Properties

Label 21.21.106...961.1
Degree $21$
Signature $[21, 0]$
Discriminant $1.068\times 10^{46}$
Root discriminant \(155.54\)
Ramified primes $3,43$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_{21}$ (as 21T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701)
 
gp: K = bnfinit(y^21 - 129*y^19 - 86*y^18 + 6579*y^17 + 7998*y^16 - 169635*y^15 - 282897*y^14 + 2351541*y^13 + 4822837*y^12 - 17262522*y^11 - 42117210*y^10 + 60860609*y^9 + 187574256*y^8 - 66235308*y^7 - 399918748*y^6 - 118626723*y^5 + 303277839*y^4 + 245278579*y^3 + 53399808*y^2 - 44247*y - 722701, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701)
 

\( x^{21} - 129 x^{19} - 86 x^{18} + 6579 x^{17} + 7998 x^{16} - 169635 x^{15} - 282897 x^{14} + 2351541 x^{13} + 4822837 x^{12} - 17262522 x^{11} - 42117210 x^{10} + \cdots - 722701 \) 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:   \(10684747015052975538074582285998174444374188961\) \(\medspace = 3^{28}\cdot 43^{20}\) 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:  \(155.54\)
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:  $3^{4/3}43^{20/21}\approx 155.5413687430282$
Ramified primes:   \(3\), \(43\) 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:  \(387=3^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{387}(64,·)$, $\chi_{387}(1,·)$, $\chi_{387}(196,·)$, $\chi_{387}(262,·)$, $\chi_{387}(145,·)$, $\chi_{387}(268,·)$, $\chi_{387}(13,·)$, $\chi_{387}(337,·)$, $\chi_{387}(283,·)$, $\chi_{387}(160,·)$, $\chi_{387}(226,·)$, $\chi_{387}(229,·)$, $\chi_{387}(103,·)$, $\chi_{387}(169,·)$, $\chi_{387}(367,·)$, $\chi_{387}(178,·)$, $\chi_{387}(310,·)$, $\chi_{387}(58,·)$, $\chi_{387}(379,·)$, $\chi_{387}(124,·)$, $\chi_{387}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{3}{7}a$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{3}$, $\frac{1}{7}a^{10}-\frac{1}{7}a^{4}$, $\frac{1}{7}a^{11}-\frac{1}{7}a^{5}$, $\frac{1}{49}a^{12}+\frac{3}{49}a^{11}-\frac{2}{49}a^{10}+\frac{3}{49}a^{9}+\frac{3}{49}a^{8}-\frac{3}{49}a^{7}-\frac{2}{49}a^{6}-\frac{22}{49}a^{5}-\frac{16}{49}a^{4}-\frac{2}{49}a^{3}+\frac{16}{49}a^{2}+\frac{3}{7}a$, $\frac{1}{49}a^{13}+\frac{3}{49}a^{11}+\frac{2}{49}a^{10}+\frac{1}{49}a^{9}+\frac{2}{49}a^{8}-\frac{2}{49}a^{6}+\frac{8}{49}a^{5}+\frac{11}{49}a^{4}+\frac{1}{49}a^{3}-\frac{13}{49}a^{2}-\frac{2}{7}a$, $\frac{1}{49}a^{14}-\frac{2}{49}a^{8}+\frac{1}{49}a^{2}$, $\frac{1}{343}a^{15}+\frac{1}{343}a^{14}+\frac{3}{343}a^{13}-\frac{2}{343}a^{12}+\frac{24}{343}a^{11}-\frac{11}{343}a^{10}-\frac{5}{343}a^{9}-\frac{9}{343}a^{8}-\frac{1}{343}a^{7}-\frac{9}{343}a^{6}-\frac{135}{343}a^{5}-\frac{89}{343}a^{4}+\frac{162}{343}a^{3}+\frac{3}{49}a^{2}-\frac{3}{7}a$, $\frac{1}{27097}a^{16}+\frac{10}{27097}a^{15}-\frac{142}{27097}a^{14}+\frac{193}{27097}a^{13}+\frac{188}{27097}a^{12}-\frac{1734}{27097}a^{11}+\frac{358}{27097}a^{10}-\frac{1202}{27097}a^{9}-\frac{117}{27097}a^{8}+\frac{1788}{27097}a^{7}-\frac{1553}{27097}a^{6}+\frac{3043}{27097}a^{5}-\frac{968}{27097}a^{4}-\frac{5136}{27097}a^{3}+\frac{375}{3871}a^{2}+\frac{31}{79}a+\frac{3}{79}$, $\frac{1}{189679}a^{17}-\frac{1}{189679}a^{16}+\frac{64}{189679}a^{15}-\frac{141}{189679}a^{14}+\frac{254}{27097}a^{13}-\frac{10}{189679}a^{12}+\frac{2131}{189679}a^{11}-\frac{8063}{189679}a^{10}+\frac{465}{189679}a^{9}-\frac{1626}{27097}a^{8}-\frac{7712}{189679}a^{7}-\frac{8709}{189679}a^{6}+\frac{52854}{189679}a^{5}-\frac{59110}{189679}a^{4}+\frac{4936}{27097}a^{3}+\frac{1219}{3871}a^{2}-\frac{22}{553}a-\frac{16}{79}$, $\frac{1}{189679}a^{18}-\frac{22}{27097}a^{15}-\frac{477}{189679}a^{14}-\frac{990}{189679}a^{13}+\frac{16}{3871}a^{12}+\frac{12065}{189679}a^{11}+\frac{10217}{189679}a^{10}-\frac{3763}{189679}a^{9}+\frac{2655}{189679}a^{8}-\frac{1875}{189679}a^{7}-\frac{10091}{189679}a^{6}-\frac{86812}{189679}a^{5}-\frac{32146}{189679}a^{4}-\frac{7141}{27097}a^{3}-\frac{19}{3871}a^{2}-\frac{33}{553}a+\frac{36}{79}$, $\frac{1}{1327753}a^{19}-\frac{2}{1327753}a^{18}-\frac{1}{189679}a^{16}+\frac{1301}{1327753}a^{15}+\frac{10058}{1327753}a^{14}+\frac{167}{1327753}a^{13}-\frac{8319}{1327753}a^{12}-\frac{67519}{1327753}a^{11}+\frac{59397}{1327753}a^{10}-\frac{65867}{1327753}a^{9}-\frac{43739}{1327753}a^{8}-\frac{10604}{1327753}a^{7}-\frac{58790}{1327753}a^{6}-\frac{181530}{1327753}a^{5}+\frac{30720}{1327753}a^{4}-\frac{80988}{189679}a^{3}-\frac{4168}{27097}a^{2}+\frac{1873}{3871}a+\frac{10}{79}$, $\frac{1}{23\!\cdots\!63}a^{20}+\frac{18\!\cdots\!36}{23\!\cdots\!63}a^{19}-\frac{41\!\cdots\!38}{23\!\cdots\!63}a^{18}-\frac{70\!\cdots\!92}{33\!\cdots\!09}a^{17}+\frac{70\!\cdots\!02}{23\!\cdots\!63}a^{16}+\frac{42\!\cdots\!92}{23\!\cdots\!63}a^{15}+\frac{15\!\cdots\!19}{23\!\cdots\!63}a^{14}+\frac{54\!\cdots\!41}{23\!\cdots\!63}a^{13}-\frac{18\!\cdots\!69}{48\!\cdots\!87}a^{12}-\frac{13\!\cdots\!05}{23\!\cdots\!63}a^{11}-\frac{42\!\cdots\!55}{23\!\cdots\!63}a^{10}-\frac{24\!\cdots\!05}{33\!\cdots\!09}a^{9}-\frac{14\!\cdots\!52}{23\!\cdots\!63}a^{8}-\frac{94\!\cdots\!47}{23\!\cdots\!63}a^{7}+\frac{58\!\cdots\!66}{23\!\cdots\!63}a^{6}-\frac{48\!\cdots\!92}{23\!\cdots\!63}a^{5}-\frac{27\!\cdots\!83}{23\!\cdots\!63}a^{4}+\frac{10\!\cdots\!13}{33\!\cdots\!09}a^{3}-\frac{19\!\cdots\!37}{48\!\cdots\!87}a^{2}+\frac{34\!\cdots\!34}{69\!\cdots\!41}a-\frac{39\!\cdots\!76}{14\!\cdots\!09}$ 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:  $7$

Class group and class number

$C_{3}$, which has order $3$ (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:   $\frac{29\!\cdots\!27}{23\!\cdots\!63}a^{20}-\frac{70\!\cdots\!33}{23\!\cdots\!63}a^{19}-\frac{38\!\cdots\!92}{23\!\cdots\!63}a^{18}-\frac{23\!\cdots\!72}{33\!\cdots\!09}a^{17}+\frac{19\!\cdots\!95}{23\!\cdots\!63}a^{16}+\frac{19\!\cdots\!46}{23\!\cdots\!63}a^{15}-\frac{51\!\cdots\!23}{23\!\cdots\!63}a^{14}-\frac{72\!\cdots\!92}{23\!\cdots\!63}a^{13}+\frac{14\!\cdots\!55}{48\!\cdots\!87}a^{12}+\frac{12\!\cdots\!03}{23\!\cdots\!63}a^{11}-\frac{54\!\cdots\!47}{23\!\cdots\!63}a^{10}-\frac{16\!\cdots\!20}{33\!\cdots\!09}a^{9}+\frac{20\!\cdots\!98}{23\!\cdots\!63}a^{8}+\frac{51\!\cdots\!84}{23\!\cdots\!63}a^{7}-\frac{31\!\cdots\!31}{23\!\cdots\!63}a^{6}-\frac{11\!\cdots\!06}{23\!\cdots\!63}a^{5}-\frac{92\!\cdots\!62}{23\!\cdots\!63}a^{4}+\frac{13\!\cdots\!56}{33\!\cdots\!09}a^{3}+\frac{10\!\cdots\!67}{48\!\cdots\!87}a^{2}+\frac{11\!