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Properties

Label 6.6.1279733.1-29.4-c1
Base field 6.6.1279733.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.1279733.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -11, 10, 10, -6, -2, 1]))
 
gp: K = nfinit(Polrev([-1, -11, 10, 10, -6, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 10, 10, -6, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-5a^{2}+3\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+3a^{2}+2a-1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-2a^{2}-5a-2\right){x}^{2}+\left(19a^{5}+33a^{4}-88a^{3}-192a^{2}-69a-4\right){x}+176a^{5}+209a^{4}-870a^{3}-1383a^{2}-286a-16\)
sage: E = EllipticCurve([K([3,0,-5,0,1,0]),K([-2,-5,-2,5,1,-1]),K([-1,2,3,-4,-1,1]),K([-4,-69,-192,-88,33,19]),K([-16,-286,-1383,-870,209,176])])
 
gp: E = ellinit([Polrev([3,0,-5,0,1,0]),Polrev([-2,-5,-2,5,1,-1]),Polrev([-1,2,3,-4,-1,1]),Polrev([-4,-69,-192,-88,33,19]),Polrev([-16,-286,-1383,-870,209,176])], K);
 
magma: E := EllipticCurve([K![3,0,-5,0,1,0],K![-2,-5,-2,5,1,-1],K![-1,2,3,-4,-1,1],K![-4,-69,-192,-88,33,19],K![-16,-286,-1383,-870,209,176]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-a-1)\) = \((a^2-a-1)\)
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: \((-7a^5+22a^4+16a^3-71a^2-4a+17)\) = \((a^2-a-1)^{5}\)
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: \( \frac{34286919698837807988428}{20511149} a^{5} - \frac{71481909750259171305404}{20511149} a^{4} - \frac{199666497045499987826938}{20511149} a^{3} + \frac{359831156125219449696111}{20511149} a^{2} + \frac{312348696775660588553712}{20511149} a - \frac{403713904104228202715024}{20511149} \)
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: \( 3.9871607907008414956247136122427783107 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.20285 \)
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\((j)_{-}\)
\((a^2-a-1)\) \(29\) \(1\) \(I_{5}\) 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
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 29.4-c consists of curves linked by isogenies of degree 5.

Base change

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

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