Properties

Label 28.0.135...944.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.352\times 10^{53}$
Root discriminant \(78.98\)
Ramified primes $2,3,29$
Class number $384$ (GRH)
Class group [2, 4, 4, 12] (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841)
 
gp: K = bnfinit(y^28 - 29*y^26 + 551*y^24 - 5916*y^22 + 45675*y^20 - 232000*y^18 + 854369*y^16 - 1923396*y^14 + 3056194*y^12 - 2610464*y^10 + 1563419*y^8 - 459186*y^6 + 96715*y^4 - 10933*y^2 + 841, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841)
 

\( x^{28} - 29 x^{26} + 551 x^{24} - 5916 x^{22} + 45675 x^{20} - 232000 x^{18} + 854369 x^{16} - 1923396 x^{14} + 3056194 x^{12} - 2610464 x^{10} + 1563419 x^{8} + \cdots + 841 \) Copy content Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(135171579942192030712001098144632895138136441638354944\) \(\medspace = 2^{28}\cdot 3^{14}\cdot 29^{26}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(78.98\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}29^{13/14}\approx 78.98263547298767$
Ramified primes:   \(2\), \(3\), \(29\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(1,·)$, $\chi_{348}(323,·)$, $\chi_{348}(197,·)$, $\chi_{348}(257,·)$, $\chi_{348}(115,·)$, $\chi_{348}(67,·)$, $\chi_{348}(277,·)$, $\chi_{348}(151,·)$, $\chi_{348}(25,·)$, $\chi_{348}(71,·)$, $\chi_{348}(283,·)$, $\chi_{348}(161,·)$, $\chi_{348}(35,·)$, $\chi_{348}(65,·)$, $\chi_{348}(167,·)$, $\chi_{348}(169,·)$, $\chi_{348}(299,·)$, $\chi_{348}(281,·)$, $\chi_{348}(49,·)$, $\chi_{348}(91,·)$, $\chi_{348}(179,·)$, $\chi_{348}(53,·)$, $\chi_{348}(233,·)$, $\chi_{348}(313,·)$, $\chi_{348}(347,·)$, $\chi_{348}(187,·)$, $\chi_{348}(295,·)$, $\chi_{348}(181,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29}a^{14}$, $\frac{1}{29}a^{15}$, $\frac{1}{29}a^{16}$, $\frac{1}{29}a^{17}$, $\frac{1}{29}a^{18}$, $\frac{1}{29}a^{19}$, $\frac{1}{493}a^{20}+\frac{5}{493}a^{18}+\frac{5}{493}a^{16}+\frac{7}{493}a^{14}+\frac{7}{17}a^{12}+\frac{7}{17}a^{10}+\frac{7}{17}a^{8}+\frac{5}{17}a^{6}-\frac{5}{17}a^{4}-\frac{5}{17}a^{2}+\frac{3}{17}$, $\frac{1}{493}a^{21}+\frac{5}{493}a^{19}+\frac{5}{493}a^{17}+\frac{7}{493}a^{15}+\frac{7}{17}a^{13}+\frac{7}{17}a^{11}+\frac{7}{17}a^{9}+\frac{5}{17}a^{7}-\frac{5}{17}a^{5}-\frac{5}{17}a^{3}+\frac{3}{17}a$, $\frac{1}{493}a^{22}-\frac{3}{493}a^{18}-\frac{1}{493}a^{16}-\frac{2}{493}a^{14}+\frac{6}{17}a^{12}+\frac{6}{17}a^{10}+\frac{4}{17}a^{8}+\frac{4}{17}a^{6}+\frac{3}{17}a^{4}-\frac{6}{17}a^{2}+\frac{2}{17}$, $\frac{1}{493}a^{23}-\frac{3}{493}a^{19}-\frac{1}{493}a^{17}-\frac{2}{493}a^{15}+\frac{6}{17}a^{13}+\frac{6}{17}a^{11}+\frac{4}{17}a^{9}+\frac{4}{17}a^{7}+\frac{3}{17}a^{5}-\frac{6}{17}a^{3}+\frac{2}{17}a$, $\frac{1}{1192567}a^{24}+\frac{75}{1192567}a^{22}-\frac{46}{1192567}a^{20}-\frac{6697}{1192567}a^{18}+\frac{3958}{1192567}a^{16}+\frac{10518}{1192567}a^{14}+\frac{4847}{41123}a^{12}+\frac{1076}{2419}a^{10}+\frac{11495}{41123}a^{8}-\frac{1119}{41123}a^{6}-\frac{14662}{41123}a^{4}+\frac{13163}{41123}a^{2}-\frac{7153}{41123}$, $\frac{1}{1192567}a^{25}+\frac{75}{1192567}a^{23}-\frac{46}{1192567}a^{21}-\frac{6697}{1192567}a^{19}+\frac{3958}{1192567}a^{17}+\frac{10518}{1192567}a^{15}+\frac{4847}{41123}a^{13}+\frac{1076}{2419}a^{11}+\frac{11495}{41123}a^{9}-\frac{1119}{41123}a^{7}-\frac{14662}{41123}a^{5}+\frac{13163}{41123}a^{3}-\frac{7153}{41123}a$, $\frac{1}{41\!\cdots\!13}a^{26}+\frac{73\!\cdots\!30}{41\!\cdots\!13}a^{24}+\frac{14\!\cdots\!70}{41\!\cdots\!13}a^{22}+\frac{41\!\cdots\!07}{41\!\cdots\!13}a^{20}+\frac{14\!\cdots\!59}{41\!\cdots\!13}a^{18}-\frac{15\!\cdots\!23}{41\!\cdots\!13}a^{16}+\frac{21\!\cdots\!61}{41\!\cdots\!13}a^{14}-\frac{66\!\cdots\!62}{14\!\cdots\!97}a^{12}+\frac{11\!\cdots\!47}{84\!\cdots\!41}a^{10}+\frac{32\!\cdots\!07}{14\!\cdots\!97}a^{8}+\frac{86\!\cdots\!13}{14\!\cdots\!97}a^{6}+\frac{41\!\cdots\!52}{14\!\cdots\!97}a^{4}+\frac{70\!\cdots\!48}{14\!\cdots\!97}a^{2}+\frac{13\!\cdots\!75}{84\!\cdots\!41}$, $\frac{1}{41\!\cdots\!13}a^{27}+\frac{73\!\cdots\!30}{41\!\cdots\!13}a^{25}+\frac{14\!\cdots\!70}{41\!\cdots\!13}a^{23}+\frac{41\!\cdots\!07}{41\!\cdots\!13}a^{21}+\frac{14\!\cdots\!59}{41\!\cdots\!13}a^{19}-\frac{15\!\cdots\!23}{41\!\cdots\!13}a^{17}+\frac{21\!\cdots\!61}{41\!\cdots\!13}a^{15}-\frac{66\!\cdots\!62}{14\!\cdots\!97}a^{13}+\frac{11\!\cdots\!47}{84\!\cdots\!41}a^{11}+\frac{32\!\cdots\!07}{14\!\cdots\!97}a^{9}+\frac{86\!\cdots\!13}{14\!\cdots\!97}a^{7}+\frac{41\!\cdots\!52}{14\!\cdots\!97}a^{5}+\frac{70\!\cdots\!48}{14\!\cdots\!97}a^{3}+\frac{13\!\cdots\!75}{84\!