Properties

Label 29.29.3835803278...3121.1
Degree $29$
Signature $[29, 0]$
Discriminant $59^{28}$
Root discriminant $51.26$
Ramified prime $59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{29}$ (as 29T1)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*x - 1)
 
gp: K = bnfinit(x^29 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 15, 105, -560, -1820, 6188, 12376, -31824, -43758, 92378, 92378, -167960, -125970, 203490, 116280, -170544, -74613, 100947, 33649, -42504, -10626, 12650, 2300, -2600, -325, 351, 27, -28, -1, 1]);
 

Normalized defining polynomial

\( x^{29} - x^{28} - 28 x^{27} + 27 x^{26} + 351 x^{25} - 325 x^{24} - 2600 x^{23} + 2300 x^{22} + 12650 x^{21} - 10626 x^{20} - 42504 x^{19} + 33649 x^{18} + 100947 x^{17} - 74613 x^{16} - 170544 x^{15} + 116280 x^{14} + 203490 x^{13} - 125970 x^{12} - 167960 x^{11} + 92378 x^{10} + 92378 x^{9} - 43758 x^{8} - 31824 x^{7} + 12376 x^{6} + 6188 x^{5} - 1820 x^{4} - 560 x^{3} + 105 x^{2} + 15 x - 1 \)

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

Invariants

Degree:  $29$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[29, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(38358032782038398419973086399760468678777161743121=59^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $51.26$
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:  $59$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $29$
This field is Galois and abelian over $\Q$.
Conductor:  \(59\)
Dirichlet character group:    $\lbrace$$\chi_{59}(1,·)$, $\chi_{59}(3,·)$, $\chi_{59}(4,·)$, $\chi_{59}(5,·)$, $\chi_{59}(7,·)$, $\chi_{59}(9,·)$, $\chi_{59}(12,·)$, $\chi_{59}(15,·)$, $\chi_{59}(16,·)$, $\chi_{59}(17,·)$, $\chi_{59}(19,·)$, $\chi_{59}(20,·)$, $\chi_{59}(21,·)$, $\chi_{59}(22,·)$, $\chi_{59}(25,·)$, $\chi_{59}(26,·)$, $\chi_{59}(27,·)$, $\chi_{59}(28,·)$, $\chi_{59}(29,·)$, $\chi_{59}(35,·)$, $\chi_{59}(36,·)$, $\chi_{59}(41,·)$, $\chi_{59}(45,·)$, $\chi_{59}(46,·)$, $\chi_{59}(48,·)$, $\chi_{59}(49,·)$, $\chi_{59}(51,·)$, $\chi_{59}(53,·)$, $\chi_{59}(57,·)$$\rbrace$
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:  $28$
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:  \( 2275980944744796.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_{29}$ (as 29T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 29
The 29 conjugacy class representatives for $C_{29}$
Character table for $C_{29}$ 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 $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ R

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
59Data not computed