Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^19 - 10*x^18 - 79*x^17 + 237*x^16 + 265*x^15 - 691*x^14 - 190*x^13 - 2867*x^12 + 8917*x^11 - 86*x^10 - 12167*x^9 - 1656*x^8 + 3245*x^7 + 25876*x^6 - 21636*x^5 - 3866*x^4 + 177*x^3 + 4752*x^2 + 2525*x - 2917)
gp: K = bnfinit(y^21 - 3*y^19 - 10*y^18 - 79*y^17 + 237*y^16 + 265*y^15 - 691*y^14 - 190*y^13 - 2867*y^12 + 8917*y^11 - 86*y^10 - 12167*y^9 - 1656*y^8 + 3245*y^7 + 25876*y^6 - 21636*y^5 - 3866*y^4 + 177*y^3 + 4752*y^2 + 2525*y - 2917, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 3*x^19 - 10*x^18 - 79*x^17 + 237*x^16 + 265*x^15 - 691*x^14 - 190*x^13 - 2867*x^12 + 8917*x^11 - 86*x^10 - 12167*x^9 - 1656*x^8 + 3245*x^7 + 25876*x^6 - 21636*x^5 - 3866*x^4 + 177*x^3 + 4752*x^2 + 2525*x - 2917);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 3*x^19 - 10*x^18 - 79*x^17 + 237*x^16 + 265*x^15 - 691*x^14 - 190*x^13 - 2867*x^12 + 8917*x^11 - 86*x^10 - 12167*x^9 - 1656*x^8 + 3245*x^7 + 25876*x^6 - 21636*x^5 - 3866*x^4 + 177*x^3 + 4752*x^2 + 2525*x - 2917)
\( x^{21} - 3 x^{19} - 10 x^{18} - 79 x^{17} + 237 x^{16} + 265 x^{15} - 691 x^{14} - 190 x^{13} + \cdots - 2917 \)
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: |
\(-99995832264130420565259872976896\)
\(\medspace = -\,2^{14}\cdot 29^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(33.40\) | 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: | $2\cdot 29^{13/14}\approx 45.60064585163551$ | ||
| Ramified primes: |
\(2\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-29}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{17}a^{19}-\frac{4}{17}a^{18}+\frac{8}{17}a^{16}+\frac{8}{17}a^{15}-\frac{1}{17}a^{14}-\frac{5}{17}a^{13}+\frac{5}{17}a^{12}+\frac{8}{17}a^{11}-\frac{6}{17}a^{10}-\frac{3}{17}a^{9}+\frac{4}{17}a^{8}-\frac{6}{17}a^{7}-\frac{1}{17}a^{6}-\frac{5}{17}a^{5}+\frac{1}{17}a^{4}-\frac{2}{17}a^{3}+\frac{5}{17}a^{2}-\frac{4}{17}a-\frac{6}{17}$, $\frac{1}{74\!\cdots\!49}a^{20}-\frac{70\!\cdots\!02}{74\!\cdots\!49}a^{19}+\frac{10\!\cdots\!61}{74\!\cdots\!49}a^{18}+\frac{23\!\cdots\!62}{74\!\cdots\!49}a^{17}-\frac{11\!\cdots\!08}{74\!\cdots\!49}a^{16}-\frac{77\!\cdots\!94}{74\!\cdots\!49}a^{15}-\frac{16\!\cdots\!60}{74\!\cdots\!49}a^{14}-\frac{13\!\cdots\!54}{74\!\cdots\!49}a^{13}-\frac{33\!\cdots\!89}{74\!\cdots\!49}a^{12}-\frac{16\!\cdots\!58}{43\!\cdots\!97}a^{11}+\frac{27\!\cdots\!31}{74\!\cdots\!49}a^{10}-\frac{32\!\cdots\!04}{74\!\cdots\!49}a^{9}+\frac{20\!\cdots\!42}{74\!\cdots\!49}a^{8}+\frac{37\!\cdots\!55}{74\!\cdots\!49}a^{7}+\frac{15\!\cdots\!66}{74\!\cdots\!49}a^{6}-\frac{14\!\cdots\!31}{74\!\cdots\!49}a^{5}+\frac{18\!\cdots\!53}{18\!\cdots\!89}a^{4}+\frac{24\!\cdots\!39}{74\!\cdots\!49}a^{3}-\frac{74\!\cdots\!62}{74\!\cdots\!49}a^{2}+\frac{22\!\cdots\!93}{74\!\cdots\!49}a+\frac{13\!\cdots\!40}{74\!\cdots\!49}$
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: |
$\frac{13\!\cdots\!75}{43\!\cdots\!97}a^{20}+\frac{24\!\cdots\!71}{74\!\cdots\!49}a^{19}-\frac{40\!\cdots\!41}{74\!\cdots\!49}a^{18}-\frac{15\!\cdots\!02}{43\!\cdots\!97}a^{17}-\frac{20\!\cdots\!85}{74\!\cdots\!49}a^{16}+\frac{31\!\cdots\!29}{74\!\cdots\!49}a^{15}+\frac{91\!\cdots\!37}{74\!\cdots\!49}a^{14}-\frac{62\!\cdots\!06}{74\!\cdots\!49}a^{13}-\frac{10\!\cdots\!81}{74\!\cdots\!49}a^{12}-\frac{73\!\cdots\!86}{74\!\cdots\!49}a^{11}+\frac{12\!\cdots\!65}{74\!\cdots\!49}a^{10}+\frac{12\!\cdots\!87}{74\!\cdots\!49}a^{9}-\frac{15\!\cdots\!68}{74\!\cdots\!49}a^{8}-\frac{18\!\cdots\!63}{74\!\cdots\!49}a^{7}-\frac{10\!\cdots\!30}{74\!\cdots\!49}a^{6}+\frac{48\!\cdots\!12}{74\!\cdots\!49}a^{5}-\frac{38\!\cdots\!39}{18\!\cdots\!89}a^{4}-\frac{15\!\cdots\!16}{74\!\cdots\!49}a^{3}-\frac{11\!\cdots\!63}{74\!\cdots\!49}a^{2}-\frac{49\!\cdots\!32}{74\!\cdots\!49}a+\frac{74\!\cdots\!25}{74\!\cdots\!49}$, $\frac{13\!\cdots\!01}{74\!\cdots\!49}a^{20}+\frac{14\!\cdots\!06}{74\!\cdots\!49}a^{19}-\frac{28\!\cdots\!25}{74\!\cdots\!49}a^{18}-\frac{17\!\cdots\!76}{74\!\cdots\!49}a^{17}-\frac{12\!\cdots\!23}{74\!\cdots\!49}a^{16}+\frac{18\!\cdots\!15}{74\!\cdots\!49}a^{15}+\frac{59\!\cdots\!21}{74\!\cdots\!49}a^{14}-\frac{15\!\cdots\!33}{43\!\cdots\!97}a^{13}-\frac{69\!\cdots\!26}{74\!\cdots\!49}a^{12}-\frac{46\!\cdots\!50}{74\!\cdots\!49}a^{11}+\frac{68\!\cdots\!54}{74\!\cdots\!49}a^{10}+\frac{84\!\cdots\!16}{74\!\cdots\!49}a^{9}-\frac{69\!\cdots\!48}{74\!\cdots\!49}a^{8}-\frac{12\!\cdots\!48}{74\!\cdots\!49}a^{7}-\frac{10\!\cdots\!72}{74\!\cdots\!49}a^{6}+\frac{25\!\cdots\!38}{74\!\cdots\!49}a^{5}+\frac{80\!\cdots\!73}{18\!\cdots\!89}a^{4}-\frac{30\!\cdots\!98}{74\!\cdots\!49}a^{3}-\frac{69\!\cdots\!19}{74\!\cdots\!49}a^{2}-\frac{18\!\cdots\!62}{74\!\cdots\!49}a+\frac{22\!\cdots\!98}{74\!\cdots\!49}$, $\frac{59\!\cdots\!85}{74\!\cdots\!49}a^{20}+\frac{86\!\cdots\!58}{74\!\cdots\!49}a^{19}-\frac{71\!\cdots\!22}{74\!\cdots\!49}a^{18}-\frac{73\!\cdots\!25}{74\!\cdots\!49}a^{17}-\frac{57\!\cdots\!22}{74\!\cdots\!49}a^{16}+\frac{59\!\cdots\!07}{74\!\cdots\!49}a^{15}+\frac{26\!\cdots\!66}{74\!\cdots\!49}a^{14}-\frac{36\!\cdots\!53}{74\!\cdots\!49}a^{13}-\frac{26\!\cdots\!73}{74\!\cdots\!49}a^{12}-\frac{21\!\cdots\!23}{74\!\cdots\!49}a^{11}+\frac{22\!\cdots\!66}{74\!\cdots\!49}a^{10}+\frac{40\!\cdots\!87}{74\!\cdots\!49}a^{9}-\frac{17\!\cdots\!38}{74\!\cdots\!49}a^{8}-\frac{52\!\cdots\!29}{74\!\cdots\!49}a^{7}-\frac{56\!\cdots\!23}{74\!\cdots\!49}a^{6}+\frac{87\!\cdots\!54}{74\!\cdots\!49}a^{5}+\frac{81\!\cdots\!59}{18\!\cdots\!89}a^{4}-\frac{64\!\cdots\!58}{74\!\cdots\!49}a^{3}-\frac{18\!\cdots\!15}{74\!\cdots\!49}a^{2}-\frac{58\!\cdots\!28}{74\!\cdots\!49}a-\frac{24\!\cdots\!57}{74\!\cdots\!49}$, $\frac{17\!\cdots\!75}{74\!\cdots\!49}a^{20}+\frac{18\!\cdots\!68}{74\!\cdots\!49}a^{19}-\frac{30\!\cdots\!19}{74\!\cdots\!49}a^{18}-\frac{20\!\cdots\!62}{74\!\cdots\!49}a^{17}-\frac{16\!\cdots\!74}{74\!\cdots\!49}a^{16}+\frac{24\!\cdots\!12}{74\!\cdots\!49}a^{15}+\frac{70\!\cdots\!33}{74\!\cdots\!49}a^{14}-\frac{45\!\cdots\!49}{74\!\cdots\!49}a^{13}-\frac{70\!\cdots\!08}{74\!\cdots\!49}a^{12}-\frac{58\!\cdots\!81}{74\!\cdots\!49}a^{11}+\frac{94\!\cdots\!76}{74\!\cdots\!49}a^{10}+\frac{89\!\cdots\!70}{74\!\cdots\!49}a^{9}-\frac{64\!\cdots\!01}{43\!\cdots\!97}a^{8}-\frac{12\!\cdots\!79}{74\!\cdots\!49}a^{7}-\frac{87\!\cdots\!34}{74\!\cdots\!49}a^{6}+\frac{33\!\cdots\!02}{74\!\cdots\!49}a^{5}-\frac{10\!\cdots\!13}{18\!\cdots\!89}a^{4}-\frac{64\!\cdots\!47}{74\!\cdots\!49}a^{3}-\frac{47\!\cdots\!83}{74\!\cdots\!49}a^{2}+\frac{47\!\cdots\!50}{74\!\cdots\!49}a+\frac{36\!\cdots\!71}{74\!\cdots\!49}$, $\frac{96\!\cdots\!57}{12\!\cdots\!11}a^{20}+\frac{14\!\cdots\!80}{12\!\cdots\!11}a^{19}-\frac{18\!\cdots\!61}{12\!\cdots\!11}a^{18}-\frac{14\!\cdots\!55}{12\!\cdots\!11}a^{17}-\frac{97\!\cdots\!44}{12\!\cdots\!11}a^{16}+\frac{91\!\cdots\!63}{12\!\cdots\!11}a^{15}+\frac{51\!\cdots\!67}{12\!\cdots\!11}a^{14}+\frac{22\!\cdots\!02}{12\!\cdots\!11}a^{13}-\frac{64\!\cdots\!25}{12\!\cdots\!11}a^{12}-\frac{21\!\cdots\!59}{74\!\cdots\!83}a^{11}+\frac{32\!\cdots\!83}{12\!\cdots\!11}a^{10}+\frac{90\!\cdots\!37}{12\!\cdots\!11}a^{9}-\frac{18\!\cdots\!43}{12\!\cdots\!11}a^{8}-\frac{11\!\cdots\!48}{12\!\cdots\!11}a^{7}-\frac{12\!\cdots\!59}{12\!\cdots\!11}a^{6}+\frac{14\!\cdots\!41}{12\!\cdots\!11}a^{5}+\frac{31\!\cdots\!09}{30\!\cdots\!71}a^{4}+\frac{20\!\cdots\!72}{12\!\cdots\!11}a^{3}-\frac{46\!\cdots\!26}{12\!\cdots\!11}a^{2}-\frac{46\!\cdots\!20}{12\!\cdots\!11}a+\frac{73\!\cdots\!92}{12\!\cdots\!11}$, $\frac{92\!\cdots\!08}{74\!\cdots\!49}a^{20}+\frac{10\!\cdots\!20}{74\!\cdots\!49}a^{19}-\frac{76\!\cdots\!60}{43\!\cdots\!97}a^{18}-\frac{10\!\cdots\!09}{74\!\cdots\!49}a^{17}-\frac{84\!\cdots\!43}{74\!\cdots\!49}a^{16}+\frac{12\!\cdots\!55}{74\!\cdots\!49}a^{15}+\frac{35\!\cdots\!36}{74\!\cdots\!49}a^{14}-\frac{20\!\cdots\!09}{74\!\cdots\!49}a^{13}-\frac{32\!\cdots\!88}{74\!\cdots\!49}a^{12}-\frac{30\!\cdots\!69}{74\!\cdots\!49}a^{11}+\frac{47\!\cdots\!82}{74\!\cdots\!49}a^{10}+\frac{44\!\cdots\!63}{74\!\cdots\!49}a^{9}-\frac{48\!\cdots\!76}{74\!\cdots\!49}a^{8}-\frac{65\!\cdots\!79}{74\!\cdots\!49}a^{7}-\frac{56\!\cdots\!05}{74\!\cdots\!49}a^{6}+\frac{17\!\cdots\!97}{74\!\cdots\!49}a^{5}-\frac{42\!\cdots\!99}{18\!\cdots\!89}a^{4}-\frac{24\!\cdots\!38}{74\!\cdots\!49}a^{3}-\frac{38\!\cdots\!33}{74\!\cdots\!49}a^{2}+\frac{28\!\cdots\!67}{74\!\cdots\!49}a+\frac{13\!\cdots\!77}{43\!\cdots\!97}$, $\frac{41\!\cdots\!45}{43\!\cdots\!13}a^{20}+\frac{21\!\cdots\!98}{43\!\cdots\!13}a^{19}-\frac{11\!\cdots\!49}{43\!\cdots\!13}a^{18}-\frac{50\!\cdots\!67}{43\!\cdots\!13}a^{17}-\frac{36\!\cdots\!85}{43\!\cdots\!13}a^{16}+\frac{78\!\cdots\!85}{43\!\cdots\!13}a^{15}+\frac{15\!\cdots\!59}{43\!\cdots\!13}a^{14}-\frac{17\!\cdots\!73}{43\!\cdots\!13}a^{13}-\frac{17\!\cdots\!35}{43\!\cdots\!13}a^{12}-\frac{14\!\cdots\!14}{43\!\cdots\!13}a^{11}+\frac{29\!\cdots\!30}{43\!\cdots\!13}a^{10}+\frac{14\!\cdots\!56}{43\!\cdots\!13}a^{9}-\frac{31\!\cdots\!85}{43\!\cdots\!13}a^{8}-\frac{29\!\cdots\!82}{43\!\cdots\!13}a^{7}-\frac{20\!\cdots\!39}{43\!\cdots\!13}a^{6}+\frac{99\!\cdots\!38}{43\!\cdots\!13}a^{5}-\frac{58\!\cdots\!38}{10\!\cdots\!93}a^{4}-\frac{17\!\cdots\!08}{43\!\cdots\!13}a^{3}-\frac{29\!\cdots\!66}{43\!\cdots\!13}a^{2}+\frac{58\!\cdots\!81}{43\!\cdots\!13}a+\frac{12\!\cdots\!33}{43\!\cdots\!13}$, $\frac{52\!\cdots\!34}{25\!\cdots\!89}a^{20}+\frac{89\!\cdots\!08}{43\!\cdots\!13}a^{19}-\frac{18\!\cdots\!32}{43\!\cdots\!13}a^{18}-\frac{62\!\cdots\!43}{25\!\cdots\!89}a^{17}-\frac{80\!\cdots\!98}{43\!\cdots\!13}a^{16}+\frac{13\!\cdots\!07}{43\!\cdots\!13}a^{15}+\frac{37\!\cdots\!46}{43\!\cdots\!13}a^{14}-\frac{26\!\cdots\!77}{43\!\cdots\!13}a^{13}-\frac{46\!\cdots\!32}{43\!\cdots\!13}a^{12}-\frac{29\!\cdots\!34}{43\!\cdots\!13}a^{11}+\frac{49\!\cdots\!54}{43\!\cdots\!13}a^{10}+\frac{52\!\cdots\!60}{43\!\cdots\!13}a^{9}-\frac{64\!\cdots\!66}{43\!\cdots\!13}a^{8}-\frac{80\!\cdots\!58}{43\!\cdots\!13}a^{7}-\frac{39\!\cdots\!65}{43\!\cdots\!13}a^{6}+\frac{19\!\cdots\!84}{43\!\cdots\!13}a^{5}+\frac{31\!\cdots\!90}{10\!\cdots\!93}a^{4}-\frac{53\!\cdots\!16}{43\!\cdots\!13}a^{3}-\frac{35\!\cdots\!04}{43\!\cdots\!13}a^{2}+\frac{27\!\cdots\!73}{43\!\cdots\!13}a+\frac{19\!\cdots\!66}{43\!\cdots\!13}$, $\frac{70\!\cdots\!01}{43\!\cdots\!13}a^{20}+\frac{63\!\cdots\!91}{43\!\cdots\!13}a^{19}-\frac{14\!\cdots\!22}{43\!\cdots\!13}a^{18}-\frac{83\!\cdots\!97}{43\!\cdots\!13}a^{17}-\frac{63\!\cdots\!37}{43\!\cdots\!13}a^{16}+\frac{10\!\cdots\!24}{43\!\cdots\!13}a^{15}+\frac{28\!\cdots\!73}{43\!\cdots\!13}a^{14}-\frac{22\!\cdots\!08}{43\!\cdots\!13}a^{13}-\frac{33\!\cdots\!84}{43\!\cdots\!13}a^{12}-\frac{23\!\cdots\!13}{43\!\cdots\!13}a^{11}+\frac{41\!\cdots\!18}{43\!\cdots\!13}a^{10}+\frac{36\!\cdots\!67}{43\!\cdots\!13}a^{9}-\frac{50\!\cdots\!77}{43\!\cdots\!13}a^{8}-\frac{58\!\cdots\!65}{43\!\cdots\!13}a^{7}-\frac{30\!\cdots\!02}{43\!\cdots\!13}a^{6}+\frac{15\!\cdots\!21}{43\!\cdots\!13}a^{5}-\frac{30\!\cdots\!05}{10\!\cdots\!93}a^{4}-\frac{32\!\cdots\!18}{43\!\cdots\!13}a^{3}-\frac{31\!\cdots\!47}{43\!\cdots\!13}a^{2}+\frac{61\!\cdots\!13}{43\!\cdots\!13}a+\frac{19\!\cdots\!59}{43\!\cdots\!13}$, $\frac{17\!\cdots\!74}{43\!\cdots\!13}a^{20}+\frac{37\!\cdots\!78}{43\!\cdots\!13}a^{19}-\frac{47\!\cdots\!01}{25\!\cdots\!89}a^{18}-\frac{28\!\cdots\!77}{43\!\cdots\!13}a^{17}-\frac{15\!\cdots\!22}{43\!\cdots\!13}a^{16}+\frac{38\!\cdots\!71}{43\!\cdots\!13}a^{15}+\frac{84\!\cdots\!73}{43\!\cdots\!13}a^{14}-\frac{80\!\cdots\!01}{43\!\cdots\!13}a^{13}-\frac{15\!\cdots\!42}{43\!\cdots\!13}a^{12}-\frac{63\!\cdots\!82}{43\!\cdots\!13}a^{11}+\frac{13\!\cdots\!30}{43\!\cdots\!13}a^{10}+\frac{12\!\cdots\!20}{43\!\cdots\!13}a^{9}-\frac{15\!\cdots\!86}{43\!\cdots\!13}a^{8}-\frac{24\!\cdots\!76}{43\!\cdots\!13}a^{7}-\frac{10\!\cdots\!89}{43\!\cdots\!13}a^{6}+\frac{48\!\cdots\!66}{43\!\cdots\!13}a^{5}+\frac{16\!\cdots\!84}{10\!\cdots\!93}a^{4}-\frac{28\!\cdots\!48}{43\!\cdots\!13}a^{3}-\frac{20\!\cdots\!57}{43\!\cdots\!13}a^{2}-\frac{63\!\cdots\!89}{43\!\cdots\!13}a+\frac{20\!\cdots\!92}{25\!\cdots\!89}$, $\frac{21\!\cdots\!10}{43\!\cdots\!13}a^{20}+\frac{15\!\cdots\!13}{43\!\cdots\!13}a^{19}-\frac{64\!\cdots\!77}{43\!\cdots\!13}a^{18}-\frac{29\!\cdots\!12}{43\!\cdots\!13}a^{17}-\frac{19\!\cdots\!23}{43\!\cdots\!13}a^{16}+\frac{38\!\cdots\!19}{43\!\cdots\!13}a^{15}+\frac{94\!\cdots\!48}{43\!\cdots\!13}a^{14}-\frac{81\!\cdots\!55}{43\!\cdots\!13}a^{13}-\frac{13\!\cdots\!11}{43\!\cdots\!13}a^{12}-\frac{74\!\cdots\!73}{43\!\cdots\!13}a^{11}+\frac{14\!\cdots\!47}{43\!\cdots\!13}a^{10}+\frac{12\!\cdots\!39}{43\!\cdots\!13}a^{9}-\frac{17\!\cdots\!36}{43\!\cdots\!13}a^{8}-\frac{12\!\cdots\!23}{25\!\cdots\!89}a^{7}-\frac{10\!\cdots\!72}{43\!\cdots\!13}a^{6}+\frac{52\!\cdots\!49}{43\!\cdots\!13}a^{5}-\frac{52\!\cdots\!97}{10\!\cdots\!93}a^{4}-\frac{65\!\cdots\!40}{43\!\cdots\!13}a^{3}-\frac{15\!\cdots\!33}{43\!\cdots\!13}a^{2}-\frac{49\!\cdots\!12}{43\!\cdots\!13}a+\frac{83\!\cdots\!42}{43\!\cdots\!13}$, $\frac{10\!\cdots\!02}{74\!\cdots\!49}a^{20}+\frac{17\!\cdots\!37}{74\!\cdots\!49}a^{19}-\frac{19\!\cdots\!29}{74\!\cdots\!49}a^{18}-\frac{10\!\cdots\!70}{74\!\cdots\!49}a^{17}-\frac{99\!\cdots\!10}{74\!\cdots\!49}a^{16}+\frac{94\!\cdots\!81}{74\!\cdots\!49}a^{15}+\frac{40\!\cdots\!36}{74\!\cdots\!49}a^{14}-\frac{93\!\cdots\!80}{74\!\cdots\!49}a^{13}-\frac{29\!\cdots\!68}{74\!\cdots\!49}a^{12}-\frac{35\!\cdots\!71}{74\!\cdots\!49}a^{11}+\frac{41\!\cdots\!62}{74\!\cdots\!49}a^{10}+\frac{55\!\cdots\!05}{74\!\cdots\!49}a^{9}-\frac{35\!\cdots\!31}{74\!\cdots\!49}a^{8}-\frac{71\!\cdots\!30}{74\!\cdots\!49}a^{7}-\frac{75\!\cdots\!46}{74\!\cdots\!49}a^{6}+\frac{16\!\cdots\!80}{74\!\cdots\!49}a^{5}+\frac{16\!\cdots\!68}{18\!\cdots\!89}a^{4}-\frac{12\!\cdots\!67}{74\!\cdots\!49}a^{3}-\frac{36\!\cdots\!57}{74\!\cdots\!49}a^{2}-\frac{33\!\cdots\!74}{74\!\cdots\!49}a+\frac{18\!\cdots\!44}{74\!\cdots\!49}$, $\frac{11\!\cdots\!34}{22\!\cdots\!33}a^{20}+\frac{13\!\cdots\!87}{22\!\cdots\!33}a^{19}-\frac{18\!\cdots\!36}{22\!\cdots\!33}a^{18}-\frac{13\!\cdots\!39}{22\!\cdots\!33}a^{17}-\frac{10\!\cdots\!60}{22\!\cdots\!33}a^{16}+\frac{14\!\cdots\!76}{22\!\cdots\!33}a^{15}+\frac{47\!\cdots\!61}{22\!\cdots\!33}a^{14}-\frac{22\!\cdots\!82}{22\!\cdots\!33}a^{13}-\frac{52\!\cdots\!71}{22\!\cdots\!33}a^{12}-\frac{37\!\cdots\!94}{22\!\cdots\!33}a^{11}+\frac{54\!\cdots\!93}{22\!\cdots\!33}a^{10}+\frac{69\!\cdots\!55}{22\!\cdots\!33}a^{9}-\frac{62\!\cdots\!69}{22\!\cdots\!33}a^{8}-\frac{96\!\cdots\!22}{22\!\cdots\!33}a^{7}-\frac{69\!\cdots\!27}{22\!\cdots\!33}a^{6}+\frac{21\!\cdots\!57}{22\!\cdots\!33}a^{5}+\frac{18\!\cdots\!66}{22\!\cdots\!33}a^{4}-\frac{54\!\cdots\!32}{22\!\cdots\!33}a^{3}-\frac{43\!\cdots\!82}{22\!\cdots\!33}a^{2}-\frac{46\!\cdots\!66}{22\!\cdots\!33}a+\frac{31\!\cdots\!26}{22\!\cdots\!33}$
| 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: | \( 109720182.39051026 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 109720182.39051026 \cdot 1}{2\cdot\sqrt{99995832264130420565259872976896}}\cr\approx \mathstrut & 0.271477989138736 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^19 - 10*x^18 - 79*x^17 + 237*x^16 + 265*x^15 - 691*x^14 - 190*x^13 - 2867*x^12 + 8917*x^11 - 86*x^10 - 12167*x^9 - 1656*x^8 + 3245*x^7 + 25876*x^6 - 21636*x^5 - 3866*x^4 + 177*x^3 + 4752*x^2 + 2525*x - 2917)
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 - 3*x^19 - 10*x^18 - 79*x^17 + 237*x^16 + 265*x^15 - 691*x^14 - 190*x^13 - 2867*x^12 + 8917*x^11 - 86*x^10 - 12167*x^9 - 1656*x^8 + 3245*x^7 + 25876*x^6 - 21636*x^5 - 3866*x^4 + 177*x^3 + 4752*x^2 + 2525*x - 2917, 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 - 3*x^19 - 10*x^18 - 79*x^17 + 237*x^16 + 265*x^15 - 691*x^14 - 190*x^13 - 2867*x^12 + 8917*x^11 - 86*x^10 - 12167*x^9 - 1656*x^8 + 3245*x^7 + 25876*x^6 - 21636*x^5 - 3866*x^4 + 177*x^3 + 4752*x^2 + 2525*x - 2917);
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 - 3*x^19 - 10*x^18 - 79*x^17 + 237*x^16 + 265*x^15 - 691*x^14 - 190*x^13 - 2867*x^12 + 8917*x^11 - 86*x^10 - 12167*x^9 - 1656*x^8 + 3245*x^7 + 25876*x^6 - 21636*x^5 - 3866*x^4 + 177*x^3 + 4752*x^2 + 2525*x - 2917);
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
$S_3\times C_7$ (as 21T6):
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 42 |
| The 21 conjugacy class representatives for $S_3\times C_7$ |
| Character table for $S_3\times C_7$ |
Intermediate fields
| 3.1.116.1, 7.7.594823321.1 |
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
| Galois closure: | 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 | R | $21$ | $21$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.7.0.1}{7} }$ | $21$ | $21$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }^{7}$ | ${\href{/padicField/19.7.0.1}{7} }^{3}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | R | $21$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }^{7}$ | $21$ | $21$ | $21$ | ${\href{/padicField/59.2.0.1}{2} }^{7}{,}\,{\href{/padicField/59.1.0.1}{1} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 2.14.14.15 | $x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ | |
|
\(29\)
| 29.7.6.2 | $x^{7} + 29$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |
| 29.14.13.1 | $x^{14} + 29$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.116.2t1.a.a | $1$ | $ 2^{2} \cdot 29 $ | \(\Q(\sqrt{-29}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.116.14t1.a.a | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.a | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| 1.116.14t1.a.b | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.a.c | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.b | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.29.7t1.a.c | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| 1.116.14t1.a.d | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.d | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.29.7t1.a.e | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| 1.116.14t1.a.e | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| 1.116.14t1.a.f | $1$ | $ 2^{2} \cdot 29 $ | 14.0.168110140833113738264576.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ | |
| * | 1.29.7t1.a.f | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 2.116.3t2.a.a | $2$ | $ 2^{2} \cdot 29 $ | 3.1.116.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.3364.21t6.a.a | $2$ | $ 2^{2} \cdot 29^{2}$ | 21.7.99995832264130420565259872976896.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.3364.21t6.a.b | $2$ | $ 2^{2} \cdot 29^{2}$ | 21.7.99995832264130420565259872976896.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.3364.21t6.a.c | $2$ | $ 2^{2} \cdot 29^{2}$ | 21.7.99995832264130420565259872976896.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.3364.21t6.a.d | $2$ | $ 2^{2} \cdot 29^{2}$ | 21.7.99995832264130420565259872976896.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.3364.21t6.a.e | $2$ | $ 2^{2} \cdot 29^{2}$ | 21.7.99995832264130420565259872976896.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
| * | 2.3364.21t6.a.f | $2$ | $ 2^{2} \cdot 29^{2}$ | 21.7.99995832264130420565259872976896.1 | $S_3\times C_7$ (as 21T6) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.