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Properties

Label 6.6.1397493.1-27.1-f1
Base field 6.6.1397493.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.1397493.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, 3, 10, -3, -3, 1]))
 
gp: K = nfinit(Polrev([1, -6, 3, 10, -3, -3, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 3, 10, -3, -3, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-2a^{3}-3a^{2}+5a+2\right){x}{y}+\left(a^{5}-2a^{4}-5a^{3}+7a^{2}+8a-4\right){y}={x}^{3}+\left(a^{5}-3a^{4}-3a^{3}+10a^{2}+2a-4\right){x}^{2}+\left(-8a^{5}+26a^{4}+29a^{3}-87a^{2}-67a+20\right){x}+26a^{5}-47a^{4}-122a^{3}+117a^{2}+147a-34\)
sage: E = EllipticCurve([K([2,5,-3,-2,1,0]),K([-4,2,10,-3,-3,1]),K([-4,8,7,-5,-2,1]),K([20,-67,-87,29,26,-8]),K([-34,147,117,-122,-47,26])])
 
gp: E = ellinit([Polrev([2,5,-3,-2,1,0]),Polrev([-4,2,10,-3,-3,1]),Polrev([-4,8,7,-5,-2,1]),Polrev([20,-67,-87,29,26,-8]),Polrev([-34,147,117,-122,-47,26])], K);
 
magma: E := EllipticCurve([K![2,5,-3,-2,1,0],K![-4,2,10,-3,-3,1],K![-4,8,7,-5,-2,1],K![20,-67,-87,29,26,-8],K![-34,147,117,-122,-47,26]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^5+3a^4+3a^3-8a^2-6a)\) = \((a^3-a^2-3a)^{3}\)
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: \((-3a^5+6a^4+15a^3-15a^2-24a-6)\) = \((a^3-a^2-3a)^{11}\)
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: \( 1820884570 a^{5} - 1592027539 a^{4} - 9268094856 a^{3} - 623375868 a^{2} + 5249738257 a - 943012354 \)
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: \( 2084.1241086472880266898447960526257533 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.76299 \)
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\((j)_{-}\)
\((a^3-a^2-3a)\) \(3\) \(1\) \(II^{*}\) Additive \(1\) \(3\) \(11\) \(0\)

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 27.1-f consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.