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Properties

Label 32.0.22351135421...0000.7
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 5^{16}$
Root discriminant $61.97$
Ramified primes $2, 3, 5$
Class number $196$ (GRH)
Class group $[14, 14]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4879681, 0, 141376, 0, -738128, 0, -2947392, 0, -1388052, 0, 918880, 0, 666768, 0, 248976, 0, 163951, 0, 97984, 0, 45184, 0, 16160, 0, 4524, 0, 960, 0, 152, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4524*x^24 + 16160*x^22 + 45184*x^20 + 97984*x^18 + 163951*x^16 + 248976*x^14 + 666768*x^12 + 918880*x^10 - 1388052*x^8 - 2947392*x^6 - 738128*x^4 + 141376*x^2 + 4879681)
 
gp: K = bnfinit(x^32 + 16*x^30 + 152*x^28 + 960*x^26 + 4524*x^24 + 16160*x^22 + 45184*x^20 + 97984*x^18 + 163951*x^16 + 248976*x^14 + 666768*x^12 + 918880*x^10 - 1388052*x^8 - 2947392*x^6 - 738128*x^4 + 141376*x^2 + 4879681, 1)
 

Normalized defining polynomial

\( x^{32} + 16 x^{30} + 152 x^{28} + 960 x^{26} + 4524 x^{24} + 16160 x^{22} + 45184 x^{20} + 97984 x^{18} + 163951 x^{16} + 248976 x^{14} + 666768 x^{12} + 918880 x^{10} - 1388052 x^{8} - 2947392 x^{6} - 738128 x^{4} + 141376 x^{2} + 4879681 \)

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:  \(2235113542185251937084439154754616201052160000000000000000=2^{128}\cdot 3^{16}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.97$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(131,·)$, $\chi_{480}(389,·)$, $\chi_{480}(11,·)$, $\chi_{480}(269,·)$, $\chi_{480}(401,·)$, $\chi_{480}(149,·)$, $\chi_{480}(281,·)$, $\chi_{480}(29,·)$, $\chi_{480}(161,·)$, $\chi_{480}(41,·)$, $\chi_{480}(439,·)$, $\chi_{480}(319,·)$, $\chi_{480}(451,·)$, $\chi_{480}(199,·)$, $\chi_{480}(331,·)$, $\chi_{480}(79,·)$, $\chi_{480}(211,·)$, $\chi_{480}(469,·)$, $\chi_{480}(91,·)$, $\chi_{480}(349,·)$, $\chi_{480}(479,·)$, $\chi_{480}(229,·)$, $\chi_{480}(359,·)$, $\chi_{480}(361,·)$, $\chi_{480}(109,·)$, $\chi_{480}(239,·)$, $\chi_{480}(241,·)$, $\chi_{480}(371,·)$, $\chi_{480}(119,·)$, $\chi_{480}(121,·)$, $\chi_{480}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{21} a^{16} + \frac{8}{21} a^{14} + \frac{2}{21} a^{12} + \frac{2}{21} a^{10} - \frac{1}{7} a^{8} - \frac{2}{7} a^{6} + \frac{2}{7} a^{4} - \frac{10}{21} a^{2} + \frac{4}{21}$, $\frac{1}{987} a^{17} + \frac{386}{987} a^{15} + \frac{254}{987} a^{13} - \frac{124}{987} a^{11} - \frac{127}{329} a^{9} + \frac{110}{329} a^{7} + \frac{135}{329} a^{5} - \frac{283}{987} a^{3} + \frac{256}{987} a$, $\frac{1}{987} a^{18} + \frac{10}{987} a^{16} + \frac{69}{329} a^{14} + \frac{37}{329} a^{12} - \frac{146}{987} a^{10} + \frac{157}{329} a^{8} - \frac{100}{329} a^{6} + \frac{422}{987} a^{4} + \frac{68}{987} a^{2} + \frac{10}{21}$, $\frac{1}{987} a^{19} + \frac{295}{987} a^{15} - \frac{65}{141} a^{13} + \frac{107}{987} a^{11} + \frac{111}{329} a^{9} + \frac{116}{329} a^{7} + \frac{320}{987} a^{5} - \frac{3}{47} a^{3} - \frac{116}{987} a$, $\frac{1}{987} a^{20} + \frac{13}{987} a^{16} + \frac{250}{987} a^{14} - \frac{457}{987} a^{12} - \frac{11}{47} a^{10} + \frac{69}{329} a^{8} + \frac{38}{987} a^{6} + \frac{73}{329} a^{4} - \frac{257}{987} a^{2} - \frac{1}{7}$, $\frac{1}{987} a^{21} + \frac{167}{987} a^{15} + \frac{9}{47} a^{13} + \frac{394}{987} a^{11} + \frac{75}{329} a^{9} - \frac{304}{987} a^{7} - \frac{37}{329} a^{5} + \frac{461}{987} a^{3} + \frac{479}{987} a$, $\frac{1}{987} a^{22} - \frac{1}{47} a^{16} - \frac{328}{987} a^{14} + \frac{6}{329} a^{12} - \frac{151}{987} a^{10} + \frac{260}{987} a^{8} + \frac{10}{329} a^{6} + \frac{320}{987} a^{4} + \frac{55}{141} a^{2} + \frac{5}{21}$, $\frac{1}{987} a^{23} - \frac{118}{987} a^{15} + \frac{139}{329} a^{13} + \frac{206}{987} a^{11} + \frac{155}{987} a^{9} + \frac{17}{329} a^{7} - \frac{58}{987} a^{5} + \frac{52}{141} a^{3} - \frac{311}{987} a$, $\frac{1}{2961} a^{24} + \frac{1}{2961} a^{18} + \frac{11}{987} a^{16} - \frac{74}{987} a^{14} + \frac{599}{2961} a^{12} + \frac{97}{987} a^{10} + \frac{11}{329} a^{8} + \frac{110}{423} a^{6} - \frac{38}{329} a^{4} + \frac{436}{987} a^{2} - \frac{20}{63}$, $\frac{1}{2961} a^{25} + \frac{1}{2961} a^{19} - \frac{124}{329} a^{15} + \frac{1100}{2961} a^{13} + \frac{158}{329} a^{11} + \frac{92}{329} a^{9} - \frac{1237}{2961} a^{7} + \frac{122}{329} a^{5} - \frac{19}{47} a^{3} - \frac{505}{2961} a$, $\frac{1}{2961} a^{26} + \frac{1}{2961} a^{20} + \frac{4}{987} a^{16} + \frac{1241}{2961} a^{14} + \frac{239}{987} a^{12} + \frac{41}{987} a^{10} + \frac{1301}{2961} a^{8} + \frac{4}{47} a^{6} - \frac{39}{329} a^{4} + \frac{59}{2961} a^{2} - \frac{10}{21}$, $\frac{1}{2961} a^{27} + \frac{1}{2961} a^{21} - \frac{430}{2961} a^{15} + \frac{10}{47} a^{13} - \frac{150}{329} a^{11} - \frac{7}{423} a^{9} - \frac{83}{329} a^{7} + \frac{79}{329} a^{5} + \frac{494}{2961} a^{3} + \frac{160}{329} a$, $\frac{1}{1086687} a^{28} - \frac{19}{1086687} a^{26} - \frac{125}{1086687} a^{24} + \frac{16}{155241} a^{22} - \frac{187}{1086687} a^{20} - \frac{32}{155241} a^{18} - \frac{21850}{1086687} a^{16} - \frac{396827}{1086687} a^{14} - \frac{86641}{1086687} a^{12} + \frac{279599}{1086687} a^{10} - \frac{479123}{1086687} a^{8} - \frac{543211}{1086687} a^{6} + \frac{440861}{1086687} a^{4} + \frac{194473}{1086687} a^{2} + \frac{328}{23121}$, $\frac{1}{1086687} a^{29} - \frac{19}{1086687} a^{27} - \frac{125}{1086687} a^{25} + \frac{16}{155241} a^{23} - \frac{187}{1086687} a^{21} - \frac{32}{155241} a^{19} + \frac{170}{1086687} a^{17} + \frac{496084}{1086687} a^{15} + \frac{73004}{1086687} a^{13} - \frac{277507}{1086687} a^{11} - \frac{175247}{1086687} a^{9} + \frac{203267}{1086687} a^{7} - \frac{421222}{1086687} a^{5} + \frac{482935}{1086687} a^{3} + \frac{219101}{1086687} a$, $\frac{1}{6280535854698834101758086791543391} a^{30} - \frac{1002047712776646392768668373}{6280535854698834101758086791543391} a^{28} - \frac{17828160679068633730273466149}{697837317188759344639787421282599} a^{26} - \frac{38994742551098842802144204642}{697837317188759344639787421282599} a^{24} - \frac{232776106205217846512493460610}{897219407814119157394012398791913} a^{22} + \frac{498235528801084463944541078465}{2093511951566278033919362263847797} a^{20} + \frac{2079324348793989999606729052024}{6280535854698834101758086791543391} a^{18} + \frac{149157783068035983906248920862375}{6280535854698834101758086791543391} a^{16} + \frac{2913725241839606656329980848125}{5864179136039994492771322867921} a^{14} - \frac{123689202686366467393330105965724}{299073135938039719131337466263971} a^{12} + \frac{1092884807902406349636187865771594}{6280535854698834101758086791543391} a^{10} + \frac{64436967453629414464377182572729}{2093511951566278033919362263847797} a^{8} - \frac{37101356314178021208485002960114}{299073135938039719131337466263971} a^{6} - \frac{2764423620062755051863750732028978}{6280535854698834101758086791543391} a^{4} - \frac{16218958139844823430249455594775}{99691045312679906377112488754657} a^{2} - \frac{572841876828141062882239463045}{2843157924263845224879170118399}$, $\frac{1}{295185185170845202782630079202539377} a^{31} - \frac{53017782411258828397175439710}{295185185170845202782630079202539377} a^{29} - \frac{4729933087794467919671034192862}{42169312167263600397518582743219911} a^{27} - \frac{29907449043852987226389989910380}{295185185170845202782630079202539377} a^{25} + \frac{119809969199286542660700887923846}{295185185170845202782630079202539377} a^{23} - \frac{92711568478625587349480862630505}{295185185170845202782630079202539377} a^{21} + \frac{73121258894765432354958732749180}{295185185170845202782630079202539377} a^{19} + \frac{51229493209026862585952439358583}{295185185170845202782630079202539377} a^{17} + \frac{6973217637222828851095968752643320}{17363834421814423693095887011914081} a^{15} - \frac{81095239182596177174759593712339110}{295185185170845202782630079202539377} a^{13} + \frac{36246384033347661347976391166752061}{295185185170845202782630079202539377} a^{11} + \frac{132465532871494165490444563186376179}{295185185170845202782630079202539377} a^{9} + \frac{33665719932367550523596648032071971}{295185185170845202782630079202539377} a^{7} + \frac{94728529216676536059843849033519890}{295185185170845202782630079202539377} a^{5} - \frac{73259819169121108788386385460571954}{295185185170845202782630079202539377} a^{3} + \frac{8593604429316634925239745221361}{44542807480133575189773665188251} a$

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

Class group and class number

$C_{14}\times C_{14}$, which has order $196$ (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:  \( \frac{91672349091000554824345648}{299073135938039719131337466263971} a^{30} + \frac{1397132381438424854505019192}{299073135938039719131337466263971} a^{28} + \frac{13025955408143871723328084096}{299073135938039719131337466263971} a^{26} + \frac{80304494149943252248321523780}{299073135938039719131337466263971} a^{24} + \frac{373909536745417401363309376288}{299073135938039719131337466263971} a^{22} + \frac{1319966345401911243987424048480}{299073135938039719131337466263971} a^{20} + \frac{1238755849223131768384946801344}{99691045312679906377112488754657} a^{18} + \frac{8236830719238851611906115520701}{299073135938039719131337466263971} a^{16} + \frac{877936303975845247739189963344}{17592537408119983478313968603763} a^{14} + \frac{26061264485938460806254446477248}{299073135938039719131337466263971} a^{12} + \frac{70365442071475939757434869518624}{299073135938039719131337466263971} a^{10} + \frac{84335754666857100790068360514940}{299073135938039719131337466263971} a^{8} - \frac{66004213142279305733309195098496}{299073135938039719131337466263971} a^{6} - \frac{109658642764432512766182295484848}{299073135938039719131337466263971} a^{4} + \frac{2736342960687391755231844878656}{299073135938039719131337466263971} a^{2} + \frac{81316119782642399015503077491}{135388472583992629756150958019} \) (order $6$)
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:  \( 34585710193758.383 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\zeta_{16})^+\), 4.4.460800.1, 4.0.18432.2, 4.0.51200.2, 8.0.3317760000.3, 8.8.849346560000.2, 8.0.849346560000.1, 8.0.339738624.2, 8.0.212336640000.6, 8.0.10485760000.2, 8.0.849346560000.3, 8.8.1342177280000.1, 8.8.173946175488.1, 8.0.108716359680000.13, 8.0.2147483648.1, 16.0.721389578983833600000000.9, 16.16.47276987448284518809600000000.2, 16.0.47276987448284518809600000000.6, 16.0.11819246862071129702400000000.4, 16.0.30257271966902092038144.2, 16.0.7205759403792793600000000.3, 16.0.47276987448284518809600000000.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed