# Properties

 Label 6.6.1387029.1-27.1-p2 Base field 6.6.1387029.1 Conductor $$(a^5-3a^4-3a^3+8a^2+2a-2)$$ Conductor norm $$27$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field6.6.1387029.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))

gp: K = nfinit(Pol(Vecrev([1, -4, -1, 9, -2, -3, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);

## Weierstrass equation

$${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(-a^{5}+3a^{4}+2a^{3}-7a^{2}+2\right){y}={x}^{3}+\left(3a^{5}-7a^{4}-9a^{3}+18a^{2}+4a-5\right){x}^{2}+\left(7a^{5}-34a^{4}+40a^{3}+16a^{2}-31a+10\right){x}+53a^{5}-232a^{4}+239a^{3}+81a^{2}-149a+29$$
sage: E = EllipticCurve([K([1,-4,0,1,0,0]),K([-5,4,18,-9,-7,3]),K([2,0,-7,2,3,-1]),K([10,-31,16,40,-34,7]),K([29,-149,81,239,-232,53])])

gp: E = ellinit([Pol(Vecrev([1,-4,0,1,0,0])),Pol(Vecrev([-5,4,18,-9,-7,3])),Pol(Vecrev([2,0,-7,2,3,-1])),Pol(Vecrev([10,-31,16,40,-34,7])),Pol(Vecrev([29,-149,81,239,-232,53]))], K);

magma: E := EllipticCurve([K![1,-4,0,1,0,0],K![-5,4,18,-9,-7,3],K![2,0,-7,2,3,-1],K![10,-31,16,40,-34,7],K![29,-149,81,239,-232,53]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^5-3a^4-3a^3+8a^2+2a-2)$$ = $$(a^5-3a^4-2a^3+8a^2+a-2)^{3}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$27$$ = $$3^{3}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-5a^5+17a^4+9a^3-54a^2+3a+23)$$ = $$(a^5-3a^4-2a^3+8a^2+a-2)^{11}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-177147$$ = $$-3^{11}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$3426222102 a^{5} - 12812570435 a^{4} + 2623723832 a^{3} + 28894116691 a^{2} - 24794656162 a + 4632631517$$ 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: not available sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: not available Tamagawa product: $$1$$ Torsion order: $$1$$ 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$$(j)_{-}$$
$$(a^5-3a^4-2a^3+8a^2+a-2)$$ $$3$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$3$$ $$11$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 27.1-p consists of curves linked by isogenies of degree 3.

## Base change

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