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Properties

Label 32.32.8052845212...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $362.71$
Ramified primes $2, 3, 5$
Class number Not computed
Class group Not computed
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![1226865178750781250, 0, -112100835937500000000, 0, 158809517578125000000, 0, -88933329843750000000, 0, 26150633894531250000, 0, -4649001581250000000, 0, 542383517812500000, 0, -43708561875000000, 0, 2513242307812500, 0, -105129090000000, 0, 3227647500000, 0, -72657000000, 0, 1184625000, 0, -13608000, 0, 104400, 0, -480, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 1226865178750781250)
 
gp: K = bnfinit(x^32 - 480*x^30 + 104400*x^28 - 13608000*x^26 + 1184625000*x^24 - 72657000000*x^22 + 3227647500000*x^20 - 105129090000000*x^18 + 2513242307812500*x^16 - 43708561875000000*x^14 + 542383517812500000*x^12 - 4649001581250000000*x^10 + 26150633894531250000*x^8 - 88933329843750000000*x^6 + 158809517578125000000*x^4 - 112100835937500000000*x^2 + 1226865178750781250, 1)
 

Normalized defining polynomial

\( x^{32} - 480 x^{30} + 104400 x^{28} - 13608000 x^{26} + 1184625000 x^{24} - 72657000000 x^{22} + 3227647500000 x^{20} - 105129090000000 x^{18} + 2513242307812500 x^{16} - 43708561875000000 x^{14} + 542383517812500000 x^{12} - 4649001581250000000 x^{10} + 26150633894531250000 x^{8} - 88933329843750000000 x^{6} + 158809517578125000000 x^{4} - 112100835937500000000 x^{2} + 1226865178750781250 \)

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:  \(8052845212573000012543979797231296934933304854055472857088000000000000000000000000=2^{191}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $362.71$
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:  \(1920=2^{7}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{1920}(1,·)$, $\chi_{1920}(1157,·)$, $\chi_{1920}(649,·)$, $\chi_{1920}(653,·)$, $\chi_{1920}(1681,·)$, $\chi_{1920}(917,·)$, $\chi_{1920}(409,·)$, $\chi_{1920}(413,·)$, $\chi_{1920}(1441,·)$, $\chi_{1920}(677,·)$, $\chi_{1920}(169,·)$, $\chi_{1920}(173,·)$, $\chi_{1920}(1201,·)$, $\chi_{1920}(437,·)$, $\chi_{1920}(1849,·)$, $\chi_{1920}(1853,·)$, $\chi_{1920}(961,·)$, $\chi_{1920}(197,·)$, $\chi_{1920}(1609,·)$, $\chi_{1920}(1613,·)$, $\chi_{1920}(721,·)$, $\chi_{1920}(1877,·)$, $\chi_{1920}(1369,·)$, $\chi_{1920}(1373,·)$, $\chi_{1920}(481,·)$, $\chi_{1920}(1637,·)$, $\chi_{1920}(1129,·)$, $\chi_{1920}(1133,·)$, $\chi_{1920}(241,·)$, $\chi_{1920}(1397,·)$, $\chi_{1920}(889,·)$, $\chi_{1920}(893,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{45} a^{4}$, $\frac{1}{45} a^{5}$, $\frac{1}{135} a^{6}$, $\frac{1}{135} a^{7}$, $\frac{1}{2025} a^{8}$, $\frac{1}{2025} a^{9}$, $\frac{1}{6075} a^{10}$, $\frac{1}{6075} a^{11}$, $\frac{1}{91125} a^{12}$, $\frac{1}{91125} a^{13}$, $\frac{1}{273375} a^{14}$, $\frac{1}{273375} a^{15}$, $\frac{1}{3538839375} a^{16} - \frac{16}{235922625} a^{14} - \frac{343}{78640875} a^{12} - \frac{34}{5242725} a^{10} + \frac{103}{1747575} a^{8} - \frac{403}{116505} a^{6} - \frac{287}{38835} a^{4} - \frac{233}{2589} a^{2} + \frac{387}{863}$, $\frac{1}{675918320625} a^{17} + \frac{46586}{45061221375} a^{15} - \frac{33137}{15020407125} a^{13} + \frac{11783}{333786825} a^{11} + \frac{21034}{111262275} a^{9} - \frac{78073}{22252455} a^{7} - \frac{63286}{7417485} a^{5} - \frac{22228}{164833} a^{3} + \frac{79783}{164833} a$, $\frac{1}{2027754961875} a^{18} - \frac{2}{15020407125} a^{16} + \frac{68857}{45061221375} a^{14} - \frac{6233}{15020407125} a^{12} + \frac{73798}{1001360475} a^{10} + \frac{3859}{333786825} a^{8} + \frac{53797}{22252455} a^{6} - \frac{5468}{824165} a^{4} - \frac{7149}{164833} a^{2} - \frac{79}{863}$, $\frac{1}{2027754961875} a^{19} - \frac{24061}{45061221375} a^{15} - \frac{21569}{15020407125} a^{13} - \frac{41452}{1001360475} a^{11} + \frac{26239}{111262275} a^{9} - \frac{49787}{22252455} a^{7} + \frac{24203}{7417485} a^{5} + \frac{75814}{494499} a^{3} + \frac{77562}{164833} a$, $\frac{1}{30416324428125} a^{20} + \frac{1}{135183664125} a^{16} + \frac{39196}{45061221375} a^{14} - \frac{10888}{3004081425} a^{12} + \frac{54631}{1001360475} a^{10} - \frac{43484}{333786825} a^{8} - \frac{1577}{22252455} a^{6} - \frac{4865}{494499} a^{4} + \frac{15141}{164833} a^{2} - \frac{429}{863}$, $\frac{1}{30416324428125} a^{21} - \frac{28901}{45061221375} a^{15} - \frac{53588}{15020407125} a^{13} + \frac{42719}{1001360475} a^{11} - \frac{9776}{111262275} a^{9} + \frac{59122}{22252455} a^{7} + \frac{78622}{7417485} a^{5} + \frac{49177}{494499} a^{3} + \frac{13645}{164833} a$, $\frac{1}{91248973284375} a^{22} + \frac{82}{675918320625} a^{16} - \frac{59318}{45061221375} a^{14} + \frac{70154}{15020407125} a^{12} - \frac{296}{1001360475} a^{10} + \frac{4687}{66757365} a^{8} - \frac{778}{4450491} a^{6} - \frac{42334}{7417485} a^{4} - \frac{16398}{164833} a^{2} + \frac{402}{863}$, $\frac{1}{91248973284375} a^{23} + \frac{76622}{45061221375} a^{15} - \frac{14773}{15020407125} a^{13} + \frac{13616}{200272095} a^{11} - \frac{13702}{111262275} a^{9} - \frac{30391}{22252455} a^{7} + \frac{7459}{1483497} a^{5} - \frac{6865}{164833} a^{3} - \frac{36937}{164833} a$, $\frac{1}{1368734599265625} a^{24} + \frac{31}{675918320625} a^{16} - \frac{63196}{45061221375} a^{14} - \frac{34487}{15020407125} a^{12} + \frac{24478}{1001360475} a^{10} - \frac{1456}{66757365} a^{8} + \frac{49861}{22252455} a^{6} + \frac{38233}{7417485} a^{4} + \frac{3385}{494499} a^{2} + \frac{153}{863}$, $\frac{1}{1368734599265625} a^{25} - \frac{1591}{3004081425} a^{15} + \frac{418}{1668934125} a^{13} - \frac{82343}{1001360475} a^{11} + \frac{14554}{333786825} a^{9} - \frac{2371}{22252455} a^{7} + \frac{22103}{7417485} a^{5} - \frac{24080}{164833} a^{3} + \frac{28445}{164833} a$, $\frac{1}{4106203797796875} a^{26} + \frac{2}{27036732825} a^{16} + \frac{71758}{45061221375} a^{14} + \frac{15743}{15020407125} a^{12} + \frac{25543}{333786825} a^{10} - \frac{583}{2472495} a^{8} + \frac{45214}{22252455} a^{6} - \frac{1379}{1483497} a^{4} + \frac{70847}{494499} a^{2} + \frac{235}{863}$, $\frac{1}{4106203797796875} a^{27} + \frac{10024}{9012244275} a^{15} + \frac{24263}{15020407125} a^{13} - \frac{42491}{1001360475} a^{11} + \frac{12571}{66757365} a^{9} - \frac{264}{824165} a^{7} + \frac{2842}{824165} a^{5} - \frac{56446}{494499} a^{3} + \frac{11727}{164833} a$, $\frac{1}{61593056966953125} a^{28} + \frac{26}{225306106875} a^{16} - \frac{51607}{45061221375} a^{14} - \frac{20717}{15020407125} a^{12} + \frac{65669}{1001360475} a^{10} - \frac{51631}{333786825} a^{8} + \frac{29449}{22252455} a^{6} - \frac{11621}{2472495} a^{4} - \frac{8045}{494499} a^{2} - \frac{423}{863}$, $\frac{1}{61593056966953125} a^{29} - \frac{19663}{15020407125} a^{15} - \frac{2717}{556311375} a^{13} - \frac{241}{4450491} a^{11} - \frac{28597}{333786825} a^{9} + \frac{2258}{2472495} a^{7} - \frac{2903}{494499} a^{5} - \frac{81349}{494499} a^{3} - \frac{40213}{164833} a$, $\frac{1}{184779170900859375} a^{30} - \frac{41}{675918320625} a^{16} - \frac{11666}{45061221375} a^{14} - \frac{50902}{15020407125} a^{12} - \frac{104}{200272095} a^{10} - \frac{17383}{333786825} a^{8} - \frac{15088}{7417485} a^{6} + \frac{1927}{1483497} a^{4} - \frac{5828}{164833} a^{2} + \frac{332}{863}$, $\frac{1}{184779170900859375} a^{31} - \frac{79636}{45061221375} a^{15} + \frac{73978}{15020407125} a^{13} - \frac{34708}{1001360475} a^{11} - \frac{67529}{333786825} a^{9} + \frac{16801}{7417485} a^{7} + \frac{52237}{7417485} a^{5} + \frac{50633}{494499} a^{3} + \frac{37855}{164833} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), 16.16.236118324143482260684800000000.1

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 R R $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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