# Properties

 Label 32.0.38897685648...7953.1 Degree $32$ Signature $[0, 16]$ Discriminant $97^{31}$ Root discriminant $84.08$ Ramified prime $97$ Class number $3457$ (GRH) Class group $[3457]$ (GRH) Galois group $C_{32}$ (as 32T33)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27583, -93555, -20144, 198989, 175914, 314657, 132437, 326343, 496535, -151090, 431212, 537573, 98317, -33297, 541045, 163941, -136853, 310800, 75534, -54402, 95912, 16583, -10509, 15802, 1854, -1071, 1385, 99, -53, 60, 2, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(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)

gp: K = bnfinit(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, 1)

## Normalizeddefining polynomial

$$x^{32} - x^{31} + 2 x^{30} + 60 x^{29} - 53 x^{28} + 99 x^{27} + 1385 x^{26} - 1071 x^{25} + 1854 x^{24} + 15802 x^{23} - 10509 x^{22} + 16583 x^{21} + 95912 x^{20} - 54402 x^{19} + 75534 x^{18} + 310800 x^{17} - 136853 x^{16} + 163941 x^{15} + 541045 x^{14} - 33297 x^{13} + 98317 x^{12} + 537573 x^{11} + 431212 x^{10} - 151090 x^{9} + 496535 x^{8} + 326343 x^{7} + 132437 x^{6} + 314657 x^{5} + 175914 x^{4} + 198989 x^{3} - 20144 x^{2} - 93555 x + 27583$$

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: $$38897685648686306961584993323026037365043690280067733586177953=97^{31}$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $84.08$ magma: Abs(Discriminant(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(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}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{3457}$, which has order $3457$ (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: $$132045958705412.02$$ (assuming GRH) magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 32 The 32 conjugacy class representatives for $C_{32}$ Character table for $C_{32}$ is not computed

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $16^{2}$ $16^{2}$ $32$ $32$ $16^{2}$ $32$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $16^{2}$ $32$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
97Data not computed