# Properties

 Label 4.4.5125.1-19.1-b1 Base field 4.4.5125.1 Conductor $$(a^3-2a^2-3a+2)$$ Conductor norm $$19$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$1$$

# 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

$${y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+6a+4\right){x}^{2}+\left(-7a^{3}-4a^{2}+32a+32\right){x}+11a^{3}+9a^{2}-43a-45$$
sage: E = EllipticCurve([K([0,0,0,0]),K([4,6,0,-1]),K([-3,0,1,0]),K([32,32,-4,-7]),K([-45,-43,9,11])])

gp: E = ellinit([Pol(Vecrev([0,0,0,0])),Pol(Vecrev([4,6,0,-1])),Pol(Vecrev([-3,0,1,0])),Pol(Vecrev([32,32,-4,-7])),Pol(Vecrev([-45,-43,9,11]))], K);

magma: E := EllipticCurve([K![0,0,0,0],K![4,6,0,-1],K![-3,0,1,0],K![32,32,-4,-7],K![-45,-43,9,11]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^3-2a^2-3a+2)$$ = $$(a^3-2a^2-3a+2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$19$$ = $$19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-a^3+3a^2+3a-13)$$ = $$(a^3-2a^2-3a+2)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$361$$ = $$19^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{97722368}{361} a^{3} - \frac{69640192}{361} a^{2} + \frac{21032960}{19} a + \frac{398553088}{361}$$ 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 $\left(a^{3} - a^{2} - 4 a + 2 : -a^{3} + 2 a^{2} + 3 a - 6 : 1\right)$ Height $$0.00899710580018581$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.00899710580018581$$ Period: $$2008.53371966228$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$2.01941234484539$$ 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$$(j)_{-}$$
$$(a^3-2a^2-3a+2)$$ $$19$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 19.1-b consists of this curve only.

## Base change

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