# Properties

 Label 5.5.24217.1-85.2-b2 Base field 5.5.24217.1 Conductor norm $$85$$ CM no Base change no Q-curve no Torsion order $$2$$ 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

$${y}^2+\left(a^{4}-a^{3}-4a^{2}+4a+2\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+4a+2\right){y}={x}^{3}+\left(-3a^{4}+a^{3}+14a^{2}-3a-6\right){x}^{2}+\left(-50a^{4}+26a^{3}+232a^{2}-74a-103\right){x}+65a^{4}-10a^{3}-330a^{2}-4a+225$$
sage: E = EllipticCurve([K([2,4,-4,-1,1]),K([-6,-3,14,1,-3]),K([2,4,-4,-1,1]),K([-103,-74,232,26,-50]),K([225,-4,-330,-10,65])])

gp: E = ellinit([Polrev([2,4,-4,-1,1]),Polrev([-6,-3,14,1,-3]),Polrev([2,4,-4,-1,1]),Polrev([-103,-74,232,26,-50]),Polrev([225,-4,-330,-10,65])], K);

magma: E := EllipticCurve([K![2,4,-4,-1,1],K![-6,-3,14,1,-3],K![2,4,-4,-1,1],K![-103,-74,232,26,-50],K![225,-4,-330,-10,65]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^4-5a^2-2a+3)$$ = $$(2a^4-a^3-9a^2+2a+3)\cdot(a^2-a-3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$85$$ = $$5\cdot17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-6a^4+3a^3+26a^2-7a-8)$$ = $$(2a^4-a^3-9a^2+2a+3)\cdot(a^2-a-3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-1445$$ = $$-5\cdot17^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{185066934807448}{1445} a^{4} - \frac{87464764407119}{1445} a^{3} - \frac{909727616081443}{1445} a^{2} + \frac{231799410298786}{1445} a + \frac{559295724774971}{1445}$$ 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 $\left(-22 a^{4} + 17 a^{3} + 98 a^{2} - 54 a - 27 : 157 a^{4} - 114 a^{3} - 703 a^{2} + 351 a + 217 : 1\right)$ Height $$0.015098821599520145625866987257809168425$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(\frac{15}{4} a^{4} - \frac{3}{2} a^{3} - \frac{71}{4} a^{2} + \frac{15}{4} a + \frac{35}{4} : \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{13}{8} a^{2} + \frac{1}{2} a + \frac{1}{4} : 1\right)$ 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: $$0.015098821599520145625866987257809168425$$ Period: $$8142.5135995202362044011302905957081506$$ Tamagawa product: $$2$$  =  $$1\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.97506496$$ 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$$(j)_{-}$$
$$(2a^4-a^3-9a^2+2a+3)$$ $$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(a^2-a-3)$$ $$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ 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.
Its isogeny class 85.2-b consists of curves linked by isogenies of degree 2.

## Base change

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

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