Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837)
gp: K = bnfinit(y^20 - 8*y^19 + 55*y^18 - 247*y^17 + 972*y^16 - 3008*y^15 + 8238*y^14 - 18705*y^13 + 37604*y^12 - 63382*y^11 + 93279*y^10 - 111873*y^9 + 112680*y^8 - 83184*y^7 + 45374*y^6 - 11082*y^5 - 5227*y^4 + 8834*y^3 - 3846*y^2 + 1831*y + 2837, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837)
\( x^{20} - 8 x^{19} + 55 x^{18} - 247 x^{17} + 972 x^{16} - 3008 x^{15} + 8238 x^{14} - 18705 x^{13} + \cdots + 2837 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(203681981950950327645213870961\)
\(\medspace = 11^{16}\cdot 1451^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.20\) | 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: | $11^{4/5}1451^{1/2}\approx 259.3867898202987$ | ||
| Ramified primes: |
\(11\), \(1451\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{33}a^{15}-\frac{2}{11}a^{14}+\frac{5}{11}a^{13}+\frac{5}{11}a^{12}-\frac{4}{33}a^{11}-\frac{1}{3}a^{10}-\frac{2}{11}a^{9}+\frac{4}{33}a^{8}-\frac{5}{11}a^{7}-\frac{5}{33}a^{6}-\frac{4}{33}a^{5}-\frac{16}{33}a^{4}-\frac{1}{33}a^{3}+\frac{1}{11}a^{2}-\frac{13}{33}a-\frac{1}{33}$, $\frac{1}{33}a^{16}+\frac{4}{11}a^{14}+\frac{2}{11}a^{13}-\frac{13}{33}a^{12}-\frac{2}{33}a^{11}-\frac{2}{11}a^{10}+\frac{1}{33}a^{9}+\frac{3}{11}a^{8}+\frac{4}{33}a^{7}-\frac{1}{33}a^{6}-\frac{7}{33}a^{5}+\frac{2}{33}a^{4}-\frac{1}{11}a^{3}+\frac{5}{33}a^{2}-\frac{13}{33}a-\frac{2}{11}$, $\frac{1}{33}a^{17}+\frac{4}{11}a^{14}+\frac{5}{33}a^{13}+\frac{16}{33}a^{12}+\frac{3}{11}a^{11}+\frac{1}{33}a^{10}+\frac{5}{11}a^{9}-\frac{1}{3}a^{8}+\frac{14}{33}a^{7}-\frac{13}{33}a^{6}-\frac{16}{33}a^{5}-\frac{3}{11}a^{4}-\frac{16}{33}a^{3}-\frac{16}{33}a^{2}-\frac{5}{11}a+\frac{4}{11}$, $\frac{1}{3597}a^{18}-\frac{14}{3597}a^{17}-\frac{10}{1199}a^{16}+\frac{10}{3597}a^{15}+\frac{314}{3597}a^{14}-\frac{31}{109}a^{13}+\frac{1531}{3597}a^{12}-\frac{173}{1199}a^{11}-\frac{457}{3597}a^{10}-\frac{833}{3597}a^{9}-\frac{127}{327}a^{8}-\frac{101}{3597}a^{7}+\frac{470}{3597}a^{6}+\frac{1588}{3597}a^{5}+\frac{16}{3597}a^{4}+\frac{375}{1199}a^{3}-\frac{838}{3597}a^{2}+\frac{118}{327}a+\frac{278}{3597}$, $\frac{1}{20\!\cdots\!21}a^{19}+\frac{20\!\cdots\!25}{20\!\cdots\!21}a^{18}+\frac{15\!\cdots\!76}{20\!\cdots\!21}a^{17}-\frac{13\!\cdots\!25}{68\!\cdots\!07}a^{16}-\frac{99\!\cdots\!74}{68\!\cdots\!07}a^{15}+\frac{41\!\cdots\!67}{20\!\cdots\!21}a^{14}+\frac{20\!\cdots\!37}{20\!\cdots\!21}a^{13}-\frac{88\!\cdots\!09}{18\!\cdots\!11}a^{12}-\frac{98\!\cdots\!39}{20\!\cdots\!21}a^{11}+\frac{82\!\cdots\!70}{68\!\cdots\!07}a^{10}+\frac{10\!\cdots\!99}{20\!\cdots\!21}a^{9}+\frac{58\!\cdots\!18}{20\!\cdots\!21}a^{8}+\frac{10\!\cdots\!25}{20\!\cdots\!21}a^{7}+\frac{24\!\cdots\!76}{20\!\cdots\!21}a^{6}+\frac{32\!\cdots\!33}{68\!\cdots\!07}a^{5}-\frac{79\!\cdots\!10}{18\!\cdots\!11}a^{4}+\frac{79\!\cdots\!88}{20\!\cdots\!21}a^{3}-\frac{26\!\cdots\!78}{18\!\cdots\!69}a^{2}-\frac{64\!\cdots\!65}{20\!\cdots\!21}a-\frac{69\!\cdots\!77}{20\!\cdots\!21}$
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 $1$
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: | $9$ | 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: |
$\frac{33\!\cdots\!55}{71\!\cdots\!97}a^{19}+\frac{10\!\cdots\!77}{78\!\cdots\!67}a^{18}-\frac{66\!\cdots\!27}{78\!\cdots\!67}a^{17}+\frac{49\!\cdots\!14}{78\!\cdots\!67}a^{16}-\frac{64\!\cdots\!75}{26\!\cdots\!89}a^{15}+\frac{74\!\cdots\!91}{78\!\cdots\!67}a^{14}-\frac{20\!\cdots\!48}{78\!\cdots\!67}a^{13}+\frac{52\!\cdots\!86}{78\!\cdots\!67}a^{12}-\frac{99\!\cdots\!50}{78\!\cdots\!67}a^{11}+\frac{54\!\cdots\!40}{23\!\cdots\!99}a^{10}-\frac{21\!\cdots\!06}{71\!\cdots\!97}a^{9}+\frac{27\!\cdots\!00}{78\!\cdots\!67}a^{8}-\frac{55\!\cdots\!49}{26\!\cdots\!89}a^{7}+\frac{22\!\cdots\!43}{26\!\cdots\!89}a^{6}+\frac{13\!\cdots\!28}{78\!\cdots\!67}a^{5}-\frac{86\!\cdots\!84}{78\!\cdots\!67}a^{4}+\frac{13\!\cdots\!67}{78\!\cdots\!67}a^{3}+\frac{89\!\cdots\!76}{71\!\cdots\!97}a^{2}-\frac{36\!\cdots\!91}{78\!\cdots\!67}a+\frac{94\!\cdots\!24}{78\!\cdots\!67}$, $\frac{11\!\cdots\!69}{78\!\cdots\!67}a^{19}-\frac{34\!\cdots\!66}{78\!\cdots\!67}a^{18}+\frac{78\!\cdots\!13}{26\!\cdots\!89}a^{17}-\frac{48\!\cdots\!18}{26\!\cdots\!89}a^{16}+\frac{58\!\cdots\!09}{78\!\cdots\!67}a^{15}-\frac{21\!\cdots\!76}{78\!\cdots\!67}a^{14}+\frac{59\!\cdots\!30}{78\!\cdots\!67}a^{13}-\frac{15\!\cdots\!24}{78\!\cdots\!67}a^{12}+\frac{10\!\cdots\!64}{26\!\cdots\!89}a^{11}-\frac{17\!\cdots\!73}{23\!\cdots\!99}a^{10}+\frac{82\!\cdots\!11}{78\!\cdots\!67}a^{9}-\frac{10\!\cdots\!79}{78\!\cdots\!67}a^{8}+\frac{32\!\cdots\!08}{26\!\cdots\!89}a^{7}-\frac{22\!\cdots\!49}{23\!\cdots\!99}a^{6}+\frac{68\!\cdots\!42}{23\!\cdots\!99}a^{5}-\frac{22\!\cdots\!11}{78\!\cdots\!67}a^{4}-\frac{54\!\cdots\!08}{78\!\cdots\!67}a^{3}+\frac{38\!\cdots\!49}{23\!\cdots\!99}a^{2}-\frac{36\!\cdots\!99}{78\!\cdots\!67}a-\frac{89\!\cdots\!49}{78\!\cdots\!67}$, $\frac{91\!\cdots\!22}{78\!\cdots\!67}a^{19}-\frac{55\!\cdots\!88}{78\!\cdots\!67}a^{18}+\frac{37\!\cdots\!86}{78\!\cdots\!67}a^{17}-\frac{13\!\cdots\!54}{78\!\cdots\!67}a^{16}+\frac{49\!\cdots\!04}{78\!\cdots\!67}a^{15}-\frac{12\!\cdots\!83}{78\!\cdots\!67}a^{14}+\frac{29\!\cdots\!78}{78\!\cdots\!67}a^{13}-\frac{46\!\cdots\!69}{78\!\cdots\!67}a^{12}+\frac{22\!\cdots\!58}{26\!\cdots\!89}a^{11}-\frac{34\!\cdots\!67}{78\!\cdots\!67}a^{10}-\frac{35\!\cdots\!88}{71\!\cdots\!97}a^{9}+\frac{25\!\cdots\!97}{78\!\cdots\!67}a^{8}-\frac{13\!\cdots\!31}{23\!\cdots\!99}a^{7}+\frac{64\!\cdots\!18}{78\!\cdots\!67}a^{6}-\frac{16\!\cdots\!50}{26\!\cdots\!89}a^{5}+\frac{26\!\cdots\!29}{78\!\cdots\!67}a^{4}+\frac{52\!\cdots\!46}{26\!\cdots\!89}a^{3}-\frac{17\!\cdots\!76}{78\!\cdots\!67}a^{2}+\frac{98\!\cdots\!48}{78\!\cdots\!67}a-\frac{97\!\cdots\!42}{26\!\cdots\!89}$, $\frac{91\!\cdots\!22}{78\!\cdots\!67}a^{19}-\frac{55\!\cdots\!88}{78\!\cdots\!67}a^{18}+\frac{37\!\cdots\!86}{78\!\cdots\!67}a^{17}-\frac{13\!\cdots\!54}{78\!\cdots\!67}a^{16}+\frac{49\!\cdots\!04}{78\!\cdots\!67}a^{15}-\frac{12\!\cdots\!83}{78\!\cdots\!67}a^{14}+\frac{29\!\cdots\!78}{78\!\cdots\!67}a^{13}-\frac{46\!\cdots\!69}{78\!\cdots\!67}a^{12}+\frac{22\!\cdots\!58}{26\!\cdots\!89}a^{11}-\frac{34\!\cdots\!67}{78\!\cdots\!67}a^{10}-\frac{35\!\cdots\!88}{71\!\cdots\!97}a^{9}+\frac{25\!\cdots\!97}{78\!\cdots\!67}a^{8}-\frac{13\!\cdots\!31}{23\!\cdots\!99}a^{7}+\frac{64\!\cdots\!18}{78\!\cdots\!67}a^{6}-\frac{16\!\cdots\!50}{26\!\cdots\!89}a^{5}+\frac{26\!\cdots\!29}{78\!\cdots\!67}a^{4}+\frac{52\!\cdots\!46}{26\!\cdots\!89}a^{3}-\frac{17\!\cdots\!76}{78\!\cdots\!67}a^{2}+\frac{98\!\cdots\!48}{78\!\cdots\!67}a+\frac{16\!\cdots\!47}{26\!\cdots\!89}$, $\frac{33\!\cdots\!75}{68\!\cdots\!07}a^{19}-\frac{19\!\cdots\!96}{68\!\cdots\!07}a^{18}+\frac{39\!\cdots\!73}{20\!\cdots\!21}a^{17}-\frac{13\!\cdots\!69}{20\!\cdots\!21}a^{16}+\frac{16\!\cdots\!09}{68\!\cdots\!07}a^{15}-\frac{35\!\cdots\!08}{62\!\cdots\!37}a^{14}+\frac{27\!\cdots\!06}{20\!\cdots\!21}a^{13}-\frac{14\!\cdots\!33}{68\!\cdots\!07}a^{12}+\frac{63\!\cdots\!23}{20\!\cdots\!21}a^{11}-\frac{41\!\cdots\!10}{20\!\cdots\!21}a^{10}+\frac{20\!\cdots\!63}{20\!\cdots\!21}a^{9}+\frac{78\!\cdots\!56}{20\!\cdots\!21}a^{8}-\frac{15\!\cdots\!36}{68\!\cdots\!07}a^{7}+\frac{12\!\cdots\!02}{20\!\cdots\!21}a^{6}+\frac{28\!\cdots\!90}{20\!\cdots\!21}a^{5}-\frac{40\!\cdots\!38}{20\!\cdots\!21}a^{4}+\frac{36\!\cdots\!89}{20\!\cdots\!21}a^{3}-\frac{86\!\cdots\!06}{20\!\cdots\!21}a^{2}+\frac{28\!\cdots\!81}{20\!\cdots\!21}a+\frac{22\!\cdots\!41}{68\!\cdots\!07}$, $\frac{27\!\cdots\!21}{20\!\cdots\!21}a^{19}-\frac{24\!\cdots\!55}{20\!\cdots\!21}a^{18}+\frac{16\!\cdots\!43}{20\!\cdots\!21}a^{17}-\frac{79\!\cdots\!52}{20\!\cdots\!21}a^{16}+\frac{10\!\cdots\!52}{68\!\cdots\!07}a^{15}-\frac{10\!\cdots\!41}{20\!\cdots\!21}a^{14}+\frac{25\!\cdots\!17}{18\!\cdots\!11}a^{13}-\frac{66\!\cdots\!11}{20\!\cdots\!21}a^{12}+\frac{13\!\cdots\!85}{20\!\cdots\!21}a^{11}-\frac{80\!\cdots\!53}{68\!\cdots\!07}a^{10}+\frac{36\!\cdots\!06}{20\!\cdots\!21}a^{9}-\frac{45\!\cdots\!20}{20\!\cdots\!21}a^{8}+\frac{15\!\cdots\!76}{68\!\cdots\!07}a^{7}-\frac{12\!\cdots\!03}{68\!\cdots\!07}a^{6}+\frac{17\!\cdots\!69}{20\!\cdots\!21}a^{5}-\frac{27\!\cdots\!20}{18\!\cdots\!11}a^{4}-\frac{48\!\cdots\!16}{20\!\cdots\!21}a^{3}+\frac{20\!\cdots\!57}{20\!\cdots\!21}a^{2}-\frac{21\!\cdots\!55}{20\!\cdots\!21}a-\frac{90\!\cdots\!20}{20\!\cdots\!21}$, $\frac{25\!\cdots\!79}{20\!\cdots\!21}a^{19}-\frac{20\!\cdots\!69}{20\!\cdots\!21}a^{18}+\frac{14\!\cdots\!99}{20\!\cdots\!21}a^{17}-\frac{65\!\cdots\!53}{20\!\cdots\!21}a^{16}+\frac{79\!\cdots\!31}{62\!\cdots\!23}a^{15}-\frac{81\!\cdots\!01}{20\!\cdots\!21}a^{14}+\frac{74\!\cdots\!93}{68\!\cdots\!07}a^{13}-\frac{51\!\cdots\!69}{20\!\cdots\!21}a^{12}+\frac{31\!\cdots\!33}{62\!\cdots\!37}a^{11}-\frac{18\!\cdots\!19}{20\!\cdots\!21}a^{10}+\frac{90\!\cdots\!33}{68\!\cdots\!07}a^{9}-\frac{11\!\cdots\!16}{68\!\cdots\!07}a^{8}+\frac{12\!\cdots\!63}{68\!\cdots\!07}a^{7}-\frac{31\!\cdots\!54}{20\!\cdots\!21}a^{6}+\frac{67\!\cdots\!88}{68\!\cdots\!07}a^{5}-\frac{40\!\cdots\!53}{68\!\cdots\!07}a^{4}+\frac{25\!\cdots\!82}{18\!\cdots\!11}a^{3}-\frac{96\!\cdots\!14}{20\!\cdots\!21}a^{2}-\frac{26\!\cdots\!31}{68\!\cdots\!07}a+\frac{27\!\cdots\!30}{20\!\cdots\!21}$, $\frac{25\!\cdots\!24}{26\!\cdots\!89}a^{19}-\frac{57\!\cdots\!76}{78\!\cdots\!67}a^{18}+\frac{37\!\cdots\!63}{78\!\cdots\!67}a^{17}-\frac{52\!\cdots\!53}{26\!\cdots\!89}a^{16}+\frac{19\!\cdots\!37}{26\!\cdots\!89}a^{15}-\frac{16\!\cdots\!19}{78\!\cdots\!67}a^{14}+\frac{12\!\cdots\!51}{26\!\cdots\!89}a^{13}-\frac{71\!\cdots\!02}{78\!\cdots\!67}a^{12}+\frac{10\!\cdots\!78}{78\!\cdots\!67}a^{11}-\frac{10\!\cdots\!04}{78\!\cdots\!67}a^{10}+\frac{11\!\cdots\!73}{78\!\cdots\!67}a^{9}+\frac{76\!\cdots\!35}{26\!\cdots\!89}a^{8}-\frac{55\!\cdots\!41}{78\!\cdots\!67}a^{7}+\frac{82\!\cdots\!98}{78\!\cdots\!67}a^{6}-\frac{77\!\cdots\!88}{78\!\cdots\!67}a^{5}+\frac{40\!\cdots\!79}{78\!\cdots\!67}a^{4}+\frac{33\!\cdots\!52}{78\!\cdots\!67}a^{3}-\frac{22\!\cdots\!15}{78\!\cdots\!67}a^{2}+\frac{56\!\cdots\!90}{26\!\cdots\!89}a-\frac{17\!\cdots\!16}{26\!\cdots\!89}$, $\frac{15\!\cdots\!92}{18\!\cdots\!11}a^{19}+\frac{15\!\cdots\!54}{20\!\cdots\!21}a^{18}-\frac{29\!\cdots\!89}{20\!\cdots\!21}a^{17}+\frac{13\!\cdots\!21}{68\!\cdots\!07}a^{16}-\frac{20\!\cdots\!02}{20\!\cdots\!21}a^{15}+\frac{93\!\cdots\!65}{20\!\cdots\!21}a^{14}-\frac{10\!\cdots\!99}{68\!\cdots\!07}a^{13}+\frac{28\!\cdots\!97}{68\!\cdots\!07}a^{12}-\frac{19\!\cdots\!43}{20\!\cdots\!21}a^{11}+\frac{12\!\cdots\!13}{62\!\cdots\!23}a^{10}-\frac{65\!\cdots\!72}{20\!\cdots\!21}a^{9}+\frac{91\!\cdots\!76}{20\!\cdots\!21}a^{8}-\frac{99\!\cdots\!98}{20\!\cdots\!21}a^{7}+\frac{83\!\cdots\!60}{20\!\cdots\!21}a^{6}-\frac{12\!\cdots\!15}{68\!\cdots\!07}a^{5}-\frac{12\!\cdots\!35}{68\!\cdots\!07}a^{4}+\frac{17\!\cdots\!49}{20\!\cdots\!21}a^{3}-\frac{76\!\cdots\!33}{20\!\cdots\!21}a^{2}+\frac{29\!\cdots\!04}{68\!\cdots\!07}a+\frac{11\!\cdots\!92}{68\!\cdots\!07}$
| 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: | \( 2227699.13388 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 2227699.13388 \cdot 1}{2\cdot\sqrt{203681981950950327645213870961}}\cr\approx \mathstrut & 0.236673110064 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837)
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 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837, 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 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837);
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 - 8*x^19 + 55*x^18 - 247*x^17 + 972*x^16 - 3008*x^15 + 8238*x^14 - 18705*x^13 + 37604*x^12 - 63382*x^11 + 93279*x^10 - 111873*x^9 + 112680*x^8 - 83184*x^7 + 45374*x^6 - 11082*x^5 - 5227*x^4 + 8834*x^3 - 3846*x^2 + 1831*x + 2837);
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
$C_2\wr C_5$ (as 20T41):
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 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.0.311034736331.1 x2, 10.10.451311402416281.1 |
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 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.0.311034736331.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 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 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(1451\)
| 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |