# Properties

 Label 2.0.3.1-71148.2-b3 Base field $$\Q(\sqrt{-3})$$ Conductor $$(-308a+154)$$ Conductor norm $$71148$$ CM no Base change yes: 1386.g2,462.c2 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / 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(Pol(Vecrev([1, -1, 1])));

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

## Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-5764a+5763\right){x}+170324$$
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([1,0]),K([5763,-5764]),K([170324,0])])

gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([5763,-5764])),Pol(Vecrev([170324,0]))], K);

magma: E := EllipticCurve([K![1,1],K![0,1],K![1,0],K![5763,-5764],K![170324,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-308a+154)$$ = $$(-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(11)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$71148$$ = $$3\cdot4\cdot7\cdot7\cdot121$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(216872764416)$$ = $$(-2a+1)^{12}\cdot(2)^{10}\cdot(-3a+1)^{4}\cdot(3a-2)^{4}\cdot(11)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$47033795945437835821056$$ = $$3^{12}\cdot4^{10}\cdot7^{4}\cdot7^{4}\cdot121^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{21184262604460873}{216872764416}$$ 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 $\left(-\frac{131}{3} a : \frac{1}{9} a - \frac{5}{9} : 1\right)$ Height $$1.08814943393762$$ Torsion structure: $$\Z/2\Z\oplus\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-\frac{191}{4} a : \frac{191}{4} a - \frac{195}{8} : 1\right)$ $\left(-41 a : 41 a - 21 : 1\right)$ 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: $$1.08814943393762$$ Period: $$0.137317003813298$$ Tamagawa product: $$160$$  =  $$2\cdot( 2 \cdot 5 )\cdot2\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$3.45073988168233$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$
$$(2)$$ $$4$$ $$10$$ $$I_{10}$$ Split multiplicative $$-1$$ $$1$$ $$10$$ $$10$$
$$(-3a+1)$$ $$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$(3a-2)$$ $$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$(11)$$ $$121$$ $$2$$ $$I_{2}$$ 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.
Its isogeny class 71148.2-b consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base change of 1386.g2, 462.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.