# Properties

 Label 2.2.41.1-361.1-a2 Base field $$\Q(\sqrt{41})$$ Conductor $$(19)$$ Conductor norm $$361$$ CM no Base change yes: 19.a2,31939.e2 Q-curve yes Torsion order $$3$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{41})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 10$$; class number $$1$$.

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

gp: K = nfinit(Pol(Vecrev([-10, -1, 1])));

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

## Weierstrass equation

$${y}^2+{y}={x}^{3}+{x}^{2}-9{x}-15$$
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([-9,0]),K([-15,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-9,0])),Pol(Vecrev([-15,0]))], K);

magma: E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![-9,0],K![-15,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(19)$$ = $$(19)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$361$$ = $$361$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-6859)$$ = $$(19)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$47045881$$ = $$361^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{89915392}{6859}$$ 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 $\left(\frac{328071349}{100600900} : \frac{173802949917}{504513513500} a - \frac{678316463417}{1009027027000} : 1\right)$ Height $$20.1099139648163$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(5 : -10 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$20.1099139648163$$ Period: $$1.84894653281620$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$3.87125142058114$$ 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$$(j)_{-}$$
$$(19)$$ $$361$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 361.1-a consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of elliptic curves 19.a2, 31939.e2, defined over $$\Q$$, so it is also a $$\Q$$-curve.