Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 + 16*x^19 + 19*x^18 - 103*x^17 + 160*x^16 - 174*x^15 + 66*x^14 - 206*x^13 + 39*x^12 + 4498*x^11 - 10908*x^10 + 7867*x^9 + 6766*x^8 - 19649*x^7 + 7824*x^6 + 13595*x^5 - 6802*x^4 - 4602*x^3 + 1044*x^2 + 735*x + 79)
gp: K = bnfinit(y^21 - 8*y^20 + 16*y^19 + 19*y^18 - 103*y^17 + 160*y^16 - 174*y^15 + 66*y^14 - 206*y^13 + 39*y^12 + 4498*y^11 - 10908*y^10 + 7867*y^9 + 6766*y^8 - 19649*y^7 + 7824*y^6 + 13595*y^5 - 6802*y^4 - 4602*y^3 + 1044*y^2 + 735*y + 79, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 8*x^20 + 16*x^19 + 19*x^18 - 103*x^17 + 160*x^16 - 174*x^15 + 66*x^14 - 206*x^13 + 39*x^12 + 4498*x^11 - 10908*x^10 + 7867*x^9 + 6766*x^8 - 19649*x^7 + 7824*x^6 + 13595*x^5 - 6802*x^4 - 4602*x^3 + 1044*x^2 + 735*x + 79);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 8*x^20 + 16*x^19 + 19*x^18 - 103*x^17 + 160*x^16 - 174*x^15 + 66*x^14 - 206*x^13 + 39*x^12 + 4498*x^11 - 10908*x^10 + 7867*x^9 + 6766*x^8 - 19649*x^7 + 7824*x^6 + 13595*x^5 - 6802*x^4 - 4602*x^3 + 1044*x^2 + 735*x + 79)
\( x^{21} - 8 x^{20} + 16 x^{19} + 19 x^{18} - 103 x^{17} + 160 x^{16} - 174 x^{15} + 66 x^{14} - 206 x^{13} + \cdots + 79 \)
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: | $[15, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-376429776359836117500418449727791\)
\(\medspace = -\,3^{7}\cdot 107^{8}\cdot 21557^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.58\) | 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: | $3^{1/2}107^{1/2}21557^{1/2}\approx 2630.550702799701$ | ||
| Ramified primes: |
\(3\), \(107\), \(21557\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-64671}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{16}+\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{19}+\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{83\!\cdots\!19}a^{20}+\frac{22\!\cdots\!22}{27\!\cdots\!73}a^{19}+\frac{47\!\cdots\!52}{83\!\cdots\!19}a^{18}-\frac{97\!\cdots\!11}{27\!\cdots\!73}a^{17}-\frac{40\!\cdots\!37}{83\!\cdots\!19}a^{16}+\frac{22\!\cdots\!87}{83\!\cdots\!19}a^{15}-\frac{32\!\cdots\!83}{83\!\cdots\!19}a^{14}-\frac{30\!\cdots\!01}{83\!\cdots\!19}a^{13}+\frac{34\!\cdots\!09}{83\!\cdots\!19}a^{12}+\frac{39\!\cdots\!48}{83\!\cdots\!19}a^{11}+\frac{77\!\cdots\!50}{27\!\cdots\!73}a^{10}+\frac{18\!\cdots\!45}{92\!\cdots\!91}a^{9}-\frac{37\!\cdots\!47}{83\!\cdots\!19}a^{8}-\frac{37\!\cdots\!49}{92\!\cdots\!91}a^{7}-\frac{36\!\cdots\!96}{83\!\cdots\!19}a^{6}+\frac{39\!\cdots\!35}{83\!\cdots\!19}a^{5}-\frac{64\!\cdots\!15}{92\!\cdots\!91}a^{4}-\frac{31\!\cdots\!94}{83\!\cdots\!19}a^{3}+\frac{39\!\cdots\!05}{83\!\cdots\!19}a^{2}-\frac{61\!\cdots\!93}{83\!\cdots\!19}a-\frac{25\!\cdots\!76}{83\!\cdots\!19}$
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: |
$\frac{16\!\cdots\!80}{83\!\cdots\!19}a^{20}-\frac{14\!\cdots\!86}{92\!\cdots\!91}a^{19}+\frac{31\!\cdots\!06}{83\!\cdots\!19}a^{18}+\frac{18\!\cdots\!49}{92\!\cdots\!91}a^{17}-\frac{17\!\cdots\!99}{83\!\cdots\!19}a^{16}+\frac{33\!\cdots\!41}{83\!\cdots\!19}a^{15}-\frac{42\!\cdots\!74}{83\!\cdots\!19}a^{14}+\frac{29\!\cdots\!25}{83\!\cdots\!19}a^{13}-\frac{46\!\cdots\!36}{83\!\cdots\!19}a^{12}+\frac{25\!\cdots\!59}{83\!\cdots\!19}a^{11}+\frac{78\!\cdots\!98}{92\!\cdots\!91}a^{10}-\frac{68\!\cdots\!64}{27\!\cdots\!73}a^{9}+\frac{21\!\cdots\!21}{83\!\cdots\!19}a^{8}+\frac{44\!\cdots\!00}{27\!\cdots\!73}a^{7}-\frac{31\!\cdots\!56}{83\!\cdots\!19}a^{6}+\frac{26\!\cdots\!66}{83\!\cdots\!19}a^{5}+\frac{11\!\cdots\!93}{92\!\cdots\!91}a^{4}-\frac{16\!\cdots\!65}{83\!\cdots\!19}a^{3}-\frac{63\!\cdots\!41}{83\!\cdots\!19}a^{2}+\frac{25\!\cdots\!03}{83\!\cdots\!19}a+\frac{17\!\cdots\!87}{83\!\cdots\!19}$, $\frac{24\!\cdots\!78}{83\!\cdots\!19}a^{20}-\frac{21\!\cdots\!02}{92\!\cdots\!91}a^{19}+\frac{39\!\cdots\!05}{83\!\cdots\!19}a^{18}+\frac{50\!\cdots\!71}{92\!\cdots\!91}a^{17}-\frac{25\!\cdots\!57}{83\!\cdots\!19}a^{16}+\frac{41\!\cdots\!49}{83\!\cdots\!19}a^{15}-\frac{43\!\cdots\!09}{83\!\cdots\!19}a^{14}+\frac{12\!\cdots\!39}{83\!\cdots\!19}a^{13}-\frac{39\!\cdots\!76}{83\!\cdots\!19}a^{12}-\frac{57\!\cdots\!75}{83\!\cdots\!19}a^{11}+\frac{12\!\cdots\!65}{92\!\cdots\!91}a^{10}-\frac{91\!\cdots\!68}{27\!\cdots\!73}a^{9}+\frac{20\!\cdots\!68}{83\!\cdots\!19}a^{8}+\frac{61\!\cdots\!46}{27\!\cdots\!73}a^{7}-\frac{55\!\cdots\!25}{83\!\cdots\!19}a^{6}+\frac{29\!\cdots\!01}{83\!\cdots\!19}a^{5}+\frac{32\!\cdots\!74}{92\!\cdots\!91}a^{4}-\frac{23\!\cdots\!57}{83\!\cdots\!19}a^{3}-\frac{42\!\cdots\!68}{83\!\cdots\!19}a^{2}+\frac{31\!\cdots\!63}{83\!\cdots\!19}a+\frac{40\!\cdots\!46}{83\!\cdots\!19}$, $\frac{60\!\cdots\!30}{83\!\cdots\!19}a^{20}-\frac{62\!\cdots\!28}{92\!\cdots\!91}a^{19}+\frac{15\!\cdots\!26}{83\!\cdots\!19}a^{18}-\frac{23\!\cdots\!55}{92\!\cdots\!91}a^{17}-\frac{68\!\cdots\!58}{83\!\cdots\!19}a^{16}+\frac{16\!\cdots\!40}{83\!\cdots\!19}a^{15}-\frac{24\!\cdots\!82}{83\!\cdots\!19}a^{14}+\frac{24\!\cdots\!34}{83\!\cdots\!19}a^{13}-\frac{29\!\cdots\!89}{83\!\cdots\!19}a^{12}+\frac{29\!\cdots\!65}{83\!\cdots\!19}a^{11}+\frac{28\!\cdots\!06}{92\!