Normalized defining polynomial
\( x^{32} - x^{31} + 2 x^{30} + 60 x^{29} - 53 x^{28} + 99 x^{27} + 1385 x^{26} - 1071 x^{25} + 1854 x^{24} + 15802 x^{23} - 10509 x^{22} + 16583 x^{21} + 95912 x^{20} - 54402 x^{19} + 75534 x^{18} + 310800 x^{17} - 136853 x^{16} + 163941 x^{15} + 541045 x^{14} - 33297 x^{13} + 98317 x^{12} + 537573 x^{11} + 431212 x^{10} - 151090 x^{9} + 496535 x^{8} + 326343 x^{7} + 132437 x^{6} + 314657 x^{5} + 175914 x^{4} + 198989 x^{3} - 20144 x^{2} - 93555 x + 27583 \)
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: | \(38897685648686306961584993323026037365043690280067733586177953=97^{31}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $84.08$ | 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: | $97$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(97\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{97}(1,·)$, $\chi_{97}(8,·)$, $\chi_{97}(12,·)$, $\chi_{97}(18,·)$, $\chi_{97}(19,·)$, $\chi_{97}(20,·)$, $\chi_{97}(22,·)$, $\chi_{97}(27,·)$, $\chi_{97}(28,·)$, $\chi_{97}(30,·)$, $\chi_{97}(33,·)$, $\chi_{97}(34,·)$, $\chi_{97}(42,·)$, $\chi_{97}(45,·)$, $\chi_{97}(46,·)$, $\chi_{97}(47,·)$, $\chi_{97}(50,·)$, $\chi_{97}(51,·)$, $\chi_{97}(52,·)$, $\chi_{97}(55,·)$, $\chi_{97}(63,·)$, $\chi_{97}(64,·)$, $\chi_{97}(67,·)$, $\chi_{97}(69,·)$, $\chi_{97}(70,·)$, $\chi_{97}(75,·)$, $\chi_{97}(77,·)$, $\chi_{97}(78,·)$, $\chi_{97}(79,·)$, $\chi_{97}(85,·)$, $\chi_{97}(89,·)$, $\chi_{97}(96,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{61} a^{22} - \frac{19}{61} a^{21} + \frac{30}{61} a^{20} + \frac{25}{61} a^{19} - \frac{26}{61} a^{18} - \frac{25}{61} a^{17} - \frac{19}{61} a^{16} - \frac{23}{61} a^{15} + \frac{7}{61} a^{14} + \frac{11}{61} a^{12} - \frac{21}{61} a^{11} - \frac{8}{61} a^{10} + \frac{7}{61} a^{9} - \frac{18}{61} a^{8} - \frac{20}{61} a^{7} + \frac{12}{61} a^{6} + \frac{3}{61} a^{5} + \frac{25}{61} a^{4} + \frac{10}{61} a^{3} + \frac{24}{61} a^{2} - \frac{19}{61} a + \frac{1}{61}$, $\frac{1}{61} a^{23} - \frac{26}{61} a^{21} - \frac{15}{61} a^{20} + \frac{22}{61} a^{19} + \frac{30}{61} a^{18} - \frac{6}{61} a^{17} - \frac{18}{61} a^{16} - \frac{3}{61} a^{15} + \frac{11}{61} a^{14} + \frac{11}{61} a^{13} + \frac{5}{61} a^{12} + \frac{20}{61} a^{11} - \frac{23}{61} a^{10} - \frac{7}{61} a^{9} + \frac{4}{61} a^{8} - \frac{2}{61} a^{7} - \frac{13}{61} a^{6} + \frac{21}{61} a^{5} - \frac{3}{61} a^{4} - \frac{30}{61} a^{3} + \frac{10}{61} a^{2} + \frac{6}{61} a + \frac{19}{61}$, $\frac{1}{61} a^{24} - \frac{21}{61} a^{21} + \frac{9}{61} a^{20} + \frac{9}{61} a^{19} - \frac{11}{61} a^{18} + \frac{3}{61} a^{17} - \frac{9}{61} a^{16} + \frac{23}{61} a^{15} + \frac{10}{61} a^{14} + \frac{5}{61} a^{13} + \frac{1}{61} a^{12} - \frac{20}{61} a^{11} + \frac{29}{61} a^{10} + \frac{3}{61} a^{9} + \frac{18}{61} a^{8} + \frac{16}{61} a^{7} + \frac{28}{61} a^{6} + \frac{14}{61} a^{5} + \frac{10}{61} a^{4} + \frac{26}{61} a^{3} + \frac{20}{61} a^{2} + \frac{13}{61} a + \frac{26}{61}$, $\frac{1}{61} a^{25} - \frac{24}{61} a^{21} + \frac{29}{61} a^{20} + \frac{26}{61} a^{19} + \frac{6}{61} a^{18} + \frac{15}{61} a^{17} - \frac{10}{61} a^{16} + \frac{15}{61} a^{15} + \frac{30}{61} a^{14} + \frac{1}{61} a^{13} + \frac{28}{61} a^{12} + \frac{15}{61} a^{11} + \frac{18}{61} a^{10} - \frac{18}{61} a^{9} + \frac{4}{61} a^{8} - \frac{26}{61} a^{7} + \frac{22}{61} a^{6} + \frac{12}{61} a^{5} + \frac{2}{61} a^{4} - \frac{14}{61} a^{3} + \frac{29}{61} a^{2} - \frac{7}{61} a + \frac{21}{61}$, $\frac{1}{61} a^{26} + \frac{14}{61} a^{20} - \frac{4}{61} a^{19} + \frac{1}{61} a^{18} - \frac{14}{61} a^{16} + \frac{27}{61} a^{15} - \frac{14}{61} a^{14} + \frac{28}{61} a^{13} - \frac{26}{61} a^{12} + \frac{2}{61} a^{11} - \frac{27}{61} a^{10} - \frac{11}{61} a^{9} + \frac{30}{61} a^{8} + \frac{30}{61} a^{7} - \frac{5}{61} a^{6} + \frac{13}{61} a^{5} - \frac{24}{61} a^{4} + \frac{25}{61} a^{3} + \frac{20}{61} a^{2} - \frac{8}{61} a + \frac{24}{61}$, $\frac{1}{61} a^{27} + \frac{14}{61} a^{21} - \frac{4}{61} a^{20} + \frac{1}{61} a^{19} - \frac{14}{61} a^{17} + \frac{27}{61} a^{16} - \frac{14}{61} a^{15} + \frac{28}{61} a^{14} - \frac{26}{61} a^{13} + \frac{2}{61} a^{12} - \frac{27}{61} a^{11} - \frac{11}{61} a^{10} + \frac{30}{61} a^{9} + \frac{30}{61} a^{8} - \frac{5}{61} a^{7} + \frac{13}{61} a^{6} - \frac{24}{61} a^{5} + \frac{25}{61} a^{4} + \frac{20}{61} a^{3} - \frac{8}{61} a^{2} + \frac{24}{61} a$, $\frac{1}{61} a^{28} + \frac{18}{61} a^{21} + \frac{8}{61} a^{20} + \frac{16}{61} a^{19} - \frac{16}{61} a^{18} + \frac{11}{61} a^{17} + \frac{8}{61} a^{16} - \frac{16}{61} a^{15} - \frac{2}{61} a^{14} + \frac{2}{61} a^{13} + \frac{2}{61} a^{12} - \frac{22}{61} a^{11} + \frac{20}{61} a^{10} - \frac{7}{61} a^{9} + \frac{3}{61} a^{8} - \frac{12}{61} a^{7} - \frac{9}{61} a^{6} - \frac{17}{61} a^{5} - \frac{25}{61} a^{4} - \frac{26}{61} a^{3} - \frac{7}{61} a^{2} + \frac{22}{61} a - \frac{14}{61}$, $\frac{1}{3721} a^{29} + \frac{16}{3721} a^{28} - \frac{29}{3721} a^{27} - \frac{26}{3721} a^{26} + \frac{18}{3721} a^{25} - \frac{30}{3721} a^{24} + \frac{6}{3721} a^{23} - \frac{13}{3721} a^{22} - \frac{943}{3721} a^{21} - \frac{1299}{3721} a^{20} - \frac{679}{3721} a^{19} + \frac{1275}{3721} a^{18} - \frac{565}{3721} a^{17} + \frac{813}{3721} a^{16} - \frac{584}{3721} a^{15} + \frac{892}{3721} a^{14} + \frac{604}{3721} a^{13} - \frac{551}{3721} a^{12} + \frac{1552}{3721} a^{11} + \frac{1630}{3721} a^{10} + \frac{464}{3721} a^{9} + \frac{1001}{3721} a^{8} + \frac{593}{3721} a^{7} - \frac{82}{3721} a^{6} - \frac{964}{3721} a^{5} + \frac{1039}{3721} a^{4} - \frac{1101}{3721} a^{3} - \frac{957}{3721} a^{2} + \frac{630}{3721} a - \frac{1045}{3721}$, $\frac{1}{3721} a^{30} + \frac{20}{3721} a^{28} + \frac{11}{3721} a^{27} + \frac{7}{3721} a^{26} - \frac{13}{3721} a^{25} - \frac{2}{3721} a^{24} + \frac{13}{3721} a^{23} - \frac{3}{3721} a^{22} - \frac{851}{3721} a^{21} - \frac{1794}{3721} a^{20} - \frac{30}{61} a^{19} - \frac{957}{3721} a^{18} - \frac{456}{3721} a^{17} - \frac{1697}{3721} a^{16} - \frac{1720}{3721} a^{15} + \frac{1643}{3721} a^{14} - \frac{89}{3721} a^{13} + \frac{730}{3721} a^{12} + \frac{771}{3721} a^{11} + \frac{1712}{3721} a^{10} - \frac{750}{3721} a^{9} - \frac{844}{3721} a^{8} + \frac{1288}{3721} a^{7} - \frac{1848}{3721} a^{6} - \frac{1044}{3721} a^{5} - \frac{950}{3721} a^{4} - \frac{1458}{3721} a^{3} + \frac{509}{3721} a^{2} + \frac{587}{3721} a - \frac{1031}{3721}$, $\frac{1}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{31} - \frac{1596555989191631884161889003296375535263449526184212570821450151993143929653228}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{30} + \frac{1073024718872595506631620961502966677873062552374733412618402252583827779074674}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{29} + \frac{8496515359357624301281352628838617241428811525493765320805847151589565909128626}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{28} + \frac{34309525677493904677977807939907995503997465770928110636506171343742707344476387}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{27} + \frac{63890496668190502349392592799531418854082399368038907875247269595774787842599841}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{26} - \frac{76601096027253857558472048877972387055436543383599407078654501582636605640532870}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{25} + \frac{20135742857490925606341419098283411671450778245912227825879598933781326136066647}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{24} - \frac{72665181871542994466413228117632284009725101886889857092121760915807055335629316}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{23} - \frac{87706777989641507486167029714517675151410119407634375898421106952686951301840005}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{22} - \frac{4780761544204073991363050759606743432076274476019979166656679665513442547705569941}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{21} - \frac{1815724852999529078545275352105965409205669495388361717101556202038444458016871132}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{20} - \frac{5269885857706556753504098612477289721592694926362636519052443542377514360591164584}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{19} + \frac{1928048203973441070223301625124081458183523785374990103571197833306336952013381742}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{18} - \frac{309044575123428533763324655869114936385922910503175357760604292118642265642878421}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{17} + \frac{3064472235621598002162576506269064824095684388739427475135131198163636869901841383}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{16} + \frac{227050740112916721142446272901154916009885651058524621912198948142767766861390357}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{15} - \frac{3783144825001813911743417901175114595467035588471096579407623985082499148990305759}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{14} + \frac{570874397643469751194674188018345601086325908068304800369544928471395414963605138}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{13} + \frac{2175214020114134105033090136382211876743215038197931103835024996716406016588264995}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{12} + \frac{4281876606865762621757645707001231959385239379954235108272836869520957787379104145}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{11} + \frac{4695890576692480872339056313398683148382509754605658775806016402411888054108009834}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{10} - \frac{228885933952951738462920665862044096163174810077108175715002066026600024437974505}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{9} - \frac{5977705537872882157614156890826245752712429708007617956605924181652228634191010017}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{8} + \frac{5311144898380694294882566168781685371362325534002869609653145165111958781325959119}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{7} + \frac{1470294319587436711304854949641841413002993294023490165247102534409635311963896038}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{6} - \frac{3915125002238079432125245767462310053248502521701349157593705966682432082415374669}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{5} + \frac{394763214178975032777410700881326144055924200610641464357704276883287500649553731}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{4} - \frac{3299622367975644187194662752457921080447168746574685768693802604321566958565874172}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{3} + \frac{4494009084595239799371582420401691161461146812850412387430116823569328663808414726}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a^{2} - \frac{2306006131960211642588444765433083404928390836704443684381237699982863288188411299}{13023445692307387581099160793606683864592598153560302232255635846452746366322664333} a - \frac{168099487696461986522525153646451756178179717743600381070072975087185516373653}{472154794340984939314039835899165568088772002811887837880420398305215037027251}$
Class group and class number
$C_{3457}$, which has order $3457$ (assuming GRH)
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: | \( 132045958705412.02 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 32 |
| The 32 conjugacy class representatives for $C_{32}$ |
| Character table for $C_{32}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1, 16.16.633251189136789386043275954593.1 |
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 | $16^{2}$ | $16^{2}$ | $32$ | $32$ | $16^{2}$ | $32$ | $32$ | $32$ | $32$ | $32$ | $16^{2}$ | $32$ | $32$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ | $16^{2}$ | $32$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 97 | Data not computed | ||||||