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Properties

Label 32.0.53818028735...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{16}\cdot 17^{30}$
Root discriminant $55.16$
Ramified primes $3, 5, 17$
Class number $13872$ (GRH)
Class group $[17, 816]$ (GRH)
Galois group $C_2\times C_{16}$ (as 32T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3188011, -781826, -601474, 527946, 20116, 500408, -113658, 904978, -115465, 637343, 119371, 205417, 418484, -49738, 602425, -122995, 574754, -112047, 395067, -70755, 198487, -31933, 72729, -10199, 19175, -2244, 3532, -322, 430, -27, 31, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 31*x^30 - 27*x^29 + 430*x^28 - 322*x^27 + 3532*x^26 - 2244*x^25 + 19175*x^24 - 10199*x^23 + 72729*x^22 - 31933*x^21 + 198487*x^20 - 70755*x^19 + 395067*x^18 - 112047*x^17 + 574754*x^16 - 122995*x^15 + 602425*x^14 - 49738*x^13 + 418484*x^12 + 205417*x^11 + 119371*x^10 + 637343*x^9 - 115465*x^8 + 904978*x^7 - 113658*x^6 + 500408*x^5 + 20116*x^4 + 527946*x^3 - 601474*x^2 - 781826*x + 3188011)
 
gp: K = bnfinit(x^32 - x^31 + 31*x^30 - 27*x^29 + 430*x^28 - 322*x^27 + 3532*x^26 - 2244*x^25 + 19175*x^24 - 10199*x^23 + 72729*x^22 - 31933*x^21 + 198487*x^20 - 70755*x^19 + 395067*x^18 - 112047*x^17 + 574754*x^16 - 122995*x^15 + 602425*x^14 - 49738*x^13 + 418484*x^12 + 205417*x^11 + 119371*x^10 + 637343*x^9 - 115465*x^8 + 904978*x^7 - 113658*x^6 + 500408*x^5 + 20116*x^4 + 527946*x^3 - 601474*x^2 - 781826*x + 3188011, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} + 31 x^{30} - 27 x^{29} + 430 x^{28} - 322 x^{27} + 3532 x^{26} - 2244 x^{25} + 19175 x^{24} - 10199 x^{23} + 72729 x^{22} - 31933 x^{21} + 198487 x^{20} - 70755 x^{19} + 395067 x^{18} - 112047 x^{17} + 574754 x^{16} - 122995 x^{15} + 602425 x^{14} - 49738 x^{13} + 418484 x^{12} + 205417 x^{11} + 119371 x^{10} + 637343 x^{9} - 115465 x^{8} + 904978 x^{7} - 113658 x^{6} + 500408 x^{5} + 20116 x^{4} + 527946 x^{3} - 601474 x^{2} - 781826 x + 3188011 \)

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:  \(53818028735688890166227692218037660486426411285400390625=3^{16}\cdot 5^{16}\cdot 17^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.16$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(131,·)$, $\chi_{255}(134,·)$, $\chi_{255}(139,·)$, $\chi_{255}(16,·)$, $\chi_{255}(146,·)$, $\chi_{255}(149,·)$, $\chi_{255}(151,·)$, $\chi_{255}(166,·)$, $\chi_{255}(41,·)$, $\chi_{255}(71,·)$, $\chi_{255}(176,·)$, $\chi_{255}(179,·)$, $\chi_{255}(184,·)$, $\chi_{255}(116,·)$, $\chi_{255}(59,·)$, $\chi_{255}(11,·)$, $\chi_{255}(196,·)$, $\chi_{255}(199,·)$, $\chi_{255}(76,·)$, $\chi_{255}(79,·)$, $\chi_{255}(56,·)$, $\chi_{255}(214,·)$, $\chi_{255}(89,·)$, $\chi_{255}(104,·)$, $\chi_{255}(106,·)$, $\chi_{255}(109,·)$, $\chi_{255}(239,·)$, $\chi_{255}(244,·)$, $\chi_{255}(121,·)$, $\chi_{255}(124,·)$, $\chi_{255}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1597} a^{17} + \frac{17}{1597} a^{15} + \frac{119}{1597} a^{13} + \frac{442}{1597} a^{11} - \frac{662}{1597} a^{9} - \frac{475}{1597} a^{7} + \frac{714}{1597} a^{5} + \frac{204}{1597} a^{3} + \frac{17}{1597} a - \frac{610}{1597}$, $\frac{1}{1597} a^{18} + \frac{17}{1597} a^{16} + \frac{119}{1597} a^{14} + \frac{442}{1597} a^{12} - \frac{662}{1597} a^{10} - \frac{475}{1597} a^{8} + \frac{714}{1597} a^{6} + \frac{204}{1597} a^{4} + \frac{17}{1597} a^{2} - \frac{610}{1597} a$, $\frac{1}{1597} a^{19} - \frac{170}{1597} a^{15} + \frac{16}{1597} a^{13} - \frac{191}{1597} a^{11} - \frac{400}{1597} a^{9} - \frac{793}{1597} a^{7} - \frac{755}{1597} a^{5} - \frac{257}{1597} a^{3} - \frac{610}{1597} a^{2} - \frac{289}{1597} a + \frac{788}{1597}$, $\frac{1}{1597} a^{20} - \frac{170}{1597} a^{16} + \frac{16}{1597} a^{14} - \frac{191}{1597} a^{12} - \frac{400}{1597} a^{10} - \frac{793}{1597} a^{8} - \frac{755}{1597} a^{6} - \frac{257}{1597} a^{4} - \frac{610}{1597} a^{3} - \frac{289}{1597} a^{2} + \frac{788}{1597} a$, $\frac{1}{1597} a^{21} - \frac{288}{1597} a^{15} - \frac{722}{1597} a^{13} - \frac{319}{1597} a^{11} + \frac{54}{1597} a^{9} - \frac{58}{1597} a^{7} - \frac{249}{1597} a^{5} - \frac{610}{1597} a^{4} - \frac{743}{1597} a^{3} + \frac{788}{1597} a^{2} - \frac{304}{1597} a + \frac{105}{1597}$, $\frac{1}{1597} a^{22} - \frac{288}{1597} a^{16} - \frac{722}{1597} a^{14} - \frac{319}{1597} a^{12} + \frac{54}{1597} a^{10} - \frac{58}{1597} a^{8} - \frac{249}{1597} a^{6} - \frac{610}{1597} a^{5} - \frac{743}{1597} a^{4} + \frac{788}{1597} a^{3} - \frac{304}{1597} a^{2} + \frac{105}{1597} a$, $\frac{1}{1597} a^{23} - \frac{617}{1597} a^{15} + \frac{416}{1597} a^{13} - \frac{410}{1597} a^{11} - \frac{671}{1597} a^{9} + \frac{293}{1597} a^{7} - \frac{610}{1597} a^{6} + \frac{473}{1597} a^{5} + \frac{788}{1597} a^{4} - \frac{641}{1597} a^{3} + \frac{105}{1597} a^{2} + \frac{105}{1597} a - \frac{10}{1597}$, $\frac{1}{3194} a^{24} - \frac{1}{3194} a^{21} - \frac{1}{3194} a^{19} - \frac{1}{3194} a^{18} - \frac{1}{3194} a^{17} - \frac{317}{1597} a^{16} + \frac{441}{3194} a^{15} + \frac{297}{3194} a^{14} + \frac{587}{3194} a^{13} + \frac{745}{3194} a^{12} - \frac{1529}{3194} a^{11} - \frac{9}{3194} a^{10} - \frac{589}{3194} a^{9} - \frac{829}{3194} a^{8} - \frac{881}{3194} a^{7} - \frac{241}{3194} a^{6} + \frac{539}{1597} a^{5} - \frac{235}{3194} a^{4} - \frac{348}{1597} a^{3} - \frac{45}{1597} a^{2} + \frac{588}{1597} a - \frac{283}{3194}$, $\frac{1}{20365011074} a^{25} - \frac{1501605}{20365011074} a^{24} - \frac{2434308}{10182505537} a^{23} - \frac{2666155}{20365011074} a^{22} + \frac{3157305}{20365011074} a^{21} + \frac{4763485}{20365011074} a^{20} + \frac{1663155}{10182505537} a^{19} + \frac{2747323}{10182505537} a^{18} + \frac{4072301}{20365011074} a^{17} - \frac{1472247375}{20365011074} a^{16} - \frac{3781587113}{10182505537} a^{15} + \frac{163100097}{10182505537} a^{14} + \frac{2037683613}{10182505537} a^{13} - \frac{4727234623}{10182505537} a^{12} - \frac{1904851391}{10182505537} a^{11} + \frac{4706013283}{10182505537} a^{10} + \frac{4552234661}{10182505537} a^{9} + \frac{52731932}{10182505537} a^{8} - \frac{2358590198}{10182505537} a^{7} + \frac{6788350383}{20365011074} a^{6} - \frac{5497040269}{20365011074} a^{5} - \frac{3147733271}{20365011074} a^{4} + \frac{1797368280}{10182505537} a^{3} + \frac{926974911}{10182505537} a^{2} + \frac{425026659}{20365011074} a - \frac{3440615917}{20365011074}$, $\frac{1}{20365011074} a^{26} - \frac{1090}{10182505537} a^{24} + \frac{4883765}{20365011074} a^{23} - \frac{2467785}{10182505537} a^{22} + \frac{5427019}{20365011074} a^{21} + \frac{821095}{20365011074} a^{20} - \frac{6228679}{20365011074} a^{19} - \frac{1177802}{10182505537} a^{18} + \frac{1456321}{20365011074} a^{17} - \frac{6022686223}{20365011074} a^{16} + \frac{7277557171}{20365011074} a^{15} - \frac{235974589}{20365011074} a^{14} - \frac{7679092991}{20365011074} a^{13} - \frac{4662210831}{20365011074} a^{12} - \frac{8166482173}{20365011074} a^{11} + \frac{2983864555}{20365011074} a^{10} + \frac{208548055}{20365011074} a^{9} - \frac{356045933}{20365011074} a^{8} - \frac{1527508613}{10182505537} a^{7} + \frac{8090238541}{20365011074} a^{6} - \frac{4429556365}{10182505537} a^{5} - \frac{2553446748}{10182505537} a^{4} + \frac{1015231621}{10182505537} a^{3} - \frac{3241691805}{20365011074} a^{2} - \frac{2555067966}{10182505537} a + \frac{2700243450}{10182505537}$, $\frac{1}{20365011074} a^{27} + \frac{1141819}{10182505537} a^{24} + \frac{1966268}{10182505537} a^{23} - \frac{4611771}{20365011074} a^{22} + \frac{2009668}{10182505537} a^{21} - \frac{1993567}{20365011074} a^{20} - \frac{535681}{20365011074} a^{19} - \frac{379429}{10182505537} a^{18} + \frac{2424285}{10182505537} a^{17} + \frac{5197387595}{20365011074} a^{16} - \frac{213122052}{10182505537} a^{15} + \frac{4993830995}{10182505537} a^{14} - \frac{4398174506}{10182505537} a^{13} - \frac{3825028706}{10182505537} a^{12} - \frac{1285900368}{10182505537} a^{11} + \frac{4859501329}{10182505537} a^{10} - \frac{1185682471}{10182505537} a^{9} - \frac{3885617675}{20365011074} a^{8} + \frac{1605010334}{10182505537} a^{7} - \frac{2617621769}{20365011074} a^{6} - \frac{2480733271}{10182505537} a^{5} - \frac{8502549421}{20365011074} a^{4} + \frac{571747251}{20365011074} a^{3} + \frac{4855830718}{10182505537} a^{2} + \frac{2506335270}{10182505537} a - \frac{2756444425}{20365011074}$, $\frac{1}{20365011074} a^{28} + \frac{30695}{10182505537} a^{24} + \frac{3530613}{20365011074} a^{23} - \frac{1036963}{10182505537} a^{22} + \frac{1150105}{20365011074} a^{21} + \frac{2525821}{20365011074} a^{20} - \frac{1652123}{10182505537} a^{19} + \frac{1118833}{10182505537} a^{18} + \frac{2160719}{20365011074} a^{17} - \frac{17975164}{10182505537} a^{16} + \frac{1050684175}{10182505537} a^{15} - \frac{560146960}{10182505537} a^{14} + \frac{849642177}{10182505537} a^{13} - \frac{159687689}{10182505537} a^{12} - \frac{1051709287}{10182505537} a^{11} + \frac{989937145}{10182505537} a^{10} - \frac{718059771}{20365011074} a^{9} + \frac{3900753526}{10182505537} a^{8} - \frac{7118875913}{20365011074} a^{7} + \frac{2136274697}{10182505537} a^{6} + \frac{2870637883}{20365011074} a^{5} + \frac{539959891}{20365011074} a^{4} - \frac{3431579605}{10182505537} a^{3} + \frac{1357260363}{10182505537} a^{2} + \frac{8256548163}{20365011074} a + \frac{2467560546}{10182505537}$, $\frac{1}{20365011074} a^{29} + \frac{2549945}{20365011074} a^{24} - \frac{49041}{10182505537} a^{23} + \frac{3946485}{20365011074} a^{22} - \frac{6141771}{20365011074} a^{21} - \frac{1910626}{10182505537} a^{20} - \frac{742344}{10182505537} a^{19} + \frac{2857763}{20365011074} a^{18} - \frac{2610612}{10182505537} a^{17} + \frac{2113223875}{10182505537} a^{16} + \frac{4573068684}{10182505537} a^{15} + \frac{1784060204}{10182505537} a^{14} - \frac{4727938424}{10182505537} a^{13} + \frac{3883092824}{10182505537} a^{12} + \frac{1302699096}{10182505537} a^{11} + \frac{1611183751}{20365011074} a^{10} + \frac{4133491760}{10182505537} a^{9} - \frac{2768532437}{20365011074} a^{8} + \frac{492000895}{10182505537} a^{7} + \frac{8486100329}{20365011074} a^{6} + \frac{8161385093}{20365011074} a^{5} - \frac{3532668196}{10182505537} a^{4} + \frac{659013046}{10182505537} a^{3} + \frac{9847901427}{20365011074} a^{2} - \frac{4673528165}{10182505537} a - \frac{2457606866}{10182505537}$, $\frac{1}{20365011074} a^{30} - \frac{477775}{10182505537} a^{24} - \frac{5260369}{20365011074} a^{23} - \frac{2344962}{10182505537} a^{22} - \frac{417462}{10182505537} a^{21} - \frac{189005}{20365011074} a^{20} - \frac{2551643}{10182505537} a^{19} - \frac{5585055}{20365011074} a^{18} - \frac{1801257}{10182505537} a^{17} - \frac{2246622713}{20365011074} a^{16} + \frac{2298129589}{20365011074} a^{15} - \frac{6362492223}{20365011074} a^{14} - \frac{6666539787}{20365011074} a^{13} + \frac{7057304539}{20365011074} a^{12} + \frac{4293602453}{10182505537} a^{11} + \frac{5443049401}{20365011074} a^{10} - \frac{3807522539}{10182505537} a^{9} + \frac{8191209541}{20365011074} a^{8} - \frac{5080159298}{10182505537} a^{7} + \frac{7914659845}{20365011074} a^{6} - \frac{9535085725}{20365011074} a^{5} - \frac{3776495140}{10182505537} a^{4} + \frac{3142002779}{20365011074} a^{3} - \frac{4840592153}{10182505537} a^{2} - \frac{9565093563}{20365011074} a - \frac{3147194315}{10182505537}$, $\frac{1}{20365011074} a^{31} + \frac{1111871}{10182505537} a^{24} + \frac{1002879}{10182505537} a^{23} - \frac{2019144}{10182505537} a^{22} + \frac{809556}{10182505537} a^{21} + \frac{1806450}{10182505537} a^{20} - \frac{1454538}{10182505537} a^{19} + \frac{754105}{20365011074} a^{18} + \frac{2858475}{10182505537} a^{17} + \frac{6285139795}{20365011074} a^{16} + \frac{1362783129}{10182505537} a^{15} + \frac{1163578261}{10182505537} a^{14} - \frac{328115923}{10182505537} a^{13} - \frac{1863025051}{20365011074} a^{12} + \frac{1111166736}{10182505537} a^{11} - \frac{1863393415}{20365011074} a^{10} - \frac{3216140499}{10182505537} a^{9} - \frac{8356542281}{20365011074} a^{8} - \frac{822329061}{10182505537} a^{7} - \frac{2158806916}{10182505537} a^{6} - \frac{1102596524}{10182505537} a^{5} + \frac{3056659710}{10182505537} a^{4} + \frac{1843404252}{10182505537} a^{3} - \frac{5027590995}{20365011074} a^{2} + \frac{4630671186}{10182505537} a - \frac{3566900837}{20365011074}$

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

Class group and class number

$C_{17}\times C_{816}$, which has order $13872$ (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:  \( -1 \) (order $2$)
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:  \( 19955290291.92932 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{16}$ (as 32T32):

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 $C_2\times C_{16}$
Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{17})\), 4.4.4913.1, 4.0.1105425.1, 8.0.1221964430625.2, \(\Q(\zeta_{17})^+\), 8.0.20773395320625.1, 16.0.431533953146964646550390625.1, \(\Q(\zeta_{51})^+\), 16.0.1118134004496021794140625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ R R $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ $16^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data not computed
5Data not computed
17Data not computed