# Properties

 Base field $$\Q(\sqrt{-3})$$ Label 2.0.3.1-3025.1-a3 Conductor $$(55)$$ Conductor norm $$3025$$ CM no base-change yes: 55.a2,495.a2 Q-curve yes Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)

gp: K = nfinit(a^2 - a + 1);

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

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y = x^{3} + \left(-29 a + 29\right) x - 52$$
sage: E = EllipticCurve(K, [a + 1, 0, 0, -29*a + 29, -52])

gp: E = ellinit([a + 1, 0, 0, -29*a + 29, -52],K)

magma: E := ChangeRing(EllipticCurve([a + 1, 0, 0, -29*a + 29, -52]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(55)$$ = $$\left(5\right) \cdot \left(11\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$3025$$ = $$25 \cdot 121$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(73205)$$ = $$\left(5\right) \cdot \left(11\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$5358972025$$ = $$25 \cdot 121^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{2749884201}{73205}$$ 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: $\left(-5 a : -6 a + 3 : 1\right)$

sage: gens = E.gens(); gens

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

Height: 0.3296579450056658

sage: [P.height() for P in gens]

magma: [Height(P):P in gens];

Regulator: 0.329657945006

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); $\left(\frac{13}{4} a : -\frac{13}{4} a + \frac{13}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(5\right)$$ $$25$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(11\right)$$ $$121$$ $$4$$ $$I_{4}$$ 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 3025.1-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 55.a2, 495.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.