Properties

Label 29.1.67146313616...1632.1
Degree $29$
Signature $[1, 14]$
Discriminant $2^{28}\cdot 3\cdot 11\cdot 563\cdot 134635615853097819303336173003083422943$
Root discriminant $56.58$
Ramified primes $2, 3, 11, 563, 134635615853097819303336173003083422943$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_{29}$ (as 29T8)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 2*x - 2)
 
gp: K = bnfinit(x^29 - 2*x - 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 

Normalized defining polynomial

\( x^{29} - 2 x - 2 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $29$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(671463136166186133768834421731274208264739599941632=2^{28}\cdot 3\cdot 11\cdot 563\cdot 134635615853097819303336173003083422943\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $56.58$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 11, 563, 134635615853097819303336173003083422943$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 167099208611677.7 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$S_{29}$ (as 29T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 8841761993739701954543616000000
The 4565 conjugacy class representatives for $S_{29}$ are not computed
Character table for $S_{29}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R $28{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $29$ R ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $25{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ $28{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $29$ $21{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $19{,}\,{\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ $20{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $23{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.7.0.1$x^{7} + x^{2} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.7.0.1$x^{7} - x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
11.14.0.1$x^{14} + x^{2} - 5 x + 8$$1$$14$$0$$C_{14}$$[\ ]^{14}$
563Data not computed
134635615853097819303336173003083422943Data not computed