\cdots\!91}a^{10}-\frac{32\!\cdots\!68}{27\!\cdots\!73}a^{9}+\frac{13\!\cdots\!45}{83\!\cdots\!19}a^{8}-\frac{64\!\cdots\!38}{92\!\cdots\!91}a^{7}-\frac{99\!\cdots\!06}{83\!\cdots\!19}a^{6}+\frac{14\!\cdots\!50}{83\!\cdots\!19}a^{5}-\frac{29\!\cdots\!95}{27\!\cdots\!73}a^{4}-\frac{66\!\cdots\!95}{83\!\cdots\!19}a^{3}+\frac{69\!\cdots\!19}{83\!\cdots\!19}a^{2}+\frac{11\!\cdots\!36}{83\!\cdots\!19}a+\frac{77\!\cdots\!94}{83\!\cdots\!19}$, $\frac{12\!\cdots\!49}{83\!\cdots\!19}a^{20}-\frac{10\!\cdots\!31}{92\!\cdots\!91}a^{19}+\frac{15\!\cdots\!43}{83\!\cdots\!19}a^{18}+\frac{12\!\cdots\!96}{27\!\cdots\!73}a^{17}-\frac{13\!\cdots\!76}{83\!\cdots\!19}a^{16}+\frac{15\!\cdots\!83}{83\!\cdots\!19}a^{15}-\frac{91\!\cdots\!13}{83\!\cdots\!19}a^{14}-\frac{14\!\cdots\!05}{83\!\cdots\!19}a^{13}-\frac{20\!\cdots\!44}{83\!\cdots\!19}a^{12}-\frac{35\!\cdots\!19}{83\!\cdots\!19}a^{11}+\frac{20\!\cdots\!76}{27\!\cdots\!73}a^{10}-\frac{38\!\cdots\!01}{27\!\cdots\!73}a^{9}+\frac{23\!\cdots\!76}{83\!\cdots\!19}a^{8}+\frac{65\!\cdots\!94}{27\!\cdots\!73}a^{7}-\frac{32\!\cdots\!80}{83\!\cdots\!19}a^{6}+\frac{79\!\cdots\!12}{83\!\cdots\!19}a^{5}+\frac{77\!\cdots\!18}{27\!\cdots\!73}a^{4}-\frac{12\!\cdots\!46}{83\!\cdots\!19}a^{3}-\frac{35\!\cdots\!49}{83\!\cdots\!19}a^{2}+\frac{15\!\cdots\!68}{83\!\cdots\!19}a+\frac{14\!\cdots\!34}{83\!\cdots\!19}$, $\frac{24\!\cdots\!27}{27\!\cdots\!73}a^{20}-\frac{22\!\cdots\!72}{27\!\cdots\!73}a^{19}+\frac{20\!\cdots\!23}{92\!\cdots\!91}a^{18}-\frac{12\!\cdots\!50}{27\!\cdots\!73}a^{17}-\frac{88\!\cdots\!52}{92\!\cdots\!91}a^{16}+\frac{67\!\cdots\!27}{27\!\cdots\!73}a^{15}-\frac{34\!\cdots\!27}{92\!\cdots\!91}a^{14}+\frac{34\!\cdots\!93}{92\!\cdots\!91}a^{13}-\frac{12\!\cdots\!39}{27\!\cdots\!73}a^{12}+\frac{12\!\cdots\!64}{27\!\cdots\!73}a^{11}+\frac{33\!\cdots\!80}{92\!\cdots\!91}a^{10}-\frac{38\!\cdots\!28}{27\!\cdots\!73}a^{9}+\frac{55\!\cdots\!40}{27\!\cdots\!73}a^{8}-\frac{93\!\cdots\!15}{92\!\cdots\!91}a^{7}-\frac{11\!\cdots\!77}{92\!\cdots\!91}a^{6}+\frac{58\!\cdots\!95}{27\!\cdots\!73}a^{5}-\frac{86\!\cdots\!69}{27\!\cdots\!73}a^{4}-\frac{76\!\cdots\!66}{92\!\cdots\!91}a^{3}+\frac{33\!\cdots\!22}{27\!\cdots\!73}a^{2}+\frac{13\!\cdots\!16}{92\!\cdots\!91}a+\frac{21\!\cdots\!28}{92\!\cdots\!91}$, $\frac{68\!\cdots\!34}{83\!\cdots\!19}a^{20}-\frac{19\!\cdots\!42}{27\!\cdots\!73}a^{19}+\frac{13\!\cdots\!79}{83\!\cdots\!19}a^{18}+\frac{74\!\cdots\!51}{92\!\cdots\!91}a^{17}-\frac{75\!\cdots\!41}{83\!\cdots\!19}a^{16}+\frac{14\!\cdots\!54}{83\!\cdots\!19}a^{15}-\frac{18\!\cdots\!45}{83\!\cdots\!19}a^{14}+\frac{13\!\cdots\!56}{83\!\cdots\!19}a^{13}-\frac{19\!\cdots\!39}{83\!\cdots\!19}a^{12}+\frac{15\!\cdots\!58}{83\!\cdots\!19}a^{11}+\frac{10\!\cdots\!84}{27\!\cdots\!73}a^{10}-\frac{30\!\cdots\!87}{27\!\cdots\!73}a^{9}+\frac{94\!\cdots\!80}{83\!\cdots\!19}a^{8}+\frac{58\!\cdots\!04}{92\!\cdots\!91}a^{7}-\frac{13\!\cdots\!46}{83\!\cdots\!19}a^{6}+\frac{11\!\cdots\!63}{83\!\cdots\!19}a^{5}+\frac{18\!\cdots\!35}{27\!\cdots\!73}a^{4}-\frac{67\!\cdots\!19}{83\!\cdots\!19}a^{3}-\frac{92\!\cdots\!83}{83\!\cdots\!19}a^{2}+\frac{99\!\cdots\!13}{83\!\cdots\!19}a+\frac{20\!\cdots\!39}{83\!\cdots\!19}$, $\frac{23\!\cdots\!50}{22\!\cdots\!33}a^{20}-\frac{18\!\cdots\!26}{22\!\cdots\!33}a^{19}+\frac{35\!\cdots\!58}{22\!\cdots\!33}a^{18}+\frac{57\!\cdots\!13}{22\!\cdots\!33}a^{17}-\frac{27\!\cdots\!97}{22\!\cdots\!33}a^{16}+\frac{35\!\cdots\!04}{22\!\cdots\!33}a^{15}-\frac{24\!\cdots\!18}{22\!\cdots\!33}a^{14}-\frac{16\!\cdots\!29}{22\!\cdots\!33}a^{13}-\frac{12\!\cdots\!16}{22\!\cdots\!33}a^{12}-\frac{27\!\cdots\!04}{22\!\cdots\!33}a^{11}+\frac{11\!\cdots\!14}{22\!\cdots\!33}a^{10}-\frac{25\!\cdots\!37}{22\!\cdots\!33}a^{9}+\frac{11\!\cdots\!96}{22\!\cdots\!33}a^{8}+\frac{35\!\cdots\!60}{22\!\cdots\!33}a^{7}-\frac{68\!\cdots\!63}{22\!\cdots\!33}a^{6}+\frac{23\!\cdots\!30}{22\!\cdots\!33}a^{5}+\frac{53\!\cdots\!60}{22\!\cdots\!33}a^{4}-\frac{34\!\cdots\!10}{22\!\cdots\!33}a^{3}-\frac{14\!\cdots\!60}{22\!\cdots\!33}a^{2}+\frac{80\!\cdots\!08}{22\!\cdots\!33}a+\frac{19\!\cdots\!60}{22\!\cdots\!33}$, $\frac{49\!\cdots\!40}{83\!\cdots\!19}a^{20}-\frac{13\!\cdots\!76}{27\!\cdots\!73}a^{19}+\frac{82\!\cdots\!08}{83\!\cdots\!19}a^{18}+\frac{27\!\cdots\!77}{27\!\cdots\!73}a^{17}-\frac{50\!\cdots\!25}{83\!\cdots\!19}a^{16}+\frac{84\!\cdots\!03}{83\!\cdots\!19}a^{15}-\frac{10\!\cdots\!07}{83\!\cdots\!19}a^{14}+\frac{52\!\cdots\!09}{83\!\cdots\!19}a^{13}-\frac{12\!\cdots\!43}{83\!\cdots\!19}a^{12}+\frac{36\!\cdots\!55}{83\!\cdots\!19}a^{11}+\frac{73\!\cdots\!80}{27\!\cdots\!73}a^{10}-\frac{61\!\cdots\!84}{92\!\cdots\!91}a^{9}+\frac{46\!\cdots\!80}{83\!\cdots\!19}a^{8}+\frac{23\!\cdots\!11}{92\!\cdots\!91}a^{7}-\frac{91\!\cdots\!61}{83\!\cdots\!19}a^{6}+\frac{47\!\cdots\!76}{83\!\cdots\!19}a^{5}+\frac{56\!\cdots\!15}{92\!\cdots\!91}a^{4}-\frac{32\!\cdots\!21}{83\!\cdots\!19}a^{3}-\frac{12\!\cdots\!94}{83\!\cdots\!19}a^{2}+\frac{46\!\cdots\!82}{83\!\cdots\!19}a+\frac{12\!\cdots\!95}{83\!\cdots\!19}$, $\frac{31\!\cdots\!59}{83\!\cdots\!19}a^{20}-\frac{87\!\cdots\!78}{27\!\cdots\!73}a^{19}+\frac{58\!\cdots\!12}{83\!\cdots\!19}a^{18}+\frac{14\!\cdots\!63}{27\!\cdots\!73}a^{17}-\frac{33\!\cdots\!25}{83\!\cdots\!19}a^{16}+\frac{61\!\cdots\!30}{83\!\cdots\!19}a^{15}-\frac{74\!\cdots\!52}{83\!\cdots\!19}a^{14}+\frac{43\!\cdots\!76}{83\!\cdots\!19}a^{13}-\frac{78\!\cdots\!93}{83\!\cdots\!19}a^{12}+\frac{32\!\cdots\!17}{83\!\cdots\!19}a^{11}+\frac{15\!\cdots\!45}{92\!\cdots\!91}a^{10}-\frac{43\!\cdots\!32}{92\!\cdots\!91}a^{9}+\frac{37\!\cdots\!88}{83\!\cdots\!19}a^{8}+\frac{32\!\cdots\!01}{27\!\cdots\!73}a^{7}-\frac{65\!\cdots\!18}{83\!\cdots\!19}a^{6}+\frac{46\!\cdots\!86}{83\!\cdots\!19}a^{5}+\frac{90\!\cdots\!53}{27\!\cdots\!73}a^{4}-\frac{30\!\cdots\!51}{83\!\cdots\!19}a^{3}-\frac{32\!\cdots\!02}{83\!\cdots\!19}a^{2}+\frac{43\!\cdots\!26}{83\!\cdots\!19}a+\frac{60\!\cdots\!98}{83\!\cdots\!19}$, $\frac{79\!\cdots\!04}{92\!\cdots\!91}a^{20}-\frac{20\!\cdots\!80}{27\!\cdots\!73}a^{19}+\frac{51\!\cdots\!00}{27\!\cdots\!73}a^{18}+\frac{10\!\cdots\!50}{27\!\cdots\!73}a^{17}-\frac{83\!\cdots\!41}{92\!\cdots\!91}a^{16}+\frac{18\!\cdots\!18}{92\!\cdots\!91}a^{15}-\frac{78\!\cdots\!20}{27\!\cdots\!73}a^{14}+\frac{69\!\cdots\!76}{27\!\cdots\!73}a^{13}-\frac{97\!\cdots\!13}{27\!\cdots\!73}a^{12}+\frac{77\!\cdots\!40}{27\!\cdots\!73}a^{11}+\frac{10\!\cdots\!51}{27\!\cdots\!73}a^{10}-\frac{32\!\cdots\!27}{27\!\cdots\!73}a^{9}+\frac{13\!\cdots\!29}{92\!\cdots\!91}a^{8}-\frac{11\!\cdots\!59}{27\!\cdots\!73}a^{7}-\frac{37\!\cdots\!54}{27\!\cdots\!73}a^{6}+\frac{42\!\cdots\!93}{27\!\cdots\!73}a^{5}+\frac{15\!\cdots\!13}{92\!\cdots\!91}a^{4}-\frac{18\!\cdots\!11}{27\!\cdots\!73}a^{3}+\frac{34\!\cdots\!11}{27\!\cdots\!73}a^{2}+\frac{82\!\cdots\!94}{92\!\cdots\!91}a+\frac{90\!\cdots\!58}{92\!\cdots\!91}$, $\frac{19\!\cdots\!45}{83\!\cdots\!19}a^{20}-\frac{54\!\cdots\!94}{27\!\cdots\!73}a^{19}+\frac{37\!\cdots\!55}{83\!\cdots\!19}a^{18}+\frac{23\!\cdots\!07}{92\!\cdots\!91}a^{17}-\frac{20\!\cdots\!17}{83\!\cdots\!19}a^{16}+\frac{39\!\cdots\!90}{83\!\cdots\!19}a^{15}-\frac{50\!\cdots\!84}{83\!\cdots\!19}a^{14}+\frac{35\!\cdots\!58}{83\!\cdots\!19}a^{13}-\frac{58\!\cdots\!03}{83\!\cdots\!19}a^{12}+\frac{35\!\cdots\!41}{83\!\cdots\!19}a^{11}+\frac{28\!\cdots\!76}{27\!\cdots\!73}a^{10}-\frac{81\!\cdots\!98}{27\!\cdots\!73}a^{9}+\frac{25\!\cdots\!53}{83\!\cdots\!19}a^{8}+\frac{43\!\cdots\!13}{27\!\cdots\!73}a^{7}-\frac{36\!\cdots\!52}{83\!\cdots\!19}a^{6}+\frac{28\!\cdots\!11}{83\!\cdots\!19}a^{5}+\frac{45\!\cdots\!79}{27\!\cdots\!73}a^{4}-\frac{16\!\cdots\!91}{83\!\cdots\!19}a^{3}-\frac{33\!\cdots\!25}{83\!\cdots\!19}a^{2}+\frac{27\!\cdots\!45}{83\!\cdots\!19}a+\frac{66\!\cdots\!96}{83\!\cdots\!19}$, $\frac{92\!\cdots\!54}{83\!\cdots\!19}a^{20}-\frac{25\!\cdots\!32}{27\!\cdots\!73}a^{19}+\frac{17\!\cdots\!83}{83\!\cdots\!19}a^{18}+\frac{11\!\cdots\!55}{92\!\cdots\!91}a^{17}-\frac{97\!\cdots\!54}{83\!\cdots\!19}a^{16}+\frac{18\!\cdots\!26}{83\!\cdots\!19}a^{15}-\frac{23\!\cdots\!62}{83\!\cdots\!19}a^{14}+\frac{16\!\cdots\!96}{83\!\cdots\!19}a^{13}-\frac{27\!\cdots\!28}{83\!\cdots\!19}a^{12}+\frac{16\!\cdots\!10}{83\!\cdots\!19}a^{11}+\frac{13\!\cdots\!97}{27\!\cdots\!73}a^{10}-\frac{38\!\cdots\!12}{27\!\cdots\!73}a^{9}+\frac{11\!\cdots\!77}{83\!\cdots\!19}a^{8}+\frac{26\!\cdots\!39}{27\!\cdots\!73}a^{7}-\frac{17\!\cdots\!61}{83\!\cdots\!19}a^{6}+\frac{13\!\cdots\!10}{83\!\cdots\!19}a^{5}+\frac{22\!\cdots\!06}{27\!\cdots\!73}a^{4}-\frac{76\!\cdots\!93}{83\!\cdots\!19}a^{3}-\frac{12\!\cdots\!70}{83\!\cdots\!19}a^{2}+\frac{11\!\cdots\!04}{83\!\cdots\!19}a+\frac{18\!\cdots\!13}{83\!\cdots\!19}$, $\frac{29\!\cdots\!43}{83\!\cdots\!19}a^{20}-\frac{79\!\cdots\!73}{27\!\cdots\!73}a^{19}+\frac{50\!\cdots\!05}{83\!\cdots\!19}a^{18}+\frac{48\!\cdots\!59}{92\!\cdots\!91}a^{17}-\frac{29\!\cdots\!31}{83\!\cdots\!19}a^{16}+\frac{51\!\cdots\!08}{83\!\cdots\!19}a^{15}-\frac{63\!\cdots\!71}{83\!\cdots\!19}a^{14}+\frac{37\!\cdots\!75}{83\!\cdots\!19}a^{13}-\frac{77\!\cdots\!29}{83\!\cdots\!19}a^{12}+\frac{32\!\cdots\!51}{83\!\cdots\!19}a^{11}+\frac{43\!\cdots\!82}{27\!\cdots\!73}a^{10}-\frac{11\!\cdots\!77}{27\!\cdots\!73}a^{9}+\frac{30\!\cdots\!75}{83\!\cdots\!19}a^{8}+\frac{31\!\cdots\!87}{27\!\cdots\!73}a^{7}-\frac{53\!\cdots\!54}{83\!\cdots\!19}a^{6}+\frac{31\!\cdots\!14}{83\!\cdots\!19}a^{5}+\frac{30\!\cdots\!18}{92\!\cdots\!91}a^{4}-\frac{19\!\cdots\!93}{83\!\cdots\!19}a^{3}-\frac{67\!\cdots\!76}{83\!\cdots\!19}a^{2}+\frac{26\!\cdots\!80}{83\!\cdots\!19}a+\frac{79\!\cdots\!51}{83\!\cdots\!19}$, $\frac{25\!\cdots\!05}{83\!\cdots\!19}a^{20}-\frac{71\!\cdots\!30}{27\!\cdots\!73}a^{19}+\frac{48\!\cdots\!99}{83\!\cdots\!19}a^{18}+\frac{98\!\cdots\!41}{27\!\cdots\!73}a^{17}-\frac{27\!\cdots\!99}{83\!\cdots\!19}a^{16}+\frac{52\!\cdots\!48}{83\!\cdots\!19}a^{15}-\frac{64\!\cdots\!71}{83\!\cdots\!19}a^{14}+\frac{40\!\cdots\!90}{83\!\cdots\!19}a^{13}-\frac{66\!\cdots\!53}{83\!\cdots\!19}a^{12}+\frac{27\!\cdots\!45}{83\!\cdots\!19}a^{11}+\frac{37\!\cdots\!85}{27\!\cdots\!73}a^{10}-\frac{35\!\cdots\!54}{92\!\cdots\!91}a^{9}+\frac{32\!\cdots\!91}{83\!\cdots\!19}a^{8}+\frac{17\!\cdots\!30}{27\!\cdots\!73}a^{7}-\frac{54\!\cdots\!72}{83\!\cdots\!19}a^{6}+\frac{43\!\cdots\!30}{83\!\cdots\!19}a^{5}+\frac{58\!\cdots\!36}{27\!\cdots\!73}a^{4}-\frac{28\!\cdots\!92}{83\!\cdots\!19}a^{3}-\frac{60\!\cdots\!04}{83\!\cdots\!19}a^{2}+\frac{42\!\cdots\!38}{83\!\cdots\!19}a+\frac{52\!\cdots\!63}{83\!\cdots\!19}$, $\frac{46\!\cdots\!30}{27\!\cdots\!73}a^{20}-\frac{38\!\cdots\!69}{27\!\cdots\!73}a^{19}+\frac{30\!\cdots\!76}{92\!\cdots\!91}a^{18}+\frac{46\!\cdots\!76}{27\!\cdots\!73}a^{17}-\frac{16\!\cdots\!21}{92\!\cdots\!91}a^{16}+\frac{31\!\cdots\!26}{92\!\cdots\!91}a^{15}-\frac{12\!\cdots\!60}{27\!\cdots\!73}a^{14}+\frac{86\!\cdots\!77}{27\!\cdots\!73}a^{13}-\frac{13\!\cdots\!57}{27\!\cdots\!73}a^{12}+\frac{83\!\cdots\!71}{27\!\cdots\!73}a^{11}+\frac{20\!\cdots\!36}{27\!\cdots\!73}a^{10}-\frac{59\!\cdots\!66}{27\!\cdots\!73}a^{9}+\frac{62\!\cdots\!24}{27\!\cdots\!73}a^{8}+\frac{88\!\cdots\!39}{92\!\cdots\!91}a^{7}-\frac{90\!\cdots\!90}{27\!\cdots\!73}a^{6}+\frac{24\!\cdots\!37}{92\!\cdots\!91}a^{5}+\frac{30\!\cdots\!24}{27\!\cdots\!73}a^{4}-\frac{43\!\cdots\!60}{27\!\cdots\!73}a^{3}-\frac{33\!\cdots\!07}{27\!\cdots\!73}a^{2}+\frac{20\!\cdots\!41}{92\!\cdots\!91}a+\frac{90\!\cdots\!93}{27\!\cdots\!73}$, $\frac{46\!\cdots\!28}{27\!\cdots\!73}a^{20}-\frac{41\!\cdots\!57}{27\!\cdots\!73}a^{19}+\frac{11\!\cdots\!91}{27\!\cdots\!73}a^{18}-\frac{24\!\cdots\!66}{27\!\cdots\!73}a^{17}-\frac{47\!\cdots\!69}{27\!\cdots\!73}a^{16}+\frac{12\!\cdots\!87}{27\!\cdots\!73}a^{15}-\frac{66\!\cdots\!91}{92\!\cdots\!91}a^{14}+\frac{21\!\cdots\!18}{27\!\cdots\!73}a^{13}-\frac{92\!\cdots\!91}{92\!\cdots\!91}a^{12}+\frac{26\!\cdots\!87}{27\!\cdots\!73}a^{11}+\frac{62\!\cdots\!58}{92\!\cdots\!91}a^{10}-\frac{70\!\cdots\!42}{27\!\cdots\!73}a^{9}+\frac{35\!\cdots\!83}{92\!\cdots\!91}a^{8}-\frac{62\!\cdots\!90}{27\!\cdots\!73}a^{7}-\frac{45\!\cdots\!94}{27\!\cdots\!73}a^{6}+\frac{93\!\cdots\!23}{27\!\cdots\!73}a^{5}-\frac{21\!\cdots\!92}{27\!\cdots\!73}a^{4}-\frac{24\!\cdots\!54}{27\!\cdots\!73}a^{3}+\frac{72\!\cdots\!64}{92\!\cdots\!91}a^{2}+\frac{35\!\cdots\!99}{27\!\cdots\!73}a+\frac{85\!\cdots\!24}{27\!\cdots\!73}$, $\frac{45\!\cdots\!64}{83\!\cdots\!19}a^{20}-\frac{12\!\cdots\!14}{27\!\cdots\!73}a^{19}+\frac{93\!\cdots\!36}{83\!\cdots\!19}a^{18}+\frac{12\!\cdots\!73}{27\!\cdots\!73}a^{17}-\frac{49\!\cdots\!82}{83\!\cdots\!19}a^{16}+\frac{98\!\cdots\!58}{83\!\cdots\!19}a^{15}-\frac{13\!\cdots\!39}{83\!\cdots\!19}a^{14}+\frac{99\!\cdots\!70}{83\!\cdots\!19}a^{13}-\frac{14\!\cdots\!74}{83\!\cdots\!19}a^{12}+\frac{10\!\cdots\!78}{83\!\cdots\!19}a^{11}+\frac{66\!\cdots\!59}{27\!\cdots\!73}a^{10}-\frac{20\!\cdots\!18}{27\!\cdots\!73}a^{9}+\frac{67\!\cdots\!45}{83\!\cdots\!19}a^{8}-\frac{14\!\cdots\!37}{27\!\cdots\!73}a^{7}-\frac{86\!\cdots\!61}{83\!\cdots\!19}a^{6}+\frac{77\!\cdots\!36}{83\!\cdots\!19}a^{5}+\frac{81\!\cdots\!66}{27\!\cdots\!73}a^{4}-\frac{42\!\cdots\!70}{83\!\cdots\!19}a^{3}-\frac{24\!\cdots\!03}{83\!\cdots\!19}a^{2}+\frac{62\!\cdots\!26}{83\!\cdots\!19}a+\frac{91\!\cdots\!69}{83\!\cdots\!19}$
| 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: | \( 875237315.142 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{3}\cdot 875237315.142 \cdot 1}{2\cdot\sqrt{376429776359836117500418449727791}}\cr\approx \mathstrut & 0.183334156502 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 + 16*x^19 + 19*x^18 - 103*x^17 + 160*x^16 - 174*x^15 + 66*x^14 - 206*x^13 + 39*x^12 + 4498*x^11 - 10908*x^10 + 7867*x^9 + 6766*x^8 - 19649*x^7 + 7824*x^6 + 13595*x^5 - 6802*x^4 - 4602*x^3 + 1044*x^2 + 735*x + 79)
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 - 8*x^20 + 16*x^19 + 19*x^18 - 103*x^17 + 160*x^16 - 174*x^15 + 66*x^14 - 206*x^13 + 39*x^12 + 4498*x^11 - 10908*x^10 + 7867*x^9 + 6766*x^8 - 19649*x^7 + 7824*x^6 + 13595*x^5 - 6802*x^4 - 4602*x^3 + 1044*x^2 + 735*x + 79, 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 - 8*x^20 + 16*x^19 + 19*x^18 - 103*x^17 + 160*x^16 - 174*x^15 + 66*x^14 - 206*x^13 + 39*x^12 + 4498*x^11 - 10908*x^10 + 7867*x^9 + 6766*x^8 - 19649*x^7 + 7824*x^6 + 13595*x^5 - 6802*x^4 - 4602*x^3 + 1044*x^2 + 735*x + 79);
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 - 8*x^20 + 16*x^19 + 19*x^18 - 103*x^17 + 160*x^16 - 174*x^15 + 66*x^14 - 206*x^13 + 39*x^12 + 4498*x^11 - 10908*x^10 + 7867*x^9 + 6766*x^8 - 19649*x^7 + 7824*x^6 + 13595*x^5 - 6802*x^4 - 4602*x^3 + 1044*x^2 + 735*x + 79);
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
$S_3\times S_7$ (as 21T74):
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 non-solvable group of order 30240 |
| The 45 conjugacy class representatives for $S_3\times S_7$ |
| Character table for $S_3\times S_7$ |
Intermediate fields
| 3.3.321.1, 7.5.2306599.1 |
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 | $21$ | R | $21$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | $21$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $21$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{10}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(107\)
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(21557\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ |