# Properties

 Label 4.4.2304.1-72.1-b6 Base field $$\Q(\sqrt{2}, \sqrt{3})$$ Conductor $$(-a^3+a^2+5a-5)$$ Conductor norm $$72$$ CM no Base change yes: 24.a2,144.b2,192.d2,576.b2 Q-curve yes Torsion order $$16$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{2}, \sqrt{3})$$

Generator $$a$$, with minimal polynomial $$x^{4} - 4 x^{2} + 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -4, 0, 1]))

gp: K = nfinit(Pol(Vecrev([1, 0, -4, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 0, 1]);

## Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(16a^{3}-40a^{2}+16a+1\right){x}-119a^{3}+249a^{2}-16a-42$$
sage: E = EllipticCurve([K([1,1,0,0]),K([-2,0,1,0]),K([-1,0,1,0]),K([1,16,-40,16]),K([-42,-16,249,-119])])

gp: E = ellinit([Pol(Vecrev([1,1,0,0])),Pol(Vecrev([-2,0,1,0])),Pol(Vecrev([-1,0,1,0])),Pol(Vecrev([1,16,-40,16])),Pol(Vecrev([-42,-16,249,-119]))], K);

magma: E := EllipticCurve([K![1,1,0,0],K![-2,0,1,0],K![-1,0,1,0],K![1,16,-40,16],K![-42,-16,249,-119]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-a^3+a^2+5a-5)$$ = $$(a^3-4a+1)^{3}\cdot(a^2-2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$72$$ = $$2^{3}\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(6)$$ = $$(a^3-4a+1)^{4}\cdot(a^2-2)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1296$$ = $$2^{4}\cdot9^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{28756228}{3}$$ 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 $\left(-13 a^{3} + 26 a^{2} + a - 5 : -193 a^{3} + 372 a^{2} + 52 a - 100 : 1\right)$ Height $$0.539636932338590$$ Torsion structure: $$\Z/2\Z\oplus\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(2 a^{3} - 4 a^{2} - 3 a + 3 : a^{3} - a^{2} : 1\right)$ $\left(3 a^{2} - 8 a + 4 : 4 a^{3} - 17 a^{2} + 21 a - 7 : 1\right)$ 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: $$0.539636932338590$$ Period: $$2768.34635211226$$ Tamagawa product: $$4$$  =  $$2\cdot2$$ Torsion order: $$16$$ Leading coefficient: $$1.94518480872993$$ 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$$(j)_{-}$$
$$(a^3-4a+1)$$ $$2$$ $$2$$ $$III$$ Additive $$1$$ $$3$$ $$4$$ $$0$$
$$(a^2-2)$$ $$9$$ $$2$$ $$I_{2}$$ 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
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8 and 16.
Its isogeny class 72.1-b consists of curves linked by isogenies of degrees dividing 32.

## Base change

This curve is the base change of 24.a2, 144.b2, 192.d2, 576.b2, defined over $$\Q$$, so it is also a $$\Q$$-curve.