# Properties

 Label 4.4.3600.1-576.1-e3 Base field $$\Q(\sqrt{3}, \sqrt{5})$$ Conductor $$(-4/7a^3+6/7a^2+38/7a+22/7)$$ Conductor norm $$576$$ CM no Base change yes: 48.a1,1200.d1,72.a1,1800.m1 Q-curve yes Torsion order $$8$$ Rank $$2$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{3}, \sqrt{5})$$

Generator $$a$$, with minimal polynomial $$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 8, -7, -2, 1]))

gp: K = nfinit(Pol(Vecrev([1, 8, -7, -2, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -7, -2, 1]);

## Weierstrass equation

$${y}^2+\left(-\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{19}{7}a-\frac{3}{7}\right){x}{y}+\left(-\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{19}{7}a-\frac{3}{7}\right){y}={x}^{3}+\left(\frac{1}{7}a^{3}-\frac{5}{7}a^{2}-\frac{6}{7}a+\frac{26}{7}\right){x}^{2}+\left(\frac{2}{7}a^{3}+\frac{5373}{7}a^{2}-\frac{5395}{7}a-\frac{42340}{7}\right){x}+\frac{1}{7}a^{3}-\frac{152808}{7}a^{2}+\frac{152797}{7}a+\frac{1203046}{7}$$
sage: E = EllipticCurve([K([-3/7,19/7,3/7,-2/7]),K([26/7,-6/7,-5/7,1/7]),K([-3/7,19/7,3/7,-2/7]),K([-42340/7,-5395/7,5373/7,2/7]),K([1203046/7,152797/7,-152808/7,1/7])])

gp: E = ellinit([Pol(Vecrev([-3/7,19/7,3/7,-2/7])),Pol(Vecrev([26/7,-6/7,-5/7,1/7])),Pol(Vecrev([-3/7,19/7,3/7,-2/7])),Pol(Vecrev([-42340/7,-5395/7,5373/7,2/7])),Pol(Vecrev([1203046/7,152797/7,-152808/7,1/7]))], K);

magma: E := EllipticCurve([K![-3/7,19/7,3/7,-2/7],K![26/7,-6/7,-5/7,1/7],K![-3/7,19/7,3/7,-2/7],K![-42340/7,-5395/7,5373/7,2/7],K![1203046/7,152797/7,-152808/7,1/7]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-4/7a^3+6/7a^2+38/7a+22/7)$$ = $$(-2/7a^3+3/7a^2+19/7a-3/7)^{3}\cdot(-2/7a^3+3/7a^2+19/7a-10/7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$576$$ = $$4^{3}\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(288)$$ = $$(-2/7a^3+3/7a^2+19/7a-3/7)^{10}\cdot(-2/7a^3+3/7a^2+19/7a-10/7)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$6879707136$$ = $$4^{10}\cdot9^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{3065617154}{9}$$ 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: $$2$$ Generators $\left(\frac{52}{7} a^{3} - \frac{85}{7} a^{2} - \frac{375}{7} a + \frac{260}{7} : -\frac{429}{7} a^{3} + \frac{367}{7} a^{2} + \frac{3435}{7} a + \frac{487}{7} : 1\right)$ $\left(\frac{39}{14} a^{3} - \frac{111}{14} a^{2} - \frac{117}{7} a + \frac{573}{14} : \frac{229}{7} a^{3} - \frac{849}{28} a^{2} - \frac{7295}{28} a - \frac{537}{28} : 1\right)$ Heights $$0.374241721443891$$ $$0.832965173865866$$ Torsion structure: $$\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(\frac{40}{7} a^{3} - \frac{123}{7} a^{2} - \frac{233}{7} a + \frac{662}{7} : 89 a^{3} - 181 a^{2} - 609 a + 723 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.311730320570371$$ Period: $$516.836305096209$$ Tamagawa product: $$8$$  =  $$2\cdot2^{2}$$ Torsion order: $$8$$ Leading coefficient: $$5.37045156900158$$ 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$$(j)_{-}$$
$$(-2/7a^3+3/7a^2+19/7a-3/7)$$ $$4$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$3$$ $$10$$ $$0$$
$$(-2/7a^3+3/7a^2+19/7a-10/7)$$ $$9$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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, 4, 8 and 16.
Its isogeny class 576.1-e consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of elliptic curves 48.a1, 1200.d1, 72.a1, 1800.m1, defined over $$\Q$$, so it is also a $$\Q$$-curve.