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Properties

Label 32.32.5456820171...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{124}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $84.97$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -1024, 0, 20096, 0, -186432, 0, 976032, 0, -3155840, 0, 6621008, 0, -9255712, 0, 8730872, 0, -5581792, 0, 2417360, 0, -705040, 0, 136570, 0, -17104, 0, 1316, 0, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 56*x^30 + 1316*x^28 - 17104*x^26 + 136570*x^24 - 705040*x^22 + 2417360*x^20 - 5581792*x^18 + 8730872*x^16 - 9255712*x^14 + 6621008*x^12 - 3155840*x^10 + 976032*x^8 - 186432*x^6 + 20096*x^4 - 1024*x^2 + 16)
 
gp: K = bnfinit(x^32 - 56*x^30 + 1316*x^28 - 17104*x^26 + 136570*x^24 - 705040*x^22 + 2417360*x^20 - 5581792*x^18 + 8730872*x^16 - 9255712*x^14 + 6621008*x^12 - 3155840*x^10 + 976032*x^8 - 186432*x^6 + 20096*x^4 - 1024*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{32} - 56 x^{30} + 1316 x^{28} - 17104 x^{26} + 136570 x^{24} - 705040 x^{22} + 2417360 x^{20} - 5581792 x^{18} + 8730872 x^{16} - 9255712 x^{14} + 6621008 x^{12} - 3155840 x^{10} + 976032 x^{8} - 186432 x^{6} + 20096 x^{4} - 1024 x^{2} + 16 \)

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:  \(54568201713507127370225565301626372096000000000000000000000000=2^{124}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.97$
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:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(131,·)$, $\chi_{480}(257,·)$, $\chi_{480}(137,·)$, $\chi_{480}(11,·)$, $\chi_{480}(17,·)$, $\chi_{480}(403,·)$, $\chi_{480}(409,·)$, $\chi_{480}(283,·)$, $\chi_{480}(289,·)$, $\chi_{480}(163,·)$, $\chi_{480}(113,·)$, $\chi_{480}(169,·)$, $\chi_{480}(427,·)$, $\chi_{480}(307,·)$, $\chi_{480}(49,·)$, $\chi_{480}(179,·)$, $\chi_{480}(59,·)$, $\chi_{480}(67,·)$, $\chi_{480}(419,·)$, $\chi_{480}(43,·)$, $\chi_{480}(377,·)$, $\chi_{480}(473,·)$, $\chi_{480}(353,·)$, $\chi_{480}(187,·)$, $\chi_{480}(361,·)$, $\chi_{480}(241,·)$, $\chi_{480}(371,·)$, $\chi_{480}(233,·)$, $\chi_{480}(121,·)$, $\chi_{480}(251,·)$, $\chi_{480}(299,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{4} a^{18}$, $\frac{1}{4} a^{19}$, $\frac{1}{124} a^{20} + \frac{9}{124} a^{18} + \frac{11}{124} a^{16} + \frac{7}{31} a^{14} - \frac{7}{62} a^{12} + \frac{1}{31} a^{10} + \frac{4}{31} a^{8} + \frac{9}{31} a^{6} - \frac{9}{31} a^{4} + \frac{13}{31} a^{2} - \frac{12}{31}$, $\frac{1}{124} a^{21} + \frac{9}{124} a^{19} + \frac{11}{124} a^{17} + \frac{7}{31} a^{15} - \frac{7}{62} a^{13} + \frac{1}{31} a^{11} + \frac{4}{31} a^{9} + \frac{9}{31} a^{7} - \frac{9}{31} a^{5} + \frac{13}{31} a^{3} - \frac{12}{31} a$, $\frac{1}{124} a^{22} - \frac{2}{31} a^{18} - \frac{9}{124} a^{16} - \frac{9}{62} a^{14} + \frac{3}{62} a^{12} - \frac{5}{31} a^{10} + \frac{4}{31} a^{8} + \frac{3}{31} a^{6} + \frac{1}{31} a^{4} - \frac{5}{31} a^{2} + \frac{15}{31}$, $\frac{1}{124} a^{23} - \frac{2}{31} a^{19} - \frac{9}{124} a^{17} - \frac{9}{62} a^{15} + \frac{3}{62} a^{13} - \frac{5}{31} a^{11} + \frac{4}{31} a^{9} + \frac{3}{31} a^{7} + \frac{1}{31} a^{5} - \frac{5}{31} a^{3} + \frac{15}{31} a$, $\frac{1}{248} a^{24} - \frac{15}{124} a^{18} + \frac{1}{31} a^{16} + \frac{11}{62} a^{14} - \frac{1}{31} a^{12} + \frac{6}{31} a^{10} + \frac{2}{31} a^{8} - \frac{10}{31} a^{6} + \frac{8}{31} a^{4} + \frac{13}{31} a^{2} + \frac{14}{31}$, $\frac{1}{248} a^{25} - \frac{15}{124} a^{19} + \frac{1}{31} a^{17} + \frac{11}{62} a^{15} - \frac{1}{31} a^{13} + \frac{6}{31} a^{11} + \frac{2}{31} a^{9} - \frac{10}{31} a^{7} + \frac{8}{31} a^{5} + \frac{13}{31} a^{3} + \frac{14}{31} a$, $\frac{1}{248} a^{26} + \frac{15}{124} a^{18} + \frac{1}{124} a^{16} - \frac{9}{62} a^{14} + \frac{3}{62} a^{10} + \frac{7}{62} a^{8} - \frac{12}{31} a^{6} + \frac{2}{31} a^{4} - \frac{8}{31} a^{2} + \frac{6}{31}$, $\frac{1}{248} a^{27} + \frac{15}{124} a^{19} + \frac{1}{124} a^{17} - \frac{9}{62} a^{15} + \frac{3}{62} a^{11} + \frac{7}{62} a^{9} - \frac{12}{31} a^{7} + \frac{2}{31} a^{5} - \frac{8}{31} a^{3} + \frac{6}{31} a$, $\frac{1}{67208} a^{28} + \frac{71}{67208} a^{26} + \frac{63}{33604} a^{24} + \frac{105}{33604} a^{22} + \frac{91}{33604} a^{20} - \frac{2685}{33604} a^{18} + \frac{701}{8401} a^{16} - \frac{1427}{8401} a^{14} + \frac{1844}{8401} a^{12} - \frac{3661}{16802} a^{10} - \frac{1128}{8401} a^{8} + \frac{966}{8401} a^{6} - \frac{2134}{8401} a^{4} - \frac{3979}{8401} a^{2} + \frac{4062}{8401}$, $\frac{1}{67208} a^{29} + \frac{71}{67208} a^{27} + \frac{63}{33604} a^{25} + \frac{105}{33604} a^{23} + \frac{91}{33604} a^{21} - \frac{2685}{33604} a^{19} + \frac{701}{8401} a^{17} - \frac{1427}{8401} a^{15} + \frac{1844}{8401} a^{13} - \frac{3661}{16802} a^{11} - \frac{1128}{8401} a^{9} + \frac{966}{8401} a^{7} - \frac{2134}{8401} a^{5} - \frac{3979}{8401} a^{3} + \frac{4062}{8401} a$, $\frac{1}{1449630218672632} a^{30} - \frac{4430714619}{1449630218672632} a^{28} + \frac{382352027305}{1449630218672632} a^{26} - \frac{1903872187799}{1449630218672632} a^{24} + \frac{449722219516}{181203777334079} a^{22} + \frac{353154054826}{181203777334079} a^{20} - \frac{670012274927}{724815109336316} a^{18} - \frac{16360846901478}{181203777334079} a^{16} - \frac{85420554008799}{362407554668158} a^{14} + \frac{6207754112327}{181203777334079} a^{12} + \frac{6323358333593}{181203777334079} a^{10} - \frac{62609806998697}{362407554668158} a^{8} + \frac{39211280462097}{181203777334079} a^{6} - \frac{58579075210306}{181203777334079} a^{4} - \frac{13417289397663}{181203777334079} a^{2} + \frac{13074487803436}{181203777334079}$, $\frac{1}{1449630218672632} a^{31} - \frac{4430714619}{1449630218672632} a^{29} + \frac{382352027305}{1449630218672632} a^{27} - \frac{1903872187799}{1449630218672632} a^{25} + \frac{449722219516}{181203777334079} a^{23} + \frac{353154054826}{181203777334079} a^{21} - \frac{670012274927}{724815109336316} a^{19} - \frac{16360846901478}{181203777334079} a^{17} - \frac{85420554008799}{362407554668158} a^{15} + \frac{6207754112327}{181203777334079} a^{13} + \frac{6323358333593}{181203777334079} a^{11} - \frac{62609806998697}{362407554668158} a^{9} + \frac{39211280462097}{181203777334079} a^{7} - \frac{58579075210306}{181203777334079} a^{5} - \frac{13417289397663}{181203777334079} a^{3} + \frac{13074487803436}{181203777334079} a$

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:  \( 370118030677416540000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.72000.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.4.2304000.2, 4.4.2304000.1, 8.8.5184000000.1, 8.8.2621440000.1, 8.8.5308416000000.1, 8.8.33554432000000.2, 8.8.33554432000000.1, 8.8.108716359680000.1, 8.8.173946175488.1, 16.16.28179280429056000000000000.2, 16.16.1125899906842624000000000000.1, 16.16.11819246862071129702400000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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