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Properties

Label 32.0.86898859856...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 5^{24}\cdot 17^{24}$
Root discriminant $55.99$
Ramified primes $2, 5, 17$
Class number $320$ (GRH)
Class group $[4, 4, 20]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -13, 0, 129, 0, -1170, 0, 10218, 0, -43485, 0, 171666, 0, -623922, 0, 1799433, 0, -623922, 0, 171666, 0, -43485, 0, 10218, 0, -1170, 0, 129, 0, -13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1)
 
gp: K = bnfinit(x^32 - 13*x^30 + 129*x^28 - 1170*x^26 + 10218*x^24 - 43485*x^22 + 171666*x^20 - 623922*x^18 + 1799433*x^16 - 623922*x^14 + 171666*x^12 - 43485*x^10 + 10218*x^8 - 1170*x^6 + 129*x^4 - 13*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{32} - 13 x^{30} + 129 x^{28} - 1170 x^{26} + 10218 x^{24} - 43485 x^{22} + 171666 x^{20} - 623922 x^{18} + 1799433 x^{16} - 623922 x^{14} + 171666 x^{12} - 43485 x^{10} + 10218 x^{8} - 1170 x^{6} + 129 x^{4} - 13 x^{2} + 1 \)

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:  \(86898859856540647588737053982976000000000000000000000000=2^{32}\cdot 5^{24}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.99$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(259,·)$, $\chi_{340}(137,·)$, $\chi_{340}(13,·)$, $\chi_{340}(271,·)$, $\chi_{340}(273,·)$, $\chi_{340}(149,·)$, $\chi_{340}(89,·)$, $\chi_{340}(157,·)$, $\chi_{340}(33,·)$, $\chi_{340}(293,·)$, $\chi_{340}(169,·)$, $\chi_{340}(171,·)$, $\chi_{340}(47,·)$, $\chi_{340}(307,·)$, $\chi_{340}(183,·)$, $\chi_{340}(191,·)$, $\chi_{340}(67,·)$, $\chi_{340}(69,·)$, $\chi_{340}(327,·)$, $\chi_{340}(203,·)$, $\chi_{340}(81,·)$, $\chi_{340}(339,·)$, $\chi_{340}(217,·)$, $\chi_{340}(123,·)$, $\chi_{340}(101,·)$, $\chi_{340}(103,·)$, $\chi_{340}(237,·)$, $\chi_{340}(239,·)$, $\chi_{340}(319,·)$, $\chi_{340}(251,·)$, $\chi_{340}(21,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{8}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{9}$, $\frac{1}{10089} a^{20} + \frac{13}{531} a^{10} - \frac{3362}{10089}$, $\frac{1}{10089} a^{21} + \frac{13}{531} a^{11} - \frac{3362}{10089} a$, $\frac{1}{10089} a^{22} + \frac{13}{531} a^{12} - \frac{3362}{10089} a^{2}$, $\frac{1}{10089} a^{23} + \frac{13}{531} a^{13} - \frac{3362}{10089} a^{3}$, $\frac{1}{80712} a^{24} + \frac{1}{24} a^{18} + \frac{68}{531} a^{14} + \frac{1}{8} a^{12} + \frac{1}{3} a^{8} + \frac{3}{8} a^{6} + \frac{2102}{10089} a^{4} + \frac{1}{8}$, $\frac{1}{80712} a^{25} + \frac{1}{24} a^{19} + \frac{68}{531} a^{15} + \frac{1}{8} a^{13} + \frac{1}{3} a^{9} + \frac{3}{8} a^{7} + \frac{2102}{10089} a^{5} + \frac{1}{8} a$, $\frac{1}{15252227352} a^{26} + \frac{2409}{423672982} a^{24} + \frac{23887}{635509473} a^{22} - \frac{339341}{15252227352} a^{20} - \frac{65069}{1133826} a^{18} + \frac{6627656}{100343601} a^{16} + \frac{5088193}{267582936} a^{14} - \frac{3788539}{66895734} a^{12} + \frac{1244789}{100343601} a^{10} - \frac{1811335}{4535304} a^{8} + \frac{1599840475}{3813056838} a^{6} + \frac{144283600}{635509473} a^{4} + \frac{341390987}{5084075784} a^{2} - \frac{33174233}{3813056838}$, $\frac{1}{15252227352} a^{27} + \frac{2409}{423672982} a^{25} + \frac{23887}{635509473} a^{23} - \frac{339341}{15252227352} a^{21} - \frac{65069}{1133826} a^{19} + \frac{6627656}{100343601} a^{17} + \frac{5088193}{267582936} a^{15} - \frac{3788539}{66895734} a^{13} + \frac{1244789}{100343601} a^{11} - \frac{1811335}{4535304} a^{9} + \frac{1599840475}{3813056838} a^{7} + \frac{144283600}{635509473} a^{5} + \frac{341390987}{5084075784} a^{3} - \frac{33174233}{3813056838} a$, $\frac{1}{15252227352} a^{28} - \frac{1}{15252227352} a^{24} - \frac{217879}{5084075784} a^{22} - \frac{23887}{635509473} a^{20} - \frac{53018921}{802748808} a^{18} - \frac{306637}{4535304} a^{16} - \frac{6627656}{100343601} a^{14} - \frac{18694105}{267582936} a^{12} + \frac{4004867}{267582936} a^{10} - \frac{506650663}{1906528419} a^{8} + \frac{1811335}{4535304} a^{6} + \frac{6946337033}{15252227352} a^{4} - \frac{117827660}{635509473} a^{2} - \frac{678694305}{1694691928}$, $\frac{1}{15252227352} a^{29} - \frac{1}{15252227352} a^{25} - \frac{217879}{5084075784} a^{23} - \frac{23887}{635509473} a^{21} - \frac{53018921}{802748808} a^{19} - \frac{306637}{4535304} a^{17} - \frac{6627656}{100343601} a^{15} - \frac{18694105}{267582936} a^{13} + \frac{4004867}{267582936} a^{11} - \frac{506650663}{1906528419} a^{9} + \frac{1811335}{4535304} a^{7} + \frac{6946337033}{15252227352} a^{5} - \frac{117827660}{635509473} a^{3} - \frac{678694305}{1694691928} a$, $\frac{1}{45756682056} a^{30} + \frac{90091}{1906528419} a^{20} + \frac{275838071}{1906528419} a^{10} - \frac{146703017}{775536984}$, $\frac{1}{45756682056} a^{31} + \frac{90091}{1906528419} a^{21} + \frac{275838071}{1906528419} a^{11} - \frac{146703017}{775536984} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{20}$, which has order $320$ (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:  \( \frac{8471948981}{22878341028} a^{31} - \frac{18355965731}{3813056838} a^{29} + \frac{364293806183}{7626113676} a^{27} - \frac{550676683765}{1271018946} a^{25} + \frac{14427729114643}{3813056838} a^{23} - \frac{122800900479595}{7626113676} a^{21} + \frac{12757240003909}{200687202} a^{19} - \frac{46366976773013}{200687202} a^{17} + \frac{89150319127063}{133791468} a^{15} - \frac{46366976773013}{200687202} a^{13} + \frac{26932325810599}{423672982} a^{11} - \frac{131358420785431}{7626113676} a^{9} + \frac{14427729114643}{3813056838} a^{7} - \frac{550676683765}{1271018946} a^{5} + \frac{364293806183}{7626113676} a^{3} - \frac{110135336753}{22878341028} a \) (order $20$)
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:  \( 19962612841466.996 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{85})\), 4.4.9826000.2, 4.0.614125.2, \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-5}, \sqrt{-17})\), 4.4.9826000.1, 4.0.614125.1, \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{-5}, \sqrt{17})\), 4.0.36125.1, 4.4.578000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.0.1965200.1, 4.4.122825.1, 4.0.78608.1, 4.4.4913.1, 8.0.96550276000000.11, 8.0.13363360000.1, 8.0.96550276000000.12, 8.8.96550276000000.1, 8.0.96550276000000.4, 8.0.377149515625.1, 8.0.96550276000000.5, 8.0.334084000000.2, \(\Q(\zeta_{20})\), 8.0.3862011040000.3, 8.0.6179217664.1, 8.0.1305015625.1, 8.8.334084000000.1, 8.0.3862011040000.4, 8.8.15085980625.1, 8.0.334084000000.1, 8.0.334084000000.3, 8.0.3862011040000.2, 8.0.3862011040000.1, 16.0.9321955795676176000000000000.1, 16.0.111612119056000000000000.1, 16.0.14915129273081881600000000.1, 16.0.9321955795676176000000000000.2, 16.16.9321955795676176000000000000.1, 16.0.142241757136172119140625.1, 16.0.9321955795676176000000000000.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
17Data not computed