# Properties

 Label 2.0.3.1-7203.3-a3 Base field $$\Q(\sqrt{-3})$$ Conductor $$(-98a+49)$$ Conductor norm $$7203$$ CM no Base change yes: 441.f4,147.a4 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands for: 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

$${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-1667{x}+72764$$
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-1667,0]),K([72764,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-1667,0])),Pol(Vecrev([72764,0]))], K);

magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-1667,0],K![72764,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-98a+49)$$ = $$(-2a+1)\cdot(-3a+1)^{2}\cdot(3a-2)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$7203$$ = $$3\cdot7^{2}\cdot7^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2034669218547)$$ = $$(-2a+1)^{2}\cdot(-3a+1)^{14}\cdot(3a-2)^{14}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4139878828902659648791209$$ = $$3^{2}\cdot7^{14}\cdot7^{14}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{4354703137}{17294403}$$ 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 $\left(\frac{37}{4} a + \frac{51}{2} : \frac{37}{2} a + \frac{1509}{8} : 1\right)$ Height $$2.63850882313416$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-\frac{221}{4} : \frac{217}{8} : 1\right)$ $\left(-28 a + 41 : 14 a - 21 : 1\right)$ 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: $$2.63850882313416$$ Period: $$0.123153847091018$$ Tamagawa product: $$32$$  =  $$2\cdot2^{2}\cdot2^{2}$$ Torsion order: $$4$$ Leading coefficient: $$1.50084517489950$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(-3a+1)$$ $$7$$ $$4$$ $$I_8^{*}$$ Additive $$-1$$ $$2$$ $$14$$ $$8$$
$$(3a-2)$$ $$7$$ $$4$$ $$I_8^{*}$$ Additive $$-1$$ $$2$$ $$14$$ $$8$$

## 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 and 8.
Its isogeny class 7203.3-a consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of elliptic curves 441.f4, 147.a4, defined over $$\Q$$, so it is also a $$\Q$$-curve.