↑

Properties

Label 32.32.7219274832...1136.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{124}\cdot 17^{24}$
Root discriminant $122.84$
Ramified primes $2, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times C_8$ (as 32T43)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2753, 49768, 552740, -6390976, -21676018, 50201368, 144420248, -187713208, -437179375, 424971976, 749016552, -627979512, -797607810, 624757920, 553415588, -425796088, -255633450, 200834584, 78677084, -65768208, -15740682, 14880024, 1894744, -2290952, -101787, 232952, -4040, -14808, 942, 528, -52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 - 52*x^30 + 528*x^29 + 942*x^28 - 14808*x^27 - 4040*x^26 + 232952*x^25 - 101787*x^24 - 2290952*x^23 + 1894744*x^22 + 14880024*x^21 - 15740682*x^20 - 65768208*x^19 + 78677084*x^18 + 200834584*x^17 - 255633450*x^16 - 425796088*x^15 + 553415588*x^14 + 624757920*x^13 - 797607810*x^12 - 627979512*x^11 + 749016552*x^10 + 424971976*x^9 - 437179375*x^8 - 187713208*x^7 + 144420248*x^6 + 50201368*x^5 - 21676018*x^4 - 6390976*x^3 + 552740*x^2 + 49768*x - 2753)
 
gp: K = bnfinit(x^32 - 8*x^31 - 52*x^30 + 528*x^29 + 942*x^28 - 14808*x^27 - 4040*x^26 + 232952*x^25 - 101787*x^24 - 2290952*x^23 + 1894744*x^22 + 14880024*x^21 - 15740682*x^20 - 65768208*x^19 + 78677084*x^18 + 200834584*x^17 - 255633450*x^16 - 425796088*x^15 + 553415588*x^14 + 624757920*x^13 - 797607810*x^12 - 627979512*x^11 + 749016552*x^10 + 424971976*x^9 - 437179375*x^8 - 187713208*x^7 + 144420248*x^6 + 50201368*x^5 - 21676018*x^4 - 6390976*x^3 + 552740*x^2 + 49768*x - 2753, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} - 52 x^{30} + 528 x^{29} + 942 x^{28} - 14808 x^{27} - 4040 x^{26} + 232952 x^{25} - 101787 x^{24} - 2290952 x^{23} + 1894744 x^{22} + 14880024 x^{21} - 15740682 x^{20} - 65768208 x^{19} + 78677084 x^{18} + 200834584 x^{17} - 255633450 x^{16} - 425796088 x^{15} + 553415588 x^{14} + 624757920 x^{13} - 797607810 x^{12} - 627979512 x^{11} + 749016552 x^{10} + 424971976 x^{9} - 437179375 x^{8} - 187713208 x^{7} + 144420248 x^{6} + 50201368 x^{5} - 21676018 x^{4} - 6390976 x^{3} + 552740 x^{2} + 49768 x - 2753 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7219274832693987941813796201388115200494782171469796274056657371136=2^{124}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.84$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(544=2^{5}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{544}(1,·)$, $\chi_{544}(137,·)$, $\chi_{544}(13,·)$, $\chi_{544}(273,·)$, $\chi_{544}(149,·)$, $\chi_{544}(89,·)$, $\chi_{544}(409,·)$, $\chi_{544}(157,·)$, $\chi_{544}(33,·)$, $\chi_{544}(293,·)$, $\chi_{544}(169,·)$, $\chi_{544}(429,·)$, $\chi_{544}(285,·)$, $\chi_{544}(305,·)$, $\chi_{544}(441,·)$, $\chi_{544}(69,·)$, $\chi_{544}(353,·)$, $\chi_{544}(205,·)$, $\chi_{544}(81,·)$, $\chi_{544}(341,·)$, $\chi_{544}(217,·)$, $\chi_{544}(477,·)$, $\chi_{544}(421,·)$, $\chi_{544}(225,·)$, $\chi_{544}(101,·)$, $\chi_{544}(361,·)$, $\chi_{544}(237,·)$, $\chi_{544}(497,·)$, $\chi_{544}(373,·)$, $\chi_{544}(489,·)$, $\chi_{544}(509,·)$, $\chi_{544}(21,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{16} a^{24} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{4} a^{20} - \frac{1}{4} a^{18} - \frac{1}{8} a^{16} - \frac{1}{2} a^{15} + \frac{3}{8} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{16}$, $\frac{1}{16} a^{25} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{4} a^{21} - \frac{1}{4} a^{19} - \frac{1}{8} a^{17} - \frac{1}{2} a^{16} + \frac{3}{8} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{16} a$, $\frac{1}{32} a^{26} - \frac{1}{32} a^{24} - \frac{1}{4} a^{23} - \frac{3}{8} a^{22} - \frac{1}{4} a^{21} - \frac{7}{16} a^{18} - \frac{1}{4} a^{17} + \frac{1}{16} a^{16} - \frac{1}{4} a^{15} + \frac{3}{16} a^{14} - \frac{1}{2} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{7}{32} a^{2} - \frac{1}{2} a + \frac{7}{32}$, $\frac{1}{32} a^{27} - \frac{1}{32} a^{25} - \frac{3}{8} a^{23} - \frac{1}{4} a^{22} - \frac{7}{16} a^{19} - \frac{1}{4} a^{18} + \frac{1}{16} a^{17} + \frac{1}{4} a^{16} + \frac{3}{16} a^{15} - \frac{1}{2} a^{14} + \frac{1}{16} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{3}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{7}{32} a^{3} - \frac{1}{2} a^{2} + \frac{7}{32} a + \frac{1}{4}$, $\frac{1}{224} a^{28} + \frac{1}{112} a^{26} + \frac{3}{112} a^{25} + \frac{3}{224} a^{24} - \frac{5}{14} a^{23} + \frac{15}{56} a^{22} + \frac{2}{7} a^{21} - \frac{27}{112} a^{20} - \frac{5}{14} a^{18} + \frac{9}{56} a^{17} - \frac{5}{28} a^{16} + \frac{9}{28} a^{15} - \frac{27}{56} a^{14} - \frac{25}{56} a^{13} + \frac{41}{112} a^{12} + \frac{5}{14} a^{11} - \frac{11}{56} a^{10} + \frac{13}{28} a^{9} - \frac{23}{56} a^{8} + \frac{3}{7} a^{7} - \frac{3}{56} a^{6} + \frac{3}{28} a^{5} + \frac{109}{224} a^{4} - \frac{5}{28} a^{3} - \frac{15}{112} a^{2} + \frac{47}{112} a + \frac{39}{224}$, $\frac{1}{224} a^{29} + \frac{1}{112} a^{27} - \frac{1}{224} a^{26} + \frac{3}{224} a^{25} - \frac{3}{224} a^{24} - \frac{27}{56} a^{23} + \frac{9}{56} a^{22} - \frac{55}{112} a^{21} - \frac{1}{4} a^{20} - \frac{5}{14} a^{19} + \frac{39}{112} a^{18} + \frac{1}{14} a^{17} - \frac{41}{112} a^{16} + \frac{15}{56} a^{15} + \frac{41}{112} a^{14} - \frac{15}{112} a^{13} + \frac{19}{112} a^{12} + \frac{3}{56} a^{11} - \frac{1}{28} a^{10} + \frac{19}{56} a^{9} + \frac{17}{56} a^{8} - \frac{3}{56} a^{7} + \frac{3}{28} a^{6} - \frac{59}{224} a^{5} + \frac{25}{56} a^{4} - \frac{43}{112} a^{3} + \frac{31}{224} a^{2} - \frac{73}{224} a + \frac{3}{32}$, $\frac{1}{62835579865409499735993938304396323244832} a^{30} + \frac{2076980401041237834708456010707553705}{1122063926168026780999891755435648629372} a^{29} - \frac{71830252900307680042841160238328427665}{62835579865409499735993938304396323244832} a^{28} + \frac{118016327213388568242922664471065070637}{7854447483176187466999242288049540405604} a^{27} + \frac{533696258922885043121761784133961759145}{62835579865409499735993938304396323244832} a^{26} - \frac{712518930573067507340744901514419285167}{31417789932704749867996969152198161622416} a^{25} + \frac{1588092889854073628963633943398523948729}{62835579865409499735993938304396323244832} a^{24} - \frac{2327098857903693068608952627971762761407}{7854447483176187466999242288049540405604} a^{23} - \frac{2128789005190940217888505664305200282051}{4488255704672107123999567021742594517488} a^{22} + \frac{547269917761548380692315073003390649137}{7854447483176187466999242288049540405604} a^{21} + \frac{988668935005284061456964120349059466341}{31417789932704749867996969152198161622416} a^{20} + \frac{1231654598355273374615104237343048714421}{3927223741588093733499621144024770202802} a^{19} - \frac{187204743742241077857875229118194142533}{1122063926168026780999891755435648629372} a^{18} - \frac{6751921422639978570056391662040402771351}{15708894966352374933998484576099080811208} a^{17} + \frac{370259625044496806480024675634364809727}{1963611870794046866749810572012385101401} a^{16} + \frac{586254882580369228623225453007539421658}{1963611870794046866749810572012385101401} a^{15} - \frac{15565405493794727158507286353830004506641}{31417789932704749867996969152198161622416} a^{14} - \frac{1170062713915351598268367103049549370823}{15708894966352374933998484576099080811208} a^{13} - \frac{4133240992909127754548511373230173426375}{31417789932704749867996969152198161622416} a^{12} + \frac{2887587573838769377788237875818029887281}{7854447483176187466999242288049540405604} a^{11} + \frac{411413629389797655338904922315525324705}{1122063926168026780999891755435648629372} a^{10} - \frac{513010951181824237496952271303157165091}{3927223741588093733499621144024770202802} a^{9} + \frac{3855682615201361059154752616721421700053}{7854447483176187466999242288049540405604} a^{8} + \frac{459660652579877379503644493592077106363}{3927223741588093733499621144024770202802} a^{7} - \frac{19856705796081189014178019655629719409223}{62835579865409499735993938304396323244832} a^{6} + \frac{2151401198505265634243132322989260029221}{7854447483176187466999242288049540405604} a^{5} + \frac{9669363482931362084197737929858220829283}{62835579865409499735993938304396323244832} a^{4} - \frac{238178946716079774536553900891033866859}{7854447483176187466999242288049540405604} a^{3} + \frac{22300519881723776697691847593634065649629}{62835579865409499735993938304396323244832} a^{2} + \frac{8034968975644500425523474493562629558717}{31417789932704749867996969152198161622416} a - \frac{24367933900260096605042009309328242293407}{62835579865409499735993938304396323244832}$, $\frac{1}{4226087720905031048686844143276366513965581225353991392} a^{31} - \frac{1527865248995}{4226087720905031048686844143276366513965581225353991392} a^{30} - \frac{2066546203673683604675616499427344262961112670765805}{2113043860452515524343422071638183256982790612676995696} a^{29} + \frac{2721825991579899294005777276706585628454456453011117}{2113043860452515524343422071638183256982790612676995696} a^{28} - \frac{44733892180385906881243377429894956815310842063238839}{4226087720905031048686844143276366513965581225353991392} a^{27} + \frac{1487259598256030984273007870106513773877807455264811}{1056521930226257762171711035819091628491395306338497848} a^{26} + \frac{8084513719434527480576317966665279035359905159912479}{2113043860452515524343422071638183256982790612676995696} a^{25} + \frac{50600363926425076953861341370486382476371067032676257}{4226087720905031048686844143276366513965581225353991392} a^{24} + \frac{1003791374822011803872612291566127266229505627419293041}{2113043860452515524343422071638183256982790612676995696} a^{23} + \frac{238666217757337655013822284483187174039936911798552991}{2113043860452515524343422071638183256982790612676995696} a^{22} - \frac{30073471316787089487603187145615261072160619087371747}{528260965113128881085855517909545814245697653169248924} a^{21} + \frac{225322170635935972423331304987857756515094661806594179}{528260965113128881085855517909545814245697653169248924} a^{20} - \frac{39511387242969814542699759321641072615958556275086687}{1056521930226257762171711035819091628491395306338497848} a^{19} - \frac{384358490382518063306296268984003886199947422115316599}{2113043860452515524343422071638183256982790612676995696} a^{18} - \frac{151384288451559682991141015103423272827139158390027659}{528260965113128881085855517909545814245697653169248924} a^{17} - \frac{746975094621085368955806619543130903230624777938575207}{2113043860452515524343422071638183256982790612676995696} a^{16} + \frac{728482411434015981640379207682107245257616039508942747}{2113043860452515524343422071638183256982790612676995696} a^{15} + \frac{176767667617959119909628803991308730311762896267502101}{528260965113128881085855517909545814245697653169248924} a^{14} - \frac{63585035786854051895001997885252644327852669202679939}{264130482556564440542927758954772907122848826584624462} a^{13} - \frac{508319824522336193892940816257089332113790859935053911}{2113043860452515524343422071638183256982790612676995696} a^{12} - \frac{304724243598959470724523868734819072500939102617620935}{1056521930226257762171711035819091628491395306338497848} a^{11} + \frac{303919637290558922724167657481166574727691768167498627}{1056521930226257762171711035819091628491395306338497848} a^{10} - \frac{350025855969356871105554469670620543309527814067853161}{1056521930226257762171711035819091628491395306338497848} a^{9} + \frac{249637758716092818750766409409166686251554128111882411}{528260965113128881085855517909545814245697653169248924} a^{8} - \frac{558332607926669757262448369387802237897872615666016691}{4226087720905031048686844143276366513965581225353991392} a^{7} - \frac{733265000327680327806086236033596538148630113045262927}{4226087720905031048686844143276366513965581225353991392} a^{6} - \frac{344683444189157749145100537986186213491722182214634593}{2113043860452515524343422071638183256982790612676995696} a^{5} - \frac{107925121348163617800841048774287808402599136056339903}{301863408636073646334774581662597608140398658953856528} a^{4} + \frac{624229880708303434253238892280657082689460352933749437}{4226087720905031048686844143276366513965581225353991392} a^{3} + \frac{170643082161178090581069830432707327704946448928303349}{528260965113128881085855517909545814245697653169248924} a^{2} - \frac{276754170676662884941409161767132485363343492682042959}{2113043860452515524343422071638183256982790612676995696} a + \frac{1107845989822302222088427859633646139144045731499959909}{4226087720905031048686844143276366513965581225353991392}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $31$
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:  \( 269323465797089430000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), 4.4.4913.1, \(\Q(\sqrt{2}, \sqrt{17})\), 4.4.314432.1, \(\Q(\zeta_{16})^+\), 4.4.591872.2, 4.4.10061824.1, 4.4.10061824.2, 8.8.98867482624.1, 8.8.350312464384.1, 8.8.101240302206976.1, 8.8.179359981764608.1, \(\Q(\zeta_{32})^+\), 8.8.51835034729971712.3, 8.8.51835034729971712.2, 16.16.10249598790959829536343064576.1, 16.16.32170003058600514289521393664.1, 16.16.2686870825457373553975116320210944.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$