Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 6*x^19 + 11*x^18 + 47*x^17 - 45*x^16 - 170*x^15 + 162*x^14 + 400*x^13 - 307*x^12 - 750*x^11 + 395*x^10 + 733*x^9 - 371*x^8 - 425*x^7 + 197*x^6 + 204*x^5 - 78*x^4 - 36*x^3 + 17*x^2 - 1)
gp: K = bnfinit(y^21 - y^20 - 6*y^19 + 11*y^18 + 47*y^17 - 45*y^16 - 170*y^15 + 162*y^14 + 400*y^13 - 307*y^12 - 750*y^11 + 395*y^10 + 733*y^9 - 371*y^8 - 425*y^7 + 197*y^6 + 204*y^5 - 78*y^4 - 36*y^3 + 17*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^20 - 6*x^19 + 11*x^18 + 47*x^17 - 45*x^16 - 170*x^15 + 162*x^14 + 400*x^13 - 307*x^12 - 750*x^11 + 395*x^10 + 733*x^9 - 371*x^8 - 425*x^7 + 197*x^6 + 204*x^5 - 78*x^4 - 36*x^3 + 17*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - x^20 - 6*x^19 + 11*x^18 + 47*x^17 - 45*x^16 - 170*x^15 + 162*x^14 + 400*x^13 - 307*x^12 - 750*x^11 + 395*x^10 + 733*x^9 - 371*x^8 - 425*x^7 + 197*x^6 + 204*x^5 - 78*x^4 - 36*x^3 + 17*x^2 - 1)
\( x^{21} - x^{20} - 6 x^{19} + 11 x^{18} + 47 x^{17} - 45 x^{16} - 170 x^{15} + 162 x^{14} + 400 x^{13} + \cdots - 1 \)
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: | $[7, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-105070765003019731854543276767\)
\(\medspace = -\,17^{3}\cdot 23^{8}\cdot 64879^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.10\) | 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: | $17^{1/2}23^{1/2}64879^{1/2}\approx 5036.634689949233$ | ||
| Ramified primes: |
\(17\), \(23\), \(64879\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1102943}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23}a^{17}+\frac{3}{23}a^{16}-\frac{5}{23}a^{15}-\frac{5}{23}a^{14}-\frac{9}{23}a^{13}+\frac{11}{23}a^{12}-\frac{7}{23}a^{11}+\frac{11}{23}a^{9}-\frac{7}{23}a^{8}+\frac{9}{23}a^{7}-\frac{5}{23}a^{6}-\frac{4}{23}a^{5}-\frac{11}{23}a^{4}+\frac{10}{23}a^{3}+\frac{1}{23}a^{2}-\frac{7}{23}a-\frac{9}{23}$, $\frac{1}{23}a^{18}+\frac{9}{23}a^{16}+\frac{10}{23}a^{15}+\frac{6}{23}a^{14}-\frac{8}{23}a^{13}+\frac{6}{23}a^{12}-\frac{2}{23}a^{11}+\frac{11}{23}a^{10}+\frac{6}{23}a^{9}+\frac{7}{23}a^{8}-\frac{9}{23}a^{7}+\frac{11}{23}a^{6}+\frac{1}{23}a^{5}-\frac{3}{23}a^{4}-\frac{6}{23}a^{3}-\frac{10}{23}a^{2}-\frac{11}{23}a+\frac{4}{23}$, $\frac{1}{23}a^{19}+\frac{6}{23}a^{16}+\frac{5}{23}a^{15}-\frac{9}{23}a^{14}-\frac{5}{23}a^{13}-\frac{9}{23}a^{12}+\frac{5}{23}a^{11}+\frac{6}{23}a^{10}+\frac{8}{23}a^{8}-\frac{1}{23}a^{7}+\frac{10}{23}a^{5}+\frac{1}{23}a^{4}-\frac{8}{23}a^{3}+\frac{3}{23}a^{2}-\frac{2}{23}a-\frac{11}{23}$, $\frac{1}{29\!