# Properties

 Label 6.6.703493.1-41.4-e2 Base field 6.6.703493.1 Conductor $$(a^5-a^4-6a^3+4a^2+7a+1)$$ Conductor norm $$41$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field6.6.703493.1

Generator $$a$$, with minimal polynomial $$x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))

gp: K = nfinit(Pol(Vecrev([1, -9, 2, 11, -5, -2, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);

## Weierstrass equation

$${y}^2+\left(2a^{5}-a^{4}-12a^{3}+5a^{2}+14a-1\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+11a^{2}+9a-5\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-7a+4\right){x}^{2}+\left(-30a^{5}+3a^{4}+169a^{3}-71a^{2}-156a+28\right){x}-115a^{5}-28a^{4}+635a^{3}-157a^{2}-561a+63$$
sage: E = EllipticCurve([K([-1,14,5,-12,-1,2]),K([4,-7,-6,6,1,-1]),K([-5,9,11,-11,-2,2]),K([28,-156,-71,169,3,-30]),K([63,-561,-157,635,-28,-115])])

gp: E = ellinit([Pol(Vecrev([-1,14,5,-12,-1,2])),Pol(Vecrev([4,-7,-6,6,1,-1])),Pol(Vecrev([-5,9,11,-11,-2,2])),Pol(Vecrev([28,-156,-71,169,3,-30])),Pol(Vecrev([63,-561,-157,635,-28,-115]))], K);

magma: E := EllipticCurve([K![-1,14,5,-12,-1,2],K![4,-7,-6,6,1,-1],K![-5,9,11,-11,-2,2],K![28,-156,-71,169,3,-30],K![63,-561,-157,635,-28,-115]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^5-a^4-6a^3+4a^2+7a+1)$$ = $$(a^5-a^4-6a^3+4a^2+7a+1)$$ 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: $$(3a^5-a^4-18a^3+4a^2+20a+1)$$ = $$(a^5-a^4-6a^3+4a^2+7a+1)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$41$$ = $$41$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{1661681620471232}{41} a^{5} + \frac{213704000156457}{41} a^{4} - \frac{7841722250015960}{41} a^{3} + \frac{1577558381739890}{41} a^{2} + \frac{6635361185868735}{41} a - \frac{778052671995566}{41}$$ 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: $$1$$ Generator $\left(7 a^{5} - 6 a^{4} - 41 a^{3} + 29 a^{2} + 36 a - 4 : -27 a^{5} + 13 a^{4} + 169 a^{3} - 60 a^{2} - 216 a + 25 : 1\right)$ Height $$0.738892988210579$$ 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: $\left(2 a^{4} - 5 a^{2} : -a^{4} - 3 a^{3} + a^{2} + 8 a + 1 : 1\right)$ 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: $$0.738892988210579$$ Period: not available Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: not available Analytic order of Ш: not available

## 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$$(j)_{-}$$
$$(a^5-a^4-6a^3+4a^2+7a+1)$$ $$41$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## 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 41.4-e 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.