\cdots\!41}a$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{12}$, which has order $384$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{17794974015976247830}{100887524644686236821031} a^{26} - \frac{513845098838430797670}{100887524644686236821031} a^{24} + \frac{9741588466385237087800}{100887524644686236821031} a^{22} - \frac{104075676595349242494679}{100887524644686236821031} a^{20} + \frac{800053248285444887451775}{100887524644686236821031} a^{18} - \frac{139003485903995192324975}{3478880160161594373139} a^{16} + \frac{14718196606973246016914140}{100887524644686236821031} a^{14} - \frac{1119818527282558012165915}{3478880160161594373139} a^{12} + \frac{1745638968351759372561100}{3478880160161594373139} a^{10} - \frac{1404909169814334023239958}{3478880160161594373139} a^{8} + \frac{815179770710311150035865}{3478880160161594373139} a^{6} - \frac{202696972803554977446945}{3478880160161594373139} a^{4} + \frac{49449697077842951634753}{3478880160161594373139} a^{2} - \frac{2088700679450172287950}{3478880160161594373139} \)  (order $6$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{12\!\cdots\!15}{41\!\cdots\!13}a^{26}-\frac{36\!\cdots\!77}{41\!\cdots\!13}a^{24}+\frac{70\!\cdots\!55}{41\!\cdots\!13}a^{22}-\frac{78\!\cdots\!25}{41\!\cdots\!13}a^{20}+\frac{62\!\cdots\!40}{41\!\cdots\!13}a^{18}-\frac{33\!\cdots\!55}{41\!\cdots\!13}a^{16}+\frac{12\!\cdots\!08}{41\!\cdots\!13}a^{14}-\frac{11\!\cdots\!02}{14\!\cdots\!97}a^{12}+\frac{19\!\cdots\!82}{14\!\cdots\!97}a^{10}-\frac{20\!\cdots\!81}{14\!\cdots\!97}a^{8}+\frac{13\!\cdots\!90}{14\!\cdots\!97}a^{6}-\frac{56\!\cdots\!05}{14\!\cdots\!97}a^{4}+\frac{90\!\cdots\!59}{14\!\cdots\!97}a^{2}-\frac{10\!\cdots\!23}{14\!\cdots\!97}$, $\frac{16\!\cdots\!77}{41\!\cdots\!13}a^{26}-\frac{46\!\cdots\!87}{41\!\cdots\!13}a^{24}+\frac{88\!\cdots\!95}{41\!\cdots\!13}a^{22}-\frac{94\!\cdots\!53}{41\!\cdots\!13}a^{20}+\frac{73\!\cdots\!05}{41\!\cdots\!13}a^{18}-\frac{12\!\cdots\!60}{14\!\cdots\!97}a^{16}+\frac{13\!\cdots\!18}{41\!\cdots\!13}a^{14}-\frac{10\!\cdots\!09}{14\!\cdots\!97}a^{12}+\frac{16\!\cdots\!25}{14\!\cdots\!97}a^{10}-\frac{13\!\cdots\!29}{14\!\cdots\!97}a^{8}+\frac{76\!\cdots\!14}{14\!\cdots\!97}a^{6}-\frac{19\!\cdots\!02}{14\!\cdots\!97}a^{4}+\frac{29\!\cdots\!77}{14\!\cdots\!97}a^{2}-\frac{33\!\cdots\!70}{24\!\cdots\!83}$, $\frac{13\!\cdots\!50}{41\!\cdots\!13}a^{26}-\frac{39\!\cdots\!95}{41\!\cdots\!13}a^{24}+\frac{74\!\cdots\!75}{41\!\cdots\!13}a^{22}-\frac{47\!\cdots\!35}{24\!\cdots\!89}a^{20}+\frac{61\!\cdots\!75}{41\!\cdots\!13}a^{18}-\frac{10\!\cdots\!50}{14\!\cdots\!97}a^{16}+\frac{11\!\cdots\!70}{41\!\cdots\!13}a^{14}-\frac{88\!\cdots\!15}{14\!\cdots\!97}a^{12}+\frac{13\!\cdots\!25}{14\!\cdots\!97}a^{10}-\frac{11\!\cdots\!97}{14\!\cdots\!97}a^{8}+\frac{65\!\cdots\!40}{14\!\cdots\!97}a^{6}-\frac{16\!\cdots\!20}{14\!\cdots\!97}a^{4}+\frac{13\!\cdots\!84}{84\!\cdots\!41}a^{2}-\frac{28\!\cdots\!50}{24\!\cdots\!83}$, $\frac{11\!\cdots\!80}{41\!\cdots\!13}a^{26}-\frac{35\!\cdots\!20}{41\!\cdots\!13}a^{24}+\frac{70\!\cdots\!05}{41\!\cdots\!13}a^{22}-\frac{80\!\cdots\!40}{41\!\cdots\!13}a^{20}+\frac{65\!\cdots\!00}{41\!\cdots\!13}a^{18}-\frac{36\!\cdots\!10}{41\!\cdots\!13}a^{16}+\frac{14\!\cdots\!20}{41\!\cdots\!13}a^{14}-\frac{13\!\cdots\!20}{14\!\cdots\!97}a^{12}+\frac{25\!\cdots\!20}{14\!\cdots\!97}a^{10}-\frac{30\!\cdots\!00}{14\!\cdots\!97}a^{8}+\frac{20\!\cdots\!80}{14\!\cdots\!97}a^{6}-\frac{93\!\cdots\!37}{14\!\cdots\!97}a^{4}+\frac{13\!\cdots\!20}{14\!\cdots\!97}a^{2}-\frac{15\!\cdots\!20}{14\!\cdots\!97}$, $\frac{16\!\cdots\!64}{70\!\cdots\!07}a^{26}-\frac{27\!\cdots\!80}{41\!\cdots\!13}a^{24}+\frac{52\!\cdots\!74}{41\!\cdots\!13}a^{22}-\frac{55\!\cdots\!70}{41\!\cdots\!13}a^{20}+\frac{42\!\cdots\!98}{41\!\cdots\!13}a^{18}-\frac{12\!\cdots\!42}{24\!\cdots\!89}a^{16}+\frac{26\!\cdots\!96}{14\!\cdots\!97}a^{14}-\frac{56\!\cdots\!63}{14\!\cdots\!97}a^{12}+\frac{86\!\cdots\!66}{14\!\cdots\!97}a^{10}-\frac{63\!\cdots\!10}{14\!\cdots\!97}a^{8}+\frac{32\!\cdots\!40}{14\!\cdots\!97}a^{6}-\frac{43\!\cdots\!20}{14\!\cdots\!97}a^{4}+\frac{49\!\cdots\!28}{14\!\cdots\!97}a^{2}+\frac{64\!\cdots\!58}{14\!\cdots\!97}$, $\frac{48\!\cdots\!07}{41\!\cdots\!13}a^{26}-\frac{14\!\cdots\!53}{41\!\cdots\!13}a^{24}+\frac{16\!\cdots\!20}{24\!\cdots\!89}a^{22}-\frac{31\!\cdots\!76}{41\!\cdots\!13}a^{20}+\frac{24\!\cdots\!25}{41\!\cdots\!13}a^{18}-\frac{13\!\cdots\!85}{41\!\cdots\!13}a^{16}+\frac{51\!\cdots\!78}{41\!\cdots\!13}a^{14}-\frac{44\!\cdots\!13}{14\!\cdots\!97}a^{12}+\frac{76\!\cdots\!45}{14\!\cdots\!97}a^{10}-\frac{82\!\cdots\!95}{14\!\cdots\!97}a^{8}+\frac{54\!\cdots\!37}{14\!\cdots\!97}a^{6}-\frac{22\!\cdots\!64}{14\!\cdots\!97}a^{4}+\frac{35\!\cdots\!78}{14\!\cdots\!97}a^{2}-\frac{40\!\cdots\!58}{14\!\cdots\!97}$, $\frac{95\!\cdots\!86}{41\!\cdots\!13}a^{27}-\frac{46\!\cdots\!35}{41\!\cdots\!13}a^{26}-\frac{27\!\cdots\!20}{41\!\cdots\!13}a^{25}+\frac{13\!\cdots\!71}{41\!\cdots\!13}a^{24}+\frac{52\!\cdots\!85}{41\!\cdots\!13}a^{23}-\frac{25\!\cdots\!53}{41\!\cdots\!13}a^{22}-\frac{55\!\cdots\!56}{41\!\cdots\!13}a^{21}+\frac{95\!\cdots\!56}{14\!\cdots\!97}a^{20}+\frac{42\!\cdots\!70}{41\!\cdots\!13}a^{19}-\frac{21\!\cdots\!48}{41\!\cdots\!13}a^{18}-\frac{21\!\cdots\!25}{41\!\cdots\!13}a^{17}+\frac{37\!\cdots\!45}{14\!\cdots\!97}a^{16}+\frac{78\!\cdots\!02}{41\!\cdots\!13}a^{15}-\frac{40\!\cdots\!93}{41\!\cdots\!13}a^{14}-\frac{35\!\cdots\!68}{84\!\cdots\!41}a^{13}+\frac{30\!\cdots\!80}{14\!\cdots\!97}a^{12}+\frac{92\!\cdots\!05}{14\!\cdots\!97}a^{11}-\frac{48\!\cdots\!83}{14\!\cdots\!97}a^{10}-\frac{72\!\cdots\!92}{14\!\cdots\!97}a^{9}+\frac{40\!\cdots\!49}{14\!\cdots\!97}a^{8}+\frac{39\!\cdots\!56}{14\!\cdots\!97}a^{7}-\frac{23\!\cdots\!97}{14\!\cdots\!97}a^{6}-\frac{82\!\cdots\!23}{14\!\cdots\!97}a^{5}+\frac{57\!\cdots\!61}{14\!\cdots\!97}a^{4}+\frac{11\!\cdots\!54}{14\!\cdots\!97}a^{3}-\frac{59\!\cdots\!45}{14\!\cdots\!97}a^{2}-\frac{48\!\cdots\!10}{14\!\cdots\!97}a+\frac{10\!\cdots\!08}{24\!\cdots\!83}$, $\frac{83\!\cdots\!48}{41\!\cdots\!13}a^{27}+\frac{13\!\cdots\!12}{70\!\cdots\!07}a^{26}-\frac{23\!\cdots\!88}{41\!\cdots\!13}a^{25}-\frac{22\!\cdots\!98}{41\!\cdots\!13}a^{24}+\frac{44\!\cdots\!56}{41\!\cdots\!13}a^{23}+\frac{24\!\cdots\!16}{24\!\cdots\!89}a^{22}-\frac{46\!\cdots\!12}{41\!\cdots\!13}a^{21}-\frac{44\!\cdots\!60}{41\!\cdots\!13}a^{20}+\frac{35\!\cdots\!30}{41\!\cdots\!13}a^{19}+\frac{34\!\cdots\!85}{41\!\cdots\!13}a^{18}-\frac{10\!\cdots\!20}{24\!\cdots\!89}a^{17}-\frac{17\!\cdots\!62}{41\!\cdots\!13}a^{16}+\frac{14\!\cdots\!74}{10\!\cdots\!93}a^{15}+\frac{21\!\cdots\!18}{14\!\cdots\!97}a^{14}-\frac{42\!\cdots\!16}{14\!\cdots\!97}a^{13}-\frac{27\!\cdots\!42}{84\!\cdots\!41}a^{12}+\frac{60\!\cdots\!22}{14\!\cdots\!97}a^{11}+\frac{69\!\cdots\!38}{14\!\cdots\!97}a^{10}-\frac{33\!\cdots\!55}{14\!\cdots\!97}a^{9}-\frac{51\!\cdots\!80}{14\!\cdots\!97}a^{8}+\frac{12\!\cdots\!82}{14\!\cdots\!97}a^{7}+\frac{26\!\cdots\!55}{14\!\cdots\!97}a^{6}+\frac{45\!\cdots\!16}{14\!\cdots\!97}a^{5}-\frac{35\!\cdots\!70}{14\!\cdots\!97}a^{4}-\frac{85\!\cdots\!58}{14\!\cdots\!97}a^{3}+\frac{40\!\cdots\!54}{14\!\cdots\!97}a^{2}+\frac{17\!\cdots\!46}{14\!\cdots\!97}a+\frac{16\!\cdots\!62}{14\!\cdots\!97}$, $\frac{25\!\cdots\!35}{41\!\cdots\!13}a^{27}-\frac{54\!\cdots\!82}{14\!\cdots\!97}a^{26}-\frac{74\!\cdots\!07}{41\!\cdots\!13}a^{25}+\frac{45\!\cdots\!34}{41\!\cdots\!13}a^{24}+\frac{48\!\cdots\!06}{14\!\cdots\!97}a^{23}-\frac{87\!\cdots\!18}{41\!\cdots\!13}a^{22}-\frac{15\!\cdots\!50}{41\!\cdots\!13}a^{21}+\frac{92\!\cdots\!90}{41\!\cdots\!13}a^{20}+\frac{11\!\cdots\!75}{41\!\cdots\!13}a^{19}-\frac{71\!\cdots\!28}{41\!\cdots\!13}a^{18}-\frac{58\!\cdots\!92}{41\!\cdots\!13}a^{17}+\frac{12\!\cdots\!10}{14\!\cdots\!97}a^{16}+\frac{21\!\cdots\!20}{41\!\cdots\!13}a^{15}-\frac{13\!\cdots\!89}{41\!\cdots\!13}a^{14}-\frac{16\!\cdots\!15}{14\!\cdots\!97}a^{13}+\frac{99\!\cdots\!56}{14\!\cdots\!97}a^{12}+\frac{24\!\cdots\!67}{14\!\cdots\!97}a^{11}-\frac{15\!\cdots\!18}{14\!\cdots\!97}a^{10}-\frac{19\!\cdots\!51}{14\!\cdots\!97}a^{9}+\frac{12\!\cdots\!26}{14\!\cdots\!97}a^{8}+\frac{10\!\cdots\!27}{14\!\cdots\!97}a^{7}-\frac{71\!\cdots\!58}{14\!\cdots\!97}a^{6}-\frac{22\!\cdots\!78}{14\!\cdots\!97}a^{5}+\frac{17\!\cdots\!34}{14\!\cdots\!97}a^{4}+\frac{33\!\cdots\!54}{14\!\cdots\!97}a^{3}-\frac{31\!\cdots\!33}{14\!\cdots\!97}a^{2}-\frac{11\!\cdots\!96}{14\!\cdots\!97}a+\frac{31\!\cdots\!48}{24\!\cdots\!83}$, $\frac{93\!\cdots\!74}{41\!\cdots\!13}a^{27}-\frac{62\!\cdots\!66}{41\!\cdots\!71}a^{26}-\frac{26\!\cdots\!31}{41\!\cdots\!13}a^{25}+\frac{17\!\cdots\!01}{41\!\cdots\!13}a^{24}+\frac{50\!\cdots\!67}{41\!\cdots\!13}a^{23}-\frac{33\!\cdots\!93}{41\!\cdots\!13}a^{22}-\frac{52\!\cdots\!35}{41\!\cdots\!13}a^{21}+\frac{20\!\cdots\!55}{24\!\cdots\!89}a^{20}+\frac{39\!\cdots\!75}{41\!\cdots\!13}a^{19}-\frac{26\!\cdots\!30}{41\!\cdots\!13}a^{18}-\frac{19\!\cdots\!16}{41\!\cdots\!13}a^{17}+\frac{13\!\cdots\!27}{41\!\cdots\!13}a^{16}+\frac{16\!\cdots\!26}{10\!\cdots\!93}a^{15}-\frac{16\!\cdots\!88}{14\!\cdots\!97}a^{14}-\frac{28\!\cdots\!83}{84\!\cdots\!41}a^{13}+\frac{34\!\cdots\!28}{14\!\cdots\!97}a^{12}+\frac{67\!\cdots\!61}{14\!\cdots\!97}a^{11}-\frac{53\!\cdots\!95}{14\!\cdots\!97}a^{10}-\frac{36\!\cdots\!33}{14\!\cdots\!97}a^{9}+\frac{39\!\cdots\!45}{14\!\cdots\!97}a^{8}+\frac{12\!\cdots\!44}{14\!\cdots\!97}a^{7}-\frac{25\!\cdots\!20}{14\!\cdots\!97}a^{6}+\frac{58\!\cdots\!88}{14\!\cdots\!97}a^{5}+\frac{26\!\cdots\!12}{14\!\cdots\!97}a^{4}-\frac{11\!\cdots\!20}{14\!\cdots\!97}a^{3}-\frac{31\!\cdots\!30}{14\!\cdots\!97}a^{2}+\frac{22\!\cdots\!66}{14\!\cdots\!97}a+\frac{90\!\cdots\!71}{14\!\cdots\!97}$, $\frac{88\!\cdots\!22}{41\!\cdots\!13}a^{27}-\frac{43\!\cdots\!28}{41\!\cdots\!13}a^{26}-\frac{25\!\cdots\!94}{41\!\cdots\!13}a^{25}+\frac{12\!\cdots\!03}{41\!\cdots\!13}a^{24}+\frac{48\!\cdots\!82}{41\!\cdots\!13}a^{23}-\frac{24\!\cdots\!15}{41\!\cdots\!13}a^{22}-\frac{51\!\cdots\!44}{41\!\cdots\!13}a^{21}+\frac{25\!\cdots\!13}{41\!\cdots\!13}a^{20}+\frac{39\!\cdots\!80}{41\!\cdots\!13}a^{19}-\frac{19\!\cdots\!55}{41\!\cdots\!13}a^{18}-\frac{11\!\cdots\!20}{24\!\cdots\!89}a^{17}+\frac{34\!\cdots\!90}{14\!\cdots\!97}a^{16}+\frac{73\!\cdots\!79}{41\!\cdots\!13}a^{15}-\frac{36\!\cdots\!86}{41\!\cdots\!13}a^{14}-\frac{55\!\cdots\!95}{14\!\cdots\!97}a^{13}+\frac{28\!\cdots\!31}{14\!\cdots\!97}a^{12}+\frac{85\!\cdots\!74}{14\!\cdots\!97}a^{11}-\frac{44\!\cdots\!85}{14\!\cdots\!97}a^{10}-\frac{67\!\cdots\!63}{14\!\cdots\!97}a^{9}+\frac{36\!\cdots\!36}{14\!\cdots\!97}a^{8}+\frac{37\!\cdots\!10}{14\!\cdots\!97}a^{7}-\frac{20\!\cdots\!16}{14\!\cdots\!97}a^{6}-\frac{77\!\cdots\!07}{14\!\cdots\!97}a^{5}+\frac{52\!\cdots\!88}{14\!\cdots\!97}a^{4}+\frac{11\!\cdots\!71}{14\!\cdots\!97}a^{3}-\frac{80\!\cdots\!58}{14\!\cdots\!97}a^{2}-\frac{39\!\cdots\!63}{14\!\cdots\!97}a+\frac{91\!\cdots\!90}{24\!\cdots\!83}$, $\frac{18\!\cdots\!37}{41\!\cdots\!13}a^{27}-\frac{16\!\cdots\!56}{17\!\cdots\!27}a^{26}-\frac{53\!\cdots\!82}{41\!\cdots\!13}a^{25}+\frac{16\!\cdots\!73}{59\!\cdots\!29}a^{24}+\frac{99\!\cdots\!82}{41\!\cdots\!13}a^{23}-\frac{52\!\cdots\!99}{10\!\cdots\!93}a^{22}-\frac{10\!\cdots\!72}{41\!\cdots\!13}a^{21}+\frac{55\!\cdots\!05}{10\!\cdots\!93}a^{20}+\frac{79\!\cdots\!63}{41\!\cdots\!13}a^{19}-\frac{42\!\cdots\!70}{10\!\cdots\!93}a^{18}-\frac{39\!\cdots\!39}{41\!\cdots\!13}a^{17}+\frac{21\!\cdots\!46}{10\!\cdots\!93}a^{16}+\frac{33\!\cdots\!28}{10\!\cdots\!93}a^{15}-\frac{26\!\cdots\!99}{34\!\cdots\!17}a^{14}-\frac{98\!\cdots\!05}{14\!\cdots\!97}a^{13}+\frac{57\!\cdots\!31}{34\!\cdots\!17}a^{12}+\frac{14\!\cdots\!20}{14\!\cdots\!97}a^{11}-\frac{86\!\cdots\!30}{34\!\cdots\!17}a^{10}-\frac{81\!\cdots\!60}{14\!\cdots\!97}a^{9}+\frac{63\!\cdots\!35}{34\!\cdots\!17}a^{8}+\frac{32\!\cdots\!23}{14\!\cdots\!97}a^{7}-\frac{32\!\cdots\!11}{34\!\cdots\!17}a^{6}+\frac{74\!\cdots\!23}{14\!\cdots\!97}a^{5}+\frac{43\!\cdots\!71}{34\!\cdots\!17}a^{4}-\frac{12\!\cdots\!82}{14\!\cdots\!97}a^{3}-\frac{50\!\cdots\!75}{34\!\cdots\!17}a^{2}+\frac{27\!\cdots\!96}{14\!\cdots\!97}a-\frac{35\!\cdots\!67}{34\!\cdots\!17}$, $\frac{81\!\cdots\!68}{41\!\cdots\!13}a^{27}-\frac{92\!\cdots\!06}{41\!\cdots\!13}a^{26}-\frac{23\!\cdots\!33}{41\!\cdots\!13}a^{25}+\frac{93\!\cdots\!36}{14\!\cdots\!97}a^{24}+\frac{45\!\cdots\!22}{41\!\cdots\!13}a^{23}-\frac{52\!\cdots\!85}{41\!\cdots\!13}a^{22}-\frac{48\!\cdots\!52}{41\!\cdots\!13}a^{21}+\frac{56\!\cdots\!54}{41\!\cdots\!13}a^{20}+\frac{37\!\cdots\!12}{41\!\cdots\!13}a^{19}-\frac{44\!\cdots\!00}{41\!\cdots\!13}a^{18}-\frac{19\!\cdots\!89}{41\!\cdots\!13}a^{17}+\frac{13\!\cdots\!07}{24\!\cdots\!89}a^{16}+\frac{71\!\cdots\!37}{41\!\cdots\!13}a^{15}-\frac{87\!\cdots\!09}{41\!\cdots\!13}a^{14}-\frac{55\!\cdots\!75}{14\!\cdots\!97}a^{13}+\frac{72\!\cdots\!79}{14\!\cdots\!97}a^{12}+\frac{88\!\cdots\!33}{14\!\cdots\!97}a^{11}-\frac{12\!\cdots\!58}{14\!\cdots\!97}a^{10}-\frac{77\!\cdots\!30}{14\!\cdots\!97}a^{9}+\frac{11\!\cdots\!07}{14\!\cdots\!97}a^{8}+\frac{45\!\cdots\!82}{14\!\cdots\!97}a^{7}-\frac{75\!\cdots\!36}{14\!\cdots\!97}a^{6}-\frac{13\!\cdots\!13}{14\!\cdots\!97}a^{5}+\frac{16\!\cdots\!66}{84\!\cdots\!41}a^{4}+\frac{21\!\cdots\!09}{14\!\cdots\!97}a^{3}-\frac{48\!\cdots\!01}{14\!\cdots\!97}a^{2}-\frac{18\!\cdots\!56}{14\!\cdots\!97}a+\frac{55\!\cdots\!35}{14\!\cdots\!97}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 4155697700585.085 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 4155697700585.085 \cdot 384}{6\cdot\sqrt{135171579942192030712001098144632895138136441638354944}}\cr\approx \mathstrut & 0.108118343698100 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 29*x^26 + 551*x^24 - 5916*x^22 + 45675*x^20 - 232000*x^18 + 854369*x^16 - 1923396*x^14 + 3056194*x^12 - 2610464*x^10 + 1563419*x^8 - 459186*x^6 + 96715*x^4 - 10933*x^2 + 841);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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

\(\Q(\sqrt{87}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-29})\), 7.7.594823321.1, 14.14.367656878002019745584627712.1, 14.0.168110140833113738264576.1, 14.0.773792930870360792667.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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 ${\href{/padicField/5.14.0.1}{14} }^{2}$ ${\href{/padicField/7.14.0.1}{14} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{14}$ ${\href{/padicField/19.7.0.1}{7} }^{4}$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ R ${\href{/padicField/31.7.0.1}{7} }^{4}$ ${\href{/padicField/37.14.0.1}{14} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{14}$ ${\href{/padicField/43.7.0.1}{7} }^{4}$ ${\href{/padicField/47.14.0.1}{14} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{14}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $28$$2$$14$$28$
\(3\) Copy content Toggle raw display 3.14.7.2$x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.2$x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(29\) Copy content Toggle raw display Deg $28$$14$$2$$26$