Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 33*x^20 + 143*x^18 - 22*x^17 + 385*x^16 - 2222*x^15 - 902*x^14 + 14586*x^13 - 3234*x^12 + 94*x^11 - 39512*x^10 - 28006*x^9 + 136862*x^8 - 99286*x^7 - 30151*x^6 + 136136*x^5 - 99561*x^4 - 13794*x^3 + 44385*x^2 - 17182*x + 2039)
gp: K = bnfinit(y^22 - 33*y^20 + 143*y^18 - 22*y^17 + 385*y^16 - 2222*y^15 - 902*y^14 + 14586*y^13 - 3234*y^12 + 94*y^11 - 39512*y^10 - 28006*y^9 + 136862*y^8 - 99286*y^7 - 30151*y^6 + 136136*y^5 - 99561*y^4 - 13794*y^3 + 44385*y^2 - 17182*y + 2039, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 33*x^20 + 143*x^18 - 22*x^17 + 385*x^16 - 2222*x^15 - 902*x^14 + 14586*x^13 - 3234*x^12 + 94*x^11 - 39512*x^10 - 28006*x^9 + 136862*x^8 - 99286*x^7 - 30151*x^6 + 136136*x^5 - 99561*x^4 - 13794*x^3 + 44385*x^2 - 17182*x + 2039);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 33*x^20 + 143*x^18 - 22*x^17 + 385*x^16 - 2222*x^15 - 902*x^14 + 14586*x^13 - 3234*x^12 + 94*x^11 - 39512*x^10 - 28006*x^9 + 136862*x^8 - 99286*x^7 - 30151*x^6 + 136136*x^5 - 99561*x^4 - 13794*x^3 + 44385*x^2 - 17182*x + 2039)
\( x^{22} - 33 x^{20} + 143 x^{18} - 22 x^{17} + 385 x^{16} - 2222 x^{15} - 902 x^{14} + 14586 x^{13} + \cdots + 2039 \)
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: | $[14, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1543118794783990130660601200150793158656\)
\(\medspace = 2^{26}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.44\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11}a^{17}+\frac{5}{11}a^{16}+\frac{4}{11}a^{15}+\frac{2}{11}a^{14}+\frac{4}{11}a^{13}+\frac{5}{11}a^{12}+\frac{1}{11}a^{11}-\frac{4}{11}a^{6}+\frac{2}{11}a^{5}-\frac{5}{11}a^{4}+\frac{3}{11}a^{3}-\frac{5}{11}a^{2}+\frac{2}{11}a-\frac{4}{11}$, $\frac{1}{11}a^{18}+\frac{1}{11}a^{16}+\frac{4}{11}a^{15}+\frac{5}{11}a^{14}-\frac{4}{11}a^{13}-\frac{2}{11}a^{12}-\frac{5}{11}a^{11}-\frac{4}{11}a^{7}-\frac{4}{11}a^{5}-\frac{5}{11}a^{4}+\frac{2}{11}a^{3}+\frac{5}{11}a^{2}-\frac{3}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{19}-\frac{1}{11}a^{16}+\frac{1}{11}a^{15}+\frac{5}{11}a^{14}+\frac{5}{11}a^{13}+\frac{1}{11}a^{12}-\frac{1}{11}a^{11}-\frac{4}{11}a^{8}+\frac{4}{11}a^{5}-\frac{4}{11}a^{4}+\frac{2}{11}a^{3}+\frac{2}{11}a^{2}-\frac{4}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{20}-\frac{5}{11}a^{16}-\frac{2}{11}a^{15}-\frac{4}{11}a^{14}+\frac{5}{11}a^{13}+\frac{4}{11}a^{12}+\frac{1}{11}a^{11}-\frac{4}{11}a^{9}-\frac{2}{11}a^{5}-\frac{3}{11}a^{4}+\frac{5}{11}a^{3}+\frac{2}{11}a^{2}-\frac{5}{11}a-\frac{4}{11}$, $\frac{1}{26\!\cdots\!51}a^{21}-\frac{70\!\cdots\!59}{24\!\cdots\!41}a^{20}-\frac{36\!\cdots\!90}{26\!\cdots\!51}a^{19}-\frac{52\!\cdots\!06}{26\!\cdots\!51}a^{18}-\frac{25\!\cdots\!27}{24\!\cdots\!41}a^{17}-\frac{95\!\cdots\!08}{26\!\cdots\!51}a^{16}-\frac{53\!\cdots\!42}{26\!\cdots\!51}a^{15}+\frac{66\!\cdots\!63}{26\!\cdots\!51}a^{14}+\frac{38\!\cdots\!84}{24\!\cdots\!41}a^{13}-\frac{88\!\cdots\!54}{26\!\cdots\!51}a^{12}+\frac{43\!\cdots\!07}{26\!\cdots\!51}a^{11}-\frac{82\!\cdots\!75}{26\!\cdots\!51}a^{10}-\frac{53\!\cdots\!99}{24\!\cdots\!41}a^{9}+\frac{73\!\cdots\!36}{26\!\cdots\!51}a^{8}+\frac{10\!\cdots\!35}{26\!\cdots\!51}a^{7}+\frac{63\!\cdots\!54}{24\!\cdots\!41}a^{6}+\frac{10\!\cdots\!70}{26\!\cdots\!51}a^{5}-\frac{11\!\cdots\!73}{26\!\cdots\!51}a^{4}+\frac{22\!\cdots\!30}{26\!\cdots\!51}a^{3}+\frac{11\!\cdots\!64}{24\!\cdots\!41}a^{2}-\frac{91\!\cdots\!33}{26\!\cdots\!51}a+\frac{15\!\cdots\!71}{26\!\cdots\!51}$
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: | $17$ | 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: |
$\frac{51\!\cdots\!04}{26\!\cdots\!51}a^{21}+\frac{32\!\cdots\!19}{26\!\cdots\!51}a^{20}-\frac{16\!\cdots\!32}{26\!\cdots\!51}a^{19}-\frac{10\!\cdots\!01}{26\!\cdots\!51}a^{18}+\frac{67\!\cdots\!59}{26\!\cdots\!51}a^{17}+\frac{27\!\cdots\!03}{24\!\cdots\!41}a^{16}+\frac{21\!\cdots\!98}{26\!\cdots\!51}a^{15}-\frac{10\!\cdots\!82}{26\!\cdots\!51}a^{14}-\frac{10\!\cdots\!58}{26\!\cdots\!51}a^{13}+\frac{68\!\cdots\!20}{26\!\cdots\!51}a^{12}+\frac{25\!\cdots\!52}{26\!\cdots\!51}a^{11}+\frac{16\!\cdots\!72}{26\!\cdots\!51}a^{10}-\frac{19\!\cdots\!54}{26\!\cdots\!51}a^{9}-\frac{26\!\cdots\!63}{26\!\cdots\!51}a^{8}+\frac{54\!\cdots\!42}{26\!\cdots\!51}a^{7}-\frac{17\!\cdots\!52}{26\!\cdots\!51}a^{6}-\frac{24\!\cdots\!04}{24\!\cdots\!41}a^{5}+\frac{54\!\cdots\!49}{26\!\cdots\!51}a^{4}-\frac{18\!\cdots\!06}{26\!\cdots\!51}a^{3}-\frac{18\!\cdots\!47}{26\!\cdots\!51}a^{2}+\frac{12\!\cdots\!29}{26\!\cdots\!51}a-\frac{18\!\cdots\!37}{26\!\cdots\!51}$, $\frac{81\!\cdots\!90}{26\!\cdots\!51}a^{21}+\frac{26\!\cdots\!60}{26\!\cdots\!51}a^{20}-\frac{26\!\cdots\!50}{26\!\cdots\!51}a^{19}-\frac{86\!\cdots\!39}{26\!\cdots\!51}a^{18}+\frac{11\!\cdots\!96}{26\!\cdots\!51}a^{17}+\frac{19\!\cdots\!45}{26\!\cdots\!51}a^{16}+\frac{31\!\cdots\!59}{26\!\cdots\!51}a^{15}-\frac{17\!\cdots\!66}{26\!\cdots\!51}a^{14}-\frac{12\!\cdots\!55}{26\!\cdots\!51}a^{13}+\frac{11\!\cdots\!07}{26\!\cdots\!51}a^{12}+\frac{10\!\cdots\!33}{24\!\cdots\!41}a^{11}+\frac{33\!\cdots\!54}{26\!\cdots\!51}a^{10}-\frac{32\!\cdots\!31}{26\!\cdots\!51}a^{9}-\frac{33\!\cdots\!57}{26\!\cdots\!51}a^{8}+\frac{10\!\cdots\!14}{26\!\cdots\!51}a^{7}-\frac{47\!\cdots\!40}{26\!\cdots\!51}a^{6}-\frac{40\!\cdots\!20}{26\!\cdots\!51}a^{5}+\frac{97\!\cdots\!07}{26\!\cdots\!51}a^{4}-\frac{48\!\cdots\!08}{26\!\cdots\!51}a^{3}-\frac{27\!\cdots\!14}{26\!\cdots\!51}a^{2}+\frac{27\!\cdots\!86}{26\!\cdots\!51}a-\frac{44\!\cdots\!70}{24\!\cdots\!41}$, $\frac{20\!\cdots\!66}{26\!\cdots\!51}a^{21}+\frac{22\!\cdots\!99}{26\!\cdots\!51}a^{20}-\frac{65\!\cdots\!38}{26\!\cdots\!51}a^{19}-\frac{73\!\cdots\!53}{26\!\cdots\!51}a^{18}+\frac{19\!\cdots\!74}{26\!\cdots\!51}a^{17}+\frac{22\!\cdots\!86}{26\!\cdots\!51}a^{16}+\frac{11\!\cdots\!61}{26\!\cdots\!51}a^{15}-\frac{36\!\cdots\!16}{26\!\cdots\!51}a^{14}-\frac{55\!\cdots\!37}{26\!\cdots\!51}a^{13}+\frac{22\!\cdots\!18}{26\!\cdots\!51}a^{12}+\frac{20\!\cdots\!23}{26\!\cdots\!51}a^{11}+\frac{32\!\cdots\!94}{26\!\cdots\!51}a^{10}-\frac{71\!\cdots\!75}{26\!\cdots\!51}a^{9}-\frac{12\!\cdots\!22}{26\!\cdots\!51}a^{8}+\frac{10\!\cdots\!70}{26\!\cdots\!51}a^{7}-\frac{34\!\cdots\!08}{26\!\cdots\!51}a^{6}+\frac{19\!\cdots\!94}{26\!\cdots\!51}a^{5}+\frac{50\!\cdots\!09}{26\!\cdots\!51}a^{4}+\frac{41\!\cdots\!70}{26\!\cdots\!51}a^{3}+\frac{26\!\cdots\!66}{26\!\cdots\!51}a^{2}-\frac{68\!\cdots\!32}{26\!\cdots\!51}a+\frac{19\!\cdots\!40}{26\!\cdots\!51}$, $\frac{46\!\cdots\!61}{26\!\cdots\!51}a^{21}+\frac{28\!\cdots\!73}{26\!\cdots\!51}a^{20}-\frac{15\!\cdots\!82}{26\!\cdots\!51}a^{19}-\frac{92\!\cdots\!18}{26\!\cdots\!51}a^{18}+\frac{60\!\cdots\!28}{26\!\cdots\!51}a^{17}+\frac{26\!\cdots\!42}{26\!\cdots\!51}a^{16}+\frac{19\!\cdots\!58}{26\!\cdots\!51}a^{15}-\frac{91\!\cdots\!34}{26\!\cdots\!51}a^{14}-\frac{97\!\cdots\!69}{26\!\cdots\!51}a^{13}+\frac{62\!\cdots\!57}{26\!\cdots\!51}a^{12}+\frac{22\!\cdots\!24}{26\!\cdots\!51}a^{11}+\frac{14\!\cdots\!16}{26\!\cdots\!51}a^{10}-\frac{17\!\cdots\!87}{26\!\cdots\!51}a^{9}-\frac{23\!\cdots\!14}{26\!\cdots\!51}a^{8}+\frac{49\!\cdots\!60}{26\!\cdots\!51}a^{7}-\frac{16\!\cdots\!71}{26\!\cdots\!51}a^{6}-\frac{23\!\cdots\!91}{26\!\cdots\!51}a^{5}+\frac{49\!\cdots\!82}{26\!\cdots\!51}a^{4}-\frac{16\!\cdots\!37}{26\!\cdots\!51}a^{3}-\frac{16\!\cdots\!40}{26\!\cdots\!51}a^{2}+\frac{10\!\cdots\!41}{26\!\cdots\!51}a-\frac{16\!\cdots\!94}{26\!\cdots\!51}$, $\frac{29\!\cdots\!06}{26\!\cdots\!51}a^{21}+\frac{14\!\cdots\!41}{24\!\cdots\!41}a^{20}-\frac{97\!\cdots\!07}{26\!\cdots\!51}a^{19}-\frac{51\!\cdots\!70}{26\!\cdots\!51}a^{18}+\frac{39\!\cdots\!60}{26\!\cdots\!51}a^{17}+\frac{14\!\cdots\!85}{26\!\cdots\!51}a^{16}+\frac{12\!\cdots\!15}{26\!\cdots\!51}a^{15}-\frac{59\!\cdots\!62}{26\!\cdots\!51}a^{14}-\frac{53\!\cdots\!55}{24\!\cdots\!41}a^{13}+\frac{40\!\cdots\!74}{26\!\cdots\!51}a^{12}+\frac{11\!\cdots\!43}{26\!\cdots\!51}a^{11}+\frac{72\!\cdots\!86}{26\!\cdots\!51}a^{10}-\frac{10\!\cdots\!57}{24\!\cdots\!41}a^{9}-\frac{14\!\cdots\!47}{26\!\cdots\!51}a^{8}+\frac{33\!\cdots\!96}{26\!\cdots\!51}a^{7}-\frac{12\!\cdots\!09}{26\!\cdots\!51}a^{6}-\frac{15\!\cdots\!69}{26\!\cdots\!51}a^{5}+\frac{32\!