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Properties

Label 28.0.39980252956...7761.1
Degree $28$
Signature $[0, 14]$
Discriminant $11^{14}\cdot 29^{26}$
Root discriminant $75.62$
Ramified primes $11, 29$
Class number $880640$ (GRH)
Class group $[16, 16, 3440]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1532409301, -1711780865, 3196869115, -2996413671, 3335457433, -2689454001, 2293844933, -1624224783, 1159745156, -732077019, 455881295, -259144734, 143721798, -74021368, 36946487, -17272242, 7789506, -3302776, 1342595, -511845, 186224, -63303, 20190, -5878, 1591, -388, 81, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 12*x^27 + 81*x^26 - 388*x^25 + 1591*x^24 - 5878*x^23 + 20190*x^22 - 63303*x^21 + 186224*x^20 - 511845*x^19 + 1342595*x^18 - 3302776*x^17 + 7789506*x^16 - 17272242*x^15 + 36946487*x^14 - 74021368*x^13 + 143721798*x^12 - 259144734*x^11 + 455881295*x^10 - 732077019*x^9 + 1159745156*x^8 - 1624224783*x^7 + 2293844933*x^6 - 2689454001*x^5 + 3335457433*x^4 - 2996413671*x^3 + 3196869115*x^2 - 1711780865*x + 1532409301)
 
gp: K = bnfinit(x^28 - 12*x^27 + 81*x^26 - 388*x^25 + 1591*x^24 - 5878*x^23 + 20190*x^22 - 63303*x^21 + 186224*x^20 - 511845*x^19 + 1342595*x^18 - 3302776*x^17 + 7789506*x^16 - 17272242*x^15 + 36946487*x^14 - 74021368*x^13 + 143721798*x^12 - 259144734*x^11 + 455881295*x^10 - 732077019*x^9 + 1159745156*x^8 - 1624224783*x^7 + 2293844933*x^6 - 2689454001*x^5 + 3335457433*x^4 - 2996413671*x^3 + 3196869115*x^2 - 1711780865*x + 1532409301, 1)
 

Normalized defining polynomial

\( x^{28} - 12 x^{27} + 81 x^{26} - 388 x^{25} + 1591 x^{24} - 5878 x^{23} + 20190 x^{22} - 63303 x^{21} + 186224 x^{20} - 511845 x^{19} + 1342595 x^{18} - 3302776 x^{17} + 7789506 x^{16} - 17272242 x^{15} + 36946487 x^{14} - 74021368 x^{13} + 143721798 x^{12} - 259144734 x^{11} + 455881295 x^{10} - 732077019 x^{9} + 1159745156 x^{8} - 1624224783 x^{7} + 2293844933 x^{6} - 2689454001 x^{5} + 3335457433 x^{4} - 2996413671 x^{3} + 3196869115 x^{2} - 1711780865 x + 1532409301 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(39980252956536233141313409161989698787991996663947761=11^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.62$
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:  $11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(319=11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(67,·)$, $\chi_{319}(197,·)$, $\chi_{319}(65,·)$, $\chi_{319}(265,·)$, $\chi_{319}(111,·)$, $\chi_{319}(78,·)$, $\chi_{319}(109,·)$, $\chi_{319}(144,·)$, $\chi_{319}(274,·)$, $\chi_{319}(23,·)$, $\chi_{319}(219,·)$, $\chi_{319}(285,·)$, $\chi_{319}(208,·)$, $\chi_{319}(34,·)$, $\chi_{319}(100,·)$, $\chi_{319}(296,·)$, $\chi_{319}(199,·)$, $\chi_{319}(45,·)$, $\chi_{319}(210,·)$, $\chi_{319}(175,·)$, $\chi_{319}(241,·)$, $\chi_{319}(254,·)$, $\chi_{319}(54,·)$, $\chi_{319}(120,·)$, $\chi_{319}(122,·)$, $\chi_{319}(252,·)$, $\chi_{319}(318,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{17} a^{24} + \frac{4}{17} a^{22} + \frac{4}{17} a^{21} - \frac{1}{17} a^{20} + \frac{6}{17} a^{19} - \frac{1}{17} a^{18} - \frac{4}{17} a^{17} - \frac{3}{17} a^{16} + \frac{2}{17} a^{15} - \frac{7}{17} a^{14} - \frac{4}{17} a^{13} - \frac{6}{17} a^{12} + \frac{5}{17} a^{11} + \frac{4}{17} a^{9} - \frac{8}{17} a^{8} - \frac{7}{17} a^{7} + \frac{6}{17} a^{5} - \frac{5}{17} a^{4} + \frac{8}{17} a^{3} - \frac{7}{17} a^{2} - \frac{4}{17}$, $\frac{1}{1003} a^{25} - \frac{5}{1003} a^{24} + \frac{208}{1003} a^{23} + \frac{494}{1003} a^{22} + \frac{336}{1003} a^{21} + \frac{215}{1003} a^{20} - \frac{337}{1003} a^{19} - \frac{152}{1003} a^{18} - \frac{7}{59} a^{17} + \frac{18}{59} a^{16} - \frac{9}{59} a^{15} - \frac{71}{1003} a^{14} + \frac{6}{17} a^{13} + \frac{188}{1003} a^{12} - \frac{416}{1003} a^{11} + \frac{259}{1003} a^{10} + \frac{482}{1003} a^{9} + \frac{101}{1003} a^{8} + \frac{239}{1003} a^{7} - \frac{436}{1003} a^{6} - \frac{409}{1003} a^{5} + \frac{101}{1003} a^{4} + \frac{38}{1003} a^{3} - \frac{135}{1003} a^{2} - \frac{463}{1003} a - \frac{6}{17}$, $\frac{1}{1003} a^{26} + \frac{6}{1003} a^{24} - \frac{8}{17} a^{23} + \frac{92}{1003} a^{22} + \frac{184}{1003} a^{21} - \frac{88}{1003} a^{20} + \frac{110}{1003} a^{19} + \frac{301}{1003} a^{18} + \frac{419}{1003} a^{17} - \frac{98}{1003} a^{16} - \frac{11}{59} a^{15} + \frac{235}{1003} a^{14} - \frac{343}{1003} a^{13} - \frac{420}{1003} a^{12} + \frac{303}{1003} a^{11} - \frac{229}{1003} a^{10} - \frac{203}{1003} a^{9} + \frac{154}{1003} a^{8} - \frac{8}{1003} a^{7} + \frac{420}{1003} a^{6} + \frac{3}{1003} a^{5} + \frac{25}{59} a^{4} - \frac{358}{1003} a^{3} + \frac{101}{1003} a^{2} + \frac{20}{59} a - \frac{1}{17}$, $\frac{1}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{27} - \frac{84587041098555302956687893200287940976462877920655685896844795800769147416387069349287657110116}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{26} - \frac{6910448814951081615979182297263743281463784922071214503528779419569020645759000825706469459054}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{25} - \frac{286764899650504768645055637742054793479963943383565722517223394781922210636601590139821247543647}{30786228841863788806595111381652411775871151314377953529752707614883969374733756250114729043571411} a^{24} - \frac{149397931507585858221719833119500703928872043399701942970353068524242210249971296616954528525156466}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{23} - \frac{55227938254264477125689986920010783023295878049906183181517364632537573418267794394537413211813046}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{22} - \frac{63356263648390109534712530755565262891712222762839242164956802111992011549672215398733813826484345}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{21} + \frac{220586606469662017537271782181837923103766145845056222365914293937608509667312065772452456643421876}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{20} - \frac{78551171933254220561918652794934004284764403759031144702456167862476782831082849792480471295816186}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{19} - \frac{195779603008650091809658663401201435967247248597308725639566723560843247800747065028956664486207555}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{18} + \frac{195246806636784743752306072691195725386451655454187362496431904003965946469573321302747270763556909}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{17} + \frac{165852076993790121262523547962612494332374116358117074877348206414093543504194156586887540491896263}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{16} - \frac{246255659025596239016821255916314765514238983123160117996120658071532943002612790782396640970084558}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{15} + \frac{58079198858919603784947068708884903984488186021870681300643843380004046068351067772724241918862331}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{14} - \frac{34444087943151854969958042038094814845824079095443281293827700250287815790768569434657388567280043}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{13} + \frac{3783631579385743681359365355022756790667994646300038893306927289184640808282265846432747939224764}{30786228841863788806595111381652411775871151314377953529752707614883969374733756250114729043571411} a^{12} - \frac{147385735625913644795823445443573269004848483244706231346219930254915406789259712969563704599864008}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{11} + \frac{205168259594341023708608060081960244103240951044984479866449968836321760645527847466674227052825867}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{10} - \frac{138029915501566477487729858103402414658798060632647401950635562431419247413621437717486451070220292}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{9} - \frac{8186544958159204379940425003330103168492638542355645101618322513981861585022901371527492343942541}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{8} + \frac{160289298689139185590582074538962470645747858928679372699323947860879497401676367322832115064560701}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{7} - \frac{57253263173670540913868467400502676999691127138660954965623683693208637819844412367625219721845749}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{6} - \frac{2718370491601366815454234591857290025258252205053506740400998828966830223069712729973583825361310}{8870608310367532368001981245560864409996772412617376440776203889034364057126675529694074470181593} a^{5} + \frac{118666373611306264173157650926219172727342621571109153976049913668144417293267383302569700980383528}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{4} + \frac{153049157604213338721438359785551656146612849264970773217419950310451174333426311034953280856808211}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{3} + \frac{204595263160113440020313727697678046134513070252500305951985372473386729039720079071534647737115707}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a^{2} - \frac{208112871518097832538532643329226596506018317939498623818619518604789733495247595575028151336484430}{523365890311684409712116893488091000189809572344425210005796029453027479370473856251950393740713987} a - \frac{580344579654858732898697470820252817472410836169287543098085388283288993784429415621601651048613}{8870608310367532368001981245560864409996772412617376440776203889034364057126675529694074470181593}$

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

Class group and class number

$C_{16}\times C_{16}\times C_{3440}$, which has order $880640$ (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:  $13$
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:  \( 487075979.1876791 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{-319}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{29})\), 7.7.594823321.1, 14.0.199950626296934196778017319.1, \(\Q(\zeta_{29})^+\), 14.0.6894849182652903337173011.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
11Data not computed
29Data not computed