# Properties

 Base field $$\Q(\sqrt{3})$$ Label 2.2.12.1-200.1-a1 Conductor $$(10 a + 10)$$ Conductor norm $$200$$ CM no base-change yes: 40.a4,720.e4 Q-curve yes Torsion order $$4$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(a^2 - 3);

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

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y = x^{3} + \left(a - 1\right) x^{2} + \left(13 a + 25\right) x - 52 a - 91$$
sage: E = EllipticCurve(K, [a + 1, a - 1, 0, 13*a + 25, -52*a - 91])

gp: E = ellinit([a + 1, a - 1, 0, 13*a + 25, -52*a - 91],K)

magma: E := ChangeRing(EllipticCurve([a + 1, a - 1, 0, 13*a + 25, -52*a - 91]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(10 a + 10)$$ = $$\left(a + 1\right)^{3} \cdot \left(5\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$200$$ = $$2^{3} \cdot 25$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(10000)$$ = $$\left(a + 1\right)^{8} \cdot \left(5\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$100000000$$ = $$2^{8} \cdot 25^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{237276}{625}$$ 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: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(3 a + 7 : -20 a - 33 : 1\right)$ 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$$(j)_{-}$$
$$\left(a + 1\right)$$ $$2$$ $$4$$ $$I_{1}^*$$ Additive $$-1$$ $$3$$ $$8$$ $$0$$
$$\left(5\right)$$ $$25$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 40.a4, 720.e4, defined over $$\Q$$, so it is also a $$\Q$$-curve.