Properties

Label 21.21.827...729.1
Degree $21$
Signature $[21, 0]$
Discriminant $8.277\times 10^{48}$
Root discriminant \(213.51\)
Ramified primes $13,71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{21}$ (as 21T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375)
 
gp: K = bnfinit(y^21 - 4*y^20 - 114*y^19 + 488*y^18 + 4946*y^17 - 23362*y^16 - 100260*y^15 + 557394*y^14 + 893184*y^13 - 6986867*y^12 - 1446343*y^11 + 45520578*y^10 - 23590055*y^9 - 154124034*y^8 + 143234027*y^7 + 261729665*y^6 - 315720051*y^5 - 199051958*y^4 + 300106465*y^3 + 36273150*y^2 - 102144250*y + 15389375, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375)
 

\( x^{21} - 4 x^{20} - 114 x^{19} + 488 x^{18} + 4946 x^{17} - 23362 x^{16} - 100260 x^{15} + 557394 x^{14} + 893184 x^{13} - 6986867 x^{12} - 1446343 x^{11} + \cdots + 15389375 \) Copy content Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[21, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(8276699911115832752222429511299889027510277198729\) \(\medspace = 13^{14}\cdot 71^{18}\) 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:  \(213.51\)
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:  $13^{2/3}71^{6/7}\approx 213.5117459255851$
Ramified primes:   \(13\), \(71\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $21$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(923=13\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{923}(640,·)$, $\chi_{923}(1,·)$, $\chi_{923}(900,·)$, $\chi_{923}(261,·)$, $\chi_{923}(321,·)$, $\chi_{923}(456,·)$, $\chi_{923}(458,·)$, $\chi_{923}(588,·)$, $\chi_{923}(529,·)$, $\chi_{923}(659,·)$, $\chi_{923}(534,·)$, $\chi_{923}(471,·)$, $\chi_{923}(542,·)$, $\chi_{923}(243,·)$, $\chi_{923}(742,·)$, $\chi_{923}(872,·)$, $\chi_{923}(711,·)$, $\chi_{923}(172,·)$, $\chi_{923}(48,·)$, $\chi_{923}(755,·)$, $\chi_{923}(250,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{3}$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{9}-\frac{1}{5}a$, $\frac{1}{25}a^{10}-\frac{2}{25}a^{6}+\frac{1}{25}a^{2}$, $\frac{1}{25}a^{11}-\frac{2}{25}a^{7}+\frac{1}{25}a^{3}$, $\frac{1}{25}a^{12}-\frac{2}{25}a^{8}+\frac{1}{25}a^{4}$, $\frac{1}{125}a^{13}+\frac{1}{125}a^{12}+\frac{1}{125}a^{11}+\frac{1}{125}a^{10}-\frac{7}{125}a^{9}+\frac{8}{125}a^{8}+\frac{8}{125}a^{7}+\frac{8}{125}a^{6}-\frac{4}{125}a^{5}-\frac{9}{125}a^{4}+\frac{41}{125}a^{3}+\frac{41}{125}a^{2}+\frac{12}{25}a+\frac{2}{5}$, $\frac{1}{125}a^{14}+\frac{2}{125}a^{10}-\frac{2}{25}a^{9}-\frac{7}{125}a^{6}-\frac{1}{25}a^{5}+\frac{2}{5}a^{4}+\frac{4}{125}a^{2}+\frac{3}{25}a-\frac{2}{5}$, $\frac{1}{625}a^{15}+\frac{2}{625}a^{14}+\frac{2}{625}a^{13}+\frac{7}{625}a^{12}-\frac{6}{625}a^{11}-\frac{4}{625}a^{10}-\frac{9}{625}a^{9}+\frac{6}{625}a^{8}-\frac{46}{625}a^{7}-\frac{28}{625}a^{6}-\frac{18}{625}a^{5}-\frac{288}{625}a^{4}+\frac{26}{625}a^{3}+\frac{51}{125}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{625}a^{16}-\frac{2}{625}a^{14}-\frac{2}{625}a^{13}+\frac{3}{625}a^{11}-\frac{6}{625}a^{10}+\frac{59}{625}a^{9}-\frac{23}{625}a^{8}+\frac{24}{625}a^{7}-\frac{2}{625}a^{6}+\frac{18}{625}a^{5}-\frac{78}{625}a^{4}-\frac{2}{625}a^{3}+\frac{57}{125}a^{2}-\frac{12}{25}a+\frac{2}{5}$, $\frac{1}{625}a^{17}+\frac{2}{625}a^{14}-\frac{1}{625}a^{13}+\frac{12}{625}a^{12}+\frac{2}{625}a^{11}-\frac{4}{625}a^{10}-\frac{6}{625}a^{9}-\frac{4}{625}a^{8}-\frac{59}{625}a^{7}+\frac{22}{625}a^{6}+\frac{31}{625}a^{5}+\frac{92}{625}a^{4}+\frac{32}{625}a^{3}-\frac{9}{125}a^{2}-\frac{2}{25}a-\frac{1}{5}$, $\frac{1}{53125}a^{18}+\frac{42}{53125}a^{17}-\frac{22}{53125}a^{16}-\frac{41}{53125}a^{15}+\frac{26}{53125}a^{14}-\frac{12}{53125}a^{13}+\frac{118}{10625}a^{12}+\frac{507}{53125}a^{11}-\frac{28}{10625}a^{10}-\frac{2937}{53125}a^{9}-\frac{4524}{53125}a^{8}-\frac{2626}{53125}a^{7}+\frac{1763}{53125}a^{6}-\frac{5268}{53125}a^{5}+\frac{23906}{53125}a^{4}-\frac{2453}{10625}a^{3}+\frac{857}{2125}a^{2}-\frac{5}{17}a-\frac{3}{17}$, $\frac{1}{53125}a^{19}-\frac{1}{53125}a^{17}+\frac{33}{53125}a^{16}-\frac{37}{53125}a^{15}+\frac{171}{53125}a^{14}-\frac{11}{53125}a^{13}-\frac{48}{53125}a^{12}-\frac{779}{53125}a^{11}-\frac{882}{53125}a^{10}-\frac{39}{625}a^{9}+\frac{3357}{53125}a^{8}+\frac{69}{2125}a^{7}+\frac{3136}{53125}a^{6}+\frac{1127}{53125}a^{5}-\frac{16292}{53125}a^{4}-\frac{1897}{10625}a^{3}-\frac{58}{425}a^{2}+\frac{194}{425}a+\frac{7}{17}$, $\frac{1}{33\!\cdots\!75}a^{20}+\frac{35\!\cdots\!63}{66\!\cdots\!75}a^{19}-\frac{19\!\cdots\!19}{33\!\cdots\!75}a^{18}-\frac{78\!\cdots\!18}{33\!\cdots\!75}a^{17}-\frac{15\!\cdots\!71}{33\!\cdots\!75}a^{16}+\frac{15\!\cdots\!34}{33\!\cdots\!75}a^{15}+\frac{73\!\cdots\!86}{33\!\cdots\!75}a^{14}+\frac{24\!\cdots\!18}{33\!\cdots\!75}a^{13}-\frac{61\!\cdots\!94}{33\!\cdots\!75}a^{12}-\frac{56\!\cdots\!33}{33\!\cdots\!75}a^{11}+\frac{12\!\cdots\!26}{13\!\cdots\!75}a^{10}+\frac{72\!\cdots\!33}{33\!\cdots\!75}a^{9}+\frac{26\!\cdots\!62}{33\!\cdots\!75}a^{8}+\frac{28\!\cdots\!19}{33\!\cdots\!75}a^{7}+\frac{23\!\cdots\!08}{33\!\cdots\!75}a^{6}+\frac{29\!\cdots\!67}{33\!\cdots\!75}a^{5}-\frac{59\!\cdots\!98}{33\!\cdots\!75}a^{4}+\frac{58\!\cdots\!63}{66\!\cdots\!75}a^{3}+\frac{44\!\cdots\!21}{13\!\cdots\!75}a^{2}-\frac{58\!\cdots\!94}{26\!\cdots\!75}a+\frac{22\!\cdots\!73}{52\!\cdots\!75}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $5$

Class group and class number

Trivial group, which has order $1$ (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:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{22\!\cdots\!74}{19\!\cdots\!75}a^{20}-\frac{18\!\cdots\!66}{77\!\cdots\!75}a^{19}-\frac{26\!\cdots\!46}{19\!\cdots\!75}a^{18}+\frac{57\!\cdots\!73}{19\!\cdots\!75}a^{17}+\frac{12\!\cdots\!21}{19\!\cdots\!75}a^{16}-\frac{28\!\cdots\!99}{19\!\cdots\!75}a^{15}-\frac{27\!\cdots\!61}{19\!\cdots\!75}a^{14}+\frac{69\!\cdots\!72}{19\!\cdots\!75}a^{13}+\frac{33\!\cdots\!24}{19\!\cdots\!75}a^{12}-\frac{89\!\cdots\!82}{19\!\cdots\!75}a^{11}-\frac{41\!\cdots\!11}{38\!\cdots\!75}a^{10}+\frac{59\!\cdots\!72}{19\!\cdots\!75}a^{9}+\frac{64\!\cdots\!53}{19\!\cdots\!75}a^{8}-\frac{21\!\cdots\!54}{19\!\cdots\!75}a^{7}-\frac{94\!\cdots\!13}{19\!\cdots\!75}a^{6}+\frac{37\!\cdots\!33}{19\!\cdots\!75}a^{5}+\frac{47\!\cdots\!53}{19\!\cdots\!75}a^{4}-\frac{64\!\cdots\!73}{38\!\cdots\!75}a^{3}+\frac{69\!\cdots\!74}{77\!\cdots\!75}a^{2}+\frac{81\!\cdots\!79}{15\!\cdots\!75}a-\frac{26\!\cdots\!73}{31\!\cdots\!75}$, $\frac{22\!\cdots\!74}{19\!\cdots\!75}a^{20}-\frac{18\!\cdots\!66}{77\!\cdots\!75}a^{19}-\frac{26\!\cdots\!46}{19\!\cdots\!75}a^{18}+\frac{57\!\cdots\!73}{19\!\cdots\!75}a^{17}+\frac{12\!\cdots\!21}{19\!\cdots\!75}a^{16}-\frac{28\!\cdots\!99}{19\!\cdots\!75}a^{15}-\frac{27\!\cdots\!61}{19\!\cdots\!75}a^{14}+\frac{69\!\cdots\!72}{19\!\cdots\!75}a^{13}+\frac{33\!\cdots\!24}{19\!\cdots\!75}a^{12}-\frac{89\!\cdots\!82}{19\!\cdots\!75}a^{11}-\frac{41\!\cdots\!11}{38\!\cdots\!75}a^{10}+\frac{59\!\cdots\!72}{19\!\cdots\!75}a^{9}+\frac{64\!\cdots\!53}{19\!\cdots\!75}a^{8}-\frac{21\!\cdots\!54}{19\!\cdots\!75}a^{7}-\frac{94\!\cdots\!13}{19\!\cdots\!75}a^{6}+\frac{37\!\cdots\!33}{19\!\cdots\!75}a^{5}+\frac{47\!\cdots\!53}{19\!\cdots\!75}a^{4}-\frac{64\!\cdots\!73}{38\!\cdots\!75}a^{3}+\frac{69\!\cdots\!74}{77\!\cdots\!75}a^{2}+\frac{81\!\cdots\!79}{15\!\cdots\!75}a-\frac{27\!\cdots\!48}{31\!\cdots\!75}$, $\frac{80\!\cdots\!39}{33\!\cdots\!75}a^{20}-\frac{32\!\cdots\!32}{66\!\cdots\!75}a^{19}-\frac{55\!\cdots\!23}{19\!\cdots\!75}a^{18}+\frac{20\!\cdots\!68}{33\!\cdots\!75}a^{17}+\frac{43\!\cdots\!46}{33\!\cdots\!75}a^{16}-\frac{10\!\cdots\!09}{33\!\cdots\!75}a^{15}-\frac{10\!\cdots\!41}{33\!\cdots\!75}a^{14}+\frac{24\!\cdots\!72}{33\!\cdots\!75}a^{13}+\frac{12\!\cdots\!69}{33\!\cdots\!75}a^{12}-\frac{32\!\cdots\!82}{33\!\cdots\!75}a^{11}-\frac{15\!\cdots\!02}{66\!\cdots\!75}a^{10}+\frac{21\!\cdots\!37}{33\!\cdots\!75}a^{9}+\frac{24\!\cdots\!53}{33\!\cdots\!75}a^{8}-\frac{75\!\cdots\!34}{33\!\cdots\!75}a^{7}-\frac{38\!\cdots\!58}{33\!\cdots\!75}a^{6}+\frac{13\!\cdots\!98}{33\!\cdots\!75}a^{5}+\frac{23\!\cdots\!93}{33\!\cdots\!75}a^{4}-\frac{13\!\cdots\!89}{38\!\cdots\!75}a^{3}+\frac{10\!\cdots\!84}{13\!\cdots\!75}a^{2}+\frac{29\!\cdots\!54}{26\!\cdots\!75}a-\frac{95\!\cdots\!58}{52\!\cdots\!75}$, $\frac{22\!\cdots\!09}{33\!\cdots\!75}a^{20}-\frac{95\!\cdots\!66}{66\!\cdots\!75}a^{19}-\frac{15\!\cdots\!63}{19\!\cdots\!75}a^{18}+\frac{60\!\cdots\!28}{33\!\cdots\!75}a^{17}+\frac{12\!\cdots\!41}{33\!\cdots\!75}a^{16}-\frac{29\!\cdots\!39}{33\!\cdots\!75}a^{15}-\frac{28\!\cdots\!91}{33\!\cdots\!75}a^{14}+\frac{73\!\cdots\!02}{33\!\cdots\!75}a^{13}+\frac{34\!\cdots\!99}{33\!\cdots\!75}a^{12}-\frac{95\!\cdots\!12}{33\!\cdots\!75}a^{11}-\frac{43\!\cdots\!84}{66\!\cdots\!75}a^{10}+\frac{64\!\cdots\!47}{33\!\cdots\!75}a^{9}+\frac{71\!\cdots\!03}{33\!\cdots\!75}a^{8}-\frac{23\!\cdots\!04}{33\!\cdots\!75}a^{7}-\frac{11\!\cdots\!18}{33\!\cdots\!75}a^{6}+\frac{44\!\cdots\!98}{33\!\cdots\!75}a^{5}+\frac{76\!\cdots\!48}{33\!\cdots\!75}a^{4}-\frac{46\!\cdots\!79}{38\!\cdots\!75}a^{3}+\frac{42\!\cdots\!34}{13\!\cdots\!75}a^{2}+\frac{10\!\cdots\!39}{26\!\cdots\!75}a-\frac{37\!\cdots\!18}{52\!\cdots\!75}$, $\frac{10\!\cdots\!56}{33\!\cdots\!75}a^{20}-\frac{41\!\cdots\!62}{66\!\cdots\!75}a^{19}-\frac{71\!\cdots\!82}{19\!\cdots\!75}a^{18}+\frac{25\!\cdots\!32}{33\!\cdots\!75}a^{17}+\frac{56\!\cdots\!09}{33\!\cdots\!75}a^{16}-\frac{12\!\cdots\!16}{33\!\cdots\!75}a^{15}-\frac{12\!\cdots\!64}{33\!\cdots\!75}a^{14}+\frac{31\!\cdots\!18}{33\!\cdots\!75}a^{13}+\frac{15\!\cdots\!36}{33\!\cdots\!75}a^{12}-\frac{40\!\cdots\!58}{33\!\cdots\!75}a^{11}-\frac{39\!\cdots\!14}{13\!\cdots\!75}a^{10}+\frac{27\!\cdots\!08}{33\!\cdots\!75}a^{9}+\frac{31\!\cdots\!67}{33\!\cdots\!75}a^{8}-\frac{97\!\cdots\!06}{33\!\cdots\!75}a^{7}-\frac{49\!\cdots\!42}{33\!\cdots\!75}a^{6}+\frac{17\!\cdots\!42}{33\!\cdots\!75}a^{5}+\frac{29\!\cdots\!82}{33\!\cdots\!75}a^{4}-\frac{17\!\cdots\!06}{38\!\cdots\!75}a^{3}+\frac{16\!\cdots\!61}{13\!\cdots\!75}a^{2}+\frac{39\!\cdots\!91}{26\!\cdots\!75}a-\frac{12\!\cdots\!67}{52\!\cdots\!75}$, $\frac{20\!\cdots\!23}{33\!\cdots\!75}a^{20}-\frac{82\!\cdots\!74}{66\!\cdots\!75}a^{19}-\frac{14\!\cdots\!11}{19\!\cdots\!75}a^{18}+\frac{51\!\cdots\!01}{33\!\cdots\!75}a^{17}+\frac{11\!\cdots\!22}{33\!\cdots\!75}a^{16}-\frac{25\!\cdots\!88}{33\!\cdots\!75}a^{15}-\frac{26\!\cdots\!62}{33\!\cdots\!75}a^{14}+\frac{63\!\cdots\!54}{33\!\cdots\!75}a^{13}+\frac{33\!\cdots\!33}{33\!\cdots\!75}a^{12}-\frac{83\!\cdots\!74}{33\!\cdots\!75}a^{11}-\frac{43\!\cdots\!99}{66\!\cdots\!75}a^{10}+\frac{57\!\cdots\!59}{33\!\cdots\!75}a^{9}+\frac{74\!\cdots\!96}{33\!\cdots\!75}a^{8}-\frac{20\!\cdots\!88}{33\!\cdots\!75}a^{7}-\frac{13\!\cdots\!31}{33\!\cdots\!75}a^{6}+\frac{38\!\cdots\!11}{33\!\cdots\!75}a^{5}+\frac{10\!\cdots\!26}{33\!\cdots\!75}a^{4}-\frac{40\!\cdots\!68}{38\!\cdots\!75}a^{3}-\frac{22\!\cdots\!62}{13\!\cdots\!75}a^{2}+\frac{91\!\cdots\!33}{26\!\cdots\!75}a-\frac{29\!\cdots\!21}{52\!\cdots\!75}$, $\frac{20\!\cdots\!47}{19\!\cdots\!75}a^{20}-\frac{86\!\cdots\!96}{38\!\cdots\!75}a^{19}-\frac{24\!\cdots\!53}{19\!\cdots\!75}a^{18}+\frac{54\!\cdots\!69}{19\!\cdots\!75}a^{17}+\frac{11\!\cdots\!03}{19\!\cdots\!75}a^{16}-\frac{26\!\cdots\!72}{19\!\cdots\!75}a^{15}-\frac{26\!\cdots\!78}{19\!\cdots\!75}a^{14}+\frac{66\!\cdots\!76}{19\!\cdots\!75}a^{13}+\frac{31\!\cdots\!37}{19\!\cdots\!75}a^{12}-\frac{85\!\cdots\!56}{19\!\cdots\!75}a^{11}-\frac{38\!\cdots\!01}{38\!\cdots\!75}a^{10}+\frac{57\!\cdots\!71}{19\!\cdots\!75}a^{9}+\frac{60\!\cdots\!09}{19\!\cdots\!75}a^{8}-\frac{20\!\cdots\!22}{19\!\cdots\!75}a^{7}-\frac{90\!\cdots\!89}{19\!\cdots\!75}a^{6}+\frac{36\!\cdots\!09}{19\!\cdots\!75}a^{5}+\frac{49\!\cdots\!04}{19\!\cdots\!75}a^{4}-\frac{62\!\cdots\!74}{38\!\cdots\!75}a^{3}+\frac{42\!\cdots\!67}{77\!\cdots\!75}a^{2}+\frac{77\!\cdots\!62}{15\!\cdots\!75}a-\frac{22\!\cdots\!79}{31\!\cdots\!75}$, $\frac{10\!\cdots\!56}{33\!\cdots\!75}a^{20}-\frac{82\!\cdots\!52}{13\!\cdots\!75}a^{19}-\frac{71\!\cdots\!82}{19\!\cdots\!75}a^{18}+\frac{26\!\cdots\!97}{33\!\cdots\!75}a^{17}+\frac{56\!\cdots\!14}{33\!\cdots\!75}a^{16}-\frac{12\!\cdots\!11}{33\!\cdots\!75}a^{15}-\frac{12\!\cdots\!29}{33\!\cdots\!75}a^{14}+\frac{31\!\cdots\!08}{33\!\cdots\!75}a^{13}+\frac{15\!\cdots\!06}{33\!\cdots\!75}a^{12}-\frac{41\!\cdots\!98}{33\!\cdots\!75}a^{11}-\frac{19\!\cdots\!29}{66\!\cdots\!75}a^{10}+\frac{27\!\cdots\!58}{33\!\cdots\!75}a^{9}+\frac{31\!\cdots\!62}{33\!\cdots\!75}a^{8}-\frac{97\!\cdots\!56}{33\!\cdots\!75}a^{7}-\frac{48\!\cdots\!07}{33\!\cdots\!75}a^{6}+\frac{17\!\cdots\!37}{33\!\cdots\!75}a^{5}+\frac{29\!\cdots\!37}{33\!\cdots\!75}a^{4}-\frac{17\!\cdots\!71}{38\!\cdots\!75}a^{3}+\frac{16\!\cdots\!76}{13\!\cdots\!75}a^{2}+\frac{38\!\cdots\!56}{26\!\cdots\!75}a-\frac{12\!\cdots\!22}{52\!\cdots\!75}$, $\frac{27\!\cdots\!77}{33\!\cdots\!75}a^{20}-\frac{12\!\cdots\!07}{66\!\cdots\!75}a^{19}-\frac{32\!\cdots\!48}{33\!\cdots\!75}a^{18}+\frac{76\!\cdots\!84}{33\!\cdots\!75}a^{17}+\frac{15\!\cdots\!58}{33\!\cdots\!75}a^{16}-\frac{37\!\cdots\!67}{33\!\cdots\!75}a^{15}-\frac{34\!\cdots\!28}{33\!\cdots\!75}a^{14}+\frac{92\!\cdots\!21}{33\!\cdots\!75}a^{13}+\frac{41\!\cdots\!32}{33\!\cdots\!75}a^{12}-\frac{11\!\cdots\!01}{33\!\cdots\!75}a^{11}-\frac{50\!\cdots\!59}{66\!\cdots\!75}a^{10}+\frac{80\!\cdots\!61}{33\!\cdots\!75}a^{9}+\frac{78\!\cdots\!59}{33\!\cdots\!75}a^{8}-\frac{28\!\cdots\!02}{33\!\cdots\!75}a^{7}-\frac{11\!\cdots\!79}{33\!\cdots\!75}a^{6}+\frac{52\!\cdots\!84}{33\!\cdots\!75}a^{5}+\frac{56\!\cdots\!74}{33\!\cdots\!75}a^{4}-\frac{90\!\cdots\!69}{66\!\cdots\!75}a^{3}+\frac{99\!\cdots\!52}{13\!\cdots\!75}a^{2}+\frac{11\!\cdots\!97}{26\!\cdots\!75}a-\frac{38\!\cdots\!99}{52\!\cdots\!75}$, $\frac{21\!\cdots\!98}{33\!\cdots\!75}a^{20}-\frac{29\!\cdots\!91}{13\!\cdots\!75}a^{19}-\frac{25\!\cdots\!62}{33\!\cdots\!75}a^{18}+\frac{89\!\cdots\!56}{33\!\cdots\!75}a^{17}+\frac{11\!\cdots\!82}{33\!\cdots\!75}a^{16}-\frac{42\!\cdots\!78}{33\!\cdots\!75}a^{15}-\frac{24\!\cdots\!92}{33\!\cdots\!75}a^{14}+\frac{60\!\cdots\!27}{19\!\cdots\!75}a^{13}+\frac{26\!\cdots\!73}{33\!\cdots\!75}a^{12}-\frac{75\!\cdots\!87}{19\!\cdots\!75}a^{11}-\frac{26\!\cdots\!02}{66\!\cdots\!75}a^{10}+\frac{85\!\cdots\!59}{33\!\cdots\!75}a^{9}+\frac{23\!\cdots\!11}{33\!\cdots\!75}a^{8}-\frac{30\!\cdots\!88}{33\!\cdots\!75}a^{7}+\frac{14\!\cdots\!89}{33\!\cdots\!75}a^{6}+\frac{59\!\cdots\!26}{33\!\cdots\!75}a^{5}-\frac{97\!\cdots\!89}{33\!\cdots\!75}a^{4}-\frac{11\!\cdots\!46}{66\!\cdots\!75}a^{3}+\frac{40\!\cdots\!78}{13\!\cdots\!75}a^{2}+\frac{16\!\cdots\!18}{26\!\cdots\!75}a-\frac{55\!\cdots\!91}{52\!\cdots\!75}$, $\frac{34\!\cdots\!81}{66\!\cdots\!75}a^{20}-\frac{70\!\cdots\!53}{66\!\cdots\!75}a^{19}-\frac{40\!\cdots\!79}{66\!\cdots\!75}a^{18}+\frac{35\!\cdots\!04}{26\!\cdots\!75}a^{17}+\frac{37\!\cdots\!27}{13\!\cdots\!75}a^{16}-\frac{87\!\cdots\!89}{13\!\cdots\!75}a^{15}-\frac{42\!\cdots\!72}{66\!\cdots\!75}a^{14}+\frac{10\!\cdots\!56}{66\!\cdots\!75}a^{13}+\frac{10\!\cdots\!11}{13\!\cdots\!75}a^{12}-\frac{13\!\cdots\!61}{66\!\cdots\!75}a^{11}-\frac{32\!\cdots\!59}{66\!\cdots\!75}a^{10}+\frac{93\!\cdots\!98}{66\!\cdots\!75}a^{9}+\frac{10\!\cdots\!46}{66\!\cdots\!75}a^{8}-\frac{33\!\cdots\!96}{66\!\cdots\!75}a^{7}-\frac{30\!\cdots\!23}{13\!\cdots\!75}a^{6}+\frac{60\!\cdots\!46}{66\!\cdots\!75}a^{5}+\frac{83\!\cdots\!58}{66\!\cdots\!75}a^{4}-\frac{10\!\cdots\!34}{13\!\cdots\!75}a^{3}+\frac{17\!\cdots\!64}{52\!\cdots\!75}a^{2}+\frac{13\!\cdots\!43}{52\!\cdots\!75}a-\frac{43\!\cdots\!42}{10\!\cdots\!35}$, $\frac{10\!\cdots\!63}{33\!\cdots\!75}a^{20}-\frac{27\!\cdots\!32}{66\!\cdots\!75}a^{19}-\frac{85\!\cdots\!47}{33\!\cdots\!75}a^{18}+\frac{15\!\cdots\!71}{33\!\cdots\!75}a^{17}+\frac{12\!\cdots\!12}{33\!\cdots\!75}a^{16}-\frac{65\!\cdots\!98}{33\!\cdots\!75}a^{15}+\frac{74\!\cdots\!63}{33\!\cdots\!75}a^{14}+\frac{13\!\cdots\!14}{33\!\cdots\!75}a^{13}-\frac{29\!\cdots\!82}{33\!\cdots\!75}a^{12}-\frac{12\!\cdots\!09}{33\!\cdots\!75}a^{11}+\frac{79\!\cdots\!32}{66\!\cdots\!75}a^{10}+\frac{21\!\cdots\!62}{19\!\cdots\!75}a^{9}-\frac{21\!\cdots\!04}{33\!\cdots\!75}a^{8}+\frac{38\!\cdots\!72}{33\!\cdots\!75}a^{7}+\frac{48\!\cdots\!24}{33\!\cdots\!75}a^{6}-\frac{29\!\cdots\!39}{33\!\cdots\!75}a^{5}-\frac{46\!\cdots\!14}{33\!\cdots\!75}a^{4}+\frac{78\!\cdots\!14}{66\!\cdots\!75}a^{3}+\frac{53\!\cdots\!78}{13\!\cdots\!75}a^{2}-\frac{12\!\cdots\!02}{26\!\cdots\!75}a+\frac{33\!\cdots\!69}{52\!\cdots\!75}$, $\frac{12\!\cdots\!86}{33\!\cdots\!75}a^{20}-\frac{63\!\cdots\!64}{38\!\cdots\!75}a^{19}-\frac{80\!\cdots\!27}{19\!\cdots\!75}a^{18}+\frac{63\!\cdots\!57}{33\!\cdots\!75}a^{17}+\frac{55\!\cdots\!54}{33\!\cdots\!75}a^{16}-\frac{29\!\cdots\!66}{33\!\cdots\!75}a^{15}-\frac{99\!\cdots\!84}{33\!\cdots\!75}a^{14}+\frac{66\!\cdots\!78}{33\!\cdots\!75}a^{13}+\frac{55\!\cdots\!81}{33\!\cdots\!75}a^{12}-\frac{74\!\cdots\!93}{33\!\cdots\!75}a^{11}+\frac{77\!\cdots\!92}{66\!\cdots\!75}a^{10}+\frac{38\!\cdots\!63}{33\!\cdots\!75}a^{9}-\frac{52\!\cdots\!78}{33\!\cdots\!75}a^{8}-\frac{45\!\cdots\!48}{19\!\cdots\!75}a^{7}+\frac{10\!\cdots\!49}{19\!\cdots\!75}a^{6}-\frac{68\!\cdots\!73}{33\!\cdots\!75}a^{5}-\frac{22\!\cdots\!43}{33\!\cdots\!75}a^{4}+\frac{25\!\cdots\!43}{66\!\cdots\!75}a^{3}+\frac{30\!\cdots\!71}{13\!\cdots\!75}a^{2}-\frac{62\!\cdots\!04}{26\!\cdots\!75}a+\frac{17\!\cdots\!73}{52\!\cdots\!75}$, $\frac{27\!\cdots\!73}{33\!\cdots\!75}a^{20}-\frac{19\!\cdots\!22}{66\!\cdots\!75}a^{19}-\frac{31\!\cdots\!32}{33\!\cdots\!75}a^{18}+\frac{11\!\cdots\!26}{33\!\cdots\!75}a^{17}+\frac{11\!\cdots\!17}{33\!\cdots\!75}a^{16}-\frac{52\!\cdots\!88}{33\!\cdots\!75}a^{15}-\frac{10\!\cdots\!97}{33\!\cdots\!75}a^{14}+\frac{12\!\cdots\!34}{33\!\cdots\!75}a^{13}-\frac{34\!\cdots\!97}{33\!\cdots\!75}a^{12}-\frac{15\!\cdots\!04}{33\!\cdots\!75}a^{11}+\frac{17\!\cdots\!82}{66\!\cdots\!75}a^{10}+\frac{10\!\cdots\!24}{33\!\cdots\!75}a^{9}-\frac{74\!\cdots\!79}{33\!\cdots\!75}a^{8}-\frac{36\!\cdots\!68}{33\!\cdots\!75}a^{7}+\frac{27\!\cdots\!69}{33\!\cdots\!75}a^{6}+\frac{73\!\cdots\!16}{33\!\cdots\!75}a^{5}-\frac{27\!\cdots\!42}{19\!\cdots\!75}a^{4}-\frac{14\!\cdots\!76}{66\!\cdots\!75}a^{3}+\frac{13\!\cdots\!93}{13\!\cdots\!75}a^{2}+\frac{23\!\cdots\!83}{26\!\cdots\!75}a-\frac{91\!\cdots\!91}{52\!\cdots\!75}$, $\frac{64\!\cdots\!14}{33\!\cdots\!75}a^{20}-\frac{27\!\cdots\!33}{66\!\cdots\!75}a^{19}-\frac{76\!\cdots\!86}{33\!\cdots\!75}a^{18}+\frac{17\!\cdots\!58}{33\!\cdots\!75}a^{17}+\frac{34\!\cdots\!96}{33\!\cdots\!75}a^{16}-\frac{84\!\cdots\!04}{33\!\cdots\!75}a^{15}-\frac{78\!\cdots\!41}{33\!\cdots\!75}a^{14}+\frac{20\!\cdots\!67}{33\!\cdots\!75}a^{13}+\frac{92\!\cdots\!84}{33\!\cdots\!75}a^{12}-\frac{26\!\cdots\!27}{33\!\cdots\!75}a^{11}-\frac{21\!\cdots\!08}{13\!\cdots\!75}a^{10}+\frac{17\!\cdots\!02}{33\!\cdots\!75}a^{9}+\frac{15\!\cdots\!98}{33\!\cdots\!75}a^{8}-\frac{36\!\cdots\!17}{19\!\cdots\!75}a^{7}-\frac{19\!\cdots\!23}{33\!\cdots\!75}a^{6}+\frac{10\!\cdots\!73}{33\!\cdots\!75}a^{5}+\frac{40\!\cdots\!08}{33\!\cdots\!75}a^{4}-\frac{18\!\cdots\!83}{66\!\cdots\!75}a^{3}+\frac{34\!\cdots\!64}{13\!\cdots\!75}a^{2}+\frac{22\!\cdots\!04}{26\!\cdots\!75}a-\frac{75\!\cdots\!33}{52\!\cdots\!75}$, $\frac{22\!\cdots\!08}{33\!\cdots\!75}a^{20}-\frac{86\!\cdots\!91}{66\!\cdots\!75}a^{19}-\frac{26\!\cdots\!92}{33\!\cdots\!75}a^{18}+\frac{54\!\cdots\!26}{33\!\cdots\!75}a^{17}+\frac{12\!\cdots\!62}{33\!\cdots\!75}a^{16}-\frac{27\!\cdots\!38}{33\!\cdots\!75}a^{15}-\frac{28\!\cdots\!02}{33\!\cdots\!75}a^{14}+\frac{67\!\cdots\!49}{33\!\cdots\!75}a^{13}+\frac{34\!\cdots\!23}{33\!\cdots\!75}a^{12}-\frac{87\!\cdots\!19}{33\!\cdots\!75}a^{11}-\frac{17\!\cdots\!32}{26\!\cdots\!75}a^{10}+\frac{58\!\cdots\!19}{33\!\cdots\!75}a^{9}+\frac{70\!\cdots\!31}{33\!\cdots\!75}a^{8}-\frac{20\!\cdots\!08}{33\!\cdots\!75}a^{7}-\frac{65\!\cdots\!93}{19\!\cdots\!75}a^{6}+\frac{37\!\cdots\!06}{33\!\cdots\!75}a^{5}+\frac{67\!\cdots\!01}{33\!\cdots\!75}a^{4}-\frac{65\!\cdots\!96}{66\!\cdots\!75}a^{3}+\frac{34\!\cdots\!28}{13\!\cdots\!75}a^{2}+\frac{84\!\cdots\!18}{26\!\cdots\!75}a-\frac{27\!\cdots\!81}{52\!\cdots\!75}$, $\frac{11\!\cdots\!34}{66\!\cdots\!75}a^{20}-\frac{22\!\cdots\!24}{66\!\cdots\!75}a^{19}-\frac{13\!\cdots\!71}{66\!\cdots\!75}a^{18}+\frac{28\!\cdots\!52}{66\!\cdots\!75}a^{17}+\frac{64\!\cdots\!29}{66\!\cdots\!75}a^{16}-\frac{14\!\cdots\!26}{66\!\cdots\!75}a^{15}-\frac{29\!\cdots\!78}{13\!\cdots\!75}a^{14}+\frac{35\!\cdots\!46}{66\!\cdots\!75}a^{13}+\frac{17\!\cdots\!91}{66\!\cdots\!75}a^{12}-\frac{45\!\cdots\!16}{66\!\cdots\!75}a^{11}-\frac{11\!\cdots\!12}{66\!\cdots\!75}a^{10}+\frac{30\!\cdots\!72}{66\!\cdots\!75}a^{9}+\frac{70\!\cdots\!18}{13\!\cdots\!75}a^{8}-\frac{10\!\cdots\!49}{66\!\cdots\!75}a^{7}-\frac{53\!\cdots\!52}{66\!\cdots\!75}a^{6}+\frac{39\!\cdots\!21}{13\!\cdots\!75}a^{5}+\frac{30\!\cdots\!06}{66\!\cdots\!75}a^{4}-\frac{34\!\cdots\!57}{13\!\cdots\!75}a^{3}+\frac{24\!\cdots\!51}{26\!\cdots\!75}a^{2}+\frac{43\!\cdots\!86}{52\!\cdots\!75}a-\frac{14\!\cdots\!36}{10\!\cdots\!35}$, $\frac{12\!\cdots\!82}{66\!\cdots\!75}a^{20}-\frac{20\!\cdots\!73}{66\!\cdots\!75}a^{19}-\frac{14\!\cdots\!18}{66\!\cdots\!75}a^{18}+\frac{26\!\cdots\!32}{66\!\cdots\!75}a^{17}+\frac{69\!\cdots\!44}{66\!\cdots\!75}a^{16}-\frac{13\!\cdots\!86}{66\!\cdots\!75}a^{15}-\frac{16\!\cdots\!26}{66\!\cdots\!75}a^{14}+\frac{32\!\cdots\!24}{66\!\cdots\!75}a^{13}+\frac{19\!\cdots\!96}{66\!\cdots\!75}a^{12}-\frac{42\!\cdots\!84}{66\!\cdots\!75}a^{11}-\frac{12\!\cdots\!94}{66\!\cdots\!75}a^{10}+\frac{28\!\cdots\!06}{66\!\cdots\!75}a^{9}+\frac{43\!\cdots\!73}{66\!\cdots\!75}a^{8}-\frac{96\!\cdots\!07}{66\!\cdots\!75}a^{7}-\frac{78\!\cdots\!12}{66\!\cdots\!75}a^{6}+\frac{16\!\cdots\!13}{66\!\cdots\!75}a^{5}+\frac{13\!\cdots\!96}{13\!\cdots\!75}a^{4}-\frac{52\!\cdots\!19}{26\!\cdots\!75}a^{3}-\frac{69\!\cdots\!87}{26\!\cdots\!75}a^{2}+\frac{28\!\cdots\!46}{52\!\cdots\!75}a-\frac{92\!\cdots\!89}{10\!\cdots\!35}$, $\frac{19\!\cdots\!69}{66\!\cdots\!75}a^{20}+\frac{11\!\cdots\!11}{66\!\cdots\!75}a^{19}-\frac{23\!\cdots\!91}{66\!\cdots\!75}a^{18}-\frac{11\!\cdots\!88}{66\!\cdots\!75}a^{17}+\frac{11\!\cdots\!99}{66\!\cdots\!75}a^{16}+\frac{48\!\cdots\!39}{66\!\cdots\!75}a^{15}-\frac{31\!\cdots\!48}{77\!\cdots\!75}a^{14}-\frac{11\!\cdots\!34}{66\!\cdots\!75}a^{13}+\frac{34\!\cdots\!46}{66\!\cdots\!75}a^{12}+\frac{18\!\cdots\!29}{66\!\cdots\!75}a^{11}-\frac{25\!\cdots\!97}{66\!\cdots\!75}a^{10}-\frac{12\!\cdots\!44}{38\!\cdots\!75}a^{9}+\frac{22\!\cdots\!11}{13\!\cdots\!75}a^{8}+\frac{12\!\cdots\!26}{66\!\cdots\!75}a^{7}-\frac{27\!\cdots\!57}{66\!\cdots\!75}a^{6}-\frac{79\!\cdots\!16}{13\!\cdots\!75}a^{5}+\frac{37\!\cdots\!56}{66\!\cdots\!75}a^{4}+\frac{11\!\cdots\!39}{13\!\cdots\!75}a^{3}-\frac{16\!\cdots\!62}{52\!\cdots\!75}a^{2}-\frac{19\!\cdots\!43}{52\!\cdots\!75}a+\frac{72\!\cdots\!73}{10\!\cdots\!35}$, $\frac{55\!\cdots\!47}{19\!\cdots\!75}a^{20}-\frac{37\!\cdots\!57}{66\!\cdots\!75}a^{19}-\frac{11\!\cdots\!31}{33\!\cdots\!75}a^{18}+\frac{23\!\cdots\!88}{33\!\cdots\!75}a^{17}+\frac{51\!\cdots\!36}{33\!\cdots\!75}a^{16}-\frac{11\!\cdots\!19}{33\!\cdots\!75}a^{15}-\frac{11\!\cdots\!31}{33\!\cdots\!75}a^{14}+\frac{28\!\cdots\!27}{33\!\cdots\!75}a^{13}+\frac{14\!\cdots\!54}{33\!\cdots\!75}a^{12}-\frac{37\!\cdots\!62}{33\!\cdots\!75}a^{11}-\frac{17\!\cdots\!07}{66\!\cdots\!75}a^{10}+\frac{25\!\cdots\!42}{33\!\cdots\!75}a^{9}+\frac{28\!\cdots\!98}{33\!\cdots\!75}a^{8}-\frac{88\!\cdots\!94}{33\!\cdots\!75}a^{7}-\frac{25\!\cdots\!84}{19\!\cdots\!75}a^{6}+\frac{16\!\cdots\!68}{33\!\cdots\!75}a^{5}+\frac{24\!\cdots\!13}{33\!\cdots\!75}a^{4}-\frac{27\!\cdots\!18}{66\!\cdots\!75}a^{3}+\frac{11\!\cdots\!37}{77\!\cdots\!75}a^{2}+\frac{35\!\cdots\!54}{26\!\cdots\!75}a-\frac{11\!\cdots\!18}{52\!\cdots\!75}$ 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:  \( 30398950792448647000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 30398950792448647000 \cdot 1}{2\cdot\sqrt{8276699911115832752222429511299889027510277198729}}\cr\approx \mathstrut & 11.0797485154435 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375)
 
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^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375, 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^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375);
 
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^21 - 4*x^20 - 114*x^19 + 488*x^18 + 4946*x^17 - 23362*x^16 - 100260*x^15 + 557394*x^14 + 893184*x^13 - 6986867*x^12 - 1446343*x^11 + 45520578*x^10 - 23590055*x^9 - 154124034*x^8 + 143234027*x^7 + 261729665*x^6 - 315720051*x^5 - 199051958*x^4 + 300106465*x^3 + 36273150*x^2 - 102144250*x + 15389375);
 
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

$C_{21}$ (as 21T1):

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)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.169.1, 7.7.128100283921.1

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 $21$ $21$ ${\href{/padicField/5.1.0.1}{1} }^{21}$ $21$ $21$ R ${\href{/padicField/17.3.0.1}{3} }^{7}$ $21$ $21$ $21$ ${\href{/padicField/31.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/padicField/47.7.0.1}{7} }^{3}$ ${\href{/padicField/53.7.0.1}{7} }^{3}$ $21$

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
\(13\) Copy content Toggle raw display 13.21.14.1$x^{21} + 91 x^{18} + 3558 x^{15} + 33 x^{14} + 71162 x^{12} - 33033 x^{11} + 1138073 x^{9} + 1991187 x^{8} + 363 x^{7} + 9603867 x^{6} - 19222203 x^{5} + 165165 x^{4} + 21288112 x^{3} + 26273544 x^{2} + 430518 x + 66151155$$3$$7$$14$$C_{21}$$[\ ]_{3}^{7}$
\(71\) Copy content Toggle raw display 71.21.18.1$x^{21} + 28 x^{19} + 448 x^{18} + 336 x^{17} + 10752 x^{16} + 88256 x^{15} + 107733 x^{14} + 1729280 x^{13} + 9738540 x^{12} + 12734400 x^{11} + 148624336 x^{10} + 647370752 x^{9} + 1195115712 x^{8} + 7139768083 x^{7} + 25022002432 x^{6} + 13389865944 x^{5} + 183749980736 x^{4} + 503460963936 x^{3} + 461875672832 x^{2} + 1914119021696 x + 4431958413847$$7$$3$$18$$C_{21}$$[\ ]_{7}^{3}$