# Properties

 Base field $$\Q(\sqrt{-11})$$ Label 2.0.11.1-207.6-a1 Conductor $$(2 a + 13)$$ Conductor norm $$207$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

Generator $$a$$, with minimal polynomial $$x^{2} - x + 3$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 3)

gp: K = nfinit(a^2 - a + 3);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);

## Weierstrass equation

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

gp: E = ellinit([a + 1, -1, a, -3*a + 2, a + 1],K)

magma: E := ChangeRing(EllipticCurve([a + 1, -1, a, -3*a + 2, a + 1]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(2 a + 13)$$ = $$\left(a - 1\right)^{2} \cdot \left(a - 5\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$207$$ = $$3^{2} \cdot 23$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(-214 a + 265)$$ = $$\left(a - 1\right)^{8} \cdot \left(a - 5\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$150903$$ = $$3^{8} \cdot 23$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{401402}{207} a + \frac{234017}{69}$$ 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: $\left(0 : 1 : 1\right)$

sage: gens = E.gens(); gens

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

Height: 0.07141236712408516

sage: [P.height() for P in gens]

magma: [Height(P):P in gens];

Regulator: 0.0714123671241

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(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$$(j)_{-}$$
$$\left(a - 1\right)$$ $$3$$ $$2$$ $$I_{2}^*$$ Additive $$-1$$ $$2$$ $$8$$ $$2$$
$$\left(a - 5\right)$$ $$23$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 207.6-a consists of this curve only.

## Base change

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