\cdots\!61}{69\!\cdots\!41}a-\frac{54\!\cdots\!73}{14\!\cdots\!09}$, $\frac{72\!\cdots\!63}{23\!\cdots\!63}a^{20}-\frac{15\!\cdots\!13}{23\!\cdots\!63}a^{19}-\frac{93\!\cdots\!94}{23\!\cdots\!63}a^{18}-\frac{60\!\cdots\!06}{33\!\cdots\!09}a^{17}+\frac{47\!\cdots\!83}{23\!\cdots\!63}a^{16}+\frac{47\!\cdots\!28}{23\!\cdots\!63}a^{15}-\frac{12\!\cdots\!44}{23\!\cdots\!63}a^{14}-\frac{17\!\cdots\!13}{23\!\cdots\!63}a^{13}+\frac{35\!\cdots\!36}{48\!\cdots\!87}a^{12}+\frac{30\!\cdots\!11}{23\!\cdots\!63}a^{11}-\frac{13\!\cdots\!55}{23\!\cdots\!63}a^{10}-\frac{39\!\cdots\!60}{33\!\cdots\!09}a^{9}+\frac{51\!\cdots\!18}{23\!\cdots\!63}a^{8}+\frac{12\!\cdots\!36}{23\!\cdots\!63}a^{7}-\frac{81\!\cdots\!74}{23\!\cdots\!63}a^{6}-\frac{27\!\cdots\!81}{23\!\cdots\!63}a^{5}-\frac{96\!\cdots\!12}{23\!\cdots\!63}a^{4}+\frac{32\!\cdots\!66}{33\!\cdots\!09}a^{3}+\frac{22\!\cdots\!78}{48\!\cdots\!87}a^{2}+\frac{19\!\cdots\!47}{69\!\cdots\!41}a-\frac{11\!\cdots\!18}{14\!\cdots\!09}$, $\frac{10\!\cdots\!50}{23\!\cdots\!63}a^{20}+\frac{91\!\cdots\!59}{23\!\cdots\!63}a^{19}-\frac{14\!\cdots\!80}{23\!\cdots\!63}a^{18}-\frac{29\!\cdots\!34}{33\!\cdots\!09}a^{17}+\frac{72\!\cdots\!19}{23\!\cdots\!63}a^{16}+\frac{14\!\cdots\!96}{23\!\cdots\!63}a^{15}-\frac{18\!\cdots\!11}{23\!\cdots\!63}a^{14}-\frac{45\!\cdots\!59}{23\!\cdots\!63}a^{13}+\frac{53\!\cdots\!48}{48\!\cdots\!87}a^{12}+\frac{74\!\cdots\!86}{23\!\cdots\!63}a^{11}-\frac{19\!\cdots\!94}{23\!\cdots\!63}a^{10}-\frac{91\!\cdots\!02}{33\!\cdots\!09}a^{9}+\frac{64\!\cdots\!00}{23\!\cdots\!63}a^{8}+\frac{28\!\cdots\!18}{23\!\cdots\!63}a^{7}-\frac{49\!\cdots\!11}{23\!\cdots\!63}a^{6}-\frac{61\!\cdots\!38}{23\!\cdots\!63}a^{5}-\frac{21\!\cdots\!97}{23\!\cdots\!63}a^{4}+\frac{70\!\cdots\!55}{33\!\cdots\!09}a^{3}+\frac{77\!\cdots\!83}{48\!\cdots\!87}a^{2}+\frac{16\!\cdots\!94}{69\!\cdots\!41}a-\frac{44\!\cdots\!88}{14\!\cdots\!09}$, $\frac{17\!\cdots\!02}{23\!\cdots\!63}a^{20}-\frac{44\!\cdots\!67}{23\!\cdots\!63}a^{19}-\frac{22\!\cdots\!99}{23\!\cdots\!63}a^{18}-\frac{13\!\cdots\!09}{33\!\cdots\!09}a^{17}+\frac{11\!\cdots\!17}{23\!\cdots\!63}a^{16}+\frac{10\!\cdots\!86}{23\!\cdots\!63}a^{15}-\frac{29\!\cdots\!29}{23\!\cdots\!63}a^{14}-\frac{41\!\cdots\!51}{23\!\cdots\!63}a^{13}+\frac{85\!\cdots\!33}{48\!\cdots\!87}a^{12}+\frac{72\!\cdots\!97}{23\!\cdots\!63}a^{11}-\frac{31\!\cdots\!01}{23\!\cdots\!63}a^{10}-\frac{92\!\cdots\!35}{33\!\cdots\!09}a^{9}+\frac{12\!\cdots\!21}{23\!\cdots\!63}a^{8}+\frac{29\!\cdots\!71}{23\!\cdots\!63}a^{7}-\frac{19\!\cdots\!12}{23\!\cdots\!63}a^{6}-\frac{64\!\cdots\!70}{23\!\cdots\!63}a^{5}-\frac{36\!\cdots\!52}{23\!\cdots\!63}a^{4}+\frac{77\!\cdots\!32}{33\!\cdots\!09}a^{3}+\frac{58\!\cdots\!58}{48\!\cdots\!87}a^{2}+\frac{48\!\cdots\!62}{69\!\cdots\!41}a-\frac{27\!\cdots\!90}{14\!\cdots\!09}$, $\frac{13\!\cdots\!56}{23\!\cdots\!63}a^{20}-\frac{32\!\cdots\!75}{23\!\cdots\!63}a^{19}-\frac{17\!\cdots\!67}{23\!\cdots\!63}a^{18}-\frac{10\!\cdots\!73}{33\!\cdots\!09}a^{17}+\frac{87\!\cdots\!86}{23\!\cdots\!63}a^{16}+\frac{84\!\cdots\!69}{23\!\cdots\!63}a^{15}-\frac{22\!\cdots\!23}{23\!\cdots\!63}a^{14}-\frac{32\!\cdots\!76}{23\!\cdots\!63}a^{13}+\frac{65\!\cdots\!52}{48\!\cdots\!87}a^{12}+\frac{56\!\cdots\!91}{23\!\cdots\!63}a^{11}-\frac{24\!\cdots\!16}{23\!\cdots\!63}a^{10}-\frac{71\!\cdots\!29}{33\!\cdots\!09}a^{9}+\frac{94\!\cdots\!38}{23\!\cdots\!63}a^{8}+\frac{22\!\cdots\!91}{23\!\cdots\!63}a^{7}-\frac{14\!\cdots\!52}{23\!\cdots\!63}a^{6}-\frac{50\!\cdots\!13}{23\!\cdots\!63}a^{5}-\frac{32\!\cdots\!90}{23\!\cdots\!63}a^{4}+\frac{59\!\cdots\!55}{33\!\cdots\!09}a^{3}+\frac{45\!\cdots\!68}{48\!\cdots\!87}a^{2}+\frac{36\!\cdots\!58}{69\!\cdots\!41}a-\frac{30\!\cdots\!22}{14\!\cdots\!09}$, $\frac{98\!\cdots\!71}{23\!\cdots\!63}a^{20}-\frac{18\!\cdots\!16}{23\!\cdots\!63}a^{19}-\frac{12\!\cdots\!78}{23\!\cdots\!63}a^{18}-\frac{85\!\cdots\!61}{33\!\cdots\!09}a^{17}+\frac{64\!\cdots\!51}{23\!\cdots\!63}a^{16}+\frac{65\!\cdots\!16}{23\!\cdots\!63}a^{15}-\frac{16\!\cdots\!92}{23\!\cdots\!63}a^{14}-\frac{24\!\cdots\!47}{23\!\cdots\!63}a^{13}+\frac{48\!\cdots\!77}{48\!\cdots\!87}a^{12}+\frac{42\!\cdots\!99}{23\!\cdots\!63}a^{11}-\frac{17\!\cdots\!41}{23\!\cdots\!63}a^{10}-\frac{53\!\cdots\!31}{33\!\cdots\!09}a^{9}+\frac{68\!\cdots\!59}{23\!\cdots\!63}a^{8}+\frac{16\!\cdots\!64}{23\!\cdots\!63}a^{7}-\frac{10\!\cdots\!68}{23\!\cdots\!63}a^{6}-\frac{36\!\cdots\!25}{23\!\cdots\!63}a^{5}-\frac{25\!\cdots\!49}{23\!\cdots\!63}a^{4}+\frac{43\!\cdots\!37}{33\!\cdots\!09}a^{3}+\frac{33\!\cdots\!60}{48\!\cdots\!87}a^{2}+\frac{28\!\cdots\!83}{69\!\cdots\!41}a-\frac{15\!\cdots\!80}{14\!\cdots\!09}$, $\frac{15\!\cdots\!76}{23\!\cdots\!63}a^{20}-\frac{11\!\cdots\!80}{23\!\cdots\!63}a^{19}-\frac{20\!\cdots\!72}{23\!\cdots\!63}a^{18}+\frac{12\!\cdots\!40}{33\!\cdots\!09}a^{17}+\frac{10\!\cdots\!20}{23\!\cdots\!63}a^{16}+\frac{53\!\cdots\!17}{23\!\cdots\!63}a^{15}-\frac{27\!\cdots\!62}{23\!\cdots\!63}a^{14}-\frac{25\!\cdots\!59}{23\!\cdots\!63}a^{13}+\frac{80\!\cdots\!90}{48\!\cdots\!87}a^{12}+\frac{49\!\cdots\!33}{23\!\cdots\!63}a^{11}-\frac{31\!\cdots\!70}{23\!\cdots\!63}a^{10}-\frac{66\!\cdots\!54}{33\!\cdots\!09}a^{9}+\frac{13\!\cdots\!76}{23\!\cdots\!63}a^{8}+\frac{22\!\cdots\!39}{23\!\cdots\!63}a^{7}-\frac{25\!\cdots\!77}{23\!\cdots\!63}a^{6}-\frac{50\!\cdots\!42}{23\!\cdots\!63}a^{5}+\frac{11\!\cdots\!19}{23\!\cdots\!63}a^{4}+\frac{62\!\cdots\!62}{33\!\cdots\!09}a^{3}+\frac{32\!\cdots\!01}{48\!\cdots\!87}a^{2}+\frac{20\!\cdots\!12}{69\!\cdots\!41}a-\frac{13\!\cdots\!33}{14\!\cdots\!09}$, $\frac{19\!\cdots\!06}{23\!\cdots\!63}a^{20}-\frac{48\!\cdots\!84}{23\!\cdots\!63}a^{19}-\frac{24\!\cdots\!10}{23\!\cdots\!63}a^{18}-\frac{14\!\cdots\!60}{33\!\cdots\!09}a^{17}+\frac{12\!\cdots\!24}{23\!\cdots\!63}a^{16}+\frac{12\!\cdots\!40}{23\!\cdots\!63}a^{15}-\frac{32\!\cdots\!50}{23\!\cdots\!63}a^{14}-\frac{46\!\cdots\!37}{23\!\cdots\!63}a^{13}+\frac{94\!\cdots\!93}{48\!\cdots\!87}a^{12}+\frac{81\!\cdots\!88}{23\!\cdots\!63}a^{11}-\frac{35\!\cdots\!44}{23\!\cdots\!63}a^{10}-\frac{10\!\cdots\!39}{33\!\cdots\!09}a^{9}+\frac{13\!\cdots\!84}{23\!\cdots\!63}a^{8}+\frac{32\!\cdots\!55}{23\!\cdots\!63}a^{7}-\frac{20\!\cdots\!98}{23\!\cdots\!63}a^{6}-\frac{72\!\cdots\!16}{23\!\cdots\!63}a^{5}-\frac{66\!\cdots\!88}{23\!\cdots\!63}a^{4}+\frac{86\!\cdots\!91}{33\!\cdots\!09}a^{3}+\frac{69\!\cdots\!00}{48\!\cdots\!87}a^{2}+\frac{83\!\cdots\!90}{69\!\cdots\!41}a-\frac{33\!\cdots\!98}{14\!\cdots\!09}$, $\frac{13\!\cdots\!14}{23\!\cdots\!63}a^{20}-\frac{29\!\cdots\!49}{23\!\cdots\!63}a^{19}-\frac{17\!\cdots\!11}{23\!\cdots\!63}a^{18}-\frac{10\!\cdots\!73}{33\!\cdots\!09}a^{17}+\frac{87\!\cdots\!71}{23\!\cdots\!63}a^{16}+\frac{86\!\cdots\!69}{23\!\cdots\!63}a^{15}-\frac{22\!\cdots\!15}{23\!\cdots\!63}a^{14}-\frac{32\!\cdots\!91}{23\!\cdots\!63}a^{13}+\frac{65\!\cdots\!84}{48\!\cdots\!87}a^{12}+\frac{56\!\cdots\!06}{23\!\cdots\!63}a^{11}-\frac{24\!\cdots\!00}{23\!\cdots\!63}a^{10}-\frac{72\!\cdots\!61}{33\!\cdots\!09}a^{9}+\frac{92\!\cdots\!53}{23\!\cdots\!63}a^{8}+\frac{22\!\cdots\!10}{23\!\cdots\!63}a^{7}-\frac{14\!\cdots\!79}{23\!\cdots\!63}a^{6}-\frac{49\!\cdots\!28}{23\!\cdots\!63}a^{5}-\frac{40\!\cdots\!77}{23\!\cdots\!63}a^{4}+\frac{58\!\cdots\!94}{33\!\cdots\!09}a^{3}+\frac{46\!\cdots\!25}{48\!\cdots\!87}a^{2}+\frac{51\!\cdots\!37}{69\!\cdots\!41}a-\frac{23\!\cdots\!68}{14\!\cdots\!09}$, $\frac{20\!\cdots\!59}{23\!\cdots\!63}a^{20}-\frac{67\!\cdots\!98}{23\!\cdots\!63}a^{19}-\frac{25\!\cdots\!46}{23\!\cdots\!63}a^{18}-\frac{12\!\cdots\!58}{33\!\cdots\!09}a^{17}+\frac{13\!\cdots\!85}{23\!\cdots\!63}a^{16}+\frac{11\!\cdots\!08}{23\!\cdots\!63}a^{15}-\frac{34\!\cdots\!90}{23\!\cdots\!63}a^{14}-\frac{45\!\cdots\!59}{23\!\cdots\!63}a^{13}+\frac{98\!\cdots\!73}{48\!\cdots\!87}a^{12}+\frac{80\!\cdots\!53}{23\!\cdots\!63}a^{11}-\frac{36\!\cdots\!13}{23\!\cdots\!63}a^{10}-\frac{10\!\cdots\!88}{33\!\cdots\!09}a^{9}+\frac{14\!\cdots\!76}{23\!\cdots\!63}a^{8}+\frac{32\!\cdots\!92}{23\!\cdots\!63}a^{7}-\frac{22\!\cdots\!00}{23\!\cdots\!63}a^{6}-\frac{71\!\cdots\!96}{23\!\cdots\!63}a^{5}-\frac{24\!\cdots\!85}{23\!\cdots\!63}a^{4}+\frac{84\!\cdots\!75}{33\!\cdots\!09}a^{3}+\frac{63\!\cdots\!67}{48\!\cdots\!87}a^{2}+\frac{62\!\cdots\!77}{69\!\cdots\!41}a-\frac{28\!\cdots\!96}{14\!\cdots\!09}$, $\frac{58\!\cdots\!45}{23\!\cdots\!63}a^{20}-\frac{10\!\cdots\!04}{23\!\cdots\!63}a^{19}-\frac{75\!\cdots\!60}{23\!\cdots\!63}a^{18}-\frac{52\!\cdots\!58}{33\!\cdots\!09}a^{17}+\frac{38\!\cdots\!14}{23\!\cdots\!63}a^{16}+\frac{39\!\cdots\!18}{23\!\cdots\!63}a^{15}-\frac{10\!\cdots\!97}{23\!\cdots\!63}a^{14}-\frac{14\!\cdots\!32}{23\!\cdots\!63}a^{13}+\frac{28\!\cdots\!72}{48\!\cdots\!87}a^{12}+\frac{25\!\cdots\!05}{23\!\cdots\!63}a^{11}-\frac{10\!\cdots\!85}{23\!\cdots\!63}a^{10}-\frac{32\!\cdots\!64}{33\!\cdots\!09}a^{9}+\frac{40\!\cdots\!31}{23\!\cdots\!63}a^{8}+\frac{10\!\cdots\!76}{23\!\cdots\!63}a^{7}-\frac{61\!\cdots\!59}{23\!\cdots\!63}a^{6}-\frac{22\!\cdots\!01}{23\!\cdots\!63}a^{5}-\frac{19\!\cdots\!13}{23\!\cdots\!63}a^{4}+\frac{26\!\cdots\!17}{33\!\cdots\!09}a^{3}+\frac{21\!\cdots\!31}{48\!\cdots\!87}a^{2}+\frac{24\!\cdots\!02}{69\!\cdots\!41}a-\frac{83\!\cdots\!73}{14\!\cdots\!09}$, $\frac{63\!\cdots\!69}{33\!\cdots\!09}a^{20}-\frac{13\!\cdots\!89}{33\!\cdots\!09}a^{19}-\frac{81\!\cdots\!81}{33\!\cdots\!09}a^{18}-\frac{52\!\cdots\!17}{48\!\cdots\!87}a^{17}+\frac{41\!\cdots\!75}{33\!\cdots\!09}a^{16}+\frac{41\!\cdots\!08}{33\!\cdots\!09}a^{15}-\frac{10\!\cdots\!68}{33\!\cdots\!09}a^{14}-\frac{15\!\cdots\!39}{33\!\cdots\!09}a^{13}+\frac{30\!\cdots\!72}{69\!\cdots\!41}a^{12}+\frac{27\!\cdots\!88}{33\!\cdots\!09}a^{11}-\frac{11\!\cdots\!72}{33\!\cdots\!09}a^{10}-\frac{34\!\cdots\!75}{48\!\cdots\!87}a^{9}+\frac{43\!\cdots\!68}{33\!\cdots\!09}a^{8}+\frac{10\!\cdots\!70}{33\!\cdots\!09}a^{7}-\frac{65\!\cdots\!77}{33\!\cdots\!09}a^{6}-\frac{23\!\cdots\!06}{33\!\cdots\!09}a^{5}-\frac{20\!\cdots\!74}{33\!\cdots\!09}a^{4}+\frac{27\!\cdots\!25}{48\!\cdots\!87}a^{3}+\frac{21\!\cdots\!23}{69\!\cdots\!41}a^{2}+\frac{22\!\cdots\!26}{98\!\cdots\!63}a-\frac{12\!\cdots\!59}{20\!\cdots\!87}$, $\frac{14\!\cdots\!01}{29\!\cdots\!97}a^{20}-\frac{38\!\cdots\!19}{29\!\cdots\!97}a^{19}-\frac{14\!\cdots\!56}{23\!\cdots\!63}a^{18}-\frac{84\!\cdots\!16}{33\!\cdots\!09}a^{17}+\frac{76\!\cdots\!90}{23\!\cdots\!63}a^{16}+\frac{71\!\cdots\!06}{23\!\cdots\!63}a^{15}-\frac{19\!\cdots\!31}{23\!\cdots\!63}a^{14}-\frac{27\!\cdots\!47}{23\!\cdots\!63}a^{13}+\frac{57\!\cdots\!59}{48\!\cdots\!87}a^{12}+\frac{48\!\cdots\!02}{23\!\cdots\!63}a^{11}-\frac{21\!\cdots\!34}{23\!\cdots\!63}a^{10}-\frac{62\!\cdots\!46}{33\!\cdots\!09}a^{9}+\frac{85\!\cdots\!16}{23\!\cdots\!63}a^{8}+\frac{20\!\cdots\!41}{23\!\cdots\!63}a^{7}-\frac{13\!\cdots\!27}{23\!\cdots\!63}a^{6}-\frac{44\!\cdots\!24}{23\!\cdots\!63}a^{5}-\frac{17\!\cdots\!04}{23\!\cdots\!63}a^{4}+\frac{54\!\cdots\!80}{33\!\cdots\!09}a^{3}+\frac{41\!\cdots\!05}{48\!\cdots\!87}a^{2}+\frac{42\!\cdots\!51}{69\!\cdots\!41}a-\frac{20\!\cdots\!69}{14\!\cdots\!09}$, $\frac{25\!\cdots\!56}{48\!\cdots\!87}a^{20}-\frac{13\!\cdots\!79}{48\!\cdots\!87}a^{19}-\frac{32\!\cdots\!96}{48\!\cdots\!87}a^{18}-\frac{70\!\cdots\!56}{69\!\cdots\!41}a^{17}+\frac{16\!\cdots\!16}{48\!\cdots\!87}a^{16}+\frac{11\!\cdots\!81}{48\!\cdots\!87}a^{15}-\frac{43\!\cdots\!52}{48\!\cdots\!87}a^{14}-\frac{48\!\cdots\!59}{48\!\cdots\!87}a^{13}+\frac{12\!\cdots\!68}{98\!\cdots\!63}a^{12}+\frac{89\!\cdots\!27}{48\!\cdots\!87}a^{11}-\frac{46\!\cdots\!25}{48\!\cdots\!87}a^{10}-\frac{11\!\cdots\!25}{69\!\cdots\!41}a^{9}+\frac{18\!\cdots\!88}{48\!\cdots\!87}a^{8}+\frac{36\!\cdots\!57}{48\!\cdots\!87}a^{7}-\frac{31\!\cdots\!34}{48\!\cdots\!87}a^{6}-\frac{78\!\cdots\!89}{48\!\cdots\!87}a^{5}+\frac{29\!\cdots\!20}{48\!\cdots\!87}a^{4}+\frac{90\!\cdots\!38}{69\!\cdots\!41}a^{3}+\frac{67\!\cdots\!46}{98\!\cdots\!63}a^{2}+\frac{11\!\cdots\!16}{14\!\cdots\!09}a-\frac{10\!\cdots\!83}{28\!\cdots\!41}$, $\frac{11\!\cdots\!00}{23\!\cdots\!63}a^{20}-\frac{50\!\cdots\!41}{23\!\cdots\!63}a^{19}-\frac{14\!\cdots\!07}{23\!\cdots\!63}a^{18}-\frac{50\!\cdots\!02}{33\!\cdots\!09}a^{17}+\frac{76\!\cdots\!49}{23\!\cdots\!63}a^{16}+\frac{60\!\cdots\!94}{23\!\cdots\!63}a^{15}-\frac{20\!\cdots\!21}{23\!\cdots\!63}a^{14}-\frac{24\!\cdots\!64}{23\!\cdots\!63}a^{13}+\frac{58\!\cdots\!19}{48\!\cdots\!87}a^{12}+\frac{44\!\cdots\!77}{23\!\cdots\!63}a^{11}-\frac{22\!\cdots\!65}{23\!\cdots\!63}a^{10}-\frac{58\!\cdots\!67}{33\!\cdots\!09}a^{9}+\frac{88\!\cdots\!96}{23\!\cdots\!63}a^{8}+\frac{18\!\cdots\!80}{23\!\cdots\!63}a^{7}-\frac{14\!\cdots\!53}{23\!\cdots\!63}a^{6}-\frac{42\!\cdots\!84}{23\!\cdots\!63}a^{5}+\frac{65\!\cdots\!21}{29\!\cdots\!97}a^{4}+\frac{50\!\cdots\!81}{33\!\cdots\!09}a^{3}+\frac{36\!\cdots\!89}{48\!\cdots\!87}a^{2}+\frac{42\!\cdots\!25}{69\!\cdots\!41}a-\frac{13\!\cdots\!55}{14\!\cdots\!09}$, $\frac{40\!\cdots\!92}{23\!\cdots\!63}a^{20}-\frac{14\!\cdots\!88}{23\!\cdots\!63}a^{19}-\frac{52\!\cdots\!39}{23\!\cdots\!63}a^{18}-\frac{22\!\cdots\!73}{33\!\cdots\!09}a^{17}+\frac{26\!\cdots\!89}{23\!\cdots\!63}a^{16}+\frac{22\!\cdots\!50}{23\!\cdots\!63}a^{15}-\frac{69\!\cdots\!23}{23\!\cdots\!63}a^{14}-\frac{90\!\cdots\!38}{23\!\cdots\!63}a^{13}+\frac{20\!\cdots\!52}{48\!\cdots\!87}a^{12}+\frac{16\!\cdots\!74}{23\!\cdots\!63}a^{11}-\frac{75\!\cdots\!47}{23\!\cdots\!63}a^{10}-\frac{20\!\cdots\!79}{33\!\cdots\!09}a^{9}+\frac{29\!\cdots\!63}{23\!\cdots\!63}a^{8}+\frac{67\!\cdots\!89}{23\!\cdots\!63}a^{7}-\frac{47\!\cdots\!29}{23\!\cdots\!63}a^{6}-\frac{14\!\cdots\!29}{23\!\cdots\!63}a^{5}-\frac{65\!\cdots\!61}{23\!\cdots\!63}a^{4}+\frac{18\!\cdots\!15}{33\!\cdots\!09}a^{3}+\frac{13\!\cdots\!91}{48\!\cdots\!87}a^{2}+\frac{13\!\cdots\!30}{69\!\cdots\!41}a-\frac{74\!\cdots\!45}{14\!\cdots\!09}$, $\frac{81\!\cdots\!78}{23\!\cdots\!63}a^{20}-\frac{44\!\cdots\!11}{23\!\cdots\!63}a^{19}-\frac{10\!\cdots\!60}{23\!\cdots\!63}a^{18}+\frac{70\!\cdots\!02}{33\!\cdots\!09}a^{17}+\frac{52\!\cdots\!52}{23\!\cdots\!63}a^{16}-\frac{22\!\cdots\!42}{23\!\cdots\!63}a^{15}-\frac{14\!\cdots\!06}{23\!\cdots\!63}a^{14}+\frac{51\!\cdots\!38}{23\!\cdots\!63}a^{13}+\frac{52\!\cdots\!32}{48\!\cdots\!87}a^{12}-\frac{67\!\cdots\!72}{23\!\cdots\!63}a^{11}-\frac{26\!\cdots\!55}{23\!\cdots\!63}a^{10}+\frac{71\!\cdots\!75}{33\!\cdots\!09}a^{9}+\frac{16\!\cdots\!34}{23\!\cdots\!63}a^{8}-\frac{19\!\cdots\!55}{23\!\cdots\!63}a^{7}-\frac{61\!\cdots\!50}{23\!\cdots\!63}a^{6}+\frac{34\!\cdots\!67}{23\!\cdots\!63}a^{5}+\frac{11\!\cdots\!30}{23\!\cdots\!63}a^{4}-\frac{11\!\cdots\!77}{33\!\cdots\!09}a^{3}-\frac{18\!\cdots\!52}{48\!\cdots\!87}a^{2}-\frac{95\!\cdots\!40}{69\!\cdots\!41}a-\frac{14\!\cdots\!74}{14\!\cdots\!09}$, $\frac{36\!\cdots\!37}{23\!\cdots\!63}a^{20}-\frac{24\!\cdots\!17}{23\!\cdots\!63}a^{19}-\frac{46\!\cdots\!33}{23\!\cdots\!63}a^{18}-\frac{39\!\cdots\!47}{33\!\cdots\!09}a^{17}+\frac{23\!\cdots\!08}{23\!\cdots\!63}a^{16}+\frac{27\!\cdots\!76}{23\!\cdots\!63}a^{15}-\frac{61\!\cdots\!26}{23\!\cdots\!63}a^{14}-\frac{97\!\cdots\!02}{23\!\cdots\!63}a^{13}+\frac{17\!\cdots\!92}{48\!\cdots\!87}a^{12}+\frac{16\!\cdots\!49}{23\!\cdots\!63}a^{11}-\frac{63\!\cdots\!43}{23\!\cdots\!63}a^{10}-\frac{20\!\cdots\!10}{33\!\cdots\!09}a^{9}+\frac{23\!\cdots\!67}{23\!\cdots\!63}a^{8}+\frac{62\!\cdots\!54}{23\!\cdots\!63}a^{7}-\frac{32\!\cdots\!17}{23\!\cdots\!63}a^{6}-\frac{13\!\cdots\!61}{23\!\cdots\!63}a^{5}-\frac{15\!\cdots\!34}{23\!\cdots\!63}a^{4}+\frac{14\!\cdots\!45}{33\!\cdots\!09}a^{3}+\frac{11\!\cdots\!61}{48\!\cdots\!87}a^{2}+\frac{93\!\cdots\!37}{69\!\cdots\!41}a-\frac{61\!\cdots\!54}{14\!\cdots\!09}$, $\frac{14\!\cdots\!85}{23\!\cdots\!63}a^{20}-\frac{57\!\cdots\!38}{23\!\cdots\!63}a^{19}-\frac{19\!\cdots\!79}{23\!\cdots\!63}a^{18}-\frac{78\!\cdots\!58}{33\!\cdots\!09}a^{17}+\frac{98\!\cdots\!75}{23\!\cdots\!63}a^{16}+\frac{82\!\cdots\!28}{23\!\cdots\!63}a^{15}-\frac{25\!\cdots\!12}{23\!\cdots\!63}a^{14}-\frac{32\!\cdots\!07}{23\!\cdots\!63}a^{13}+\frac{73\!\cdots\!18}{48\!\cdots\!87}a^{12}+\frac{58\!\cdots\!24}{23\!\cdots\!63}a^{11}-\frac{27\!\cdots\!44}{23\!\cdots\!63}a^{10}-\frac{74\!\cdots\!97}{33\!\cdots\!09}a^{9}+\frac{10\!\cdots\!12}{23\!\cdots\!63}a^{8}+\frac{23\!\cdots\!81}{23\!\cdots\!63}a^{7}-\frac{16\!\cdots\!17}{23\!\cdots\!63}a^{6}-\frac{51\!\cdots\!43}{23\!\cdots\!63}a^{5}-\frac{19\!\cdots\!89}{23\!\cdots\!63}a^{4}+\frac{61\!\cdots\!54}{33\!\cdots\!09}a^{3}+\frac{46\!\cdots\!35}{48\!\cdots\!87}a^{2}+\frac{51\!\cdots\!23}{69\!\cdots\!41}a-\frac{22\!\cdots\!02}{14\!\cdots\!09}$, $\frac{11\!\cdots\!62}{23\!\cdots\!63}a^{20}+\frac{10\!\cdots\!24}{23\!\cdots\!63}a^{19}-\frac{15\!\cdots\!97}{23\!\cdots\!63}a^{18}-\frac{16\!\cdots\!43}{33\!\cdots\!09}a^{17}+\frac{79\!\cdots\!87}{23\!\cdots\!63}a^{16}+\frac{10\!\cdots\!02}{23\!\cdots\!63}a^{15}-\frac{20\!\cdots\!79}{23\!\cdots\!63}a^{14}-\frac{35\!\cdots\!48}{23\!\cdots\!63}a^{13}+\frac{59\!\cdots\!79}{48\!\cdots\!87}a^{12}+\frac{60\!\cdots\!30}{23\!\cdots\!63}a^{11}-\frac{22\!\cdots\!71}{23\!\cdots\!63}a^{10}-\frac{75\!\cdots\!57}{33\!\cdots\!09}a^{9}+\frac{83\!\cdots\!32}{23\!\cdots\!63}a^{8}+\frac{23\!\cdots\!52}{23\!\cdots\!63}a^{7}-\frac{12\!\cdots\!83}{23\!\cdots\!63}a^{6}-\frac{52\!\cdots\!00}{23\!\cdots\!63}a^{5}-\frac{64\!\cdots\!51}{23\!\cdots\!63}a^{4}+\frac{62\!\cdots\!64}{33\!\cdots\!09}a^{3}+\frac{49\!\cdots\!74}{48\!\cdots\!87}a^{2}+\frac{44\!\cdots\!92}{69\!\cdots\!41}a-\frac{25\!\cdots\!46}{14\!\cdots\!09}$ 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:  \( 104145016649537420 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 104145016649537420 \cdot 3}{2\cdot\sqrt{10684747015052975538074582285998174444374188961}}\cr\approx \mathstrut & 3.16940354879221 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701)
 
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 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701, 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 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701);
 
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 - 129*x^19 - 86*x^18 + 6579*x^17 + 7998*x^16 - 169635*x^15 - 282897*x^14 + 2351541*x^13 + 4822837*x^12 - 17262522*x^11 - 42117210*x^10 + 60860609*x^9 + 187574256*x^8 - 66235308*x^7 - 399918748*x^6 - 118626723*x^5 + 303277839*x^4 + 245278579*x^3 + 53399808*x^2 - 44247*x - 722701);
 
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

3.3.149769.1, 7.7.6321363049.1

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 $21$ R ${\href{/padicField/5.7.0.1}{7} }^{3}$ ${\href{/padicField/7.1.0.1}{1} }^{21}$ $21$ $21$ $21$ $21$ ${\href{/padicField/23.7.0.1}{7} }^{3}$ ${\href{/padicField/29.7.0.1}{7} }^{3}$ ${\href{/padicField/31.7.0.1}{7} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ $21$ R $21$ $21$ $21$

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
\(3\) Copy content Toggle raw display Deg $21$$3$$7$$28$
\(43\) Copy content Toggle raw display 43.21.20.15$x^{21} + 258$$21$$1$$20$$C_{21}$$[\ ]_{21}$