Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 - 11*x^19 - 5*x^18 + 22*x^17 + 22*x^16 + 62*x^15 - 83*x^14 - 148*x^13 + 44*x^12 + 72*x^11 + 52*x^10 + 25*x^9 + 50*x^8 - 75*x^7 - 65*x^6 + 60*x^5 - 17*x^3 + 17*x^2 - 7*x + 1)
gp: K = bnfinit(y^22 - y^21 + y^20 - 11*y^19 - 5*y^18 + 22*y^17 + 22*y^16 + 62*y^15 - 83*y^14 - 148*y^13 + 44*y^12 + 72*y^11 + 52*y^10 + 25*y^9 + 50*y^8 - 75*y^7 - 65*y^6 + 60*y^5 - 17*y^3 + 17*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 + x^20 - 11*x^19 - 5*x^18 + 22*x^17 + 22*x^16 + 62*x^15 - 83*x^14 - 148*x^13 + 44*x^12 + 72*x^11 + 52*x^10 + 25*x^9 + 50*x^8 - 75*x^7 - 65*x^6 + 60*x^5 - 17*x^3 + 17*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 + x^20 - 11*x^19 - 5*x^18 + 22*x^17 + 22*x^16 + 62*x^15 - 83*x^14 - 148*x^13 + 44*x^12 + 72*x^11 + 52*x^10 + 25*x^9 + 50*x^8 - 75*x^7 - 65*x^6 + 60*x^5 - 17*x^3 + 17*x^2 - 7*x + 1)
\( x^{22} - x^{21} + x^{20} - 11 x^{19} - 5 x^{18} + 22 x^{17} + 22 x^{16} + 62 x^{15} - 83 x^{14} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10787252710010591209601111089\)
\(\medspace = 167^{10}\cdot 639361\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.80\) | 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: | $167^{1/2}639361^{1/2}\approx 10333.116035349647$ | ||
| Ramified primes: |
\(167\), \(639361\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{639361}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13}a^{19}-\frac{1}{13}a^{18}+\frac{4}{13}a^{16}-\frac{6}{13}a^{15}+\frac{5}{13}a^{14}+\frac{6}{13}a^{13}-\frac{1}{13}a^{12}-\frac{6}{13}a^{11}+\frac{2}{13}a^{10}-\frac{3}{13}a^{9}-\frac{1}{13}a^{8}+\frac{5}{13}a^{7}-\frac{3}{13}a^{6}+\frac{5}{13}a^{5}-\frac{2}{13}a^{4}+\frac{5}{13}a^{3}+\frac{2}{13}a^{2}+\frac{6}{13}a-\frac{1}{13}$, $\frac{1}{65}a^{20}-\frac{2}{65}a^{19}+\frac{1}{65}a^{18}+\frac{17}{65}a^{17}+\frac{16}{65}a^{16}-\frac{28}{65}a^{15}-\frac{12}{65}a^{14}-\frac{4}{13}a^{13}+\frac{21}{65}a^{12}+\frac{21}{65}a^{11}+\frac{21}{65}a^{10}-\frac{11}{65}a^{9}+\frac{6}{65}a^{8}+\frac{31}{65}a^{7}-\frac{18}{65}a^{6}+\frac{6}{65}a^{5}+\frac{4}{13}a^{4}+\frac{23}{65}a^{3}+\frac{17}{65}a^{2}-\frac{4}{13}a-\frac{12}{65}$, $\frac{1}{31\!\cdots\!05}a^{21}-\frac{302913351064}{633541645732021}a^{20}-\frac{66024453997718}{31\!\cdots\!05}a^{19}-\frac{116393543809706}{31\!\cdots\!05}a^{18}+\frac{47839410711119}{633541645732021}a^{17}-\frac{21842281057607}{243669863743085}a^{16}+\frac{685846579688552}{31\!\cdots\!05}a^{15}+\frac{15\!\cdots\!66}{31\!\cdots\!05}a^{14}+\frac{12\!\cdots\!76}{31\!\cdots\!05}a^{13}-\frac{610595261462517}{31\!\cdots\!05}a^{12}-\frac{986381573550677}{31\!\cdots\!05}a^{11}-\frac{12\!\cdots\!49}{31\!\cdots\!05}a^{10}-\frac{763214669518231}{31\!\cdots\!05}a^{9}-\frac{647747073020577}{31\!\cdots\!05}a^{8}+\frac{717763764960674}{31\!\cdots\!05}a^{7}+\frac{52588438474753}{633541645732021}a^{6}+\frac{921026431721457}{31\!\cdots\!05}a^{5}+\frac{12\!\cdots\!88}{31\!\cdots\!05}a^{4}-\frac{567553927296647}{31\!\cdots\!05}a^{3}+\frac{170434785117459}{31\!\cdots\!05}a^{2}+\frac{155678779812538}{31\!\cdots\!05}a-\frac{836651604066679}{31\!\cdots\!05}$
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: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{135421738384594}{243669863743085}a^{21}-\frac{23832200691905}{48733972748617}a^{20}+\frac{107704712365598}{243669863743085}a^{19}-\frac{14\!\cdots\!39}{243669863743085}a^{18}-\frac{172600877729764}{48733972748617}a^{17}+\frac{231989527951667}{18743835672545}a^{16}+\frac{35\!\cdots\!93}{243669863743085}a^{15}+\frac{87\!\cdots\!94}{243669863743085}a^{14}-\frac{10\!\cdots\!81}{243669863743085}a^{13}-\frac{22\!\cdots\!68}{243669863743085}a^{12}+\frac{28\!\cdots\!42}{243669863743085}a^{11}+\frac{12\!\cdots\!44}{243669863743085}a^{10}+\frac{11\!\cdots\!76}{243669863743085}a^{9}+\frac{49\!\cdots\!87}{243669863743085}a^{8}+\frac{57\!\cdots\!76}{243669863743085}a^{7}-\frac{23\!\cdots\!73}{48733972748617}a^{6}-\frac{933311634855894}{18743835672545}a^{5}+\frac{62\!\cdots\!47}{243669863743085}a^{4}+\frac{20\!\cdots\!22}{243669863743085}a^{3}-\frac{12\!\cdots\!79}{243669863743085}a^{2}+\frac{19\!\cdots\!47}{243669863743085}a-\frac{786623240674331}{243669863743085}$, $\frac{655636982813530}{633541645732021}a^{21}-\frac{22\!\cdots\!26}{31\!\cdots\!05}a^{20}+\frac{24\!\cdots\!67}{31\!\cdots\!05}a^{19}-\frac{35\!\cdots\!36}{31\!\cdots\!05}a^{18}-\frac{27\!\cdots\!37}{31\!\cdots\!05}a^{17}+\frac{49\!\cdots\!13}{243669863743085}a^{16}+\frac{94\!\cdots\!28}{31\!\cdots\!05}a^{15}+\frac{23\!\cdots\!32}{31\!\cdots\!05}a^{14}-\frac{40\!\cdots\!38}{633541645732021}a^{13}-\frac{56\!\cdots\!76}{31\!\cdots\!05}a^{12}-\frac{36\!\cdots\!06}{31\!\cdots\!05}a^{11}+\frac{24\!\cdots\!94}{31\!\cdots\!05}a^{10}+\frac{26\!\cdots\!06}{31\!\cdots\!05}a^{9}+\frac{16\!\cdots\!39}{31\!\cdots\!05}a^{8}+\frac{21\!\cdots\!79}{31\!\cdots\!05}a^{7}-\frac{19\!\cdots\!47}{31\!\cdots\!05}a^{6}-\frac{29\!\cdots\!51}{31\!\cdots\!05}a^{5}+\frac{18\!\cdots\!70}{633541645732021}a^{4}+\frac{40\!\cdots\!97}{31\!\cdots\!05}a^{3}-\frac{38\!\cdots\!17}{31\!\cdots\!05}a^{2}+\frac{86\!\cdots\!99}{633541645732021}a-\frac{82\!\cdots\!63}{31\!\cdots\!05}$, $\frac{961966074732258}{633541645732021}a^{21}-\frac{33\!\cdots\!66}{31\!\cdots\!05}a^{20}+\frac{37\!\cdots\!97}{31\!\cdots\!05}a^{19}-\frac{51\!\cdots\!36}{31\!\cdots\!05}a^{18}-\frac{39\!\cdots\!72}{31\!\cdots\!05}a^{17}+\frac{73\!\cdots\!08}{243669863743085}a^{16}+\frac{13\!\cdots\!18}{31\!\cdots\!05}a^{15}+\frac{33\!\cdots\!57}{31\!\cdots\!05}a^{14}-\frac{60\!\cdots\!94}{633541645732021}a^{13}-\frac{80\!\cdots\!66}{31\!\cdots\!05}a^{12}-\frac{14\!\cdots\!41}{31\!\cdots\!05}a^{11}+\frac{35\!\cdots\!19}{31\!\cdots\!05}a^{10}+\frac{33\!\cdots\!31}{31\!\cdots\!05}a^{9}+\frac{20\!\cdots\!69}{31\!\cdots\!05}a^{8}+\frac{29\!\cdots\!89}{31\!\cdots\!05}a^{7}-\frac{26\!\cdots\!97}{31\!\cdots\!05}a^{6}-\frac{38\!\cdots\!06}{31\!\cdots\!05}a^{5}+\frac{36\!\cdots\!45}{633541645732021}a^{4}+\frac{62\!\cdots\!77}{31\!\cdots\!05}a^{3}-\frac{75\!\cdots\!67}{31\!\cdots\!05}a^{2}+\frac{11\!\cdots\!18}{633541645732021}a-\frac{13\!\cdots\!18}{31\!\cdots\!05}$, $\frac{63\!\cdots\!62}{31\!\cdots\!05}a^{21}-\frac{37\!\cdots\!11}{31\!\cdots\!05}a^{20}+\frac{45\!\cdots\!16}{31\!\cdots\!05}a^{19}-\frac{67\!\cdots\!33}{31\!\cdots\!05}a^{18}-\frac{59\!\cdots\!67}{31\!\cdots\!05}a^{17}+\frac{90\!\cdots\!64}{243669863743085}a^{16}+\frac{19\!\cdots\!77}{31\!\cdots\!05}a^{15}+\frac{46\!\cdots\!04}{31\!\cdots\!05}a^{14}-\frac{34\!\cdots\!63}{31\!\cdots\!05}a^{13}-\frac{21\!\cdots\!18}{633541645732021}a^{12}-\frac{32\!\cdots\!43}{633541645732021}a^{11}+\frac{42\!\cdots\!31}{31\!\cdots\!05}a^{10}+\frac{51\!\cdots\!49}{31\!\cdots\!05}a^{9}+\frac{72\!\cdots\!94}{633541645732021}a^{8}+\frac{44\!\cdots\!02}{31\!\cdots\!05}a^{7}-\frac{30\!\cdots\!32}{31\!\cdots\!05}a^{6}-\frac{55\!\cdots\!87}{31\!\cdots\!05}a^{5}+\frac{15\!\cdots\!96}{31\!\cdots\!05}a^{4}+\frac{80\!\cdots\!48}{31\!\cdots\!05}a^{3}-\frac{77\!\cdots\!34}{31\!\cdots\!05}a^{2}+\frac{69\!\cdots\!26}{31\!\cdots\!05}a-\frac{11\!\cdots\!06}{31\!\cdots\!05}$, $\frac{11\!\cdots\!97}{633541645732021}a^{21}-\frac{42\!\cdots\!84}{31\!\cdots\!05}a^{20}+\frac{47\!\cdots\!78}{31\!\cdots\!05}a^{19}-\frac{63\!\cdots\!29}{31\!\cdots\!05}a^{18}-\frac{47\!\cdots\!53}{31\!\cdots\!05}a^{17}+\frac{88\!\cdots\!72}{243669863743085}a^{16}+\frac{16\!\cdots\!47}{31\!\cdots\!05}a^{15}+\frac{41\!\cdots\!28}{31\!\cdots\!05}a^{14}-\frac{73\!\cdots\!03}{633541645732021}a^{13}-\frac{97\!\cdots\!74}{31\!\cdots\!05}a^{12}-\frac{27\!\cdots\!99}{31\!\cdots\!05}a^{11}+\frac{40\!\cdots\!11}{31\!\cdots\!05}a^{10}+\frac{42\!\cdots\!24}{31\!\cdots\!05}a^{9}+\frac{27\!\cdots\!71}{31\!\cdots\!05}a^{8}+\frac{38\!\cdots\!36}{31\!\cdots\!05}a^{7}-\frac{33\!\cdots\!98}{31\!\cdots\!05}a^{6}-\frac{47\!\cdots\!24}{31\!\cdots\!05}a^{5}+\frac{41\!\cdots\!76}{633541645732021}a^{4}+\frac{50\!\cdots\!58}{31\!\cdots\!05}a^{3}-\frac{80\!\cdots\!18}{31\!\cdots\!05}a^{2}+\frac{15\!\cdots\!88}{633541645732021}a-\frac{20\!\cdots\!42}{31\!\cdots\!05}$, $a$, $\frac{33\!\cdots\!79}{31\!\cdots\!05}a^{21}-\frac{25\!\cdots\!46}{31\!\cdots\!05}a^{20}+\frac{535713880553834}{633541645732021}a^{19}-\frac{71\!\cdots\!76}{633541645732021}a^{18}-\frac{24\!\cdots\!42}{31\!\cdots\!05}a^{17}+\frac{10\!\cdots\!39}{48733972748617}a^{16}+\frac{87\!\cdots\!01}{31\!\cdots\!05}a^{15}+\frac{22\!\cdots\!01}{31\!\cdots\!05}a^{14}-\frac{22\!\cdots\!21}{31\!\cdots\!05}a^{13}-\frac{54\!\cdots\!89}{31\!\cdots\!05}a^{12}+\frac{33\!\cdots\!91}{31\!\cdots\!05}a^{11}+\frac{24\!\cdots\!18}{31\!\cdots\!05}a^{10}+\frac{20\!\cdots\!82}{31\!\cdots\!05}a^{9}+\frac{12\!\cdots\!36}{31\!\cdots\!05}a^{8}+\frac{39\!\cdots\!53}{633541645732021}a^{7}-\frac{19\!\cdots\!92}{31\!\cdots\!05}a^{6}-\frac{26\!\cdots\!28}{31\!\cdots\!05}a^{5}+\frac{14\!\cdots\!67}{31\!\cdots\!05}a^{4}+\frac{32\!\cdots\!34}{31\!\cdots\!05}a^{3}-\frac{64\!\cdots\!61}{31\!\cdots\!05}a^{2}+\frac{49\!\cdots\!92}{31\!\cdots\!05}a-\frac{11\!\cdots\!34}{31\!\cdots\!05}$, $\frac{10\!\cdots\!59}{633541645732021}a^{21}-\frac{794464536838443}{633541645732021}a^{20}+\frac{870054984265516}{633541645732021}a^{19}-\frac{11\!\cdots\!48}{633541645732021}a^{18}-\frac{81\!\cdots\!36}{633541645732021}a^{17}+\frac{16\!\cdots\!28}{48733972748617}a^{16}+\frac{28\!\cdots\!44}{633541645732021}a^{15}+\frac{72\!\cdots\!60}{633541645732021}a^{14}-\frac{69\!\cdots\!96}{633541645732021}a^{13}-\frac{17\!\cdots\!29}{633541645732021}a^{12}+\frac{25\!\cdots\!78}{633541645732021}a^{11}+\frac{75\!\cdots\!97}{633541645732021}a^{10}+\frac{73\!\cdots\!28}{633541645732021}a^{9}+\frac{45\!\cdots\!26}{633541645732021}a^{8}+\frac{64\!\cdots\!96}{633541645732021}a^{7}-\frac{62\!\cdots\!25}{633541645732021}a^{6}-\frac{83\!\cdots\!73}{633541645732021}a^{5}+\frac{43\!\cdots\!06}{633541645732021}a^{4}+\frac{10\!\cdots\!61}{633541645732021}a^{3}-\frac{15\!\cdots\!01}{633541645732021}a^{2}+\frac{13\!\cdots\!06}{633541645732021}a-\frac{42\!\cdots\!42}{633541645732021}$, $\frac{59682029392789}{633541645732021}a^{21}-\frac{381320134537951}{31\!\cdots\!05}a^{20}+\frac{434238951639482}{31\!\cdots\!05}a^{19}-\frac{33\!\cdots\!71}{31\!\cdots\!05}a^{18}-\frac{580799247989972}{31\!\cdots\!05}a^{17}+\frac{501114651475508}{243669863743085}a^{16}+\frac{36\!\cdots\!13}{31\!\cdots\!05}a^{15}+\frac{17\!\cdots\!02}{31\!\cdots\!05}a^{14}-\frac{55\!\cdots\!18}{633541645732021}a^{13}-\frac{32\!\cdots\!76}{31\!\cdots\!05}a^{12}+\frac{26\!\cdots\!24}{31\!\cdots\!05}a^{11}+\frac{62\!\cdots\!14}{31\!\cdots\!05}a^{10}+\frac{44\!\cdots\!06}{31\!\cdots\!05}a^{9}+\frac{49\!\cdots\!19}{31\!\cdots\!05}a^{8}+\frac{16\!\cdots\!19}{31\!\cdots\!05}a^{7}-\frac{22\!\cdots\!37}{31\!\cdots\!05}a^{6}-\frac{90\!\cdots\!26}{31\!\cdots\!05}a^{5}+\frac{51\!\cdots\!51}{633541645732021}a^{4}-\frac{12\!\cdots\!48}{31\!\cdots\!05}a^{3}-\frac{51\!\cdots\!47}{31\!\cdots\!05}a^{2}+\frac{21\!\cdots\!25}{633541645732021}a-\frac{44\!\cdots\!38}{31\!\cdots\!05}$, $\frac{38\!\cdots\!67}{31\!\cdots\!05}a^{21}-\frac{26\!\cdots\!61}{31\!\cdots\!05}a^{20}+\frac{31\!\cdots\!56}{31\!\cdots\!05}a^{19}-\frac{41\!\cdots\!78}{31\!\cdots\!05}a^{18}-\frac{31\!\cdots\!57}{31\!\cdots\!05}a^{17}+\frac{56\!\cdots\!24}{243669863743085}a^{16}+\frac{10\!\cdots\!02}{31\!\cdots\!05}a^{15}+\frac{27\!\cdots\!94}{31\!\cdots\!05}a^{14}-\frac{23\!\cdots\!38}{31\!\cdots\!05}a^{13}-\frac{12\!\cdots\!20}{633541645732021}a^{12}-\frac{66\!\cdots\!07}{633541645732021}a^{11}+\frac{22\!\cdots\!86}{31\!\cdots\!05}a^{10}+\frac{26\!\cdots\!89}{31\!\cdots\!05}a^{9}+\frac{41\!\cdots\!35}{633541645732021}a^{8}+\frac{27\!\cdots\!22}{31\!\cdots\!05}a^{7}-\frac{19\!\cdots\!32}{31\!\cdots\!05}a^{6}-\frac{30\!\cdots\!22}{31\!\cdots\!05}a^{5}+\frac{12\!\cdots\!26}{31\!\cdots\!05}a^{4}+\frac{11\!\cdots\!63}{31\!\cdots\!05}a^{3}-\frac{58\!\cdots\!29}{31\!\cdots\!05}a^{2}+\frac{56\!\cdots\!71}{31\!\cdots\!05}a-\frac{11\!\cdots\!86}{31\!\cdots\!05}$, $\frac{343532022871344}{633541645732021}a^{21}-\frac{268815150172018}{31\!\cdots\!05}a^{20}+\frac{10\!\cdots\!11}{31\!\cdots\!05}a^{19}-\frac{17\!\cdots\!13}{31\!\cdots\!05}a^{18}-\frac{24\!\cdots\!66}{31\!\cdots\!05}a^{17}+\frac{16\!\cdots\!34}{243669863743085}a^{16}+\frac{61\!\cdots\!69}{31\!\cdots\!05}a^{15}+\frac{15\!\cdots\!16}{31\!\cdots\!05}a^{14}-\frac{55\!\cdots\!65}{633541645732021}a^{13}-\frac{31\!\cdots\!68}{31\!\cdots\!05}a^{12}-\frac{17\!\cdots\!93}{31\!\cdots\!05}a^{11}+\frac{42\!\cdots\!92}{31\!\cdots\!05}a^{10}+\frac{17\!\cdots\!98}{31\!\cdots\!05}a^{9}+\frac{16\!\cdots\!47}{31\!\cdots\!05}a^{8}+\frac{17\!\cdots\!37}{31\!\cdots\!05}a^{7}-\frac{51\!\cdots\!81}{31\!\cdots\!05}a^{6}-\frac{15\!\cdots\!88}{31\!\cdots\!05}a^{5}-\frac{31\!\cdots\!02}{633541645732021}a^{4}+\frac{15\!\cdots\!06}{31\!\cdots\!05}a^{3}-\frac{99\!\cdots\!21}{31\!\cdots\!05}a^{2}+\frac{28\!\cdots\!82}{633541645732021}a-\frac{25\!\cdots\!99}{31\!\cdots\!05}$
| 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: | \( 116549.459796 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 116549.459796 \cdot 1}{2\cdot\sqrt{10787252710010591209601111089}}\cr\approx \mathstrut & 0.215220441362 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 - 11*x^19 - 5*x^18 + 22*x^17 + 22*x^16 + 62*x^15 - 83*x^14 - 148*x^13 + 44*x^12 + 72*x^11 + 52*x^10 + 25*x^9 + 50*x^8 - 75*x^7 - 65*x^6 + 60*x^5 - 17*x^3 + 17*x^2 - 7*x + 1)
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^22 - x^21 + x^20 - 11*x^19 - 5*x^18 + 22*x^17 + 22*x^16 + 62*x^15 - 83*x^14 - 148*x^13 + 44*x^12 + 72*x^11 + 52*x^10 + 25*x^9 + 50*x^8 - 75*x^7 - 65*x^6 + 60*x^5 - 17*x^3 + 17*x^2 - 7*x + 1, 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^22 - x^21 + x^20 - 11*x^19 - 5*x^18 + 22*x^17 + 22*x^16 + 62*x^15 - 83*x^14 - 148*x^13 + 44*x^12 + 72*x^11 + 52*x^10 + 25*x^9 + 50*x^8 - 75*x^7 - 65*x^6 + 60*x^5 - 17*x^3 + 17*x^2 - 7*x + 1);
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^22 - x^21 + x^20 - 11*x^19 - 5*x^18 + 22*x^17 + 22*x^16 + 62*x^15 - 83*x^14 - 148*x^13 + 44*x^12 + 72*x^11 + 52*x^10 + 25*x^9 + 50*x^8 - 75*x^7 - 65*x^6 + 60*x^5 - 17*x^3 + 17*x^2 - 7*x + 1);
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
$C_2^{10}.D_{22}$ (as 22T32):
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 45056 |
| The 200 conjugacy class representatives for $C_2^{10}.D_{22}$ are not computed |
| Character table for $C_2^{10}.D_{22}$ is not computed |
Intermediate fields
| 11.1.129891985607.1 |
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 22 siblings: | data not computed |
| Degree 44 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{5}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/13.2.0.1}{2} }^{11}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $22$ | $22$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | $22$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(167\)
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 167.4.2.1 | $x^{4} + 332 x^{3} + 27900 x^{2} + 57104 x + 4628096$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(639361\)
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{639361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |