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Properties

Label 32.0.13895699721...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $45.78$
Ramified primes $2, 5, 13$
Class number $64$ (GRH)
Class group $[4, 4, 4]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 0, -2916, 0, -11826, 0, 13275, 0, 16051, 0, 33462, 0, -41235, 0, -44722, 0, 111717, 0, 22329, 0, -2090, 0, -1521, 0, -114, 0, -85, 0, -9, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561)
 
gp: K = bnfinit(x^32 + 3*x^30 - 9*x^28 - 85*x^26 - 114*x^24 - 1521*x^22 - 2090*x^20 + 22329*x^18 + 111717*x^16 - 44722*x^14 - 41235*x^12 + 33462*x^10 + 16051*x^8 + 13275*x^6 - 11826*x^4 - 2916*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{32} + 3 x^{30} - 9 x^{28} - 85 x^{26} - 114 x^{24} - 1521 x^{22} - 2090 x^{20} + 22329 x^{18} + 111717 x^{16} - 44722 x^{14} - 41235 x^{12} + 33462 x^{10} + 16051 x^{8} + 13275 x^{6} - 11826 x^{4} - 2916 x^{2} + 6561 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(138956997215838851269528412416000000000000000000000000=2^{32}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.78$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(260=2^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(259,·)$, $\chi_{260}(129,·)$, $\chi_{260}(131,·)$, $\chi_{260}(21,·)$, $\chi_{260}(151,·)$, $\chi_{260}(27,·)$, $\chi_{260}(157,·)$, $\chi_{260}(31,·)$, $\chi_{260}(161,·)$, $\chi_{260}(47,·)$, $\chi_{260}(177,·)$, $\chi_{260}(51,·)$, $\chi_{260}(53,·)$, $\chi_{260}(183,·)$, $\chi_{260}(57,·)$, $\chi_{260}(187,·)$, $\chi_{260}(181,·)$, $\chi_{260}(73,·)$, $\chi_{260}(203,·)$, $\chi_{260}(77,·)$, $\chi_{260}(79,·)$, $\chi_{260}(209,·)$, $\chi_{260}(83,·)$, $\chi_{260}(213,·)$, $\chi_{260}(207,·)$, $\chi_{260}(99,·)$, $\chi_{260}(229,·)$, $\chi_{260}(103,·)$, $\chi_{260}(233,·)$, $\chi_{260}(109,·)$, $\chi_{260}(239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{7}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{16} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{2}{9} a^{8} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{19} - \frac{2}{27} a^{17} + \frac{1}{9} a^{15} - \frac{4}{27} a^{13} - \frac{4}{27} a^{11} - \frac{4}{27} a^{9} + \frac{4}{9} a^{7} - \frac{11}{27} a^{5} + \frac{1}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{108} a^{20} + \frac{1}{27} a^{18} - \frac{1}{27} a^{14} + \frac{2}{27} a^{12} - \frac{1}{108} a^{10} - \frac{1}{9} a^{8} + \frac{4}{27} a^{6} + \frac{13}{27} a^{4} + \frac{2}{9} a^{2} - \frac{1}{4}$, $\frac{1}{108} a^{21} + \frac{2}{27} a^{17} - \frac{4}{27} a^{15} - \frac{1}{9} a^{13} + \frac{5}{36} a^{11} + \frac{1}{27} a^{9} - \frac{8}{27} a^{7} - \frac{1}{9} a^{5} - \frac{4}{27} a^{3} + \frac{5}{12} a$, $\frac{1}{108} a^{22} - \frac{1}{27} a^{18} + \frac{2}{27} a^{16} - \frac{1}{9} a^{14} - \frac{1}{12} a^{12} + \frac{4}{27} a^{10} + \frac{13}{27} a^{8} + \frac{2}{9} a^{6} - \frac{7}{27} a^{4} + \frac{11}{36} a^{2}$, $\frac{1}{108} a^{23} + \frac{11}{108} a^{13} + \frac{1}{108} a^{3}$, $\frac{1}{108} a^{24} + \frac{11}{108} a^{14} + \frac{1}{108} a^{4}$, $\frac{1}{108} a^{25} + \frac{11}{108} a^{15} + \frac{1}{108} a^{5}$, $\frac{1}{1456295763348} a^{26} + \frac{436656233}{242715960558} a^{24} - \frac{8598646}{40452660093} a^{22} + \frac{162482033}{40452660093} a^{20} + \frac{491600858}{13484220031} a^{18} + \frac{123874714055}{1456295763348} a^{16} - \frac{18370971659}{242715960558} a^{14} + \frac{5004295789}{40452660093} a^{12} + \frac{1877518436}{13484220031} a^{10} + \frac{14849052893}{40452660093} a^{8} - \frac{229642858259}{1456295763348} a^{6} + \frac{115372803131}{242715960558} a^{4} + \frac{1952193478}{13484220031} a^{2} - \frac{1226994442}{13484220031}$, $\frac{1}{1456295763348} a^{27} + \frac{436656233}{242715960558} a^{25} - \frac{8598646}{40452660093} a^{23} + \frac{162482033}{40452660093} a^{21} - \frac{210996865}{364073940837} a^{19} + \frac{231748474303}{1456295763348} a^{17} + \frac{35565908465}{242715960558} a^{15} - \frac{22382438054}{364073940837} a^{13} - \frac{16728102383}{364073940837} a^{11} - \frac{55137604397}{364073940837} a^{9} + \frac{93978422485}{1456295763348} a^{7} - \frac{328092592157}{728147881674} a^{5} + \frac{160582984154}{364073940837} a^{3} - \frac{17165203357}{40452660093} a$, $\frac{1}{4368887290044} a^{28} - \frac{756717713}{1456295763348} a^{24} - \frac{8397050963}{2184443645022} a^{22} - \frac{3173183095}{1456295763348} a^{20} + \frac{5400358661}{161810640372} a^{18} - \frac{168127493225}{1092221822511} a^{16} - \frac{171885519311}{1456295763348} a^{14} + \frac{1736565031}{728147881674} a^{12} + \frac{496758606893}{4368887290044} a^{10} - \frac{260557156309}{1456295763348} a^{8} - \frac{41468490176}{121357980279} a^{6} - \frac{1786851985187}{4368887290044} a^{4} + \frac{17199500173}{242715960558} a^{2} + \frac{11770936847}{53936880124}$, $\frac{1}{13106661870132} a^{29} + \frac{1}{4368887290044} a^{27} - \frac{1936833391}{728147881674} a^{25} - \frac{58175415787}{13106661870132} a^{23} - \frac{5404024969}{2184443645022} a^{21} - \frac{20038173253}{1456295763348} a^{19} + \frac{346356730753}{13106661870132} a^{17} + \frac{45123612001}{728147881674} a^{15} + \frac{119640879623}{1456295763348} a^{13} + \frac{491858299939}{6553330935066} a^{11} - \frac{588981334145}{4368887290044} a^{9} - \frac{86800180055}{485431921116} a^{7} - \frac{2059740739483}{6553330935066} a^{5} + \frac{147261155443}{485431921116} a^{3} - \frac{16794850507}{80905320186} a$, $\frac{1}{39319985610396} a^{30} + \frac{1}{13106661870132} a^{28} - \frac{1}{4368887290044} a^{26} + \frac{9257801039}{9829996402599} a^{24} - \frac{29349668189}{13106661870132} a^{22} + \frac{11712482075}{4368887290044} a^{20} + \frac{2117068222525}{39319985610396} a^{18} - \frac{17551756493}{161810640372} a^{16} + \frac{89087782901}{1092221822511} a^{14} - \frac{5001641332921}{39319985610396} a^{12} - \frac{706711017953}{13106661870132} a^{10} - \frac{1981734423185}{4368887290044} a^{8} - \frac{15464536204685}{39319985610396} a^{6} + \frac{324155339690}{1092221822511} a^{4} + \frac{73440071909}{485431921116} a^{2} - \frac{5866965635}{13484220031}$, $\frac{1}{117959956831188} a^{31} + \frac{1}{39319985610396} a^{29} - \frac{1}{13106661870132} a^{27} + \frac{9257801039}{29489989207797} a^{25} + \frac{46004156045}{19659992805198} a^{23} + \frac{11712482075}{13106661870132} a^{21} + \frac{660772459177}{117959956831188} a^{19} - \frac{535523969305}{4368887290044} a^{17} - \frac{32270197378}{3276665467533} a^{15} + \frac{2414177534839}{58979978415594} a^{13} + \frac{1235016666511}{39319985610396} a^{11} + \frac{121803901651}{13106661870132} a^{9} - \frac{19833423494729}{117959956831188} a^{7} + \frac{1133208541550}{3276665467533} a^{5} + \frac{191788566311}{728147881674} a^{3} + \frac{7617254396}{40452660093} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1620997322}{29489989207797} a^{31} - \frac{6359036717}{39319985610396} a^{29} + \frac{1620997322}{3276665467533} a^{27} + \frac{137784772370}{29489989207797} a^{25} + \frac{61597898236}{9829996402599} a^{23} + \frac{273948547418}{3276665467533} a^{21} + \frac{12532743803623}{117959956831188} a^{19} - \frac{1340564785294}{1092221822511} a^{17} - \frac{20121439757986}{3276665467533} a^{15} + \frac{72494242234484}{29489989207797} a^{13} + \frac{22280608190890}{9829996402599} a^{11} + \frac{66474293300125}{13106661870132} a^{9} - \frac{26018628015422}{29489989207797} a^{7} - \frac{2390971049950}{3276665467533} a^{5} + \frac{236665609012}{364073940837} a^{3} + \frac{6483989288}{40452660093} a \) (order $20$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4622596654413.537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(i, \sqrt{65})\), 4.0.4394000.2, 4.4.274625.1, \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), 4.0.4394000.1, 4.4.274625.2, \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), 4.0.21125.1, 4.4.338000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.4.35152.1, 4.0.2197.1, 4.4.878800.1, 4.0.54925.1, 8.0.19307236000000.1, 8.0.4569760000.1, 8.0.19307236000000.4, 8.0.19307236000000.5, 8.0.19307236000000.3, 8.8.75418890625.1, 8.0.19307236000000.2, 8.0.114244000000.2, \(\Q(\zeta_{20})\), 8.0.1235663104.1, 8.0.772289440000.3, 8.0.446265625.1, 8.8.114244000000.1, 8.8.772289440000.1, 8.0.3016755625.1, 8.0.114244000000.1, 8.0.114244000000.3, 8.0.772289440000.1, 8.0.772289440000.2, 16.0.372769361959696000000000000.1, 16.0.13051691536000000000000.1, 16.0.596430979135513600000000.1, 16.0.372769361959696000000000000.3, 16.0.372769361959696000000000000.2, 16.0.5688009063105712890625.1, 16.16.372769361959696000000000000.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
13Data not computed