# Properties

 Label 4.4.5125.1-55.2-d3 Base field 4.4.5125.1 Conductor $$(-3a^2+4a+10)$$ Conductor norm $$55$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field4.4.5125.1

Generator $$a$$, with minimal polynomial $$x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))

gp: K = nfinit(Pol(Vecrev([11, 7, -6, -2, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);

## Weierstrass equation

$${y}^2+a{x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(189a^{3}-180a^{2}-756a-392\right){x}+2259a^{3}-2017a^{2}-9352a-4703$$
sage: E = EllipticCurve([K([0,1,0,0]),K([1,-1,0,0]),K([0,-3,-1,1]),K([-392,-756,-180,189]),K([-4703,-9352,-2017,2259])])

gp: E = ellinit([Pol(Vecrev([0,1,0,0])),Pol(Vecrev([1,-1,0,0])),Pol(Vecrev([0,-3,-1,1])),Pol(Vecrev([-392,-756,-180,189])),Pol(Vecrev([-4703,-9352,-2017,2259]))], K);

magma: E := EllipticCurve([K![0,1,0,0],K![1,-1,0,0],K![0,-3,-1,1],K![-392,-756,-180,189],K![-4703,-9352,-2017,2259]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-3a^2+4a+10)$$ = $$(-a^3+a^2+4a+1)\cdot(a-1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$55$$ = $$5\cdot11$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-3a^3+12a^2+14a-34)$$ = $$(-a^3+a^2+4a+1)^{3}\cdot(a-1)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$166375$$ = $$5^{3}\cdot11^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{133990069789633150033955762628}{6655} a^{3} + \frac{496615724716951900546923293583}{6655} a^{2} - \frac{8693172568676612045861200086}{1331} a - \frac{78523816804016589414106934741}{605}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(4 a^{3} - \frac{25}{4} a^{2} - 13 a + 1 : -\frac{11}{8} a^{3} - 5 a^{2} + 15 a + 22 : 1\right)$ 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: $$8.25979024593038$$ Tamagawa product: $$1$$  =  $$1\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$1.03839979400277$$ Analytic order of Ш: $$36$$ (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$$(j)_{-}$$
$$(-a^3+a^2+4a+1)$$ $$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(a-1)$$ $$11$$ $$1$$ $$I_{3}$$ Non-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
$$2$$ 2B
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 55.2-d consists of curves linked by isogenies of degrees dividing 6.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.