# Properties

 Label 4.4.5125.1-41.1-c1 Base field 4.4.5125.1 Conductor $$(a^3-a^2-4a-3)$$ Conductor norm $$41$$ CM no Base change no Q-curve no Torsion order $$7$$ 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

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

gp: E = ellinit([Pol(Vecrev([0,0,0,0])),Pol(Vecrev([-4,-1,1,0])),Pol(Vecrev([-4,-1,1,0])),Pol(Vecrev([0,0,0,0])),Pol(Vecrev([0,0,0,0]))], K);

magma: E := EllipticCurve([K![0,0,0,0],K![-4,-1,1,0],K![-4,-1,1,0],K![0,0,0,0],K![0,0,0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^3-a^2-4a-3)$$ = $$(a^3-a^2-4a-3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$41$$ = $$41$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(a^2-a-10)$$ = $$(a^3-a^2-4a-3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1681$$ = $$41^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{176128}{41} a^{2} - \frac{176128}{41} a - \frac{815104}{41}$$ 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: $$\Z/7\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(1 : 1 : 1\right)$ 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: $$2140.06887671813$$ Tamagawa product: $$2$$ Torsion order: $$7$$ Leading coefficient: $$1.22015423553756$$ 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$$(j)_{-}$$
$$(a^3-a^2-4a-3)$$ $$41$$ $$2$$ $$I_{2}$$ Non-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
$$7$$ 7B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 41.1-c consists of curves linked by isogenies of degree 7.

## Base change

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