Properties

Label 6.6.485125.1-41.1-b3
Base field 6.6.485125.1
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
 
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(-a^{5}+2a^{4}+4a^{3}-6a^{2}-3a+2\right){x}{y}+\left(-a^{5}+2a^{4}+4a^{3}-7a^{2}-2a+3\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+5a-2\right){x}^{2}+\left(7a^{5}+a^{4}-46a^{3}+59a-26\right){x}+18a^{5}-20a^{4}-50a^{3}+53a^{2}-44a+23\)
sage: E = EllipticCurve([K([2,-3,-6,4,2,-1]),K([-2,5,4,-5,-1,1]),K([3,-2,-7,4,2,-1]),K([-26,59,0,-46,1,7]),K([23,-44,53,-50,-20,18])])
 
gp: E = ellinit([Polrev([2,-3,-6,4,2,-1]),Polrev([-2,5,4,-5,-1,1]),Polrev([3,-2,-7,4,2,-1]),Polrev([-26,59,0,-46,1,7]),Polrev([23,-44,53,-50,-20,18])], K);
 
magma: E := EllipticCurve([K![2,-3,-6,4,2,-1],K![-2,5,4,-5,-1,1],K![3,-2,-7,4,2,-1],K![-26,59,0,-46,1,7],K![23,-44,53,-50,-20,18]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-5a^3+5a)\) = \((a^5-5a^3+5a)\)
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: \((-19a^5+22a^4+58a^3-69a^2+12a+26)\) = \((a^5-5a^3+5a)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -4750104241 \) = \(-41^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5160746097604466171083}{4750104241} a^{5} - \frac{15116996144924874322550}{4750104241} a^{4} - \frac{6596324090888058193728}{4750104241} a^{3} + \frac{47417622246192743893408}{4750104241} a^{2} - \frac{33743571908962038596076}{4750104241} a + \frac{5553513136786228568492}{4750104241} \)
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 $\left(-20 a^{5} + 9 a^{4} + 95 a^{3} - 14 a^{2} - 59 a + 17 : 308 a^{5} - 87 a^{4} - 1376 a^{3} + 107 a^{2} + 791 a - 179 : 1\right)$
Height \(0.0089115643658937453907062171653012132681\)
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: $\left(-\frac{5}{4} a^{5} + \frac{3}{4} a^{4} + \frac{13}{2} a^{3} - \frac{17}{4} a^{2} - 8 a + 4 : \frac{7}{4} a^{5} - \frac{5}{4} a^{4} - \frac{25}{4} a^{3} + \frac{47}{8} a^{2} + \frac{3}{2} a - \frac{21}{8} : 1\right)$
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: \( 0.0089115643658937453907062171653012132681 \)
Period: \( 19378.268634232349756069043160243624743 \)
Tamagawa product: \( 6 \)
Torsion order: \(2\)
Leading coefficient: \( 2.23144 \)
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\((j)_{-}\)
\((a^5-5a^3+5a)\) \(41\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 41.1-b consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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