^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(4960 a - 27760)$$ = $$\left(a + 3\right)^{8} \cdot \left(a - 4\right) \cdot \left(-a - 4\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$500000000$$ = $$2^{8} \cdot 5^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{179425843432}{390625} a + \frac{595817255308}{390625}$$ 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); $\left(\frac{5}{2} a - 10 : \frac{15}{4} a - \frac{35}{4} : 1\right)$ 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$$(j)_{-}$$
$$\left(a + 3\right)$$ $$2$$ $$2$$ $$I_{1}^*$$ Additive $$1$$ $$3$$ $$8$$ $$0$$
$$\left(a - 4\right)$$ $$5$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(-a - 4\right)$$ $$5$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$

## 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 and 4.
Its isogeny class 200.1-e consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.