Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3)
gp: K = bnfinit(y^21 - 9*y^19 + 3*y^17 + 66*y^15 - 75*y^14 - 90*y^13 + 198*y^12 - 21*y^11 + 63*y^9 - 177*y^8 + 54*y^7 + 45*y^6 - 51*y^5 + 18*y^4 - 8*y^3 - 3*y^2 + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3)
\( x^{21} - 9 x^{19} + 3 x^{17} + 66 x^{15} - 75 x^{14} - 90 x^{13} + 198 x^{12} - 21 x^{11} + 63 x^{9} + \cdots + 3 \)
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: | $[9, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2523828389200110188894232441\)
\(\medspace = 3^{34}\cdot 73^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.18\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(3\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29}a^{18}+\frac{6}{29}a^{17}-\frac{2}{29}a^{16}-\frac{7}{29}a^{15}-\frac{9}{29}a^{14}-\frac{6}{29}a^{13}-\frac{5}{29}a^{12}-\frac{5}{29}a^{11}-\frac{5}{29}a^{10}-\frac{2}{29}a^{9}+\frac{4}{29}a^{7}-\frac{10}{29}a^{6}-\frac{5}{29}a^{5}-\frac{14}{29}a^{4}-\frac{2}{29}a^{3}-\frac{1}{29}a^{2}+\frac{11}{29}$, $\frac{1}{29}a^{19}-\frac{9}{29}a^{17}+\frac{5}{29}a^{16}+\frac{4}{29}a^{15}-\frac{10}{29}a^{14}+\frac{2}{29}a^{13}-\frac{4}{29}a^{12}-\frac{4}{29}a^{11}-\frac{1}{29}a^{10}+\frac{12}{29}a^{9}+\frac{4}{29}a^{8}-\frac{5}{29}a^{7}-\frac{3}{29}a^{6}-\frac{13}{29}a^{5}-\frac{5}{29}a^{4}+\frac{11}{29}a^{3}+\frac{6}{29}a^{2}+\frac{11}{29}a-\frac{8}{29}$, $\frac{1}{55\!\cdots\!31}a^{20}+\frac{15\!\cdots\!66}{55\!\cdots\!31}a^{19}-\frac{93\!\cdots\!08}{55\!\cdots\!31}a^{18}-\frac{38\!\cdots\!29}{55\!\cdots\!31}a^{17}-\frac{99\!\cdots\!65}{55\!\cdots\!31}a^{16}+\frac{74\!\cdots\!71}{55\!\cdots\!31}a^{15}+\frac{21\!\cdots\!97}{55\!\cdots\!31}a^{14}+\frac{12\!\cdots\!04}{55\!\cdots\!31}a^{13}-\frac{20\!\cdots\!27}{55\!\cdots\!31}a^{12}+\frac{24\!\cdots\!97}{55\!\cdots\!31}a^{11}+\frac{16\!\cdots\!64}{55\!\cdots\!31}a^{10}-\frac{75\!\cdots\!73}{55\!\cdots\!31}a^{9}+\frac{24\!\cdots\!30}{55\!\cdots\!31}a^{8}+\frac{20\!\cdots\!84}{55\!\cdots\!31}a^{7}-\frac{21\!\cdots\!67}{55\!\cdots\!31}a^{6}+\frac{21\!\cdots\!47}{55\!\cdots\!31}a^{5}+\frac{10\!\cdots\!48}{55\!\cdots\!31}a^{4}-\frac{21\!\cdots\!93}{55\!\cdots\!31}a^{3}+\frac{17\!\cdots\!72}{55\!\cdots\!31}a^{2}+\frac{34\!\cdots\!92}{55\!\cdots\!31}a+\frac{10\!\cdots\!60}{55\!\cdots\!31}$
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: | $14$ | 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: |
$\frac{71\!\cdots\!50}{55\!\cdots\!31}a^{20}+\frac{70\!\cdots\!02}{55\!\cdots\!31}a^{19}-\frac{64\!\cdots\!91}{55\!\cdots\!31}a^{18}-\frac{61\!\cdots\!43}{55\!\cdots\!31}a^{17}+\frac{18\!\cdots\!91}{55\!\cdots\!31}a^{16}+\frac{36\!\cdots\!77}{55\!\cdots\!31}a^{15}+\frac{46\!\cdots\!49}{55\!\cdots\!31}a^{14}-\frac{81\!\cdots\!53}{55\!\cdots\!31}a^{13}-\frac{11\!\cdots\!83}{55\!\cdots\!31}a^{12}+\frac{89\!\cdots\!12}{55\!\cdots\!31}a^{11}+\frac{10\!\cdots\!20}{55\!\cdots\!31}a^{10}-\frac{19\!\cdots\!32}{55\!\cdots\!31}a^{9}+\frac{68\!\cdots\!02}{55\!\cdots\!31}a^{8}-\frac{83\!\cdots\!47}{55\!\cdots\!31}a^{7}-\frac{57\!\cdots\!55}{55\!\cdots\!31}a^{6}+\frac{64\!\cdots\!93}{55\!\cdots\!31}a^{5}-\frac{26\!\cdots\!42}{55\!\cdots\!31}a^{4}-\frac{11\!\cdots\!82}{55\!\cdots\!31}a^{3}-\frac{84\!\cdots\!98}{55\!\cdots\!31}a^{2}-\frac{36\!\cdots\!96}{55\!\cdots\!31}a-\frac{97\!\cdots\!26}{55\!\cdots\!31}$, $\frac{22\!\cdots\!87}{55\!\cdots\!31}a^{20}+\frac{23\!\cdots\!47}{55\!\cdots\!31}a^{19}-\frac{21\!\cdots\!41}{55\!\cdots\!31}a^{18}-\frac{21\!\cdots\!81}{55\!\cdots\!31}a^{17}+\frac{21\!\cdots\!35}{55\!\cdots\!31}a^{16}+\frac{65\!\cdots\!11}{55\!\cdots\!31}a^{15}+\frac{14\!\cdots\!49}{55\!\cdots\!31}a^{14}-\frac{51\!\cdots\!03}{55\!\cdots\!31}a^{13}-\frac{48\!\cdots\!05}{55\!\cdots\!31}a^{12}+\frac{35\!\cdots\!99}{55\!\cdots\!31}a^{11}+\frac{52\!\cdots\!86}{55\!\cdots\!31}a^{10}-\frac{37\!\cdots\!26}{55\!\cdots\!31}a^{9}+\frac{21\!\cdots\!52}{55\!\cdots\!31}a^{8}-\frac{28\!\cdots\!76}{55\!\cdots\!31}a^{7}-\frac{41\!\cdots\!16}{55\!\cdots\!31}a^{6}+\frac{39\!\cdots\!42}{55\!\cdots\!31}a^{5}-\frac{21\!\cdots\!28}{55\!\cdots\!31}a^{4}-\frac{11\!\cdots\!97}{55\!\cdots\!31}a^{3}+\frac{13\!\cdots\!84}{55\!\cdots\!31}a^{2}-\frac{32\!\cdots\!97}{55\!\cdots\!31}a+\frac{39\!\cdots\!17}{55\!\cdots\!31}$, $\frac{51\!\cdots\!41}{55\!\cdots\!31}a^{20}-\frac{31\!\cdots\!42}{55\!\cdots\!31}a^{19}-\frac{39\!\cdots\!95}{55\!\cdots\!31}a^{18}+\frac{24\!\cdots\!29}{55\!\cdots\!31}a^{17}-\frac{41\!\cdots\!82}{55\!\cdots\!31}a^{16}+\frac{18\!\cdots\!47}{55\!\cdots\!31}a^{15}+\frac{32\!\cdots\!10}{55\!\cdots\!31}a^{14}-\frac{58\!\cdots\!80}{55\!\cdots\!31}a^{13}+\frac{20\!\cdots\!67}{55\!\cdots\!31}a^{12}+\frac{58\!\cdots\!54}{55\!\cdots\!31}a^{11}-\frac{82\!\cdots\!36}{55\!\cdots\!31}a^{10}+\frac{42\!\cdots\!62}{19\!\cdots\!39}a^{9}-\frac{44\!\cdots\!80}{55\!\cdots\!31}a^{8}-\frac{39\!\cdots\!33}{55\!\cdots\!31}a^{7}+\frac{71\!\cdots\!73}{55\!\cdots\!31}a^{6}-\frac{80\!\cdots\!24}{55\!\cdots\!31}a^{5}+\frac{26\!\cdots\!06}{55\!\cdots\!31}a^{4}+\frac{61\!\cdots\!60}{55\!\cdots\!31}a^{3}-\frac{15\!\cdots\!32}{55\!\cdots\!31}a^{2}+\frac{38\!\cdots\!26}{55\!\cdots\!31}a-\frac{12\!\cdots\!83}{55\!\cdots\!31}$, $\frac{65\!\cdots\!71}{55\!\cdots\!31}a^{20}+\frac{23\!\cdots\!28}{55\!\cdots\!31}a^{19}-\frac{55\!\cdots\!86}{55\!\cdots\!31}a^{18}-\frac{22\!\cdots\!45}{55\!\cdots\!31}a^{17}-\frac{16\!\cdots\!28}{55\!\cdots\!31}a^{16}+\frac{19\!\cdots\!18}{55\!\cdots\!31}a^{15}+\frac{44\!\cdots\!49}{55\!\cdots\!31}a^{14}-\frac{32\!\cdots\!18}{55\!\cdots\!31}a^{13}-\frac{49\!\cdots\!42}{55\!\cdots\!31}a^{12}+\frac{68\!\cdots\!13}{55\!\cdots\!31}a^{11}+\frac{12\!\cdots\!65}{55\!\cdots\!31}a^{10}+\frac{70\!\cdots\!69}{55\!\cdots\!31}a^{9}+\frac{15\!\cdots\!35}{55\!\cdots\!31}a^{8}-\frac{80\!\cdots\!82}{55\!\cdots\!31}a^{7}-\frac{14\!\cdots\!37}{55\!\cdots\!31}a^{6}-\frac{19\!\cdots\!40}{55\!\cdots\!31}a^{5}-\frac{33\!\cdots\!98}{55\!\cdots\!31}a^{4}+\frac{11\!\cdots\!88}{55\!\cdots\!31}a^{3}-\frac{49\!\cdots\!57}{55\!\cdots\!31}a^{2}-\frac{49\!\cdots\!06}{55\!\cdots\!31}a+\frac{11\!\cdots\!00}{55\!\cdots\!31}$, $\frac{46\!\cdots\!45}{55\!\cdots\!31}a^{20}-\frac{61\!\cdots\!03}{55\!\cdots\!31}a^{19}-\frac{44\!\cdots\!05}{55\!\cdots\!31}a^{18}+\frac{52\!\cdots\!90}{55\!\cdots\!31}a^{17}+\frac{37\!\cdots\!08}{55\!\cdots\!31}a^{16}+\frac{68\!\cdots\!25}{55\!\cdots\!31}a^{15}+\frac{31\!\cdots\!74}{55\!\cdots\!31}a^{14}-\frac{75\!\cdots\!96}{55\!\cdots\!31}a^{13}-\frac{14\!\cdots\!76}{55\!\cdots\!31}a^{12}+\frac{14\!\cdots\!06}{55\!\cdots\!31}a^{11}-\frac{91\!\cdots\!95}{55\!\cdots\!31}a^{10}-\frac{17\!\cdots\!09}{55\!\cdots\!31}a^{9}+\frac{25\!\cdots\!38}{55\!\cdots\!31}a^{8}-\frac{13\!\cdots\!19}{55\!\cdots\!31}a^{7}+\frac{90\!\cdots\!28}{55\!\cdots\!31}a^{6}+\frac{38\!\cdots\!55}{55\!\cdots\!31}a^{5}-\frac{39\!\cdots\!70}{55\!\cdots\!31}a^{4}+\frac{30\!\cdots\!23}{55\!\cdots\!31}a^{3}-\frac{31\!\cdots\!95}{55\!\cdots\!31}a^{2}-\frac{10\!\cdots\!70}{55\!\cdots\!31}a+\frac{67\!\cdots\!98}{55\!\cdots\!31}$, $\frac{48\!\cdots\!24}{55\!\cdots\!31}a^{20}-\frac{48\!\cdots\!46}{55\!\cdots\!31}a^{19}-\frac{40\!\cdots\!29}{55\!\cdots\!31}a^{18}+\frac{39\!\cdots\!47}{55\!\cdots\!31}a^{17}-\frac{14\!\cdots\!20}{55\!\cdots\!31}a^{16}+\frac{25\!\cdots\!91}{55\!\cdots\!31}a^{15}+\frac{30\!\cdots\!39}{55\!\cdots\!31}a^{14}-\frac{67\!\cdots\!22}{55\!\cdots\!31}a^{13}+\frac{14\!\cdots\!33}{55\!\cdots\!31}a^{12}+\frac{83\!\cdots\!55}{55\!\cdots\!31}a^{11}-\frac{82\!\cdots\!00}{55\!\cdots\!31}a^{10}+\frac{81\!\cdots\!42}{55\!\cdots\!31}a^{9}-\frac{55\!\cdots\!53}{55\!\cdots\!31}a^{8}-\frac{62\!\cdots\!47}{55\!\cdots\!31}a^{7}+\frac{79\!\cdots\!24}{55\!\cdots\!31}a^{6}-\frac{34\!\cdots\!26}{55\!\cdots\!31}a^{5}+\frac{19\!\cdots\!86}{55\!\cdots\!31}a^{4}+\frac{16\!\cdots\!71}{55\!\cdots\!31}a^{3}-\frac{14\!\cdots\!70}{55\!\cdots\!31}a^{2}+\frac{27\!\cdots\!56}{55\!\cdots\!31}a+\frac{23\!\cdots\!36}{55\!\cdots\!31}$, $\frac{67\!\cdots\!20}{55\!\cdots\!31}a^{20}-\frac{21\!\cdots\!12}{55\!\cdots\!31}a^{19}-\frac{24\!\cdots\!68}{55\!\cdots\!31}a^{18}+\frac{18\!\cdots\!77}{55\!\cdots\!31}a^{17}-\frac{28\!\cdots\!17}{55\!\cdots\!31}a^{16}+\frac{13\!\cdots\!36}{55\!\cdots\!31}a^{15}+\frac{35\!\cdots\!50}{55\!\cdots\!31}a^{14}-\frac{18\!\cdots\!47}{55\!\cdots\!31}a^{13}+\frac{33\!\cdots\!48}{55\!\cdots\!31}a^{12}-\frac{11\!\cdots\!00}{55\!\cdots\!31}a^{11}-\frac{55\!\cdots\!12}{55\!\cdots\!31}a^{10}+\frac{21\!\cdots\!31}{19\!\cdots\!39}a^{9}-\frac{28\!\cdots\!69}{55\!\cdots\!31}a^{8}+\frac{16\!\cdots\!91}{55\!\cdots\!31}a^{7}+\frac{46\!\cdots\!16}{55\!\cdots\!31}a^{6}-\frac{59\!\cdots\!63}{55\!\cdots\!31}a^{5}+\frac{19\!\cdots\!22}{55\!\cdots\!31}a^{4}+\frac{50\!\cdots\!33}{55\!\cdots\!31}a^{3}-\frac{17\!\cdots\!30}{55\!\cdots\!31}a^{2}+\frac{46\!\cdots\!05}{55\!\cdots\!31}a-\frac{17\!\cdots\!14}{55\!\cdots\!31}$, $\frac{46\!\cdots\!70}{55\!\cdots\!31}a^{20}+\frac{38\!\cdots\!75}{55\!\cdots\!31}a^{19}-\frac{45\!\cdots\!31}{55\!\cdots\!31}a^{18}-\frac{33\!\cdots\!42}{55\!\cdots\!31}a^{17}+\frac{38\!\cdots\!41}{55\!\cdots\!31}a^{16}+\frac{92\!\cdots\!99}{55\!\cdots\!31}a^{15}+\frac{31\!\cdots\!68}{55\!\cdots\!31}a^{14}-\frac{11\!\cdots\!62}{55\!\cdots\!31}a^{13}-\frac{89\!\cdots\!67}{55\!\cdots\!31}a^{12}+\frac{82\!\cdots\!13}{55\!\cdots\!31}a^{11}+\frac{76\!\cdots\!60}{55\!\cdots\!31}a^{10}-\frac{40\!\cdots\!48}{55\!\cdots\!31}a^{9}+\frac{31\!\cdots\!20}{55\!\cdots\!31}a^{8}-\frac{94\!\cdots\!74}{55\!\cdots\!31}a^{7}-\frac{96\!\cdots\!85}{55\!\cdots\!31}a^{6}+\frac{64\!\cdots\!61}{55\!\cdots\!31}a^{5}-\frac{11\!\cdots\!94}{55\!\cdots\!31}a^{4}+\frac{21\!\cdots\!73}{55\!\cdots\!31}a^{3}-\frac{13\!\cdots\!13}{55\!\cdots\!31}a^{2}+\frac{12\!\cdots\!51}{55\!\cdots\!31}a+\frac{96\!\cdots\!21}{55\!\cdots\!31}$, $\frac{45\!\cdots\!38}{55\!\cdots\!31}a^{20}-\frac{74\!\cdots\!12}{55\!\cdots\!31}a^{19}-\frac{36\!\cdots\!66}{55\!\cdots\!31}a^{18}+\frac{62\!\cdots\!25}{55\!\cdots\!31}a^{17}-\frac{22\!\cdots\!07}{55\!\cdots\!31}a^{16}+\frac{12\!\cdots\!85}{55\!\cdots\!31}a^{15}+\frac{26\!\cdots\!24}{55\!\cdots\!31}a^{14}-\frac{82\!\cdots\!55}{55\!\cdots\!31}a^{13}+\frac{44\!\cdots\!93}{55\!\cdots\!31}a^{12}+\frac{33\!\cdots\!07}{19\!\cdots\!39}a^{11}-\frac{13\!\cdots\!88}{55\!\cdots\!31}a^{10}+\frac{87\!\cdots\!86}{55\!\cdots\!31}a^{9}-\frac{72\!\cdots\!43}{55\!\cdots\!31}a^{8}-\frac{30\!\cdots\!06}{55\!\cdots\!31}a^{7}+\frac{12\!\cdots\!98}{55\!\cdots\!31}a^{6}-\frac{54\!\cdots\!68}{55\!\cdots\!31}a^{5}+\frac{17\!\cdots\!34}{55\!\cdots\!31}a^{4}-\frac{18\!\cdots\!00}{55\!\cdots\!31}a^{3}-\frac{10\!\cdots\!76}{55\!\cdots\!31}a^{2}+\frac{48\!\cdots\!16}{55\!\cdots\!31}a+\frac{43\!\cdots\!82}{55\!\cdots\!31}$, $\frac{15\!\cdots\!33}{19\!\cdots\!39}a^{20}-\frac{78\!\cdots\!15}{19\!\cdots\!39}a^{19}-\frac{11\!\cdots\!91}{19\!\cdots\!39}a^{18}+\frac{65\!\cdots\!63}{19\!\cdots\!39}a^{17}-\frac{15\!\cdots\!73}{19\!\cdots\!39}a^{16}+\frac{10\!\cdots\!05}{19\!\cdots\!39}a^{15}+\frac{94\!\cdots\!51}{19\!\cdots\!39}a^{14}-\frac{15\!\cdots\!03}{19\!\cdots\!39}a^{13}+\frac{78\!\cdots\!87}{19\!\cdots\!39}a^{12}+\frac{16\!\cdots\!01}{19\!\cdots\!39}a^{11}-\frac{26\!\cdots\!95}{19\!\cdots\!39}a^{10}+\frac{33\!\cdots\!81}{19\!\cdots\!39}a^{9}-\frac{50\!\cdots\!80}{19\!\cdots\!39}a^{8}-\frac{30\!\cdots\!58}{19\!\cdots\!39}a^{7}+\frac{13\!\cdots\!54}{19\!\cdots\!39}a^{6}-\frac{27\!\cdots\!70}{19\!\cdots\!39}a^{5}+\frac{22\!\cdots\!12}{19\!\cdots\!39}a^{4}-\frac{43\!\cdots\!86}{19\!\cdots\!39}a^{3}+\frac{81\!\cdots\!42}{19\!\cdots\!39}a^{2}+\frac{77\!\cdots\!29}{19\!\cdots\!39}a-\frac{16\!\cdots\!80}{19\!\cdots\!39}$, $\frac{20\!\cdots\!53}{55\!\cdots\!31}a^{20}-\frac{38\!\cdots\!72}{55\!\cdots\!31}a^{19}-\frac{18\!\cdots\!41}{55\!\cdots\!31}a^{18}-\frac{49\!\cdots\!77}{55\!\cdots\!31}a^{17}+\frac{47\!\cdots\!29}{55\!\cdots\!31}a^{16}+\frac{43\!\cdots\!73}{55\!\cdots\!31}a^{15}+\frac{13\!\cdots\!64}{55\!\cdots\!31}a^{14}-\frac{15\!\cdots\!64}{55\!\cdots\!31}a^{13}-\frac{17\!\cdots\!42}{55\!\cdots\!31}a^{12}+\frac{36\!\cdots\!79}{55\!\cdots\!31}a^{11}+\frac{18\!\cdots\!69}{55\!\cdots\!31}a^{10}+\frac{36\!\cdots\!94}{55\!\cdots\!31}a^{9}+\frac{29\!\cdots\!50}{55\!\cdots\!31}a^{8}-\frac{28\!\cdots\!98}{55\!\cdots\!31}a^{7}+\frac{67\!\cdots\!63}{55\!\cdots\!31}a^{6}+\frac{79\!\cdots\!27}{55\!\cdots\!31}a^{5}-\frac{39\!\cdots\!10}{55\!\cdots\!31}a^{4}-\frac{19\!\cdots\!03}{55\!\cdots\!31}a^{3}-\frac{92\!\cdots\!26}{55\!\cdots\!31}a^{2}+\frac{41\!\cdots\!96}{55\!\cdots\!31}a+\frac{44\!\cdots\!69}{55\!\cdots\!31}$, $\frac{42\!\cdots\!20}{55\!\cdots\!31}a^{20}-\frac{80\!\cdots\!79}{55\!\cdots\!31}a^{19}-\frac{36\!\cdots\!62}{55\!\cdots\!31}a^{18}+\frac{72\!\cdots\!78}{55\!\cdots\!31}a^{17}-\frac{25\!\cdots\!25}{55\!\cdots\!31}a^{16}-\frac{30\!\cdots\!82}{55\!\cdots\!31}a^{15}+\frac{28\!\cdots\!42}{55\!\cdots\!31}a^{14}-\frac{84\!\cdots\!21}{55\!\cdots\!31}a^{13}+\frac{34\!\cdots\!88}{55\!\cdots\!31}a^{12}+\frac{14\!\cdots\!60}{55\!\cdots\!31}a^{11}-\frac{18\!\cdots\!10}{55\!\cdots\!31}a^{10}+\frac{14\!\cdots\!18}{19\!\cdots\!39}a^{9}+\frac{29\!\cdots\!12}{55\!\cdots\!31}a^{8}-\frac{11\!\cdots\!72}{55\!\cdots\!31}a^{7}+\frac{17\!\cdots\!47}{55\!\cdots\!31}a^{6}-\frac{69\!\cdots\!72}{55\!\cdots\!31}a^{5}-\frac{63\!\cdots\!49}{55\!\cdots\!31}a^{4}+\frac{51\!\cdots\!98}{55\!\cdots\!31}a^{3}-\frac{18\!\cdots\!52}{55\!\cdots\!31}a^{2}+\frac{11\!\cdots\!63}{55\!\cdots\!31}a-\frac{55\!\cdots\!16}{55\!\cdots\!31}$, $\frac{38\!\cdots\!07}{55\!\cdots\!31}a^{20}+\frac{11\!\cdots\!45}{55\!\cdots\!31}a^{19}-\frac{36\!\cdots\!23}{55\!\cdots\!31}a^{18}-\frac{98\!\cdots\!13}{55\!\cdots\!31}a^{17}+\frac{23\!\cdots\!35}{55\!\cdots\!31}a^{16}+\frac{12\!\cdots\!43}{55\!\cdots\!31}a^{15}+\frac{26\!\cdots\!25}{55\!\cdots\!31}a^{14}+\frac{45\!\cdots\!95}{55\!\cdots\!31}a^{13}-\frac{12\!\cdots\!57}{55\!\cdots\!31}a^{12}+\frac{21\!\cdots\!46}{55\!\cdots\!31}a^{11}+\frac{19\!\cdots\!72}{55\!\cdots\!31}a^{10}-\frac{55\!\cdots\!06}{55\!\cdots\!31}a^{9}+\frac{77\!\cdots\!00}{55\!\cdots\!31}a^{8}-\frac{28\!\cdots\!86}{55\!\cdots\!31}a^{7}-\frac{16\!\cdots\!18}{55\!\cdots\!31}a^{6}+\frac{82\!\cdots\!76}{55\!\cdots\!31}a^{5}-\frac{10\!\cdots\!29}{55\!\cdots\!31}a^{4}-\frac{22\!\cdots\!39}{55\!\cdots\!31}a^{3}+\frac{24\!\cdots\!04}{55\!\cdots\!31}a^{2}-\frac{76\!\cdots\!65}{55\!\cdots\!31}a-\frac{26\!\cdots\!92}{55\!\cdots\!31}$, $\frac{26\!\cdots\!39}{19\!\cdots\!39}a^{20}+\frac{89\!\cdots\!84}{55\!\cdots\!31}a^{19}-\frac{68\!\cdots\!34}{55\!\cdots\!31}a^{18}-\frac{86\!\cdots\!44}{55\!\cdots\!31}a^{17}+\frac{17\!\cdots\!55}{55\!\cdots\!31}a^{16}+\frac{97\!\cdots\!19}{55\!\cdots\!31}a^{15}+\frac{49\!\cdots\!21}{55\!\cdots\!31}a^{14}-\frac{53\!\cdots\!82}{55\!\cdots\!31}a^{13}-\frac{70\!\cdots\!77}{55\!\cdots\!31}a^{12}+\frac{13\!\cdots\!89}{55\!\cdots\!31}a^{11}+\frac{98\!\cdots\!43}{55\!\cdots\!31}a^{10}+\frac{18\!\cdots\!14}{55\!\cdots\!31}a^{9}+\frac{93\!\cdots\!25}{55\!\cdots\!31}a^{8}-\frac{11\!\cdots\!62}{55\!\cdots\!31}a^{7}+\frac{54\!\cdots\!31}{55\!\cdots\!31}a^{6}+\frac{28\!\cdots\!85}{55\!\cdots\!31}a^{5}-\frac{70\!\cdots\!94}{55\!\cdots\!31}a^{4}-\frac{14\!\cdots\!29}{55\!\cdots\!31}a^{3}-\frac{21\!\cdots\!46}{55\!\cdots\!31}a^{2}+\frac{34\!\cdots\!96}{55\!\cdots\!31}a+\frac{36\!\cdots\!24}{55\!\cdots\!31}$
| 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: | \( 823613.327719 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{6}\cdot 823613.327719 \cdot 1}{2\cdot\sqrt{2523828389200110188894232441}}\cr\approx \mathstrut & 0.258233529183 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3)
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 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3, 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 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3);
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 - 9*x^19 + 3*x^17 + 66*x^15 - 75*x^14 - 90*x^13 + 198*x^12 - 21*x^11 + 63*x^9 - 177*x^8 + 54*x^7 + 45*x^6 - 51*x^5 + 18*x^4 - 8*x^3 - 3*x^2 + 3);
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
$C_3\times A_7$ (as 21T44):
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 7560 |
| The 27 conjugacy class representatives for $C_3\times A_7$ |
| Character table for $C_3\times A_7$ is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 7.3.3884841.1 |
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 45 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 | $21$ | $21$ | $21$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.7.0.1}{7} }^{3}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{7}$ | ${\href{/padicField/37.7.0.1}{7} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $21$ | ${\href{/padicField/53.7.0.1}{7} }^{3}$ | $15{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| Deg $18$ | $9$ | $2$ | $30$ | ||||
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 73.3.0.1 | $x^{3} + 2 x + 68$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 73.3.0.1 | $x^{3} + 2 x + 68$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 73.3.2.2 | $x^{3} + 292$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.0.1 | $x^{3} + 2 x + 68$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 73.3.2.2 | $x^{3} + 292$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.2 | $x^{3} + 292$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |