# Properties

 Base field $$\Q(\sqrt{10})$$ Label 2.2.40.1-225.1-j2 Conductor $$(15)$$ Conductor norm $$225$$ CM no base-change yes: 4800.bq2,75.a2 Q-curve yes Torsion order $$5$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{10})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 10$$; class number $$2$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 10)

gp: K = nfinit(a^2 - 10);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, 0, 1]);

## Weierstrass equation

$$y^2 + y = x^{3} + x^{2} + 2 x + 4$$
sage: E = EllipticCurve(K, [0, 1, 1, 2, 4])

gp: E = ellinit([0, 1, 1, 2, 4],K)

magma: E := ChangeRing(EllipticCurve([0, 1, 1, 2, 4]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(15)$$ = $$\left(3, a + 1\right) \cdot \left(3, a + 2\right) \cdot \left(5, a\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$225$$ = $$3^{2} \cdot 5^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(6075)$$ = $$\left(3, a + 1\right)^{5} \cdot \left(3, a + 2\right)^{5} \cdot \left(5, a\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$36905625$$ = $$3^{10} \cdot 5^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{20480}{243}$$ 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: $$\Z/5\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-1 : -2 : 1\right)$ 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$$(j)_{-}$$
$$\left(3, a + 1\right)$$ $$3$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$
$$\left(3, a + 2\right)$$ $$3$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$
$$\left(5, a\right)$$ $$5$$ $$3$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 225.1-j consists of curves linked by isogenies of degree 5.

## Base change

This curve is the base-change of elliptic curves 4800.bq2, 75.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.