Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^19 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459)
gp: K = bnfinit(y^21 - 9*y^19 - y^18 + 90*y^17 + 33*y^16 - 517*y^15 + 792*y^14 + 2043*y^13 - 5479*y^12 + 1746*y^11 + 30675*y^10 - 29471*y^9 - 28809*y^8 + 255240*y^7 - 240039*y^6 + 419256*y^5 + 41958*y^4 + 4131*y^3 + 176418*y^2 - 546750*y + 270459, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 9*x^19 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 9*x^19 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459)
\( x^{21} - 9 x^{19} - x^{18} + 90 x^{17} + 33 x^{16} - 517 x^{15} + 792 x^{14} + 2043 x^{13} + \cdots + 270459 \)
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: | $[1, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(140982304436617562370493756640408961\)
\(\medspace = 3^{28}\cdot 151^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(47.18\) | 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: | $3^{4/3}151^{1/2}\approx 53.16797829076763$ | ||
| Ramified primes: |
\(3\), \(151\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{12}+\frac{1}{3}a^{10}+\frac{4}{9}a^{9}-\frac{1}{3}a^{7}-\frac{2}{9}a^{6}-\frac{1}{3}a^{4}+\frac{1}{9}a^{3}$, $\frac{1}{9}a^{13}+\frac{4}{9}a^{10}+\frac{1}{3}a^{8}-\frac{2}{9}a^{7}+\frac{1}{3}a^{5}+\frac{1}{9}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{27}a^{14}+\frac{4}{27}a^{11}+\frac{1}{3}a^{10}+\frac{1}{9}a^{9}+\frac{7}{27}a^{8}-\frac{1}{3}a^{7}-\frac{2}{9}a^{6}+\frac{1}{27}a^{5}+\frac{1}{3}a^{4}+\frac{2}{9}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{189}a^{15}-\frac{1}{63}a^{14}-\frac{2}{63}a^{13}+\frac{1}{27}a^{12}-\frac{1}{9}a^{11}+\frac{5}{63}a^{10}+\frac{64}{189}a^{9}+\frac{5}{63}a^{8}+\frac{17}{63}a^{7}+\frac{67}{189}a^{6}+\frac{8}{63}a^{5}+\frac{2}{21}a^{4}+\frac{25}{63}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a$, $\frac{1}{189}a^{16}-\frac{1}{189}a^{14}+\frac{10}{189}a^{13}+\frac{8}{189}a^{11}-\frac{59}{189}a^{10}+\frac{20}{63}a^{9}+\frac{68}{189}a^{8}+\frac{52}{189}a^{7}-\frac{16}{63}a^{6}-\frac{22}{189}a^{5}+\frac{29}{63}a^{4}+\frac{13}{63}a^{3}-\frac{8}{21}a^{2}-\frac{2}{7}a$, $\frac{1}{189}a^{17}-\frac{2}{63}a^{13}-\frac{2}{63}a^{12}+\frac{2}{21}a^{11}-\frac{17}{63}a^{10}+\frac{1}{7}a^{9}-\frac{5}{21}a^{8}-\frac{20}{63}a^{7}-\frac{20}{63}a^{6}+\frac{41}{189}a^{5}+\frac{19}{63}a^{4}-\frac{20}{63}a^{3}-\frac{8}{21}a^{2}-\frac{3}{7}a$, $\frac{1}{567}a^{18}-\frac{1}{567}a^{15}-\frac{1}{189}a^{14}-\frac{1}{27}a^{13}-\frac{31}{567}a^{12}-\frac{31}{189}a^{11}+\frac{20}{63}a^{10}-\frac{88}{567}a^{9}-\frac{4}{189}a^{8}-\frac{44}{189}a^{7}+\frac{247}{567}a^{6}-\frac{31}{189}a^{5}+\frac{8}{21}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{7}a$, $\frac{1}{60669}a^{19}+\frac{26}{60669}a^{18}+\frac{5}{6741}a^{17}-\frac{130}{60669}a^{16}-\frac{143}{60669}a^{15}+\frac{152}{20223}a^{14}-\frac{1579}{60669}a^{13}+\frac{412}{8667}a^{12}-\frac{326}{2247}a^{11}-\frac{22357}{60669}a^{10}+\frac{100}{8667}a^{9}+\frac{1423}{20223}a^{8}-\frac{24959}{60669}a^{7}+\frac{5018}{60669}a^{6}-\frac{3085}{6741}a^{5}+\frac{1331}{6741}a^{4}-\frac{2873}{6741}a^{3}-\frac{1097}{2247}a^{2}-\frac{302}{749}a+\frac{35}{107}$, $\frac{1}{98\!\cdots\!93}a^{20}+\frac{29\!\cdots\!56}{36\!\cdots\!59}a^{19}-\frac{71\!\cdots\!13}{32\!\cdots\!31}a^{18}-\frac{19\!\cdots\!83}{98\!\cdots\!93}a^{17}+\frac{50\!\cdots\!15}{47\!\cdots\!99}a^{16}+\frac{25\!\cdots\!57}{25\!\cdots\!87}a^{15}-\frac{38\!\cdots\!31}{98\!\cdots\!93}a^{14}+\frac{93\!\cdots\!03}{36\!\cdots\!59}a^{13}-\frac{12\!\cdots\!77}{32\!\cdots\!31}a^{12}-\frac{15\!\cdots\!43}{98\!\cdots\!93}a^{11}+\frac{13\!\cdots\!43}{10\!\cdots\!77}a^{10}+\frac{10\!\cdots\!60}{36\!\cdots\!59}a^{9}+\frac{59\!\cdots\!01}{14\!\cdots\!99}a^{8}-\frac{12\!\cdots\!23}{36\!\cdots\!23}a^{7}-\frac{13\!\cdots\!59}{32\!\cdots\!31}a^{6}+\frac{10\!\cdots\!05}{10\!\cdots\!77}a^{5}+\frac{48\!\cdots\!60}{36\!\cdots\!59}a^{4}-\frac{18\!\cdots\!06}{52\!\cdots\!37}a^{3}+\frac{28\!\cdots\!90}{40\!\cdots\!51}a^{2}+\frac{82\!\cdots\!47}{40\!\cdots\!51}a+\frac{74\!\cdots\!69}{58\!\cdots\!93}$
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: | $10$ | 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: |
$\frac{92\!\cdots\!18}{17\!\cdots\!57}a^{20}+\frac{22\!\cdots\!30}{53\!\cdots\!71}a^{19}-\frac{24\!\cdots\!70}{53\!\cdots\!71}a^{18}-\frac{25\!\cdots\!71}{59\!\cdots\!19}a^{17}+\frac{10\!\cdots\!98}{23\!\cdots\!77}a^{16}+\frac{22\!\cdots\!12}{41\!\cdots\!67}a^{15}-\frac{65\!\cdots\!02}{25\!\cdots\!51}a^{14}+\frac{10\!\cdots\!86}{53\!\cdots\!71}a^{13}+\frac{75\!\cdots\!88}{53\!\cdots\!71}a^{12}-\frac{38\!\cdots\!48}{19\!\cdots\!73}a^{11}-\frac{82\!\cdots\!59}{53\!\cdots\!71}a^{10}+\frac{88\!\cdots\!03}{53\!\cdots\!71}a^{9}-\frac{27\!\cdots\!53}{17\!\cdots\!57}a^{8}-\frac{71\!\cdots\!06}{25\!\cdots\!47}a^{7}+\frac{63\!\cdots\!58}{53\!\cdots\!71}a^{6}-\frac{19\!\cdots\!83}{19\!\cdots\!73}a^{5}+\frac{10\!\cdots\!81}{85\!\cdots\!17}a^{4}+\frac{10\!\cdots\!46}{59\!\cdots\!19}a^{3}+\frac{52\!\cdots\!65}{66\!\cdots\!91}a^{2}+\frac{39\!\cdots\!73}{66\!\cdots\!91}a-\frac{19\!\cdots\!28}{95\!\cdots\!13}$, $\frac{22\!\cdots\!62}{19\!\cdots\!73}a^{20}+\frac{37\!\cdots\!70}{53\!\cdots\!71}a^{19}-\frac{49\!\cdots\!88}{53\!\cdots\!71}a^{18}-\frac{20\!\cdots\!99}{25\!\cdots\!51}a^{17}+\frac{22\!\cdots\!90}{23\!\cdots\!77}a^{16}+\frac{44\!\cdots\!27}{41\!\cdots\!67}a^{15}-\frac{10\!\cdots\!39}{19\!\cdots\!73}a^{14}+\frac{23\!\cdots\!54}{53\!\cdots\!71}a^{13}+\frac{14\!\cdots\!83}{53\!\cdots\!71}a^{12}-\frac{59\!\cdots\!29}{17\!\cdots\!57}a^{11}-\frac{14\!\cdots\!57}{53\!\cdots\!71}a^{10}+\frac{16\!\cdots\!71}{53\!\cdots\!71}a^{9}+\frac{88\!\cdots\!29}{25\!\cdots\!51}a^{8}-\frac{77\!\cdots\!37}{18\!\cdots\!29}a^{7}+\frac{10\!\cdots\!95}{53\!\cdots\!71}a^{6}+\frac{24\!\cdots\!26}{85\!\cdots\!17}a^{5}+\frac{32\!\cdots\!36}{59\!\cdots\!19}a^{4}-\frac{74\!\cdots\!65}{19\!\cdots\!73}a^{3}+\frac{54\!\cdots\!10}{66\!\cdots\!91}a^{2}+\frac{18\!\cdots\!51}{66\!\cdots\!91}a-\frac{76\!\cdots\!29}{95\!\cdots\!13}$, $\frac{83\!\cdots\!93}{17\!\cdots\!57}a^{20}+\frac{64\!\cdots\!59}{53\!\cdots\!71}a^{19}-\frac{21\!\cdots\!34}{53\!\cdots\!71}a^{18}-\frac{69\!\cdots\!56}{59\!\cdots\!19}a^{17}+\frac{13\!\cdots\!25}{33\!\cdots\!11}a^{16}+\frac{92\!\cdots\!99}{41\!\cdots\!67}a^{15}-\frac{56\!\cdots\!24}{25\!\cdots\!51}a^{14}+\frac{18\!\cdots\!01}{53\!\cdots\!71}a^{13}+\frac{75\!\cdots\!08}{77\!\cdots\!53}a^{12}-\frac{14\!\cdots\!81}{59\!\cdots\!19}a^{11}+\frac{56\!\cdots\!94}{77\!\cdots\!53}a^{10}+\frac{78\!\cdots\!50}{53\!\cdots\!71}a^{9}-\frac{20\!\cdots\!68}{17\!\cdots\!57}a^{8}-\frac{21\!\cdots\!16}{18\!\cdots\!29}a^{7}+\frac{66\!\cdots\!17}{53\!\cdots\!71}a^{6}-\frac{53\!\cdots\!03}{59\!\cdots\!19}a^{5}+\frac{12\!\cdots\!26}{66\!\cdots\!91}a^{4}+\frac{65\!\cdots\!44}{59\!\cdots\!19}a^{3}+\frac{34\!\cdots\!25}{66\!\cdots\!91}a^{2}+\frac{10\!\cdots\!51}{66\!