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Properties

Label 21.1.990...104.1
Degree $21$
Signature $[1, 10]$
Discriminant $9.901\times 10^{27}$
Root discriminant \(21.53\)
Ramified primes $2,11,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{21}:C_6$ (as 21T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38)
 
gp: K = bnfinit(y^21 - 3*y^20 - 2*y^19 + 25*y^18 - 40*y^17 - 24*y^16 + 156*y^15 - 215*y^14 + 131*y^13 - 57*y^12 + 294*y^11 - 907*y^10 + 1374*y^9 - 1171*y^8 + 439*y^7 + 387*y^6 - 847*y^5 + 669*y^4 - 179*y^3 - 142*y^2 + 156*y - 38, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38)
 

\( x^{21} - 3 x^{20} - 2 x^{19} + 25 x^{18} - 40 x^{17} - 24 x^{16} + 156 x^{15} - 215 x^{14} + 131 x^{13} + \cdots - 38 \) 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:  $[1, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(9900725472657912423302447104\) \(\medspace = 2^{14}\cdot 11^{10}\cdot 13^{12}\) 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:  \(21.53\)
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:  $2^{2/3}11^{1/2}13^{2/3}\approx 29.107969279297173$
Ramified primes:   \(2\), \(11\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{10}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{14}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{8}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{39}a^{19}-\frac{2}{13}a^{17}-\frac{2}{13}a^{16}+\frac{5}{39}a^{15}-\frac{11}{39}a^{14}-\frac{1}{39}a^{13}-\frac{6}{13}a^{12}+\frac{1}{13}a^{11}+\frac{11}{39}a^{10}-\frac{10}{39}a^{9}-\frac{2}{13}a^{8}-\frac{8}{39}a^{7}-\frac{14}{39}a^{6}+\frac{1}{3}a^{5}+\frac{14}{39}a^{4}-\frac{4}{13}a^{3}-\frac{8}{39}a^{2}+\frac{14}{39}a-\frac{16}{39}$, $\frac{1}{54\!\cdots\!09}a^{20}-\frac{47\!\cdots\!37}{18\!\cdots\!03}a^{19}+\frac{63\!\cdots\!19}{54\!\cdots\!09}a^{18}-\frac{36\!\cdots\!18}{54\!\cdots\!09}a^{17}-\frac{48\!\cdots\!36}{54\!\cdots\!09}a^{16}+\frac{35\!\cdots\!57}{18\!\cdots\!03}a^{15}+\frac{81\!\cdots\!91}{18\!\cdots\!03}a^{14}-\frac{25\!\cdots\!87}{18\!\cdots\!03}a^{13}+\frac{10\!\cdots\!40}{54\!\cdots\!09}a^{12}-\frac{13\!\cdots\!96}{41\!\cdots\!93}a^{11}+\frac{26\!\cdots\!91}{54\!\cdots\!09}a^{10}+\frac{26\!\cdots\!90}{54\!\cdots\!09}a^{9}-\frac{16\!\cdots\!95}{54\!\cdots\!09}a^{8}-\frac{12\!\cdots\!77}{54\!\cdots\!09}a^{7}+\frac{25\!\cdots\!45}{54\!\cdots\!09}a^{6}-\frac{21\!\cdots\!54}{54\!\cdots\!09}a^{5}-\frac{39\!\cdots\!10}{18\!\cdots\!03}a^{4}+\frac{19\!\cdots\!51}{54\!\cdots\!09}a^{3}-\frac{16\!\cdots\!62}{41\!\cdots\!93}a^{2}-\frac{26\!\cdots\!52}{54\!\cdots\!09}a-\frac{21\!\cdots\!84}{54\!\cdots\!09}$ 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

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:  $10$
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:   $\frac{13\!\cdots\!20}{13\!\cdots\!31}a^{20}-\frac{16\!\cdots\!05}{41\!\cdots\!93}a^{19}-\frac{38\!\cdots\!86}{41\!\cdots\!93}a^{18}+\frac{81\!\cdots\!56}{41\!\cdots\!93}a^{17}+\frac{81\!\cdots\!84}{41\!\cdots\!93}a^{16}-\frac{50\!\cdots\!06}{41\!\cdots\!93}a^{15}+\frac{15\!\cdots\!52}{13\!\cdots\!31}a^{14}+\frac{17\!\cdots\!75}{13\!\cdots\!31}a^{13}-\frac{56\!\cdots\!37}{13\!\cdots\!31}a^{12}+\frac{18\!\cdots\!95}{41\!\cdots\!93}a^{11}-\frac{50\!\cdots\!11}{41\!\cdots\!93}a^{10}+\frac{36\!\cdots\!50}{41\!\cdots\!93}a^{9}-\frac{46\!\cdots\!54}{41\!\cdots\!93}a^{8}+\frac{11\!\cdots\!72}{41\!\cdots\!93}a^{7}-\frac{49\!\cdots\!83}{13\!\cdots\!31}a^{6}+\frac{13\!\cdots\!24}{41\!\cdots\!93}a^{5}-\frac{74\!\cdots\!89}{41\!\cdots\!93}a^{4}-\frac{33\!\cdots\!52}{13\!\cdots\!31}a^{3}+\frac{42\!\cdots\!68}{41\!\cdots\!93}a^{2}-\frac{13\!\cdots\!95}{13\!\cdots\!31}a+\frac{12\!\cdots\!63}{41\!\cdots\!93}$, $\frac{16\!\cdots\!93}{18\!\cdots\!03}a^{20}-\frac{12\!\cdots\!94}{54\!\cdots\!09}a^{19}-\frac{50\!\cdots\!46}{18\!\cdots\!03}a^{18}+\frac{38\!\cdots\!49}{18\!\cdots\!03}a^{17}-\frac{14\!\cdots\!92}{54\!\cdots\!09}a^{16}-\frac{56\!\cdots\!66}{18\!\cdots\!03}a^{15}+\frac{66\!\cdots\!48}{54\!\cdots\!09}a^{14}-\frac{77\!\cdots\!55}{54\!\cdots\!09}a^{13}+\frac{12\!\cdots\!35}{18\!\cdots\!03}a^{12}-\frac{17\!\cdots\!90}{41\!\cdots\!93}a^{11}+\frac{14\!\cdots\!21}{54\!\cdots\!09}a^{10}-\frac{38\!\cdots\!35}{54\!\cdots\!09}a^{9}+\frac{51\!\cdots\!73}{54\!\cdots\!09}a^{8}-\frac{39\!\cdots\!78}{54\!\cdots\!09}a^{7}+\frac{11\!\cdots\!28}{54\!\cdots\!09}a^{6}+\frac{57\!\cdots\!92}{18\!\cdots\!03}a^{5}-\frac{99\!\cdots\!14}{18\!\cdots\!03}a^{4}+\frac{19\!\cdots\!64}{54\!\cdots\!09}a^{3}-\frac{21\!\cdots\!67}{41\!\cdots\!93}a^{2}-\frac{61\!\cdots\!15}{54\!\cdots\!09}a+\frac{41\!\cdots\!81}{54\!\cdots\!09}$, $\frac{47\!\cdots\!27}{54\!\cdots\!09}a^{20}-\frac{72\!\cdots\!05}{54\!\cdots\!09}a^{19}-\frac{72\!\cdots\!01}{18\!\cdots\!03}a^{18}+\frac{89\!\cdots\!42}{54\!\cdots\!09}a^{17}-\frac{52\!\cdots\!80}{54\!\cdots\!09}a^{16}-\frac{22\!\cdots\!77}{54\!\cdots\!09}a^{15}+\frac{45\!\cdots\!63}{54\!\cdots\!09}a^{14}-\frac{31\!\cdots\!10}{54\!\cdots\!09}a^{13}-\frac{15\!\cdots\!20}{41\!\cdots\!93}a^{12}-\frac{43\!\cdots\!96}{54\!\cdots\!09}a^{11}+\frac{11\!\cdots\!23}{54\!\cdots\!09}a^{10}-\frac{82\!\cdots\!86}{18\!\cdots\!03}a^{9}+\frac{78\!\cdots\!03}{18\!\cdots\!03}a^{8}-\frac{83\!\cdots\!62}{54\!\cdots\!09}a^{7}-\frac{95\!\cdots\!38}{54\!\cdots\!09}a^{6}+\frac{72\!\cdots\!66}{18\!\cdots\!03}a^{5}-\frac{18\!\cdots\!80}{54\!\cdots\!09}a^{4}+\frac{55\!\cdots\!23}{54\!\cdots\!09}a^{3}+\frac{13\!\cdots\!41}{18\!\cdots\!03}a^{2}-\frac{19\!\cdots\!91}{18\!\cdots\!03}a+\frac{56\!\cdots\!73}{18\!\cdots\!03}$, $\frac{34\!\cdots\!02}{54\!\cdots\!09}a^{20}-\frac{89\!\cdots\!02}{54\!\cdots\!09}a^{19}-\frac{29\!\cdots\!28}{18\!\cdots\!03}a^{18}+\frac{79\!\cdots\!98}{54\!\cdots\!09}a^{17}-\frac{11\!\cdots\!81}{54\!\cdots\!09}a^{16}-\frac{94\!\cdots\!67}{54\!\cdots\!09}a^{15}+\frac{15\!\cdots\!96}{18\!\cdots\!03}a^{14}-\frac{20\!\cdots\!76}{18\!\cdots\!03}a^{13}+\frac{38\!\cdots\!25}{54\!\cdots\!09}a^{12}-\frac{89\!\cdots\!43}{18\!\cdots\!03}a^{11}+\frac{35\!\cdots\!59}{18\!\cdots\!03}a^{10}-\frac{95\!\cdots\!03}{18\!\cdots\!03}a^{9}+\frac{40\!\cdots\!72}{54\!\cdots\!09}a^{8}-\frac{35\!\cdots\!17}{54\!\cdots\!09}a^{7}+\frac{17\!\cdots\!67}{54\!\cdots\!09}a^{6}+\frac{49\!\cdots\!95}{54\!\cdots\!09}a^{5}-\frac{18\!\cdots\!02}{54\!\cdots\!09}a^{4}+\frac{14\!\cdots\!64}{54\!\cdots\!09}a^{3}-\frac{14\!\cdots\!10}{18\!\cdots\!03}a^{2}-\frac{20\!\cdots\!25}{54\!\cdots\!09}a+\frac{22\!\cdots\!29}{54\!\cdots\!09}$, $\frac{38\!\cdots\!42}{18\!\cdots\!03}a^{20}-\frac{21\!\cdots\!31}{54\!\cdots\!09}a^{19}-\frac{15\!\cdots\!25}{18\!\cdots\!03}a^{18}+\frac{77\!\cdots\!42}{18\!\cdots\!03}a^{17}-\frac{19\!\cdots\!58}{54\!\cdots\!09}a^{16}-\frac{46\!\cdots\!93}{54\!\cdots\!09}a^{15}+\frac{12\!\cdots\!65}{54\!\cdots\!09}a^{14}-\frac{11\!\cdots\!38}{54\!\cdots\!09}a^{13}+\frac{12\!\cdots\!96}{18\!\cdots\!03}a^{12}-\frac{40\!\cdots\!98}{54\!\cdots\!09}a^{11}+\frac{10\!\cdots\!75}{18\!\cdots\!03}a^{10}-\frac{54\!\cdots\!40}{41\!\cdots\!93}a^{9}+\frac{81\!\cdots\!30}{54\!\cdots\!09}a^{8}-\frac{52\!\cdots\!99}{54\!\cdots\!09}a^{7}+\frac{13\!\cdots\!80}{18\!\cdots\!03}a^{6}+\frac{12\!\cdots\!20}{18\!\cdots\!03}a^{5}-\frac{47\!\cdots\!92}{54\!\cdots\!09}a^{4}+\frac{80\!\cdots\!88}{18\!\cdots\!03}a^{3}+\frac{50\!\cdots\!47}{18\!\cdots\!03}a^{2}-\frac{11\!\cdots\!24}{54\!\cdots\!09}a+\frac{13\!\cdots\!63}{18\!\cdots\!03}$, $\frac{17\!\cdots\!85}{54\!\cdots\!09}a^{20}-\frac{17\!\cdots\!08}{54\!\cdots\!09}a^{19}-\frac{27\!\cdots\!74}{18\!\cdots\!03}a^{18}+\frac{28\!\cdots\!50}{54\!\cdots\!09}a^{17}-\frac{31\!\cdots\!90}{18\!\cdots\!03}a^{16}-\frac{76\!\cdots\!10}{54\!\cdots\!09}a^{15}+\frac{13\!\cdots\!34}{54\!\cdots\!09}a^{14}-\frac{95\!\cdots\!47}{54\!\cdots\!09}a^{13}-\frac{70\!\cdots\!21}{41\!\cdots\!93}a^{12}-\frac{13\!\cdots\!28}{54\!\cdots\!09}a^{11}+\frac{35\!\cdots\!79}{54\!\cdots\!09}a^{10}-\frac{73\!\cdots\!71}{54\!\cdots\!09}a^{9}+\frac{67\!\cdots\!81}{54\!\cdots\!09}a^{8}-\frac{19\!\cdots\!53}{54\!\cdots\!09}a^{7}-\frac{14\!\cdots\!12}{18\!\cdots\!03}a^{6}+\frac{83\!\cdots\!91}{54\!\cdots\!09}a^{5}-\frac{64\!\cdots\!13}{54\!\cdots\!09}a^{4}+\frac{23\!\cdots\!91}{54\!\cdots\!09}a^{3}+\frac{17\!\cdots\!00}{54\!\cdots\!09}a^{2}-\frac{69\!\cdots\!04}{18\!\cdots\!03}a+\frac{34\!\cdots\!43}{18\!\cdots\!03}$, $\frac{33\!\cdots\!33}{68\!\cdots\!81}a^{20}-\frac{23\!\cdots\!80}{20\!\cdots\!43}a^{19}-\frac{13\!\cdots\!10}{68\!\cdots\!81}a^{18}+\frac{80\!\cdots\!74}{68\!\cdots\!81}a^{17}-\frac{79\!\cdots\!15}{68\!\cdots\!81}a^{16}-\frac{51\!\cdots\!36}{20\!\cdots\!43}a^{15}+\frac{47\!\cdots\!96}{68\!\cdots\!81}a^{14}-\frac{11\!\cdots\!40}{20\!\cdots\!43}a^{13}-\frac{23\!\cdots\!99}{53\!\cdots\!37}a^{12}+\frac{12\!\cdots\!34}{68\!\cdots\!81}a^{11}+\frac{80\!\cdots\!22}{68\!\cdots\!81}a^{10}-\frac{69\!\cdots\!90}{20\!\cdots\!43}a^{9}+\frac{25\!\cdots\!31}{68\!\cdots\!81}a^{8}-\frac{75\!\cdots\!80}{68\!\cdots\!81}a^{7}-\frac{14\!\cdots\!68}{68\!\cdots\!81}a^{6}+\frac{83\!\cdots\!58}{20\!\cdots\!43}a^{5}-\frac{26\!\cdots\!02}{68\!\cdots\!81}a^{4}+\frac{31\!\cdots\!06}{20\!\cdots\!43}a^{3}+\frac{48\!\cdots\!04}{68\!\cdots\!81}a^{2}-\frac{24\!\cdots\!00}{20\!\cdots\!43}a+\frac{30\!\cdots\!63}{68\!\cdots\!81}$, $\frac{36\!\cdots\!50}{18\!\cdots\!03}a^{20}+\frac{37\!\cdots\!62}{18\!\cdots\!03}a^{19}-\frac{96\!\cdots\!34}{54\!\cdots\!09}a^{18}+\frac{90\!\cdots\!45}{54\!\cdots\!09}a^{17}+\frac{32\!\cdots\!85}{41\!\cdots\!93}a^{16}-\frac{91\!\cdots\!53}{54\!\cdots\!09}a^{15}-\frac{16\!\cdots\!91}{54\!\cdots\!09}a^{14}+\frac{20\!\cdots\!27}{54\!\cdots\!09}a^{13}-\frac{23\!\cdots\!82}{54\!\cdots\!09}a^{12}+\frac{32\!\cdots\!07}{54\!\cdots\!09}a^{11}+\frac{16\!\cdots\!59}{54\!\cdots\!09}a^{10}+\frac{16\!\cdots\!25}{54\!\cdots\!09}a^{9}-\frac{27\!\cdots\!60}{13\!\cdots\!31}a^{8}+\frac{51\!\cdots\!69}{18\!\cdots\!03}a^{7}-\frac{11\!\cdots\!32}{54\!\cdots\!09}a^{6}+\frac{15\!\cdots\!60}{18\!\cdots\!03}a^{5}+\frac{39\!\cdots\!31}{54\!\cdots\!09}a^{4}-\frac{25\!\cdots\!39}{18\!\cdots\!03}a^{3}+\frac{46\!\cdots\!65}{54\!\cdots\!09}a^{2}+\frac{14\!\cdots\!70}{41\!\cdots\!93}a-\frac{16\!\cdots\!09}{54\!\cdots\!09}$, $\frac{26\!\cdots\!78}{54\!\cdots\!09}a^{20}-\frac{20\!\cdots\!34}{18\!\cdots\!03}a^{19}-\frac{44\!\cdots\!81}{18\!\cdots\!03}a^{18}+\frac{23\!\cdots\!48}{18\!\cdots\!03}a^{17}-\frac{19\!\cdots\!45}{18\!\cdots\!03}a^{16}-\frac{19\!\cdots\!88}{54\!\cdots\!09}a^{15}+\frac{11\!\cdots\!96}{13\!\cdots\!31}a^{14}-\frac{87\!\cdots\!66}{18\!\cdots\!03}a^{13}-\frac{12\!\cdots\!68}{18\!\cdots\!03}a^{12}+\frac{58\!\cdots\!47}{54\!\cdots\!09}a^{11}+\frac{36\!\cdots\!67}{54\!\cdots\!09}a^{10}-\frac{50\!\cdots\!18}{18\!\cdots\!03}a^{9}+\frac{11\!\cdots\!13}{54\!\cdots\!09}a^{8}+\frac{15\!\cdots\!57}{54\!\cdots\!09}a^{7}-\frac{44\!\cdots\!24}{54\!\cdots\!09}a^{6}+\frac{51\!\cdots\!40}{54\!\cdots\!09}a^{5}-\frac{36\!\cdots\!33}{54\!\cdots\!09}a^{4}+\frac{11\!\cdots\!06}{54\!\cdots\!09}a^{3}+\frac{15\!\cdots\!53}{54\!\cdots\!09}a^{2}-\frac{19\!\cdots\!19}{54\!\cdots\!09}a+\frac{58\!\cdots\!17}{18\!\cdots\!03}$, $\frac{17\!\cdots\!89}{41\!\cdots\!93}a^{20}-\frac{46\!\cdots\!85}{54\!\cdots\!09}a^{19}-\frac{19\!\cdots\!78}{13\!\cdots\!31}a^{18}+\frac{15\!\cdots\!70}{18\!\cdots\!03}a^{17}-\frac{16\!\cdots\!22}{18\!\cdots\!03}a^{16}-\frac{81\!\cdots\!90}{54\!\cdots\!09}a^{15}+\frac{26\!\cdots\!49}{54\!\cdots\!09}a^{14}-\frac{93\!\cdots\!16}{18\!\cdots\!03}a^{13}+\frac{42\!\cdots\!08}{18\!\cdots\!03}a^{12}-\frac{12\!\cdots\!81}{54\!\cdots\!09}a^{11}+\frac{21\!\cdots\!82}{18\!\cdots\!03}a^{10}-\frac{15\!\cdots\!88}{54\!\cdots\!09}a^{9}+\frac{19\!\cdots\!08}{54\!\cdots\!09}a^{8}-\frac{14\!\cdots\!48}{54\!\cdots\!09}a^{7}+\frac{16\!\cdots\!28}{18\!\cdots\!03}a^{6}+\frac{37\!\cdots\!79}{41\!\cdots\!93}a^{5}-\frac{91\!\cdots\!88}{54\!\cdots\!09}a^{4}+\frac{20\!\cdots\!65}{18\!\cdots\!03}a^{3}-\frac{59\!\cdots\!46}{54\!\cdots\!09}a^{2}-\frac{65\!\cdots\!42}{18\!\cdots\!03}a+\frac{32\!\cdots\!77}{54\!\cdots\!09}$ 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:  \( 637026.16318 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 637026.16318 \cdot 1}{2\cdot\sqrt{9900725472657912423302447104}}\cr\approx \mathstrut & 0.61393507198 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38)
 
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 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38, 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 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38);
 
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 - 3*x^20 - 2*x^19 + 25*x^18 - 40*x^17 - 24*x^16 + 156*x^15 - 215*x^14 + 131*x^13 - 57*x^12 + 294*x^11 - 907*x^10 + 1374*x^9 - 1171*x^8 + 439*x^7 + 387*x^6 - 847*x^5 + 669*x^4 - 179*x^3 - 142*x^2 + 156*x - 38);
 
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

$C_{21}:C_6$ (as 21T10):

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 solvable group of order 126
The 12 conjugacy class representatives for $C_{21}:C_6$
Character table for $C_{21}:C_6$

Intermediate fields

3.1.44.1, 7.1.38014691.1

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]
 

Sibling fields

Degree 42 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{7}$ $21$ ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ R R ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ $21$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.7.0.1}{7} }^{3}$ ${\href{/padicField/53.7.0.1}{7} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{7}$

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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.18.12.1$x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$$3$$6$$12$$S_3 \times C_3$$[\ ]_{3}^{6}$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$