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Properties

Label 32.32.5981643090...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}$
Root discriminant $79.30$
Ramified primes $2, 3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (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![3721, 0, -945954, 0, 59165083, 0, -352066990, 0, 934419815, 0, -1431708442, 0, 1409169330, 0, -941151044, 0, 439508851, 0, -145742210, 0, 34483830, 0, -5793716, 0, 680298, 0, -54104, 0, 2752, 0, -80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 80*x^30 + 2752*x^28 - 54104*x^26 + 680298*x^24 - 5793716*x^22 + 34483830*x^20 - 145742210*x^18 + 439508851*x^16 - 941151044*x^14 + 1409169330*x^12 - 1431708442*x^10 + 934419815*x^8 - 352066990*x^6 + 59165083*x^4 - 945954*x^2 + 3721)
 
gp: K = bnfinit(x^32 - 80*x^30 + 2752*x^28 - 54104*x^26 + 680298*x^24 - 5793716*x^22 + 34483830*x^20 - 145742210*x^18 + 439508851*x^16 - 941151044*x^14 + 1409169330*x^12 - 1431708442*x^10 + 934419815*x^8 - 352066990*x^6 + 59165083*x^4 - 945954*x^2 + 3721, 1)
 

Normalized defining polynomial

\( x^{32} - 80 x^{30} + 2752 x^{28} - 54104 x^{26} + 680298 x^{24} - 5793716 x^{22} + 34483830 x^{20} - 145742210 x^{18} + 439508851 x^{16} - 941151044 x^{14} + 1409169330 x^{12} - 1431708442 x^{10} + 934419815 x^{8} - 352066990 x^{6} + 59165083 x^{4} - 945954 x^{2} + 3721 \)

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:  \(5981643090147991811559885370844487936000000000000000000000000=2^{32}\cdot 3^{16}\cdot 5^{24}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.30$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(780=2^{2}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{780}(1,·)$, $\chi_{780}(131,·)$, $\chi_{780}(649,·)$, $\chi_{780}(599,·)$, $\chi_{780}(151,·)$, $\chi_{780}(281,·)$, $\chi_{780}(31,·)$, $\chi_{780}(161,·)$, $\chi_{780}(547,·)$, $\chi_{780}(677,·)$, $\chi_{780}(47,·)$, $\chi_{780}(181,·)$, $\chi_{780}(311,·)$, $\chi_{780}(697,·)$, $\chi_{780}(53,·)$, $\chi_{780}(577,·)$, $\chi_{780}(707,·)$, $\chi_{780}(73,·)$, $\chi_{780}(203,·)$, $\chi_{780}(77,·)$, $\chi_{780}(83,·)$, $\chi_{780}(469,·)$, $\chi_{780}(727,·)$, $\chi_{780}(733,·)$, $\chi_{780}(103,·)$, $\chi_{780}(779,·)$, $\chi_{780}(233,·)$, $\chi_{780}(619,·)$, $\chi_{780}(749,·)$, $\chi_{780}(499,·)$, $\chi_{780}(629,·)$, $\chi_{780}(703,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{20} - \frac{1}{2} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{21} - \frac{1}{2} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} + \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{23} - \frac{1}{4} a^{15} - \frac{1}{2} a^{13} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a$, $\frac{1}{24} a^{24} - \frac{1}{24} a^{22} + \frac{1}{24} a^{20} - \frac{1}{8} a^{18} - \frac{1}{12} a^{16} - \frac{1}{24} a^{14} - \frac{5}{12} a^{12} - \frac{5}{24} a^{10} + \frac{5}{12} a^{8} - \frac{7}{24} a^{6} + \frac{1}{8} a^{4} + \frac{1}{12} a^{2} + \frac{7}{24}$, $\frac{1}{24} a^{25} - \frac{1}{24} a^{23} + \frac{1}{24} a^{21} - \frac{1}{8} a^{19} - \frac{1}{12} a^{17} - \frac{1}{24} a^{15} - \frac{5}{12} a^{13} - \frac{5}{24} a^{11} + \frac{5}{12} a^{9} - \frac{7}{24} a^{7} + \frac{1}{8} a^{5} + \frac{1}{12} a^{3} + \frac{7}{24} a$, $\frac{1}{24} a^{26} - \frac{1}{12} a^{20} + \frac{1}{24} a^{18} - \frac{1}{8} a^{16} + \frac{1}{24} a^{14} - \frac{1}{8} a^{12} - \frac{1}{24} a^{10} - \frac{3}{8} a^{8} - \frac{5}{12} a^{6} - \frac{1}{24} a^{4} - \frac{3}{8} a^{2} + \frac{7}{24}$, $\frac{1}{1464} a^{27} + \frac{9}{488} a^{25} - \frac{43}{488} a^{23} - \frac{71}{1464} a^{21} - \frac{91}{732} a^{19} + \frac{55}{488} a^{17} + \frac{65}{183} a^{15} + \frac{65}{488} a^{13} + \frac{35}{366} a^{11} + \frac{119}{488} a^{9} + \frac{497}{1464} a^{7} - \frac{77}{183} a^{5} - \frac{45}{488} a^{3} - \frac{313}{732} a$, $\frac{1}{2043986586264} a^{28} - \frac{2202749249}{340664431044} a^{26} + \frac{19221316055}{2043986586264} a^{24} - \frac{6539112937}{291998083752} a^{22} + \frac{2110306579}{28388702587} a^{20} - \frac{7499892676}{85166107761} a^{18} - \frac{2166377495}{227109620696} a^{16} + \frac{244931586089}{1021993293132} a^{14} + \frac{21584428069}{681328862088} a^{12} - \frac{463893860093}{1021993293132} a^{10} - \frac{13629457666}{36499760469} a^{8} - \frac{148331771303}{340664431044} a^{6} + \frac{58432644875}{170332215522} a^{4} - \frac{88436628445}{291998083752} a^{2} + \frac{8844223547}{33507976824}$, $\frac{1}{2043986586264} a^{29} - \frac{217001395}{681328862088} a^{27} + \frac{8912575177}{1021993293132} a^{25} + \frac{14780871523}{145999041876} a^{23} - \frac{19626315721}{681328862088} a^{21} + \frac{3669432932}{85166107761} a^{19} + \frac{15012858149}{170332215522} a^{17} + \frac{90203334818}{255498323283} a^{15} - \frac{35048559815}{340664431044} a^{13} - \frac{265638330551}{1021993293132} a^{11} + \frac{45140922511}{291998083752} a^{9} + \frac{65084057571}{227109620696} a^{7} + \frac{4783422670}{85166107761} a^{5} - \frac{68052857261}{145999041876} a^{3} + \frac{1019778637511}{2043986586264} a$, $\frac{1}{369348821827323649611139209384} a^{30} + \frac{34388177716710151}{369348821827323649611139209384} a^{28} - \frac{586288948913196324131309377}{92337205456830912402784802346} a^{26} - \frac{650427821949456491375342477}{92337205456830912402784802346} a^{24} + \frac{42716830374558276838884298343}{369348821827323649611139209384} a^{22} - \frac{165304832297567076818107585}{10259689495203434711420533594} a^{20} - \frac{1206053394844608006255408883}{10259689495203434711420533594} a^{18} - \frac{870762575636082948626328131}{13191029350975844628969257478} a^{16} - \frac{79999085996252466713244957289}{184674410913661824805569604692} a^{14} - \frac{11766208030655145386965551311}{26382058701951689257938514956} a^{12} + \frac{16260110483748163866332872853}{41038757980813738845682134376} a^{10} + \frac{3010047848938728590831791693}{6054898718480715567395724744} a^{8} + \frac{10053470947299369234564177551}{30779068485610304134261600782} a^{6} + \frac{34369112373109400592781137967}{184674410913661824805569604692} a^{4} + \frac{77551890967426462473056398267}{369348821827323649611139209384} a^{2} + \frac{1250806643998409284427613703}{3027449359240357783697862372}$, $\frac{1}{369348821827323649611139209384} a^{31} + \frac{34388177716710151}{369348821827323649611139209384} a^{29} - \frac{74568776222516958751840729}{369348821827323649611139209384} a^{27} - \frac{2853998734401189114142858439}{369348821827323649611139209384} a^{25} - \frac{989086811765976414947769973}{92337205456830912402784802346} a^{23} - \frac{14682126133273416736772213747}{123116273942441216537046403128} a^{21} + \frac{139479653706662119832529949}{10259689495203434711420533594} a^{19} + \frac{1670821820924372527742238895}{52764117403903378515877029912} a^{17} - \frac{12890625199757869174609008043}{184674410913661824805569604692} a^{15} - \frac{4250446928053250127760193467}{52764117403903378515877029912} a^{13} + \frac{41884474577419232202797932045}{123116273942441216537046403128} a^{11} - \frac{41527456137246202036657041997}{184674410913661824805569604692} a^{9} + \frac{36765955352284847240156366947}{123116273942441216537046403128} a^{7} - \frac{18611251413596860621931453543}{184674410913661824805569604692} a^{5} + \frac{8627595614611334973870319679}{184674410913661824805569604692} a^{3} + \frac{27355440642850515513635660869}{184674410913661824805569604692} a$

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:  \( 278879965707408670000 \) (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

\(\Q(\sqrt{195}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{3}, \sqrt{65})\), 4.4.274625.1, 4.4.39546000.2, \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{39})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{15}, \sqrt{39})\), 4.4.274625.2, 4.4.39546000.1, \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{5})\), 4.4.19773.1, 4.4.878800.1, 4.4.494325.1, 4.4.35152.1, \(\Q(\zeta_{15})^+\), 4.4.338000.1, 4.4.190125.1, \(\Q(\zeta_{20})^+\), 8.8.1563886116000000.5, 8.8.370150560000.1, 8.8.1563886116000000.2, 8.8.75418890625.1, 8.8.1563886116000000.3, 8.8.1563886116000000.7, 8.8.1563886116000000.1, 8.8.62555444640000.2, 8.8.62555444640000.3, 8.8.9253764000000.3, 8.8.9253764000000.2, 8.8.244357205625.1, 8.8.772289440000.1, 8.8.36147515625.1, 8.8.114244000000.1, 8.8.100088711424.1, 8.8.62555444640000.4, \(\Q(\zeta_{60})^+\), 8.8.9253764000000.1, 16.16.2445739783817565456000000000000.2, 16.16.3913183654108104729600000000.1, 16.16.85632148167696000000000000.1, 16.16.37319027463036582275390625.1, 16.16.372769361959696000000000000.2, 16.16.2445739783817565456000000000000.3, 16.16.2445739783817565456000000000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
13Data not computed