# Properties

 Base field $$\Q(\sqrt{-11})$$ Label 2.0.11.1-25344.2-z4 Conductor $$(-96 a + 48)$$ Conductor norm $$25344$$ CM no base-change yes: 528.i1,5808.be1 Q-curve yes Torsion order $$4$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

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

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

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

gp: K = nfinit(a^2 - a + 3);

## Weierstrass equation

$$y^2 = x^{3} + x^{2} - 472 x - 4108$$
magma: E := ChangeRing(EllipticCurve([0, 1, 0, -472, -4108]),K);

sage: E = EllipticCurve(K, [0, 1, 0, -472, -4108])

gp: E = ellinit([0, 1, 0, -472, -4108],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-96 a + 48)$$ = $$\left(2\right)^{4} \cdot \left(-a\right) \cdot \left(a - 1\right) \cdot \left(-2 a + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$25344$$ = $$3^{2} \cdot 4^{4} \cdot 11$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(1824768)$$ = $$\left(2\right)^{11} \cdot \left(-a\right)^{4} \cdot \left(a - 1\right)^{4} \cdot \left(-2 a + 1\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$3329778253824$$ = $$3^{8} \cdot 4^{11} \cdot 11^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{5690357426}{891}$$ 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: $$\Z/4\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-16 : -12 a + 6 : 1\right)$ 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$$(j)_{-}$$
$$\left(-a\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a - 1\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(-2 a + 1\right)$$ $$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(2\right)$$ $$4$$ $$4$$ $$I_{3}^*$$ Additive $$1$$ $$4$$ $$11$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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 25344.2-z consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 528.i1, 5808.be1, defined over $$\Q$$, so it is also a $$\Q$$-curve.