# Properties

 Label 4.4.5125.1-29.2-a2 Base field 4.4.5125.1 Conductor $$(a^2-6)$$ Conductor norm $$29$$ CM no Base change no Q-curve no Torsion order $$1$$ 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

$${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a^{3}-5a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(63a^{3}-159a^{2}-264a+319\right){x}-43a^{3}-362a^{2}+1021a+2586$$
sage: E = EllipticCurve([K([1,-3,-1,1]),K([-1,3,1,-1]),K([-3,-5,0,1]),K([319,-264,-159,63]),K([2586,1021,-362,-43])])

gp: E = ellinit([Pol(Vecrev([1,-3,-1,1])),Pol(Vecrev([-1,3,1,-1])),Pol(Vecrev([-3,-5,0,1])),Pol(Vecrev([319,-264,-159,63])),Pol(Vecrev([2586,1021,-362,-43]))], K);

magma: E := EllipticCurve([K![1,-3,-1,1],K![-1,3,1,-1],K![-3,-5,0,1],K![319,-264,-159,63],K![2586,1021,-362,-43]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^2-6)$$ = $$(a^2-6)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$29$$ = $$29$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(172a^3-290a^2-595a+100)$$ = $$(a^2-6)^{7}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-17249876309$$ = $$-29^{7}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{3402532667241798182971657}{17249876309} a^{3} - \frac{2331998286962811335379747}{17249876309} a^{2} + \frac{13839248119673699533928308}{17249876309} a + \frac{13659457823323246388468044}{17249876309}$$ 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: 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$125.226023376231$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.74923297089418$$ 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$$(j)_{-}$$
$$(a^2-6)$$ $$29$$ $$1$$ $$I_{7}$$ Non-split multiplicative $$1$$ $$1$$ $$7$$ $$7$$

## 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.6.3

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 7.
Its isogeny class 29.2-a 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.