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Properties

Label 32.32.1674417821...2113.1
Degree $32$
Signature $[32, 0]$
Discriminant $3^{16}\cdot 97^{31}$
Root discriminant $145.63$
Ramified primes $3, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![705613, -9898412, -2768445, 243218618, 354398962, -1377789078, -3490824231, 994686632, 10369480703, 8342709494, -7847282396, -15570255798, -3964294981, 8233352408, 6403713290, -835566428, -2706325012, -666000824, 503619218, 277610329, -29135620, -48952218, -4671359, 4686255, 1027629, -247839, -86885, 6404, 3924, -37, -95, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 95*x^30 - 37*x^29 + 3924*x^28 + 6404*x^27 - 86885*x^26 - 247839*x^25 + 1027629*x^24 + 4686255*x^23 - 4671359*x^22 - 48952218*x^21 - 29135620*x^20 + 277610329*x^19 + 503619218*x^18 - 666000824*x^17 - 2706325012*x^16 - 835566428*x^15 + 6403713290*x^14 + 8233352408*x^13 - 3964294981*x^12 - 15570255798*x^11 - 7847282396*x^10 + 8342709494*x^9 + 10369480703*x^8 + 994686632*x^7 - 3490824231*x^6 - 1377789078*x^5 + 354398962*x^4 + 243218618*x^3 - 2768445*x^2 - 9898412*x + 705613)
 
gp: K = bnfinit(x^32 - x^31 - 95*x^30 - 37*x^29 + 3924*x^28 + 6404*x^27 - 86885*x^26 - 247839*x^25 + 1027629*x^24 + 4686255*x^23 - 4671359*x^22 - 48952218*x^21 - 29135620*x^20 + 277610329*x^19 + 503619218*x^18 - 666000824*x^17 - 2706325012*x^16 - 835566428*x^15 + 6403713290*x^14 + 8233352408*x^13 - 3964294981*x^12 - 15570255798*x^11 - 7847282396*x^10 + 8342709494*x^9 + 10369480703*x^8 + 994686632*x^7 - 3490824231*x^6 - 1377789078*x^5 + 354398962*x^4 + 243218618*x^3 - 2768445*x^2 - 9898412*x + 705613, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 95 x^{30} - 37 x^{29} + 3924 x^{28} + 6404 x^{27} - 86885 x^{26} - 247839 x^{25} + 1027629 x^{24} + 4686255 x^{23} - 4671359 x^{22} - 48952218 x^{21} - 29135620 x^{20} + 277610329 x^{19} + 503619218 x^{18} - 666000824 x^{17} - 2706325012 x^{16} - 835566428 x^{15} + 6403713290 x^{14} + 8233352408 x^{13} - 3964294981 x^{12} - 15570255798 x^{11} - 7847282396 x^{10} + 8342709494 x^{9} + 10369480703 x^{8} + 994686632 x^{7} - 3490824231 x^{6} - 1377789078 x^{5} + 354398962 x^{4} + 243218618 x^{3} - 2768445 x^{2} - 9898412 x + 705613 \)

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:  \(1674417821664703472295706925363164706188610888296487588786531799142113=3^{16}\cdot 97^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $145.63$
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:  $3, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(291=3\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{291}(1,·)$, $\chi_{291}(130,·)$, $\chi_{291}(131,·)$, $\chi_{291}(257,·)$, $\chi_{291}(143,·)$, $\chi_{291}(272,·)$, $\chi_{291}(20,·)$, $\chi_{291}(149,·)$, $\chi_{291}(22,·)$, $\chi_{291}(152,·)$, $\chi_{291}(283,·)$, $\chi_{291}(164,·)$, $\chi_{291}(263,·)$, $\chi_{291}(172,·)$, $\chi_{291}(116,·)$, $\chi_{291}(64,·)$, $\chi_{291}(193,·)$, $\chi_{291}(70,·)$, $\chi_{291}(202,·)$, $\chi_{291}(77,·)$, $\chi_{291}(79,·)$, $\chi_{291}(85,·)$, $\chi_{291}(224,·)$, $\chi_{291}(236,·)$, $\chi_{291}(109,·)$, $\chi_{291}(239,·)$, $\chi_{291}(241,·)$, $\chi_{291}(115,·)$, $\chi_{291}(244,·)$, $\chi_{291}(245,·)$, $\chi_{291}(124,·)$, $\chi_{291}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{61} a^{23} + \frac{15}{61} a^{22} - \frac{13}{61} a^{21} + \frac{26}{61} a^{20} - \frac{2}{61} a^{19} + \frac{24}{61} a^{17} + \frac{30}{61} a^{16} - \frac{8}{61} a^{15} + \frac{10}{61} a^{14} + \frac{5}{61} a^{13} + \frac{9}{61} a^{12} - \frac{1}{61} a^{11} + \frac{28}{61} a^{10} - \frac{28}{61} a^{9} + \frac{13}{61} a^{8} + \frac{19}{61} a^{7} + \frac{2}{61} a^{6} + \frac{15}{61} a^{5} + \frac{26}{61} a^{4} - \frac{13}{61} a^{3} - \frac{12}{61} a + \frac{6}{61}$, $\frac{1}{61} a^{24} + \frac{6}{61} a^{22} - \frac{23}{61} a^{21} - \frac{26}{61} a^{20} + \frac{30}{61} a^{19} + \frac{24}{61} a^{18} - \frac{25}{61} a^{17} + \frac{30}{61} a^{16} + \frac{8}{61} a^{15} - \frac{23}{61} a^{14} - \frac{5}{61} a^{13} - \frac{14}{61} a^{12} - \frac{18}{61} a^{11} - \frac{21}{61} a^{10} + \frac{6}{61} a^{9} + \frac{7}{61} a^{8} + \frac{22}{61} a^{7} - \frac{15}{61} a^{6} - \frac{16}{61} a^{5} + \frac{24}{61} a^{4} + \frac{12}{61} a^{3} - \frac{12}{61} a^{2} + \frac{3}{61} a - \frac{29}{61}$, $\frac{1}{61} a^{25} + \frac{9}{61} a^{22} - \frac{9}{61} a^{21} - \frac{4}{61} a^{20} - \frac{25}{61} a^{19} - \frac{25}{61} a^{18} + \frac{8}{61} a^{17} + \frac{11}{61} a^{16} + \frac{25}{61} a^{15} - \frac{4}{61} a^{14} + \frac{17}{61} a^{13} - \frac{11}{61} a^{12} - \frac{15}{61} a^{11} + \frac{21}{61} a^{10} - \frac{8}{61} a^{9} + \frac{5}{61} a^{8} - \frac{7}{61} a^{7} - \frac{28}{61} a^{6} - \frac{5}{61} a^{5} - \frac{22}{61} a^{4} + \frac{5}{61} a^{3} + \frac{3}{61} a^{2} - \frac{18}{61} a + \frac{25}{61}$, $\frac{1}{61} a^{26} - \frac{22}{61} a^{22} - \frac{9}{61} a^{21} - \frac{15}{61} a^{20} - \frac{7}{61} a^{19} + \frac{8}{61} a^{18} - \frac{22}{61} a^{17} - \frac{1}{61} a^{16} + \frac{7}{61} a^{15} - \frac{12}{61} a^{14} + \frac{5}{61} a^{13} + \frac{26}{61} a^{12} + \frac{30}{61} a^{11} - \frac{16}{61} a^{10} + \frac{13}{61} a^{9} - \frac{2}{61} a^{8} - \frac{16}{61} a^{7} - \frac{23}{61} a^{6} + \frac{26}{61} a^{5} + \frac{15}{61} a^{4} - \frac{2}{61} a^{3} - \frac{18}{61} a^{2} + \frac{11}{61} a + \frac{7}{61}$, $\frac{1}{61} a^{27} + \frac{16}{61} a^{22} + \frac{4}{61} a^{21} + \frac{16}{61} a^{20} + \frac{25}{61} a^{19} - \frac{22}{61} a^{18} - \frac{22}{61} a^{17} - \frac{4}{61} a^{16} - \frac{5}{61} a^{15} - \frac{19}{61} a^{14} + \frac{14}{61} a^{13} - \frac{16}{61} a^{12} + \frac{23}{61} a^{11} + \frac{19}{61} a^{10} - \frac{8}{61} a^{9} + \frac{26}{61} a^{8} + \frac{29}{61} a^{7} + \frac{9}{61} a^{6} - \frac{21}{61} a^{5} + \frac{21}{61} a^{4} + \frac{1}{61} a^{3} + \frac{11}{61} a^{2} - \frac{13}{61} a + \frac{10}{61}$, $\frac{1}{61} a^{28} + \frac{8}{61} a^{22} - \frac{20}{61} a^{21} - \frac{25}{61} a^{20} + \frac{10}{61} a^{19} - \frac{22}{61} a^{18} - \frac{22}{61} a^{17} + \frac{3}{61} a^{16} - \frac{13}{61} a^{15} - \frac{24}{61} a^{14} + \frac{26}{61} a^{13} + \frac{1}{61} a^{12} - \frac{26}{61} a^{11} - \frac{29}{61} a^{10} - \frac{14}{61} a^{9} + \frac{4}{61} a^{8} + \frac{10}{61} a^{7} + \frac{8}{61} a^{6} + \frac{25}{61} a^{5} + \frac{12}{61} a^{4} - \frac{25}{61} a^{3} - \frac{13}{61} a^{2} + \frac{19}{61} a + \frac{26}{61}$, $\frac{1}{61} a^{29} - \frac{18}{61} a^{22} + \frac{18}{61} a^{21} - \frac{15}{61} a^{20} - \frac{6}{61} a^{19} - \frac{22}{61} a^{18} - \frac{6}{61} a^{17} - \frac{9}{61} a^{16} - \frac{21}{61} a^{15} + \frac{7}{61} a^{14} + \frac{22}{61} a^{13} + \frac{24}{61} a^{12} - \frac{21}{61} a^{11} + \frac{6}{61} a^{10} - \frac{16}{61} a^{9} + \frac{28}{61} a^{8} - \frac{22}{61} a^{7} + \frac{9}{61} a^{6} + \frac{14}{61} a^{5} + \frac{11}{61} a^{4} + \frac{30}{61} a^{3} + \frac{19}{61} a^{2} + \frac{13}{61}$, $\frac{1}{26839573} a^{30} + \frac{147026}{26839573} a^{29} - \frac{186057}{26839573} a^{28} + \frac{18921}{26839573} a^{27} - \frac{80367}{26839573} a^{26} - \frac{118740}{26839573} a^{25} + \frac{22489}{26839573} a^{24} + \frac{101385}{26839573} a^{23} - \frac{6603888}{26839573} a^{22} + \frac{77738}{439993} a^{21} + \frac{1753799}{26839573} a^{20} - \frac{5713182}{26839573} a^{19} - \frac{2501438}{26839573} a^{18} + \frac{12637288}{26839573} a^{17} - \frac{1768446}{26839573} a^{16} - \frac{12311725}{26839573} a^{15} + \frac{12876224}{26839573} a^{14} - \frac{5886736}{26839573} a^{13} + \frac{11790140}{26839573} a^{12} - \frac{3956040}{26839573} a^{11} - \frac{13108915}{26839573} a^{10} + \frac{126263}{26839573} a^{9} + \frac{10185023}{26839573} a^{8} + \frac{11060310}{26839573} a^{7} - \frac{3970379}{26839573} a^{6} + \frac{11821532}{26839573} a^{5} - \frac{4386647}{26839573} a^{4} - \frac{8696281}{26839573} a^{3} + \frac{12303186}{26839573} a^{2} + \frac{3055197}{26839573} a - \frac{8943733}{26839573}$, $\frac{1}{4718443554547776012355807666438412106185189008532946132594897239} a^{31} + \frac{75418382990363529315351944112039247994820339560722442150}{4718443554547776012355807666438412106185189008532946132594897239} a^{30} + \frac{26581110356542692681928489829461401725700085167093237080921834}{4718443554547776012355807666438412106185189008532946132594897239} a^{29} + \frac{9179861142984862752518275306221772923678686618526385601682860}{4718443554547776012355807666438412106185189008532946132594897239} a^{28} - \frac{26882048875153875755168932647387917246911800231868059333530548}{4718443554547776012355807666438412106185189008532946132594897239} a^{27} + \frac{26787233370979070292806262792941565594072854537538339088887229}{4718443554547776012355807666438412106185189008532946132594897239} a^{26} - \frac{32805438238501973302325540694651921356180111765382897061679130}{4718443554547776012355807666438412106185189008532946132594897239} a^{25} + \frac{33698799266636198028295077560551327348580294652066115043576413}{4718443554547776012355807666438412106185189008532946132594897239} a^{24} - \frac{1332804051724081309601036083601869550212349637536418278395897}{4718443554547776012355807666438412106185189008532946132594897239} a^{23} + \frac{1863117365435339880861556751415196269799790067120843749663978335}{4718443554547776012355807666438412106185189008532946132594897239} a^{22} - \frac{749007305611157436958599560751645567182730242562918959639597980}{4718443554547776012355807666438412106185189008532946132594897239} a^{21} - \frac{1144557863461443633729761303736597381062176513118462902265459649}{4718443554547776012355807666438412106185189008532946132594897239} a^{20} - \frac{1404683557147131129865659876632695464107170314904755642967009970}{4718443554547776012355807666438412106185189008532946132594897239} a^{19} - \frac{827518707147162897687754863436740735565326388757991159974313970}{4718443554547776012355807666438412106185189008532946132594897239} a^{18} + \frac{1281112159408956979985068378181282874509626516431834235155779963}{4718443554547776012355807666438412106185189008532946132594897239} a^{17} + \frac{1265267149665027169002685739823722453797255882975279353224477998}{4718443554547776012355807666438412106185189008532946132594897239} a^{16} + \frac{251234687651709500672818955684854324466958372710221731096445199}{4718443554547776012355807666438412106185189008532946132594897239} a^{15} - \frac{1575509344617781595635040039442193216412962297372759811386466651}{4718443554547776012355807666438412106185189008532946132594897239} a^{14} + \frac{1160719596613723757919717238202184438516400732899951295971477930}{4718443554547776012355807666438412106185189008532946132594897239} a^{13} - \frac{150084571368152445156450343057806500689619804172862609512886351}{4718443554547776012355807666438412106185189008532946132594897239} a^{12} + \frac{1983383210022566696575575570657465081383200652246641641986458525}{4718443554547776012355807666438412106185189008532946132594897239} a^{11} - \frac{282498791955844678471242867620575151767236385988129757927579187}{4718443554547776012355807666438412106185189008532946132594897239} a^{10} - \frac{1214764434539717717686016916206246288101913987668878110834200761}{4718443554547776012355807666438412106185189008532946132594897239} a^{9} - \frac{1577868381889766958510041566111134608862463630893410238509660490}{4718443554547776012355807666438412106185189008532946132594897239} a^{8} - \frac{993797123152016705448565282528738713995412173314562397324082137}{4718443554547776012355807666438412106185189008532946132594897239} a^{7} + \frac{1242899177246643528166738804275132923119650180596781484468276066}{4718443554547776012355807666438412106185189008532946132594897239} a^{6} + \frac{605557012081753704272248288374943479307140920800935313406690272}{4718443554547776012355807666438412106185189008532946132594897239} a^{5} + \frac{678267624034771036790774960824807281331150632928017224416978879}{4718443554547776012355807666438412106185189008532946132594897239} a^{4} + \frac{237348760888972957688435704745952069767451123038981929573370344}{4718443554547776012355807666438412106185189008532946132594897239} a^{3} + \frac{1637297221535755522539736174969138862438544441793903340198075913}{4718443554547776012355807666438412106185189008532946132594897239} a^{2} - \frac{1109099691383841954676724259747006483476158042702268924794114913}{4718443554547776012355807666438412106185189008532946132594897239} a - \frac{386497249879936513913677374280505128414040606260460561025955482}{4718443554547776012355807666438412106185189008532946132594897239}$

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:  \( 3823661354055936000000000 \) (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

\(\Q(\sqrt{97}) \), 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 $16^{2}$ R $32$ $32$ $16^{2}$ $32$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $16^{2}$ $32$

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
97Data not computed