\cdots\!63}a^{20}-\frac{48\!\cdots\!88}{29\!\cdots\!63}a^{19}+\frac{12\!\cdots\!24}{29\!\cdots\!63}a^{18}+\frac{50\!\cdots\!21}{29\!\cdots\!63}a^{17}-\frac{17\!\cdots\!49}{17\!\cdots\!39}a^{16}+\frac{70\!\cdots\!41}{29\!\cdots\!63}a^{15}+\frac{10\!\cdots\!86}{29\!\cdots\!63}a^{14}-\frac{14\!\cdots\!67}{29\!\cdots\!63}a^{13}-\frac{52\!\cdots\!89}{29\!\cdots\!63}a^{12}+\frac{14\!\cdots\!02}{29\!\cdots\!63}a^{11}+\frac{52\!\cdots\!66}{12\!\cdots\!81}a^{10}-\frac{69\!\cdots\!94}{29\!\cdots\!63}a^{9}+\frac{12\!\cdots\!36}{29\!\cdots\!63}a^{8}+\frac{65\!\cdots\!81}{29\!\cdots\!63}a^{7}-\frac{83\!\cdots\!57}{29\!\cdots\!63}a^{6}-\frac{68\!\cdots\!56}{29\!\cdots\!63}a^{5}+\frac{12\!\cdots\!16}{29\!\cdots\!63}a^{4}-\frac{28\!\cdots\!65}{29\!\cdots\!63}a^{3}+\frac{29\!\cdots\!46}{29\!\cdots\!63}a^{2}+\frac{11\!\cdots\!76}{29\!\cdots\!63}a+\frac{12\!\cdots\!82}{29\!\cdots\!63}$
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: | $13$ | 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: |
$\frac{64\!\cdots\!52}{29\!\cdots\!63}a^{20}-\frac{37\!\cdots\!21}{29\!\cdots\!63}a^{19}-\frac{27\!\cdots\!74}{29\!\cdots\!63}a^{18}+\frac{27\!\cdots\!72}{29\!\cdots\!63}a^{17}+\frac{49\!\cdots\!30}{17\!\cdots\!39}a^{16}-\frac{19\!\cdots\!06}{29\!\cdots\!63}a^{15}-\frac{67\!\cdots\!38}{29\!\cdots\!63}a^{14}+\frac{68\!\cdots\!43}{29\!\cdots\!63}a^{13}+\frac{97\!\cdots\!55}{29\!\cdots\!63}a^{12}-\frac{16\!\cdots\!24}{29\!\cdots\!63}a^{11}-\frac{33\!\cdots\!68}{29\!\cdots\!63}a^{10}+\frac{28\!\cdots\!78}{29\!\cdots\!63}a^{9}+\frac{70\!\cdots\!27}{29\!\cdots\!63}a^{8}-\frac{27\!\cdots\!97}{29\!\cdots\!63}a^{7}-\frac{43\!\cdots\!30}{29\!\cdots\!63}a^{6}+\frac{16\!\cdots\!52}{29\!\cdots\!63}a^{5}+\frac{20\!\cdots\!25}{29\!\cdots\!63}a^{4}-\frac{71\!\cdots\!40}{29\!\cdots\!63}a^{3}-\frac{14\!\cdots\!05}{29\!\cdots\!63}a^{2}+\frac{11\!\cdots\!93}{29\!\cdots\!63}a-\frac{13\!\cdots\!76}{29\!\cdots\!63}$, $\frac{30\!\cdots\!69}{29\!\cdots\!63}a^{20}-\frac{10\!\cdots\!38}{29\!\cdots\!63}a^{19}-\frac{19\!\cdots\!00}{29\!\cdots\!63}a^{18}+\frac{21\!\cdots\!34}{29\!\cdots\!63}a^{17}+\frac{41\!\cdots\!26}{74\!\cdots\!93}a^{16}-\frac{42\!\cdots\!02}{29\!\cdots\!63}a^{15}-\frac{25\!\cdots\!53}{12\!\cdots\!81}a^{14}+\frac{16\!\cdots\!45}{29\!\cdots\!63}a^{13}+\frac{14\!\cdots\!60}{29\!\cdots\!63}a^{12}-\frac{15\!\cdots\!32}{29\!\cdots\!63}a^{11}-\frac{26\!\cdots\!38}{29\!\cdots\!63}a^{10}-\frac{23\!\cdots\!11}{29\!\cdots\!63}a^{9}+\frac{25\!\cdots\!05}{29\!\cdots\!63}a^{8}+\frac{15\!\cdots\!30}{29\!\cdots\!63}a^{7}-\frac{15\!\cdots\!08}{29\!\cdots\!63}a^{6}-\frac{77\!\cdots\!17}{29\!\cdots\!63}a^{5}+\frac{66\!\cdots\!84}{29\!\cdots\!63}a^{4}+\frac{11\!\cdots\!33}{29\!\cdots\!63}a^{3}-\frac{44\!\cdots\!83}{12\!\cdots\!81}a^{2}+\frac{13\!\cdots\!76}{29\!\cdots\!63}a-\frac{64\!\cdots\!52}{29\!\cdots\!63}$, $\frac{49\!\cdots\!96}{29\!\cdots\!63}a^{20}-\frac{25\!\cdots\!51}{29\!\cdots\!63}a^{19}-\frac{33\!\cdots\!06}{29\!\cdots\!63}a^{18}+\frac{26\!\cdots\!28}{29\!\cdots\!63}a^{17}+\frac{16\!\cdots\!27}{17\!\cdots\!39}a^{16}+\frac{15\!\cdots\!21}{29\!\cdots\!63}a^{15}-\frac{97\!\cdots\!16}{29\!\cdots\!63}a^{14}+\frac{14\!\cdots\!68}{12\!\cdots\!81}a^{13}+\frac{24\!\cdots\!62}{29\!\cdots\!63}a^{12}+\frac{26\!\cdots\!33}{29\!\cdots\!63}a^{11}-\frac{44\!\cdots\!33}{29\!\cdots\!63}a^{10}-\frac{12\!\cdots\!36}{29\!\cdots\!63}a^{9}+\frac{41\!\cdots\!94}{29\!\cdots\!63}a^{8}+\frac{84\!\cdots\!09}{29\!\cdots\!63}a^{7}-\frac{25\!\cdots\!12}{29\!\cdots\!63}a^{6}-\frac{45\!\cdots\!53}{29\!\cdots\!63}a^{5}+\frac{11\!\cdots\!75}{29\!\cdots\!63}a^{4}+\frac{29\!\cdots\!31}{29\!\cdots\!63}a^{3}-\frac{19\!\cdots\!36}{29\!\cdots\!63}a^{2}+\frac{19\!\cdots\!61}{29\!\cdots\!63}a+\frac{57\!\cdots\!69}{29\!\cdots\!63}$, $\frac{80\!\cdots\!65}{29\!\cdots\!63}a^{20}-\frac{13\!\cdots\!89}{29\!\cdots\!63}a^{19}-\frac{53\!\cdots\!06}{29\!\cdots\!63}a^{18}+\frac{48\!\cdots\!62}{29\!\cdots\!63}a^{17}+\frac{25\!\cdots\!25}{17\!\cdots\!39}a^{16}-\frac{41\!\cdots\!81}{29\!\cdots\!63}a^{15}-\frac{15\!\cdots\!35}{29\!\cdots\!63}a^{14}+\frac{19\!\cdots\!09}{29\!\cdots\!63}a^{13}+\frac{39\!\cdots\!22}{29\!\cdots\!63}a^{12}+\frac{11\!\cdots\!01}{29\!\cdots\!63}a^{11}-\frac{70\!\cdots\!71}{29\!\cdots\!63}a^{10}-\frac{14\!\cdots\!47}{29\!\cdots\!63}a^{9}+\frac{66\!\cdots\!99}{29\!\cdots\!63}a^{8}+\frac{99\!\cdots\!39}{29\!\cdots\!63}a^{7}-\frac{41\!\cdots\!20}{29\!\cdots\!63}a^{6}-\frac{52\!\cdots\!70}{29\!\cdots\!63}a^{5}+\frac{77\!\cdots\!33}{12\!\cdots\!81}a^{4}+\frac{41\!\cdots\!64}{29\!\cdots\!63}a^{3}-\frac{30\!\cdots\!45}{29\!\cdots\!63}a^{2}+\frac{32\!\cdots\!37}{29\!\cdots\!63}a+\frac{34\!\cdots\!80}{29\!\cdots\!63}$, $\frac{37\!\cdots\!65}{29\!\cdots\!63}a^{20}-\frac{38\!\cdots\!21}{29\!\cdots\!63}a^{19}+\frac{15\!\cdots\!27}{29\!\cdots\!63}a^{18}+\frac{23\!\cdots\!45}{29\!\cdots\!63}a^{17}-\frac{12\!\cdots\!72}{17\!\cdots\!39}a^{16}-\frac{16\!\cdots\!05}{29\!\cdots\!63}a^{15}+\frac{90\!\cdots\!81}{29\!\cdots\!63}a^{14}+\frac{57\!\cdots\!32}{29\!\cdots\!63}a^{13}-\frac{39\!\cdots\!03}{29\!\cdots\!63}a^{12}-\frac{12\!\cdots\!17}{29\!\cdots\!63}a^{11}+\frac{66\!\cdots\!74}{29\!\cdots\!63}a^{10}+\frac{22\!\cdots\!69}{29\!\cdots\!63}a^{9}-\frac{90\!\cdots\!55}{29\!\cdots\!63}a^{8}-\frac{17\!\cdots\!29}{29\!\cdots\!63}a^{7}+\frac{82\!\cdots\!11}{29\!\cdots\!63}a^{6}+\frac{76\!\cdots\!55}{29\!\cdots\!63}a^{5}-\frac{36\!\cdots\!79}{29\!\cdots\!63}a^{4}-\frac{40\!\cdots\!91}{29\!\cdots\!63}a^{3}+\frac{15\!\cdots\!27}{29\!\cdots\!63}a^{2}-\frac{18\!\cdots\!95}{29\!\cdots\!63}a+\frac{29\!\cdots\!57}{29\!\cdots\!63}$, $\frac{49\!\cdots\!96}{29\!\cdots\!63}a^{20}-\frac{25\!\cdots\!51}{29\!\cdots\!63}a^{19}-\frac{33\!\cdots\!06}{29\!\cdots\!63}a^{18}+\frac{26\!\cdots\!28}{29\!\cdots\!63}a^{17}+\frac{16\!\cdots\!27}{17\!\cdots\!39}a^{16}+\frac{15\!\cdots\!21}{29\!\cdots\!63}a^{15}-\frac{97\!\cdots\!16}{29\!\cdots\!63}a^{14}+\frac{14\!\cdots\!68}{12\!\cdots\!81}a^{13}+\frac{24\!\cdots\!62}{29\!\cdots\!63}a^{12}+\frac{26\!\cdots\!33}{29\!\cdots\!63}a^{11}-\frac{44\!\cdots\!33}{29\!\cdots\!63}a^{10}-\frac{12\!\cdots\!36}{29\!\cdots\!63}a^{9}+\frac{41\!\cdots\!94}{29\!\cdots\!63}a^{8}+\frac{84\!\cdots\!09}{29\!\cdots\!63}a^{7}-\frac{25\!\cdots\!12}{29\!\cdots\!63}a^{6}-\frac{45\!\cdots\!53}{29\!\cdots\!63}a^{5}+\frac{11\!\cdots\!75}{29\!\cdots\!63}a^{4}+\frac{29\!\cdots\!31}{29\!\cdots\!63}a^{3}-\frac{19\!\cdots\!36}{29\!\cdots\!63}a^{2}+\frac{19\!\cdots\!61}{29\!\cdots\!63}a+\frac{34\!\cdots\!32}{29\!\cdots\!63}$, $\frac{80\!\cdots\!65}{29\!\cdots\!63}a^{20}-\frac{13\!\cdots\!89}{29\!\cdots\!63}a^{19}-\frac{53\!\cdots\!06}{29\!\cdots\!63}a^{18}+\frac{48\!\cdots\!62}{29\!\cdots\!63}a^{17}+\frac{25\!\cdots\!25}{17\!\cdots\!39}a^{16}-\frac{41\!\cdots\!81}{29\!\cdots\!63}a^{15}-\frac{15\!\cdots\!35}{29\!\cdots\!63}a^{14}+\frac{19\!\cdots\!09}{29\!\cdots\!63}a^{13}+\frac{39\!\cdots\!22}{29\!\cdots\!63}a^{12}+\frac{11\!\cdots\!01}{29\!\cdots\!63}a^{11}-\frac{70\!\cdots\!71}{29\!\cdots\!63}a^{10}-\frac{14\!\cdots\!47}{29\!\cdots\!63}a^{9}+\frac{66\!\cdots\!99}{29\!\cdots\!63}a^{8}+\frac{99\!\cdots\!39}{29\!\cdots\!63}a^{7}-\frac{41\!\cdots\!20}{29\!\cdots\!63}a^{6}-\frac{52\!\cdots\!70}{29\!\cdots\!63}a^{5}+\frac{77\!\cdots\!33}{12\!\cdots\!81}a^{4}+\frac{41\!\cdots\!64}{29\!\cdots\!63}a^{3}-\frac{30\!\cdots\!45}{29\!\cdots\!63}a^{2}+\frac{32\!\cdots\!37}{29\!\cdots\!63}a+\frac{50\!\cdots\!17}{29\!\cdots\!63}$, $\frac{20\!\cdots\!84}{29\!\cdots\!63}a^{20}-\frac{74\!\cdots\!48}{29\!\cdots\!63}a^{19}-\frac{95\!\cdots\!18}{29\!\cdots\!63}a^{18}+\frac{55\!\cdots\!43}{29\!\cdots\!63}a^{17}+\frac{32\!\cdots\!63}{17\!\cdots\!39}a^{16}-\frac{36\!\cdots\!14}{29\!\cdots\!63}a^{15}-\frac{24\!\cdots\!14}{29\!\cdots\!63}a^{14}+\frac{12\!\cdots\!48}{29\!\cdots\!63}a^{13}+\frac{43\!\cdots\!08}{29\!\cdots\!63}a^{12}-\frac{28\!\cdots\!05}{29\!\cdots\!63}a^{11}-\frac{10\!\cdots\!87}{29\!\cdots\!63}a^{10}+\frac{48\!\cdots\!62}{29\!\cdots\!63}a^{9}+\frac{15\!\cdots\!70}{29\!\cdots\!63}a^{8}-\frac{42\!\cdots\!85}{29\!\cdots\!63}a^{7}-\frac{83\!\cdots\!61}{29\!\cdots\!63}a^{6}+\frac{22\!\cdots\!66}{29\!\cdots\!63}a^{5}+\frac{50\!\cdots\!42}{29\!\cdots\!63}a^{4}-\frac{96\!\cdots\!75}{29\!\cdots\!63}a^{3}-\frac{16\!\cdots\!01}{29\!\cdots\!63}a^{2}+\frac{68\!\cdots\!68}{29\!\cdots\!63}a-\frac{22\!\cdots\!85}{29\!\cdots\!63}$, $\frac{24\!\cdots\!99}{29\!\cdots\!63}a^{20}-\frac{29\!\cdots\!00}{29\!\cdots\!63}a^{19}-\frac{14\!\cdots\!85}{29\!\cdots\!63}a^{18}+\frac{29\!\cdots\!75}{29\!\cdots\!63}a^{17}+\frac{64\!\cdots\!73}{17\!\cdots\!39}a^{16}-\frac{13\!\cdots\!99}{29\!\cdots\!63}a^{15}-\frac{38\!\cdots\!29}{29\!\cdots\!63}a^{14}+\frac{47\!\cdots\!36}{29\!\cdots\!63}a^{13}+\frac{88\!\cdots\!06}{29\!\cdots\!63}a^{12}-\frac{92\!\cdots\!75}{29\!\cdots\!63}a^{11}-\frac{16\!\cdots\!16}{29\!\cdots\!63}a^{10}+\frac{12\!\cdots\!76}{29\!\cdots\!63}a^{9}+\frac{14\!\cdots\!62}{29\!\cdots\!63}a^{8}-\frac{12\!\cdots\!26}{29\!\cdots\!63}a^{7}-\frac{76\!\cdots\!21}{29\!\cdots\!63}a^{6}+\frac{66\!\cdots\!71}{29\!\cdots\!63}a^{5}+\frac{34\!\cdots\!06}{29\!\cdots\!63}a^{4}-\frac{27\!\cdots\!92}{29\!\cdots\!63}a^{3}-\frac{23\!\cdots\!49}{29\!\cdots\!63}a^{2}+\frac{58\!\cdots\!24}{29\!\cdots\!63}a-\frac{14\!\cdots\!09}{29\!\cdots\!63}$, $\frac{12\!\cdots\!33}{29\!\cdots\!63}a^{20}-\frac{11\!\cdots\!36}{29\!\cdots\!63}a^{19}-\frac{70\!\cdots\!80}{29\!\cdots\!63}a^{18}+\frac{13\!\cdots\!88}{29\!\cdots\!63}a^{17}+\frac{32\!\cdots\!67}{17\!\cdots\!39}a^{16}-\frac{53\!\cdots\!08}{29\!\cdots\!63}a^{15}-\frac{19\!\cdots\!47}{29\!\cdots\!63}a^{14}+\frac{19\!\cdots\!61}{29\!\cdots\!63}a^{13}+\frac{45\!\cdots\!79}{29\!\cdots\!63}a^{12}-\frac{36\!\cdots\!17}{29\!\cdots\!63}a^{11}-\frac{83\!\cdots\!90}{29\!\cdots\!63}a^{10}+\frac{46\!\cdots\!45}{29\!\cdots\!63}a^{9}+\frac{76\!\cdots\!21}{29\!\cdots\!63}a^{8}-\frac{45\!\cdots\!85}{29\!\cdots\!63}a^{7}-\frac{40\!\cdots\!02}{29\!\cdots\!63}a^{6}+\frac{23\!\cdots\!21}{29\!\cdots\!63}a^{5}+\frac{18\!\cdots\!71}{29\!\cdots\!63}a^{4}-\frac{91\!\cdots\!49}{29\!\cdots\!63}a^{3}-\frac{12\!\cdots\!21}{29\!\cdots\!63}a^{2}+\frac{19\!\cdots\!70}{29\!\cdots\!63}a-\frac{48\!\cdots\!34}{29\!\cdots\!63}$, $\frac{65\!\cdots\!51}{29\!\cdots\!63}a^{20}+\frac{19\!\cdots\!76}{29\!\cdots\!63}a^{19}-\frac{77\!\cdots\!97}{29\!\cdots\!63}a^{18}-\frac{66\!\cdots\!35}{29\!\cdots\!63}a^{17}+\frac{38\!\cdots\!29}{17\!\cdots\!39}a^{16}+\frac{77\!\cdots\!00}{29\!\cdots\!63}a^{15}-\frac{27\!\cdots\!46}{29\!\cdots\!63}a^{14}-\frac{25\!\cdots\!89}{29\!\cdots\!63}a^{13}+\frac{84\!\cdots\!25}{29\!\cdots\!63}a^{12}+\frac{53\!\cdots\!06}{29\!\cdots\!63}a^{11}-\frac{16\!\cdots\!16}{29\!\cdots\!63}a^{10}-\frac{10\!\cdots\!17}{29\!\cdots\!63}a^{9}+\frac{20\!\cdots\!43}{29\!\cdots\!63}a^{8}+\frac{68\!\cdots\!60}{29\!\cdots\!63}a^{7}-\frac{16\!\cdots\!45}{29\!\cdots\!63}a^{6}-\frac{64\!\cdots\!41}{29\!\cdots\!63}a^{5}+\frac{75\!\cdots\!58}{29\!\cdots\!63}a^{4}+\frac{38\!\cdots\!26}{29\!\cdots\!63}a^{3}-\frac{29\!\cdots\!06}{29\!\cdots\!63}a^{2}+\frac{90\!\cdots\!36}{29\!\cdots\!63}a+\frac{12\!\cdots\!38}{29\!\cdots\!63}$, $\frac{25\!\cdots\!00}{29\!\cdots\!63}a^{20}-\frac{19\!\cdots\!87}{29\!\cdots\!63}a^{19}-\frac{15\!\cdots\!24}{29\!\cdots\!63}a^{18}+\frac{24\!\cdots\!10}{29\!\cdots\!63}a^{17}+\frac{73\!\cdots\!77}{17\!\cdots\!39}a^{16}-\frac{84\!\cdots\!31}{29\!\cdots\!63}a^{15}-\frac{44\!\cdots\!36}{29\!\cdots\!63}a^{14}+\frac{30\!\cdots\!60}{29\!\cdots\!63}a^{13}+\frac{10\!\cdots\!51}{29\!\cdots\!63}a^{12}-\frac{51\!\cdots\!70}{29\!\cdots\!63}a^{11}-\frac{19\!\cdots\!67}{29\!\cdots\!63}a^{10}+\frac{50\!\cdots\!83}{29\!\cdots\!63}a^{9}+\frac{18\!\cdots\!11}{29\!\cdots\!63}a^{8}-\frac{47\!\cdots\!62}{29\!\cdots\!63}a^{7}-\frac{10\!\cdots\!10}{29\!\cdots\!63}a^{6}+\frac{23\!\cdots\!78}{29\!\cdots\!63}a^{5}+\frac{48\!\cdots\!45}{29\!\cdots\!63}a^{4}-\frac{73\!\cdots\!26}{29\!\cdots\!63}a^{3}-\frac{69\!\cdots\!52}{29\!\cdots\!63}a^{2}+\frac{21\!\cdots\!94}{29\!\cdots\!63}a+\frac{53\!\cdots\!21}{12\!\cdots\!81}$, $\frac{29\!\cdots\!97}{29\!\cdots\!63}a^{20}-\frac{17\!\cdots\!65}{29\!\cdots\!63}a^{19}-\frac{18\!\cdots\!05}{29\!\cdots\!63}a^{18}+\frac{25\!\cdots\!35}{29\!\cdots\!63}a^{17}+\frac{87\!\cdots\!66}{17\!\cdots\!39}a^{16}-\frac{33\!\cdots\!72}{12\!\cdots\!81}a^{15}-\frac{53\!\cdots\!17}{29\!\cdots\!63}a^{14}+\frac{27\!\cdots\!42}{29\!\cdots\!63}a^{13}+\frac{13\!\cdots\!52}{29\!\cdots\!63}a^{12}-\frac{43\!\cdots\!77}{29\!\cdots\!63}a^{11}-\frac{24\!\cdots\!47}{29\!\cdots\!63}a^{10}+\frac{29\!\cdots\!71}{29\!\cdots\!63}a^{9}+\frac{24\!\cdots\!11}{29\!\cdots\!63}a^{8}-\frac{24\!\cdots\!95}{29\!\cdots\!63}a^{7}-\frac{15\!\cdots\!10}{29\!\cdots\!63}a^{6}+\frac{85\!\cdots\!92}{29\!\cdots\!63}a^{5}+\frac{75\!\cdots\!57}{29\!\cdots\!63}a^{4}+\frac{78\!\cdots\!63}{29\!\cdots\!63}a^{3}-\frac{15\!\cdots\!99}{29\!\cdots\!63}a^{2}+\frac{79\!\cdots\!79}{29\!\cdots\!63}a+\frac{10\!\cdots\!33}{29\!\cdots\!63}$
| 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: | \( 2726200.25314 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 2726200.25314 \cdot 1}{2\cdot\sqrt{105070765003019731854543276767}}\cr\approx \mathstrut & 0.208092227523 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 6*x^19 + 11*x^18 + 47*x^17 - 45*x^16 - 170*x^15 + 162*x^14 + 400*x^13 - 307*x^12 - 750*x^11 + 395*x^10 + 733*x^9 - 371*x^8 - 425*x^7 + 197*x^6 + 204*x^5 - 78*x^4 - 36*x^3 + 17*x^2 - 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^21 - x^20 - 6*x^19 + 11*x^18 + 47*x^17 - 45*x^16 - 170*x^15 + 162*x^14 + 400*x^13 - 307*x^12 - 750*x^11 + 395*x^10 + 733*x^9 - 371*x^8 - 425*x^7 + 197*x^6 + 204*x^5 - 78*x^4 - 36*x^3 + 17*x^2 - 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^21 - x^20 - 6*x^19 + 11*x^18 + 47*x^17 - 45*x^16 - 170*x^15 + 162*x^14 + 400*x^13 - 307*x^12 - 750*x^11 + 395*x^10 + 733*x^9 - 371*x^8 - 425*x^7 + 197*x^6 + 204*x^5 - 78*x^4 - 36*x^3 + 17*x^2 - 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^21 - x^20 - 6*x^19 + 11*x^18 + 47*x^17 - 45*x^16 - 170*x^15 + 162*x^14 + 400*x^13 - 307*x^12 - 750*x^11 + 395*x^10 + 733*x^9 - 371*x^8 - 425*x^7 + 197*x^6 + 204*x^5 - 78*x^4 - 36*x^3 + 17*x^2 - 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
$S_3\times S_7$ (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 $S_3\times S_7$ is not computed |
Intermediate fields
| 3.1.23.1, 7.7.25367689.1 |
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 | $21$ | $21$ | ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | R | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | $15{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.3.0.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(64879\)
| $\Q_{64879}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |