Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*x + 1)
gp: K = bnfinit(y^20 - 4*y^19 + 3*y^18 - 21*y^17 + 27*y^16 - 77*y^15 + 60*y^14 - 106*y^13 + 71*y^12 - 162*y^11 + 205*y^10 - 413*y^9 + 548*y^8 - 703*y^7 + 696*y^6 - 572*y^5 + 386*y^4 - 205*y^3 + 86*y^2 - 20*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*x + 1)
\( x^{20} - 4 x^{19} + 3 x^{18} - 21 x^{17} + 27 x^{16} - 77 x^{15} + 60 x^{14} - 106 x^{13} + 71 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3598368619012125638152362241\)
\(\medspace = 11^{16}\cdot 23^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.87\) | 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: | $11^{4/5}23^{1/2}\approx 32.65713384043754$ | ||
| Ramified primes: |
\(11\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6}a^{15}-\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{2}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{6}a^{16}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{2}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{2}-\frac{1}{6}a$, $\frac{1}{6}a^{17}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{2}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}$, $\frac{1}{66}a^{18}-\frac{1}{66}a^{17}-\frac{5}{66}a^{16}+\frac{1}{33}a^{15}+\frac{7}{33}a^{14}-\frac{2}{11}a^{13}+\frac{10}{33}a^{12}-\frac{5}{11}a^{11}+\frac{13}{66}a^{10}+\frac{5}{66}a^{9}+\frac{23}{66}a^{8}+\frac{8}{33}a^{7}+\frac{19}{66}a^{6}-\frac{1}{6}a^{5}+\frac{3}{11}a^{4}+\frac{5}{33}a^{3}-\frac{2}{33}a^{2}-\frac{1}{22}a+\frac{10}{33}$, $\frac{1}{18\!\cdots\!38}a^{19}-\frac{20267615738053}{31\!\cdots\!23}a^{18}-\frac{565599225167281}{94\!\cdots\!69}a^{17}-\frac{39474093925372}{31\!\cdots\!23}a^{16}+\frac{319876137990509}{63\!\cdots\!46}a^{15}-\frac{15\!\cdots\!91}{31\!\cdots\!23}a^{14}-\frac{31\!\cdots\!45}{94\!\cdots\!69}a^{13}-\frac{705392634521349}{31\!\cdots\!23}a^{12}+\frac{41\!\cdots\!93}{18\!\cdots\!38}a^{11}-\frac{41\!\cdots\!95}{94\!\cdots\!69}a^{10}-\frac{27\!\cdots\!45}{94\!\cdots\!69}a^{9}-\frac{948462612611955}{31\!\cdots\!23}a^{8}-\frac{927809049955432}{94\!\cdots\!69}a^{7}+\frac{79182244307674}{286581989426393}a^{6}+\frac{48\!\cdots\!31}{18\!\cdots\!38}a^{5}-\frac{42\!\cdots\!55}{18\!\cdots\!38}a^{4}-\frac{48\!\cdots\!79}{18\!\cdots\!38}a^{3}-\frac{26\!\cdots\!61}{94\!\cdots\!69}a^{2}+\frac{66\!\cdots\!95}{18\!\cdots\!38}a+\frac{65333108471271}{573163978852786}$
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
$C_{2}$, which has order $2$
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: |
$\frac{32\!\cdots\!07}{18\!\cdots\!38}a^{19}-\frac{18\!\cdots\!55}{31\!\cdots\!23}a^{18}+\frac{35\!\cdots\!59}{18\!\cdots\!38}a^{17}-\frac{11\!\cdots\!04}{31\!\cdots\!23}a^{16}+\frac{85\!\cdots\!99}{31\!\cdots\!23}a^{15}-\frac{11\!\cdots\!35}{94\!\cdots\!69}a^{14}+\frac{36\!\cdots\!38}{94\!\cdots\!69}a^{13}-\frac{15\!\cdots\!32}{94\!\cdots\!69}a^{12}+\frac{65\!\cdots\!05}{18\!\cdots\!38}a^{11}-\frac{81\!\cdots\!94}{31\!\cdots\!23}a^{10}+\frac{39\!\cdots\!27}{18\!\cdots\!38}a^{9}-\frac{56\!\cdots\!67}{94\!\cdots\!69}a^{8}+\frac{38\!\cdots\!31}{63\!\cdots\!46}a^{7}-\frac{82\!\cdots\!36}{94\!\cdots\!69}a^{6}+\frac{68\!\cdots\!79}{94\!\cdots\!69}a^{5}-\frac{10\!\cdots\!31}{17\!\cdots\!58}a^{4}+\frac{65\!\cdots\!03}{18\!\cdots\!38}a^{3}-\frac{10\!\cdots\!91}{63\!\cdots\!46}a^{2}+\frac{54\!\cdots\!28}{94\!\cdots\!69}a-\frac{10\!\cdots\!80}{31\!\cdots\!23}$, $\frac{873983935115033}{94\!\cdots\!69}a^{19}-\frac{15\!\cdots\!88}{94\!\cdots\!69}a^{18}-\frac{13\!\cdots\!20}{31\!\cdots\!23}a^{17}-\frac{17\!\cdots\!15}{94\!\cdots\!69}a^{16}-\frac{96\!\cdots\!13}{63\!\cdots\!46}a^{15}-\frac{37\!\cdots\!48}{94\!\cdots\!69}a^{14}-\frac{21\!\cdots\!48}{31\!\cdots\!23}a^{13}-\frac{17\!\cdots\!88}{31\!\cdots\!23}a^{12}-\frac{27\!\cdots\!91}{31\!\cdots\!23}a^{11}-\frac{86\!\cdots\!78}{94\!\cdots\!69}a^{10}-\frac{55\!\cdots\!67}{94\!\cdots\!69}a^{9}-\frac{34\!\cdots\!25}{31\!\cdots\!23}a^{8}-\frac{59\!\cdots\!97}{63\!\cdots\!46}a^{7}+\frac{47\!\cdots\!85}{859745968279179}a^{6}-\frac{19\!\cdots\!38}{94\!\cdots\!69}a^{5}+\frac{21\!\cdots\!20}{94\!\cdots\!69}a^{4}-\frac{29\!\cdots\!79}{18\!\cdots\!38}a^{3}+\frac{12\!\cdots\!20}{94\!\cdots\!69}a^{2}-\frac{96\!\cdots\!63}{18\!\cdots\!38}a+\frac{25\!\cdots\!41}{17\!\cdots\!58}$, $\frac{362617068840773}{94\!\cdots\!69}a^{19}-\frac{28\!\cdots\!75}{18\!\cdots\!38}a^{18}+\frac{807685588837810}{94\!\cdots\!69}a^{17}-\frac{12\!\cdots\!55}{17\!\cdots\!58}a^{16}+\frac{59\!\cdots\!75}{63\!\cdots\!46}a^{15}-\frac{23\!\cdots\!46}{94\!\cdots\!69}a^{14}+\frac{14\!\cdots\!53}{94\!\cdots\!69}a^{13}-\frac{756550410345691}{286581989426393}a^{12}+\frac{13\!\cdots\!23}{94\!\cdots\!69}a^{11}-\frac{92\!\cdots\!65}{18\!\cdots\!38}a^{10}+\frac{58\!\cdots\!21}{94\!\cdots\!69}a^{9}-\frac{82\!\cdots\!57}{63\!\cdots\!46}a^{8}+\frac{30\!\cdots\!05}{18\!\cdots\!38}a^{7}-\frac{35\!\cdots\!05}{18\!\cdots\!38}a^{6}+\frac{50\!\cdots\!32}{31\!\cdots\!23}a^{5}-\frac{31\!\cdots\!30}{31\!\cdots\!23}a^{4}+\frac{12\!\cdots\!74}{286581989426393}a^{3}-\frac{74\!\cdots\!55}{18\!\cdots\!38}a^{2}-\frac{21\!\cdots\!08}{94\!\cdots\!69}a+\frac{14\!\cdots\!71}{18\!\cdots\!38}$, $\frac{21794524770232}{286581989426393}a^{19}-\frac{34\!\cdots\!51}{18\!\cdots\!38}a^{18}-\frac{19\!\cdots\!04}{94\!\cdots\!69}a^{17}-\frac{45\!\cdots\!47}{31\!\cdots\!23}a^{16}-\frac{33\!\cdots\!06}{94\!\cdots\!69}a^{15}-\frac{34\!\cdots\!21}{94\!\cdots\!69}a^{14}-\frac{31\!\cdots\!73}{94\!\cdots\!69}a^{13}-\frac{14\!\cdots\!34}{31\!\cdots\!23}a^{12}-\frac{41\!\cdots\!62}{94\!\cdots\!69}a^{11}-\frac{52\!\cdots\!87}{63\!\cdots\!46}a^{10}-\frac{29\!\cdots\!49}{94\!\cdots\!69}a^{9}-\frac{12\!\cdots\!48}{94\!\cdots\!69}a^{8}+\frac{26\!\cdots\!60}{94\!\cdots\!69}a^{7}-\frac{43\!\cdots\!83}{63\!\cdots\!46}a^{6}-\frac{30\!\cdots\!84}{859745968279179}a^{5}+\frac{54\!\cdots\!99}{63\!\cdots\!46}a^{4}-\frac{45\!\cdots\!49}{63\!\cdots\!46}a^{3}+\frac{78\!\cdots\!15}{94\!\cdots\!69}a^{2}-\frac{30\!\cdots\!60}{94\!\cdots\!69}a+\frac{17\!\cdots\!74}{94\!\cdots\!69}$, $\frac{40\!\cdots\!34}{31\!\cdots\!23}a^{19}-\frac{88\!\cdots\!47}{18\!\cdots\!38}a^{18}+\frac{22\!\cdots\!88}{94\!\cdots\!69}a^{17}-\frac{24\!\cdots\!04}{94\!\cdots\!69}a^{16}+\frac{48\!\cdots\!31}{18\!\cdots\!38}a^{15}-\frac{86\!\cdots\!78}{94\!\cdots\!69}a^{14}+\frac{14\!\cdots\!44}{31\!\cdots\!23}a^{13}-\frac{11\!\cdots\!85}{94\!\cdots\!69}a^{12}+\frac{14\!\cdots\!84}{31\!\cdots\!23}a^{11}-\frac{37\!\cdots\!73}{18\!\cdots\!38}a^{10}+\frac{17\!\cdots\!72}{94\!\cdots\!69}a^{9}-\frac{14\!\cdots\!52}{31\!\cdots\!23}a^{8}+\frac{10\!\cdots\!81}{18\!\cdots\!38}a^{7}-\frac{12\!\cdots\!87}{17\!\cdots\!58}a^{6}+\frac{60\!\cdots\!69}{94\!\cdots\!69}a^{5}-\frac{98\!\cdots\!95}{18\!\cdots\!38}a^{4}+\frac{96\!\cdots\!09}{31\!\cdots\!23}a^{3}-\frac{48\!\cdots\!02}{31\!\cdots\!23}a^{2}+\frac{10\!\cdots\!01}{18\!\cdots\!38}a-\frac{22\!\cdots\!07}{573163978852786}$, $\frac{154764617697377}{94\!\cdots\!69}a^{19}+\frac{117344123883725}{63\!\cdots\!46}a^{18}-\frac{20\!\cdots\!56}{94\!\cdots\!69}a^{17}-\frac{314789884600834}{859745968279179}a^{16}-\frac{22\!\cdots\!27}{18\!\cdots\!38}a^{15}-\frac{25\!\cdots\!27}{94\!\cdots\!69}a^{14}-\frac{34\!\cdots\!71}{94\!\cdots\!69}a^{13}-\frac{259291795403314}{286581989426393}a^{12}-\frac{40\!\cdots\!11}{94\!\cdots\!69}a^{11}-\frac{16\!\cdots\!95}{18\!\cdots\!38}a^{10}-\frac{17\!\cdots\!06}{31\!\cdots\!23}a^{9}+\frac{25\!\cdots\!73}{94\!\cdots\!69}a^{8}-\frac{23\!\cdots\!11}{18\!\cdots\!38}a^{7}+\frac{23\!\cdots\!19}{18\!\cdots\!38}a^{6}-\frac{15\!\cdots\!14}{94\!\cdots\!69}a^{5}+\frac{27\!\cdots\!43}{18\!\cdots\!38}a^{4}-\frac{70\!\cdots\!56}{859745968279179}a^{3}+\frac{51\!\cdots\!05}{94\!\cdots\!69}a^{2}-\frac{11\!\cdots\!81}{63\!\cdots\!46}a+\frac{11\!\cdots\!41}{18\!\cdots\!38}$, $\frac{97\!\cdots\!13}{94\!\cdots\!69}a^{19}-\frac{10\!\cdots\!74}{286581989426393}a^{18}+\frac{11\!\cdots\!56}{94\!\cdots\!69}a^{17}-\frac{39\!\cdots\!79}{18\!\cdots\!38}a^{16}+\frac{16\!\cdots\!62}{94\!\cdots\!69}a^{15}-\frac{22\!\cdots\!26}{31\!\cdots\!23}a^{14}+\frac{24\!\cdots\!15}{94\!\cdots\!69}a^{13}-\frac{90\!\cdots\!24}{94\!\cdots\!69}a^{12}+\frac{23\!\cdots\!53}{94\!\cdots\!69}a^{11}-\frac{14\!\cdots\!03}{94\!\cdots\!69}a^{10}+\frac{41\!\cdots\!66}{31\!\cdots\!23}a^{9}-\frac{67\!\cdots\!37}{18\!\cdots\!38}a^{8}+\frac{12\!\cdots\!28}{31\!\cdots\!23}a^{7}-\frac{50\!\cdots\!30}{94\!\cdots\!69}a^{6}+\frac{42\!\cdots\!13}{94\!\cdots\!69}a^{5}-\frac{68\!\cdots\!27}{18\!\cdots\!38}a^{4}+\frac{20\!\cdots\!83}{94\!\cdots\!69}a^{3}-\frac{65\!\cdots\!39}{63\!\cdots\!46}a^{2}+\frac{23\!\cdots\!31}{63\!\cdots\!46}a-\frac{17\!\cdots\!31}{94\!\cdots\!69}$, $\frac{45\!\cdots\!01}{18\!\cdots\!38}a^{19}-\frac{84\!\cdots\!49}{94\!\cdots\!69}a^{18}+\frac{10\!\cdots\!85}{18\!\cdots\!38}a^{17}-\frac{16\!\cdots\!87}{31\!\cdots\!23}a^{16}+\frac{85\!\cdots\!87}{17\!\cdots\!58}a^{15}-\frac{15\!\cdots\!46}{859745968279179}a^{14}+\frac{92\!\cdots\!74}{94\!\cdots\!69}a^{13}-\frac{82\!\cdots\!41}{31\!\cdots\!23}a^{12}+\frac{17\!\cdots\!55}{18\!\cdots\!38}a^{11}-\frac{38\!\cdots\!25}{94\!\cdots\!69}a^{10}+\frac{70\!\cdots\!33}{18\!\cdots\!38}a^{9}-\frac{91\!\cdots\!10}{94\!\cdots\!69}a^{8}+\frac{10\!\cdots\!01}{94\!\cdots\!69}a^{7}-\frac{48\!\cdots\!74}{31\!\cdots\!23}a^{6}+\frac{12\!\cdots\!45}{94\!\cdots\!69}a^{5}-\frac{21\!\cdots\!97}{18\!\cdots\!38}a^{4}+\frac{65\!\cdots\!04}{94\!\cdots\!69}a^{3}-\frac{22\!\cdots\!03}{63\!\cdots\!46}a^{2}+\frac{24\!\cdots\!25}{18\!\cdots\!38}a-\frac{28\!\cdots\!39}{18\!\cdots\!38}$, $\frac{102396411517369}{573163978852786}a^{19}-\frac{882211453950712}{31\!\cdots\!23}a^{18}-\frac{15\!\cdots\!11}{18\!\cdots\!38}a^{17}-\frac{34\!\cdots\!47}{94\!\cdots\!69}a^{16}-\frac{38\!\cdots\!97}{94\!\cdots\!69}a^{15}-\frac{86\!\cdots\!10}{94\!\cdots\!69}a^{14}-\frac{16\!\cdots\!59}{94\!\cdots\!69}a^{13}-\frac{49\!\cdots\!09}{31\!\cdots\!23}a^{12}-\frac{52\!\cdots\!15}{18\!\cdots\!38}a^{11}-\frac{88\!\cdots\!82}{31\!\cdots\!23}a^{10}-\frac{54\!\cdots\!15}{18\!\cdots\!38}a^{9}-\frac{32\!\cdots\!98}{94\!\cdots\!69}a^{8}-\frac{25\!\cdots\!49}{63\!\cdots\!46}a^{7}-\frac{11\!\cdots\!33}{94\!\cdots\!69}a^{6}-\frac{57\!\cdots\!87}{859745968279179}a^{5}+\frac{80\!\cdots\!67}{18\!\cdots\!38}a^{4}-\frac{10\!\cdots\!67}{18\!\cdots\!38}a^{3}+\frac{21\!\cdots\!55}{63\!\cdots\!46}a^{2}-\frac{57\!\cdots\!58}{31\!\cdots\!23}a+\frac{82\!\cdots\!76}{94\!\cdots\!69}$, $\frac{10\!\cdots\!75}{18\!\cdots\!38}a^{19}-\frac{12\!\cdots\!27}{63\!\cdots\!46}a^{18}+\frac{15\!\cdots\!61}{18\!\cdots\!38}a^{17}-\frac{10\!\cdots\!91}{94\!\cdots\!69}a^{16}+\frac{19\!\cdots\!55}{18\!\cdots\!38}a^{15}-\frac{36\!\cdots\!12}{94\!\cdots\!69}a^{14}+\frac{55\!\cdots\!99}{31\!\cdots\!23}a^{13}-\frac{48\!\cdots\!57}{94\!\cdots\!69}a^{12}+\frac{12\!\cdots\!95}{63\!\cdots\!46}a^{11}-\frac{15\!\cdots\!75}{18\!\cdots\!38}a^{10}+\frac{15\!\cdots\!93}{18\!\cdots\!38}a^{9}-\frac{61\!\cdots\!50}{31\!\cdots\!23}a^{8}+\frac{19\!\cdots\!49}{859745968279179}a^{7}-\frac{18\!\cdots\!79}{63\!\cdots\!46}a^{6}+\frac{24\!\cdots\!41}{94\!\cdots\!69}a^{5}-\frac{66\!\cdots\!57}{31\!\cdots\!23}a^{4}+\frac{24\!\cdots\!37}{18\!\cdots\!38}a^{3}-\frac{12\!\cdots\!79}{18\!\cdots\!38}a^{2}+\frac{44\!\cdots\!71}{18\!\cdots\!38}a-\frac{46\!\cdots\!33}{18\!\cdots\!38}$, $\frac{174839108756801}{859745968279179}a^{19}-\frac{15\!\cdots\!37}{18\!\cdots\!38}a^{18}+\frac{55\!\cdots\!88}{94\!\cdots\!69}a^{17}-\frac{79\!\cdots\!05}{18\!\cdots\!38}a^{16}+\frac{10\!\cdots\!83}{18\!\cdots\!38}a^{15}-\frac{14\!\cdots\!03}{94\!\cdots\!69}a^{14}+\frac{35\!\cdots\!88}{31\!\cdots\!23}a^{13}-\frac{64\!\cdots\!74}{31\!\cdots\!23}a^{12}+\frac{39\!\cdots\!92}{31\!\cdots\!23}a^{11}-\frac{60\!\cdots\!11}{18\!\cdots\!38}a^{10}+\frac{37\!\cdots\!59}{94\!\cdots\!69}a^{9}-\frac{15\!\cdots\!81}{18\!\cdots\!38}a^{8}+\frac{20\!\cdots\!11}{18\!\cdots\!38}a^{7}-\frac{25\!\cdots\!03}{18\!\cdots\!38}a^{6}+\frac{11\!\cdots\!02}{859745968279179}a^{5}-\frac{99\!\cdots\!28}{94\!\cdots\!69}a^{4}+\frac{63\!\cdots\!34}{94\!\cdots\!69}a^{3}-\frac{65\!\cdots\!17}{18\!\cdots\!38}a^{2}+\frac{43\!\cdots\!09}{31\!\cdots\!23}a-\frac{51\!\cdots\!31}{18\!\cdots\!38}$
| 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: | \( 436538.984162 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 436538.984162 \cdot 2}{2\cdot\sqrt{3598368619012125638152362241}}\cr\approx \mathstrut & 0.282832386255 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*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^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*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^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*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^20 - 4*x^19 + 3*x^18 - 21*x^17 + 27*x^16 - 77*x^15 + 60*x^14 - 106*x^13 + 71*x^12 - 162*x^11 + 205*x^10 - 413*x^9 + 548*x^8 - 703*x^7 + 696*x^6 - 572*x^5 + 386*x^4 - 205*x^3 + 86*x^2 - 20*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
$C_2^4:C_5$ (as 20T23):
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 80 |
| The 8 conjugacy class representatives for $C_2^4:C_5$ |
| Character table for $C_2^4:C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.2.113395848049.1, 10.6.113395848049.1, 10.6.59986403617921.1 |
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 10 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.113395848049.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |