Properties

Label 32.0.80528452125...0000.2
Degree $32$
Signature $[0, 16]$
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![25046768244100781250, 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 + 25046768244100781250)
 
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 + 25046768244100781250, 1)
 

Normalized defining polynomial

\( x^{32} + 480 x^{30} + 104400 x^{28} + 13608000 x^{26} + 1184625000 x^{24} + 72657000000 x^{22} + 3227647500000 x^{20} + 105129090000000 x^{18} + 2513242307812500 x^{16} + 43708561875000000 x^{14} + 542383517812500000 x^{12} + 4649001581250000000 x^{10} + 26150633894531250000 x^{8} + 88933329843750000000 x^{6} + 158809517578125000000 x^{4} + 112100835937500000000 x^{2} + 25046768244100781250 \)

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:  \(8052845212573000012543979797231296934933304854055472857088000000000000000000000000=2^{191}\cdot 3^{16}\cdot 5^{24}\)
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$$\chi_{1920}(1,·)$, $\chi_{1920}(1667,·)$, $\chi_{1920}(649,·)$, $\chi_{1920}(1163,·)$, $\chi_{1920}(1681,·)$, $\chi_{1920}(1427,·)$, $\chi_{1920}(409,·)$, $\chi_{1920}(923,·)$, $\chi_{1920}(1441,·)$, $\chi_{1920}(1187,·)$, $\chi_{1920}(169,·)$, $\chi_{1920}(683,·)$, $\chi_{1920}(1201,·)$, $\chi_{1920}(947,·)$, $\chi_{1920}(1849,·)$, $\chi_{1920}(443,·)$, $\chi_{1920}(961,·)$, $\chi_{1920}(707,·)$, $\chi_{1920}(1609,·)$, $\chi_{1920}(203,·)$, $\chi_{1920}(721,·)$, $\chi_{1920}(467,·)$, $\chi_{1920}(1369,·)$, $\chi_{1920}(1883,·)$, $\chi_{1920}(481,·)$, $\chi_{1920}(227,·)$, $\chi_{1920}(1129,·)$, $\chi_{1920}(1643,·)$, $\chi_{1920}(241,·)$, $\chi_{1920}(1907,·)$, $\chi_{1920}(889,·)$, $\chi_{1920}(1403,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{45} a^{4}$, $\frac{1}{45} a^{5}$, $\frac{1}{135} a^{6}$, $\frac{1}{135} a^{7}$, $\frac{1}{2025} a^{8}$, $\frac{1}{2025} a^{9}$, $\frac{1}{6075} a^{10}$, $\frac{1}{6075} a^{11}$, $\frac{1}{91125} a^{12}$, $\frac{1}{91125} a^{13}$, $\frac{1}{273375} a^{14}$, $\frac{1}{273375} a^{15}$, $\frac{1}{783219375} a^{16} + \frac{16}{52214625} a^{14} - \frac{53}{17404875} a^{12} + \frac{41}{1160325} a^{10} + \frac{74}{386775} a^{8} - \frac{8}{25785} a^{6} - \frac{4}{1719} a^{4} - \frac{22}{573} a^{2} - \frac{87}{191}$, $\frac{1}{675918320625} a^{17} + \frac{1036}{1001360475} a^{15} + \frac{34327}{15020407125} a^{13} + \frac{39769}{1001360475} a^{11} - \frac{16352}{333786825} a^{9} - \frac{21973}{22252455} a^{7} + \frac{25651}{2472495} a^{5} - \frac{5228}{164833} a^{3} - \frac{58533}{164833} a$, $\frac{1}{2027754961875} a^{18} + \frac{2}{15020407125} a^{16} - \frac{67507}{45061221375} a^{14} + \frac{21067}{15020407125} a^{12} - \frac{9448}{1001360475} a^{10} - \frac{9028}{66757365} a^{8} - \frac{1369}{824165} a^{6} - \frac{33544}{7417485} a^{4} - \frac{42136}{494499} a^{2} - \frac{3}{191}$, $\frac{1}{2027754961875} a^{19} + \frac{22351}{45061221375} a^{15} + \frac{63464}{15020407125} a^{13} + \frac{12556}{333786825} a^{11} - \frac{56957}{333786825} a^{9} - \frac{12463}{7417485} a^{7} - \frac{36328}{7417485} a^{5} + \frac{16920}{164833} a^{3} - \frac{9275}{164833} a$, $\frac{1}{30416324428125} a^{20} - \frac{29}{225306106875} a^{16} + \frac{5476}{5006802375} a^{14} + \frac{73051}{15020407125} a^{12} + \frac{24682}{1001360475} a^{10} - \frac{49471}{333786825} a^{8} + \frac{40372}{22252455} a^{6} + \frac{5021}{7417485} a^{4} - \frac{906}{164833} a^{2} - \frac{30}{191}$, $\frac{1}{30416324428125} a^{21} - \frac{15601}{45061221375} a^{15} - \frac{24109}{5006802375} a^{13} + \frac{23092}{1001360475} a^{11} + \frac{11402}{333786825} a^{9} - \frac{19372}{7417485} a^{7} - \frac{6469}{824165} a^{5} - \frac{48562}{494499} a^{3} - \frac{8438}{164833} a$, $\frac{1}{91248973284375} a^{22} - \frac{67}{135183664125} a^{16} + \frac{16562}{45061221375} a^{14} - \frac{45058}{15020407125} a^{12} + \frac{12809}{200272095} a^{10} + \frac{5837}{111262275} a^{8} - \frac{20249}{22252455} a^{6} + \frac{5651}{2472495} a^{4} - \frac{68848}{494499} a^{2} + \frac{1}{191}$, $\frac{1}{91248973284375} a^{23} - \frac{8291}{15020407125} a^{15} + \frac{16202}{3004081425} a^{13} + \frac{11729}{333786825} a^{11} - \frac{4184}{66757365} a^{9} + \frac{36281}{22252455} a^{7} + \frac{1828}{164833} a^{5} - \frac{48332}{494499} a^{3} + \frac{7435}{164833} a$, $\frac{1}{1368734599265625} a^{24} + \frac{154}{675918320625} a^{16} - \frac{5191}{3004081425} a^{14} + \frac{1828}{1001360475} a^{12} + \frac{439}{13351473} a^{10} + \frac{75116}{333786825} a^{8} - \frac{18931}{22252455} a^{6} - \frac{54373}{7417485} a^{4} - \frac{54608}{494499} a^{2} - \frac{40}{191}$, $\frac{1}{1368734599265625} a^{25} - \frac{4693}{45061221375} a^{15} + \frac{15718}{15020407125} a^{13} + \frac{488}{66757365} a^{11} - \frac{14668}{111262275} a^{9} + \frac{68251}{22252455} a^{7} - \frac{37159}{7417485} a^{5} + \frac{53066}{494499} a^{3} + \frac{78580}{164833} a$, $\frac{1}{4106203797796875} a^{26} - \frac{164}{675918320625} a^{16} + \frac{58868}{45061221375} a^{14} - \frac{44522}{15020407125} a^{12} - \frac{8629}{111262275} a^{10} - \frac{3092}{13351473} a^{8} - \frac{6526}{2472495} a^{6} - \frac{35857}{7417485} a^{4} + \frac{60457}{494499} a^{2} - \frac{57}{191}$, $\frac{1}{4106203797796875} a^{27} - \frac{14201}{15020407125} a^{15} - \frac{19216}{15020407125} a^{13} + \frac{15968}{1001360475} a^{11} + \frac{43133}{333786825} a^{9} - \frac{7196}{4450491} a^{7} + \frac{57127}{7417485} a^{5} - \frac{39224}{494499} a^{3} + \frac{76544}{164833} a$, $\frac{1}{61593056966953125} a^{28} - \frac{316}{675918320625} a^{16} + \frac{1058}{600816285} a^{14} - \frac{82414}{15020407125} a^{12} + \frac{61097}{1001360475} a^{10} - \frac{38569}{333786825} a^{8} + \frac{35762}{22252455} a^{6} - \frac{60799}{7417485} a^{4} - \frac{24907}{494499} a^{2} - \frac{61}{191}$, $\frac{1}{61593056966953125} a^{29} - \frac{316}{600816285} a^{15} + \frac{50773}{15020407125} a^{13} - \frac{12808}{200272095} a^{11} + \frac{13771}{66757365} a^{9} + \frac{16}{23301} a^{7} + \frac{25898}{7417485} a^{5} - \frac{36061}{494499} a^{3} + \frac{77058}{164833} a$, $\frac{1}{184779170900859375} a^{30} - \frac{269}{675918320625} a^{16} - \frac{35527}{45061221375} a^{14} + \frac{6877}{15020407125} a^{12} - \frac{28664}{1001360475} a^{10} - \frac{75488}{333786825} a^{8} + \frac{7672}{2472495} a^{6} - \frac{14486}{1483497} a^{4} - \frac{51529}{494499} a^{2} - \frac{77}{191}$, $\frac{1}{184779170900859375} a^{31} - \frac{4411}{9012244275} a^{15} + \frac{10192}{15020407125} a^{13} - \frac{44948}{1001360475} a^{11} - \frac{1579}{22252455} a^{9} - \frac{24178}{7417485} a^{7} + \frac{7934}{2472495} a^{5} + \frac{15133}{494499} a^{3} + \frac{12140}{164833} a$

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:  $15$
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:  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