Properties

Label 6.6.434581.1-29.1-a1
Base field 6.6.434581.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.434581.1

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

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

Weierstrass equation

\({y}^2+\left(a^{5}-a^{4}-5a^{3}+a^{2}+4a+1\right){x}{y}+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+a-3\right){x}^{2}+\left(-34a^{5}+81a^{4}+65a^{3}-117a^{2}-27a+6\right){x}-56a^{5}+163a^{4}+31a^{3}-264a^{2}-13a+30\)
sage: E = EllipticCurve([K([1,4,1,-5,-1,1]),K([-3,1,2,-1,0,0]),K([-2,5,12,-10,-6,3]),K([6,-27,-117,65,81,-34]),K([30,-13,-264,31,163,-56])])
 
gp: E = ellinit([Polrev([1,4,1,-5,-1,1]),Polrev([-3,1,2,-1,0,0]),Polrev([-2,5,12,-10,-6,3]),Polrev([6,-27,-117,65,81,-34]),Polrev([30,-13,-264,31,163,-56])], K);
 
magma: E := EllipticCurve([K![1,4,1,-5,-1,1],K![-3,1,2,-1,0,0],K![-2,5,12,-10,-6,3],K![6,-27,-117,65,81,-34],K![30,-13,-264,31,163,-56]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-a^3-4a^2+a+3)\) = \((a^4-a^3-4a^2+a+3)\)
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: \((-10a^5+26a^4+33a^3-75a^2-27a+19)\) = \((a^4-a^3-4a^2+a+3)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 20511149 \) = \(29^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{114595298439097102868526844523731}{20511149} a^{5} + \frac{151675552625626526718826727580063}{20511149} a^{4} + \frac{560978222901143343582252842098390}{20511149} a^{3} - \frac{193517191812238007197441347247967}{20511149} a^{2} - \frac{589280924297514314247975153893832}{20511149} a - \frac{169413338416695988765174446920769}{20511149} \)
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: \( 1.4813316253083485410290468295415282899 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.40442 \)
Analytic order of Ш: \( 625 \) (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\((j)_{-}\)
\((a^4-a^3-4a^2+a+3)\) \(29\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 29.1-a consists of curves linked by isogenies of degrees dividing 15.

Base change

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

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