# Properties

 Label 2.2.40.1-225.1-g1 Base field $$\Q(\sqrt{10})$$ Conductor norm $$225$$ CM no Base change yes Q-curve yes Torsion order $$1$$ Rank $$1$$

# Learn more

Show commands: Magma / PariGP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-10, 0, 1]))

gp: K = nfinit(Polrev([-10, 0, 1]));

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

## Weierstrass equation

$${y}^2+{y}={x}^{3}-{x}^{2}-8{x}-7$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([1,0]),K([-8,0]),K([-7,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-8,0]),Polrev([-7,0])], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![1,0],K![-8,0],K![-7,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(15)$$ = $$(3,a+1)\cdot(3,a+2)\cdot(5,a)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$225$$ = $$3\cdot3\cdot5^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-1875)$$ = $$(3,a+1)\cdot(3,a+2)\cdot(5,a)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$3515625$$ = $$3\cdot3\cdot5^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{102400}{3}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$1$$ Generator $\left(\frac{2041}{490} : -\frac{41791}{34300} a - \frac{1}{2} : 1\right)$ Height $$7.9005437144427785448843110923353544712$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$7.9005437144427785448843110923353544712$$ Period: $$1.9671182839017211129536106600184551766$$ Tamagawa product: $$1$$  =  $$1\cdot1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$4.9145918428357401773961135235037324200$$ Analytic order of Ш: $$1$$ (rounded)

## 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$$(j)_{-}$$
$$(3,a+1)$$ $$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(3,a+2)$$ $$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(5,a)$$ $$5$$ $$1$$ $$IV^{*}$$ Additive $$-1$$ $$2$$ $$8$$ $$0$$

## 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.1.3

## Isogenies and isogeny class

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

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 75.c1
$$\Q$$ 4800.bb1