\cdots\!50}{26\!\cdots\!51}a^{4}-\frac{12\!\cdots\!84}{26\!\cdots\!51}a^{3}-\frac{94\!\cdots\!06}{24\!\cdots\!41}a^{2}+\frac{79\!\cdots\!88}{26\!\cdots\!51}a-\frac{12\!\cdots\!59}{26\!\cdots\!51}$, $\frac{14\!\cdots\!42}{26\!\cdots\!51}a^{21}+\frac{46\!\cdots\!33}{24\!\cdots\!41}a^{20}-\frac{46\!\cdots\!15}{26\!\cdots\!51}a^{19}-\frac{16\!\cdots\!04}{26\!\cdots\!51}a^{18}+\frac{19\!\cdots\!82}{26\!\cdots\!51}a^{17}+\frac{39\!\cdots\!76}{26\!\cdots\!51}a^{16}+\frac{55\!\cdots\!72}{26\!\cdots\!51}a^{15}-\frac{29\!\cdots\!26}{26\!\cdots\!51}a^{14}-\frac{23\!\cdots\!76}{26\!\cdots\!51}a^{13}+\frac{19\!\cdots\!30}{26\!\cdots\!51}a^{12}+\frac{25\!\cdots\!44}{26\!\cdots\!51}a^{11}+\frac{11\!\cdots\!83}{26\!\cdots\!51}a^{10}-\frac{49\!\cdots\!70}{24\!\cdots\!41}a^{9}-\frac{59\!\cdots\!63}{26\!\cdots\!51}a^{8}+\frac{17\!\cdots\!47}{26\!\cdots\!51}a^{7}-\frac{77\!\cdots\!59}{26\!\cdots\!51}a^{6}-\frac{70\!\cdots\!50}{26\!\cdots\!51}a^{5}+\frac{16\!\cdots\!88}{26\!\cdots\!51}a^{4}-\frac{79\!\cdots\!01}{26\!\cdots\!51}a^{3}-\frac{47\!\cdots\!51}{26\!\cdots\!51}a^{2}+\frac{44\!\cdots\!55}{26\!\cdots\!51}a-\frac{79\!\cdots\!54}{26\!\cdots\!51}$, $\frac{14\!\cdots\!13}{26\!\cdots\!51}a^{21}+\frac{65\!\cdots\!88}{26\!\cdots\!51}a^{20}-\frac{48\!\cdots\!65}{26\!\cdots\!51}a^{19}-\frac{21\!\cdots\!57}{26\!\cdots\!51}a^{18}+\frac{20\!\cdots\!51}{26\!\cdots\!51}a^{17}+\frac{55\!\cdots\!90}{26\!\cdots\!51}a^{16}+\frac{54\!\cdots\!55}{24\!\cdots\!41}a^{15}-\frac{30\!\cdots\!57}{26\!\cdots\!51}a^{14}-\frac{26\!\cdots\!46}{26\!\cdots\!51}a^{13}+\frac{20\!\cdots\!79}{26\!\cdots\!51}a^{12}+\frac{41\!\cdots\!50}{26\!\cdots\!51}a^{11}+\frac{23\!\cdots\!43}{26\!\cdots\!51}a^{10}-\frac{57\!\cdots\!22}{26\!\cdots\!51}a^{9}-\frac{66\!\cdots\!11}{26\!\cdots\!51}a^{8}+\frac{17\!\cdots\!46}{26\!\cdots\!51}a^{7}-\frac{73\!\cdots\!14}{26\!\cdots\!51}a^{6}-\frac{74\!\cdots\!01}{26\!\cdots\!51}a^{5}+\frac{15\!\cdots\!82}{24\!\cdots\!41}a^{4}-\frac{75\!\cdots\!75}{26\!\cdots\!51}a^{3}-\frac{50\!\cdots\!40}{26\!\cdots\!51}a^{2}+\frac{43\!\cdots\!57}{26\!\cdots\!51}a-\frac{75\!\cdots\!51}{26\!\cdots\!51}$, $\frac{18\!\cdots\!17}{26\!\cdots\!51}a^{21}+\frac{63\!\cdots\!57}{26\!\cdots\!51}a^{20}-\frac{62\!\cdots\!36}{26\!\cdots\!51}a^{19}-\frac{20\!\cdots\!87}{26\!\cdots\!51}a^{18}+\frac{26\!\cdots\!28}{26\!\cdots\!51}a^{17}+\frac{40\!\cdots\!93}{24\!\cdots\!41}a^{16}+\frac{75\!\cdots\!05}{26\!\cdots\!51}a^{15}-\frac{39\!\cdots\!56}{26\!\cdots\!51}a^{14}-\frac{30\!\cdots\!61}{26\!\cdots\!51}a^{13}+\frac{26\!\cdots\!21}{26\!\cdots\!51}a^{12}+\frac{26\!\cdots\!67}{26\!\cdots\!51}a^{11}+\frac{17\!\cdots\!73}{26\!\cdots\!51}a^{10}-\frac{73\!\cdots\!35}{26\!\cdots\!51}a^{9}-\frac{77\!\cdots\!01}{26\!\cdots\!51}a^{8}+\frac{23\!\cdots\!81}{26\!\cdots\!51}a^{7}-\frac{11\!\cdots\!95}{26\!\cdots\!51}a^{6}-\frac{82\!\cdots\!93}{24\!\cdots\!41}a^{5}+\frac{22\!\cdots\!86}{26\!\cdots\!51}a^{4}-\frac{11\!\cdots\!70}{26\!\cdots\!51}a^{3}-\frac{60\!\cdots\!60}{26\!\cdots\!51}a^{2}+\frac{63\!\cdots\!85}{26\!\cdots\!51}a-\frac{12\!\cdots\!24}{26\!\cdots\!51}$, $\frac{38\!\cdots\!63}{26\!\cdots\!51}a^{21}+\frac{30\!\cdots\!00}{26\!\cdots\!51}a^{20}-\frac{12\!\cdots\!07}{26\!\cdots\!51}a^{19}-\frac{98\!\cdots\!17}{26\!\cdots\!51}a^{18}+\frac{47\!\cdots\!65}{26\!\cdots\!51}a^{17}+\frac{27\!\cdots\!53}{24\!\cdots\!41}a^{16}+\frac{15\!\cdots\!38}{24\!\cdots\!41}a^{15}-\frac{72\!\cdots\!36}{26\!\cdots\!51}a^{14}-\frac{92\!\cdots\!63}{26\!\cdots\!51}a^{13}+\frac{48\!\cdots\!80}{26\!\cdots\!51}a^{12}+\frac{26\!\cdots\!96}{26\!\cdots\!51}a^{11}+\frac{20\!\cdots\!89}{26\!\cdots\!51}a^{10}-\frac{13\!\cdots\!90}{26\!\cdots\!51}a^{9}-\frac{21\!\cdots\!54}{26\!\cdots\!51}a^{8}+\frac{35\!\cdots\!87}{26\!\cdots\!51}a^{7}-\frac{84\!\cdots\!22}{26\!\cdots\!51}a^{6}-\frac{16\!\cdots\!58}{24\!\cdots\!41}a^{5}+\frac{32\!\cdots\!88}{24\!\cdots\!41}a^{4}-\frac{76\!\cdots\!95}{26\!\cdots\!51}a^{3}-\frac{12\!\cdots\!45}{26\!\cdots\!51}a^{2}+\frac{55\!\cdots\!09}{26\!\cdots\!51}a-\frac{65\!\cdots\!86}{26\!\cdots\!51}$, $\frac{39\!\cdots\!94}{26\!\cdots\!51}a^{21}+\frac{35\!\cdots\!34}{26\!\cdots\!51}a^{20}-\frac{12\!\cdots\!95}{26\!\cdots\!51}a^{19}-\frac{11\!\cdots\!75}{26\!\cdots\!51}a^{18}+\frac{46\!\cdots\!48}{26\!\cdots\!51}a^{17}+\frac{36\!\cdots\!43}{26\!\cdots\!51}a^{16}+\frac{17\!\cdots\!71}{26\!\cdots\!51}a^{15}-\frac{71\!\cdots\!42}{26\!\cdots\!51}a^{14}-\frac{10\!\cdots\!41}{26\!\cdots\!51}a^{13}+\frac{48\!\cdots\!67}{26\!\cdots\!51}a^{12}+\frac{33\!\cdots\!10}{26\!\cdots\!51}a^{11}+\frac{24\!\cdots\!35}{26\!\cdots\!51}a^{10}-\frac{13\!\cdots\!46}{26\!\cdots\!51}a^{9}-\frac{23\!\cdots\!78}{26\!\cdots\!51}a^{8}+\frac{33\!\cdots\!65}{26\!\cdots\!51}a^{7}-\frac{47\!\cdots\!21}{26\!\cdots\!51}a^{6}-\frac{20\!\cdots\!43}{26\!\cdots\!51}a^{5}+\frac{34\!\cdots\!52}{26\!\cdots\!51}a^{4}-\frac{42\!\cdots\!97}{26\!\cdots\!51}a^{3}-\frac{13\!\cdots\!21}{26\!\cdots\!51}a^{2}+\frac{45\!\cdots\!42}{26\!\cdots\!51}a-\frac{28\!\cdots\!65}{26\!\cdots\!51}$, $\frac{15\!\cdots\!36}{26\!\cdots\!51}a^{21}+\frac{83\!\cdots\!06}{26\!\cdots\!51}a^{20}-\frac{49\!\cdots\!83}{26\!\cdots\!51}a^{19}-\frac{27\!\cdots\!87}{26\!\cdots\!51}a^{18}+\frac{19\!\cdots\!57}{26\!\cdots\!51}a^{17}+\frac{75\!\cdots\!04}{26\!\cdots\!51}a^{16}+\frac{64\!\cdots\!60}{26\!\cdots\!51}a^{15}-\frac{30\!\cdots\!24}{26\!\cdots\!51}a^{14}-\frac{29\!\cdots\!45}{26\!\cdots\!51}a^{13}+\frac{20\!\cdots\!46}{26\!\cdots\!51}a^{12}+\frac{61\!\cdots\!17}{26\!\cdots\!51}a^{11}+\frac{57\!\cdots\!01}{26\!\cdots\!51}a^{10}-\frac{57\!\cdots\!88}{26\!\cdots\!51}a^{9}-\frac{72\!\cdots\!25}{26\!\cdots\!51}a^{8}+\frac{16\!\cdots\!81}{26\!\cdots\!51}a^{7}-\frac{65\!\cdots\!05}{26\!\cdots\!51}a^{6}-\frac{63\!\cdots\!81}{26\!\cdots\!51}a^{5}+\frac{15\!\cdots\!94}{26\!\cdots\!51}a^{4}-\frac{58\!\cdots\!59}{26\!\cdots\!51}a^{3}-\frac{44\!\cdots\!09}{26\!\cdots\!51}a^{2}+\frac{31\!\cdots\!56}{26\!\cdots\!51}a-\frac{45\!\cdots\!33}{26\!\cdots\!51}$, $\frac{52\!\cdots\!24}{26\!\cdots\!51}a^{21}+\frac{68\!\cdots\!58}{26\!\cdots\!51}a^{20}-\frac{15\!\cdots\!43}{24\!\cdots\!41}a^{19}-\frac{23\!\cdots\!43}{26\!\cdots\!51}a^{18}+\frac{76\!\cdots\!75}{26\!\cdots\!51}a^{17}-\frac{18\!\cdots\!09}{24\!\cdots\!41}a^{16}+\frac{19\!\cdots\!51}{26\!\cdots\!51}a^{15}-\frac{11\!\cdots\!48}{26\!\cdots\!51}a^{14}-\frac{64\!\cdots\!14}{26\!\cdots\!51}a^{13}+\frac{76\!\cdots\!23}{26\!\cdots\!51}a^{12}-\frac{58\!\cdots\!85}{26\!\cdots\!51}a^{11}-\frac{51\!\cdots\!99}{26\!\cdots\!51}a^{10}-\frac{21\!\cdots\!98}{26\!\cdots\!51}a^{9}-\frac{16\!\cdots\!02}{24\!\cdots\!41}a^{8}+\frac{70\!\cdots\!00}{26\!\cdots\!51}a^{7}-\frac{40\!\cdots\!07}{26\!\cdots\!51}a^{6}-\frac{22\!\cdots\!85}{24\!\cdots\!41}a^{5}+\frac{68\!\cdots\!63}{26\!\cdots\!51}a^{4}-\frac{40\!\cdots\!30}{26\!\cdots\!51}a^{3}-\frac{16\!\cdots\!52}{26\!\cdots\!51}a^{2}+\frac{21\!\cdots\!55}{26\!\cdots\!51}a-\frac{45\!\cdots\!89}{26\!\cdots\!51}$, $\frac{44\!\cdots\!33}{26\!\cdots\!51}a^{21}+\frac{29\!\cdots\!92}{26\!\cdots\!51}a^{20}-\frac{14\!\cdots\!59}{26\!\cdots\!51}a^{19}-\frac{96\!\cdots\!96}{26\!\cdots\!51}a^{18}+\frac{57\!\cdots\!38}{26\!\cdots\!51}a^{17}+\frac{28\!\cdots\!35}{26\!\cdots\!51}a^{16}+\frac{19\!\cdots\!62}{26\!\cdots\!51}a^{15}-\frac{86\!\cdots\!37}{26\!\cdots\!51}a^{14}-\frac{97\!\cdots\!42}{26\!\cdots\!51}a^{13}+\frac{58\!\cdots\!70}{26\!\cdots\!51}a^{12}+\frac{24\!\cdots\!80}{26\!\cdots\!51}a^{11}+\frac{16\!\cdots\!73}{26\!\cdots\!51}a^{10}-\frac{16\!\cdots\!04}{26\!\cdots\!51}a^{9}-\frac{23\!\cdots\!43}{26\!\cdots\!51}a^{8}+\frac{45\!\cdots\!12}{26\!\cdots\!51}a^{7}-\frac{14\!\cdots\!67}{26\!\cdots\!51}a^{6}-\frac{22\!\cdots\!93}{26\!\cdots\!51}a^{5}+\frac{45\!\cdots\!33}{26\!\cdots\!51}a^{4}-\frac{14\!\cdots\!53}{26\!\cdots\!51}a^{3}-\frac{15\!\cdots\!11}{26\!\cdots\!51}a^{2}+\frac{94\!\cdots\!27}{26\!\cdots\!51}a-\frac{13\!\cdots\!57}{26\!\cdots\!51}$, $\frac{40\!\cdots\!45}{26\!\cdots\!51}a^{21}+\frac{13\!\cdots\!19}{26\!\cdots\!51}a^{20}-\frac{13\!\cdots\!23}{26\!\cdots\!51}a^{19}-\frac{44\!\cdots\!61}{26\!\cdots\!51}a^{18}+\frac{56\!\cdots\!33}{26\!\cdots\!51}a^{17}+\frac{96\!\cdots\!14}{26\!\cdots\!51}a^{16}+\frac{16\!\cdots\!29}{26\!\cdots\!51}a^{15}-\frac{85\!\cdots\!41}{26\!\cdots\!51}a^{14}-\frac{63\!\cdots\!09}{26\!\cdots\!51}a^{13}+\frac{56\!\cdots\!73}{26\!\cdots\!51}a^{12}+\frac{55\!\cdots\!70}{26\!\cdots\!51}a^{11}+\frac{48\!\cdots\!84}{26\!\cdots\!51}a^{10}-\frac{16\!\cdots\!46}{26\!\cdots\!51}a^{9}-\frac{16\!\cdots\!69}{26\!\cdots\!51}a^{8}+\frac{49\!\cdots\!33}{26\!\cdots\!51}a^{7}-\frac{24\!\cdots\!26}{26\!\cdots\!51}a^{6}-\frac{18\!\cdots\!08}{26\!\cdots\!51}a^{5}+\frac{47\!\cdots\!22}{26\!\cdots\!51}a^{4}-\frac{24\!\cdots\!73}{26\!\cdots\!51}a^{3}-\frac{12\!\cdots\!54}{26\!\cdots\!51}a^{2}+\frac{12\!\cdots\!49}{26\!\cdots\!51}a-\frac{23\!\cdots\!24}{26\!\cdots\!51}$, $\frac{22\!\cdots\!03}{26\!\cdots\!51}a^{21}+\frac{15\!\cdots\!71}{26\!\cdots\!51}a^{20}-\frac{72\!\cdots\!63}{26\!\cdots\!51}a^{19}-\frac{49\!\cdots\!70}{26\!\cdots\!51}a^{18}+\frac{27\!\cdots\!43}{26\!\cdots\!51}a^{17}+\frac{14\!\cdots\!97}{26\!\cdots\!51}a^{16}+\frac{97\!\cdots\!50}{26\!\cdots\!51}a^{15}-\frac{42\!\cdots\!40}{26\!\cdots\!51}a^{14}-\frac{49\!\cdots\!31}{26\!\cdots\!51}a^{13}+\frac{28\!\cdots\!95}{26\!\cdots\!51}a^{12}+\frac{11\!\cdots\!85}{24\!\cdots\!41}a^{11}+\frac{10\!\cdots\!86}{26\!\cdots\!51}a^{10}-\frac{81\!\cdots\!40}{26\!\cdots\!51}a^{9}-\frac{11\!\cdots\!92}{26\!\cdots\!51}a^{8}+\frac{21\!\cdots\!51}{26\!\cdots\!51}a^{7}-\frac{72\!\cdots\!76}{26\!\cdots\!51}a^{6}-\frac{10\!\cdots\!96}{26\!\cdots\!51}a^{5}+\frac{21\!\cdots\!71}{26\!\cdots\!51}a^{4}-\frac{67\!\cdots\!28}{26\!\cdots\!51}a^{3}-\frac{68\!\cdots\!36}{26\!\cdots\!51}a^{2}+\frac{40\!\cdots\!10}{26\!\cdots\!51}a-\frac{50\!\cdots\!95}{24\!\cdots\!41}$, $\frac{59\!\cdots\!35}{26\!\cdots\!51}a^{21}+\frac{18\!\cdots\!78}{26\!\cdots\!51}a^{20}-\frac{19\!\cdots\!31}{26\!\cdots\!51}a^{19}-\frac{56\!\cdots\!34}{24\!\cdots\!41}a^{18}+\frac{83\!\cdots\!93}{26\!\cdots\!51}a^{17}+\frac{11\!\cdots\!89}{24\!\cdots\!41}a^{16}+\frac{23\!\cdots\!53}{26\!\cdots\!51}a^{15}-\frac{12\!\cdots\!97}{26\!\cdots\!51}a^{14}-\frac{93\!\cdots\!79}{26\!\cdots\!51}a^{13}+\frac{84\!\cdots\!39}{26\!\cdots\!51}a^{12}+\frac{69\!\cdots\!92}{26\!\cdots\!51}a^{11}+\frac{35\!\cdots\!17}{26\!\cdots\!51}a^{10}-\frac{23\!\cdots\!58}{26\!\cdots\!51}a^{9}-\frac{24\!\cdots\!17}{26\!\cdots\!51}a^{8}+\frac{67\!\cdots\!60}{24\!\cdots\!41}a^{7}-\frac{36\!\cdots\!88}{26\!\cdots\!51}a^{6}-\frac{26\!\cdots\!04}{24\!\cdots\!41}a^{5}+\frac{72\!\cdots\!73}{26\!\cdots\!51}a^{4}-\frac{37\!\cdots\!69}{26\!\cdots\!51}a^{3}-\frac{19\!\cdots\!13}{26\!\cdots\!51}a^{2}+\frac{20\!\cdots\!23}{26\!\cdots\!51}a-\frac{40\!\cdots\!19}{26\!\cdots\!51}$, $\frac{34\!\cdots\!97}{24\!\cdots\!41}a^{21}+\frac{23\!\cdots\!98}{26\!\cdots\!51}a^{20}-\frac{11\!\cdots\!95}{24\!\cdots\!41}a^{19}-\frac{76\!\cdots\!30}{26\!\cdots\!51}a^{18}+\frac{49\!\cdots\!42}{26\!\cdots\!51}a^{17}+\frac{22\!\cdots\!40}{26\!\cdots\!51}a^{16}+\frac{16\!\cdots\!77}{26\!\cdots\!51}a^{15}-\frac{74\!\cdots\!49}{26\!\cdots\!51}a^{14}-\frac{80\!\cdots\!67}{26\!\cdots\!51}a^{13}+\frac{50\!\cdots\!41}{26\!\cdots\!51}a^{12}+\frac{18\!\cdots\!44}{26\!\cdots\!51}a^{11}+\frac{11\!\cdots\!11}{24\!\cdots\!41}a^{10}-\frac{14\!\cdots\!43}{26\!\cdots\!51}a^{9}-\frac{17\!\cdots\!38}{24\!\cdots\!41}a^{8}+\frac{40\!\cdots\!34}{26\!\cdots\!51}a^{7}-\frac{13\!\cdots\!35}{26\!\cdots\!51}a^{6}-\frac{19\!\cdots\!81}{26\!\cdots\!51}a^{5}+\frac{39\!\cdots\!65}{26\!\cdots\!51}a^{4}-\frac{13\!\cdots\!18}{26\!\cdots\!51}a^{3}-\frac{13\!\cdots\!79}{26\!\cdots\!51}a^{2}+\frac{86\!\cdots\!23}{26\!\cdots\!51}a-\frac{12\!\cdots\!78}{26\!\cdots\!51}$
| 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: | \( 4502089440930 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 4502089440930 \cdot 1}{2\cdot\sqrt{1543118794783990130660601200150793158656}}\cr\approx \mathstrut & 1.46326793302552 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 33*x^20 + 143*x^18 - 22*x^17 + 385*x^16 - 2222*x^15 - 902*x^14 + 14586*x^13 - 3234*x^12 + 94*x^11 - 39512*x^10 - 28006*x^9 + 136862*x^8 - 99286*x^7 - 30151*x^6 + 136136*x^5 - 99561*x^4 - 13794*x^3 + 44385*x^2 - 17182*x + 2039)
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 - 33*x^20 + 143*x^18 - 22*x^17 + 385*x^16 - 2222*x^15 - 902*x^14 + 14586*x^13 - 3234*x^12 + 94*x^11 - 39512*x^10 - 28006*x^9 + 136862*x^8 - 99286*x^7 - 30151*x^6 + 136136*x^5 - 99561*x^4 - 13794*x^3 + 44385*x^2 - 17182*x + 2039, 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 - 33*x^20 + 143*x^18 - 22*x^17 + 385*x^16 - 2222*x^15 - 902*x^14 + 14586*x^13 - 3234*x^12 + 94*x^11 - 39512*x^10 - 28006*x^9 + 136862*x^8 - 99286*x^7 - 30151*x^6 + 136136*x^5 - 99561*x^4 - 13794*x^3 + 44385*x^2 - 17182*x + 2039);
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 - 33*x^20 + 143*x^18 - 22*x^17 + 385*x^16 - 2222*x^15 - 902*x^14 + 14586*x^13 - 3234*x^12 + 94*x^11 - 39512*x^10 - 28006*x^9 + 136862*x^8 - 99286*x^7 - 30151*x^6 + 136136*x^5 - 99561*x^4 - 13794*x^3 + 44385*x^2 - 17182*x + 2039);
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
$C_2^{10}.F_{11}$ (as 22T34):
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 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ is not computed |
Intermediate fields
| 11.11.4910318845910094848.1 |
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 sibling: | 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 | R | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | $20{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $22$ | $1$ | $26$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |