Properties

 Base field $$\Q(\sqrt{3})$$ Label 2.2.12.1-1452.1-d2 Conductor $$(22 a)$$ Conductor norm $$1452$$ CM no base-change yes: 528.e1,396.a1 Q-curve yes Torsion order $$4$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

Base field $$\Q(\sqrt{3})$$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);

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

gp: K = nfinit(a^2 - 3);

Weierstrass equation

$$y^2 + \left(a + 1\right) x y = x^{3} + x^{2} + \left(13 a - 20\right) x - 10 a + 18$$
magma: E := ChangeRing(EllipticCurve([a + 1, 1, 0, 13*a - 20, -10*a + 18]),K);

sage: E = EllipticCurve(K, [a + 1, 1, 0, 13*a - 20, -10*a + 18])

gp: E = ellinit([a + 1, 1, 0, 13*a - 20, -10*a + 18],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(22 a)$$ = $$\left(a + 1\right)^{2} \cdot \left(a\right) \cdot \left(-2 a + 1\right) \cdot \left(2 a + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$1452$$ = $$2^{2} \cdot 3 \cdot 11^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(1452)$$ = $$\left(a + 1\right)^{4} \cdot \left(a\right)^{2} \cdot \left(-2 a + 1\right)^{2} \cdot \left(2 a + 1\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$2108304$$ = $$2^{4} \cdot 3^{2} \cdot 11^{4}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{810448}{363}$$ 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: $$\Z/2\Z\times\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-2 a + 2 : 2 : 1\right)$,$\left(-\frac{1}{2} a : \frac{1}{4} a + \frac{3}{4} : 1\right)$ 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$$(j)_{-}$$
$$\left(a + 1\right)$$ $$2$$ $$3$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$
$$\left(a\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(-2 a + 1\right)$$ $$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(2 a + 1\right)$$ $$11$$ $$2$$ $$I_{2}$$ Non-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
$$2$$ 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 1452.1-d consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 528.e1, 396.a1, defined over $$\Q$$, so it is also a $$\Q$$-curve.