# Properties

 Label 2.0.3.1-5929.2-b2 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$5929$$ CM no Base change yes Q-curve yes Torsion order $$9$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / SageMath

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

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

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

gp: K = nfinit(Polrev([1, -1, 1]));

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

## Weierstrass equation

$${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+49a{x}+600$$
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([1,0]),K([0,49]),K([600,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([1,0]),Polrev([0,49]),Polrev([600,0])], K);

magma: E := EllipticCurve([K![0,0],K![-1,1],K![1,0],K![0,49],K![600,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(77)$$ = $$(-3a+1)\cdot(3a-2)\cdot(11)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5929$$ = $$7\cdot7\cdot121$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-156590819)$$ = $$(-3a+1)^{6}\cdot(3a-2)^{6}\cdot(11)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$24520684595090761$$ = $$7^{6}\cdot7^{6}\cdot121^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{13278380032}{156590819}$$ 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: $$0$$ Torsion structure: $$\Z/3\Z\oplus\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(2 a - 13 : -44 a + 16 : 1\right)$ $\left(0 : -25 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.60132907418281650513706266056939016620$$ Tamagawa product: $$108$$  =  $$( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot3$$ Torsion order: $$9$$ Leading coefficient: $$0.92580667426932679955204861122285511996$$ 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$$(j)_{-}$$
$$(-3a+1)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$(3a-2)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$(11)$$ $$121$$ $$3$$ $$I_{3}$$ 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[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 5929.2-b consists of curves linked by isogenies of degrees dividing 9.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 77.b2
$$\Q$$ 693.b2