# Properties

 Label 5.5.24217.1-43.1-b1 Base field 5.5.24217.1 Conductor norm $$43$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field5.5.24217.1

Generator $$a$$, with minimal polynomial $$x^{5} - 5 x^{3} - x^{2} + 3 x + 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 3, -1, -5, 0, 1]))

gp: K = nfinit(Polrev([1, 3, -1, -5, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -1, -5, 0, 1]);

## Weierstrass equation

$${y}^2+\left(2a^{4}-a^{3}-9a^{2}+3a+4\right){x}{y}+\left(-a^{4}+a^{3}+5a^{2}-3a-2\right){y}={x}^{3}+\left(4a^{4}-2a^{3}-19a^{2}+6a+7\right){x}^{2}+\left(-2a^{4}+3a^{3}+9a^{2}-13a-4\right){x}-3a^{3}-a^{2}+13a+5$$
sage: E = EllipticCurve([K([4,3,-9,-1,2]),K([7,6,-19,-2,4]),K([-2,-3,5,1,-1]),K([-4,-13,9,3,-2]),K([5,13,-1,-3,0])])

gp: E = ellinit([Polrev([4,3,-9,-1,2]),Polrev([7,6,-19,-2,4]),Polrev([-2,-3,5,1,-1]),Polrev([-4,-13,9,3,-2]),Polrev([5,13,-1,-3,0])], K);

magma: E := EllipticCurve([K![4,3,-9,-1,2],K![7,6,-19,-2,4],K![-2,-3,5,1,-1],K![-4,-13,9,3,-2],K![5,13,-1,-3,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(3a^4-a^3-14a^2+3a+6)$$ = $$(3a^4-a^3-14a^2+3a+6)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$43$$ = $$43$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-4a^4+3a^3+18a^2-9a-11)$$ = $$(3a^4-a^3-14a^2+3a+6)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-1849$$ = $$-43^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{673276224}{1849} a^{4} + \frac{251558661}{1849} a^{3} + \frac{3275824811}{1849} a^{2} - \frac{548956180}{1849} a - \frac{1829937870}{1849}$$ 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 $\left(-a^{4} + 6 a^{2} - 2 a - 2 : -a^{4} + 3 a^{3} - 4 a - 1 : 1\right)$ Height $$0.0013956060388066603632085360759871885049$$ 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: $$0.0013956060388066603632085360759871885049$$ Period: $$16404.266367643609770763198008809907217$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$1.47115856$$ 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$$(j)_{-}$$
$$(3a^4-a^3-14a^2+3a+6)$$ $$43$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 43.1-b consists of this curve only.

## Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.