\cdots\!91}a-\frac{18\!\cdots\!46}{95\!\cdots\!13}$, $\frac{55\!\cdots\!99}{14\!\cdots\!99}a^{20}-\frac{13\!\cdots\!41}{32\!\cdots\!31}a^{19}-\frac{60\!\cdots\!46}{15\!\cdots\!11}a^{18}+\frac{41\!\cdots\!92}{98\!\cdots\!93}a^{17}+\frac{56\!\cdots\!77}{14\!\cdots\!97}a^{16}-\frac{74\!\cdots\!54}{25\!\cdots\!87}a^{15}-\frac{24\!\cdots\!21}{98\!\cdots\!93}a^{14}+\frac{19\!\cdots\!57}{32\!\cdots\!31}a^{13}+\frac{85\!\cdots\!83}{12\!\cdots\!53}a^{12}-\frac{49\!\cdots\!06}{14\!\cdots\!99}a^{11}+\frac{95\!\cdots\!42}{32\!\cdots\!31}a^{10}+\frac{49\!\cdots\!99}{32\!\cdots\!31}a^{9}-\frac{25\!\cdots\!87}{92\!\cdots\!99}a^{8}-\frac{11\!\cdots\!84}{15\!\cdots\!67}a^{7}+\frac{53\!\cdots\!09}{36\!\cdots\!59}a^{6}-\frac{21\!\cdots\!45}{10\!\cdots\!77}a^{5}+\frac{67\!\cdots\!24}{40\!\cdots\!51}a^{4}+\frac{14\!\cdots\!21}{12\!\cdots\!53}a^{3}-\frac{10\!\cdots\!72}{40\!\cdots\!51}a^{2}+\frac{41\!\cdots\!53}{40\!\cdots\!51}a-\frac{28\!\cdots\!81}{58\!\cdots\!93}$, $\frac{23\!\cdots\!00}{98\!\cdots\!93}a^{20}+\frac{24\!\cdots\!46}{10\!\cdots\!77}a^{19}+\frac{18\!\cdots\!85}{40\!\cdots\!51}a^{18}-\frac{12\!\cdots\!04}{98\!\cdots\!93}a^{17}-\frac{23\!\cdots\!58}{47\!\cdots\!99}a^{16}+\frac{23\!\cdots\!33}{36\!\cdots\!41}a^{15}+\frac{44\!\cdots\!34}{98\!\cdots\!93}a^{14}+\frac{41\!\cdots\!65}{10\!\cdots\!77}a^{13}+\frac{13\!\cdots\!29}{15\!\cdots\!11}a^{12}+\frac{14\!\cdots\!76}{98\!\cdots\!93}a^{11}-\frac{22\!\cdots\!33}{10\!\cdots\!77}a^{10}-\frac{35\!\cdots\!62}{32\!\cdots\!31}a^{9}+\frac{30\!\cdots\!05}{98\!\cdots\!93}a^{8}+\frac{36\!\cdots\!75}{36\!\cdots\!23}a^{7}+\frac{11\!\cdots\!80}{10\!\cdots\!77}a^{6}+\frac{31\!\cdots\!30}{10\!\cdots\!77}a^{5}+\frac{75\!\cdots\!32}{36\!\cdots\!59}a^{4}-\frac{17\!\cdots\!99}{12\!\cdots\!53}a^{3}+\frac{19\!\cdots\!77}{40\!\cdots\!51}a^{2}-\frac{17\!\cdots\!86}{40\!\cdots\!51}a+\frac{14\!\cdots\!18}{58\!\cdots\!93}$, $\frac{14\!\cdots\!00}{76\!\cdots\!61}a^{20}-\frac{49\!\cdots\!42}{25\!\cdots\!87}a^{19}-\frac{15\!\cdots\!28}{84\!\cdots\!29}a^{18}+\frac{11\!\cdots\!21}{76\!\cdots\!61}a^{17}+\frac{19\!\cdots\!66}{11\!\cdots\!69}a^{16}-\frac{28\!\cdots\!10}{25\!\cdots\!87}a^{15}-\frac{80\!\cdots\!42}{76\!\cdots\!61}a^{14}+\frac{65\!\cdots\!71}{25\!\cdots\!87}a^{13}+\frac{18\!\cdots\!12}{84\!\cdots\!29}a^{12}-\frac{11\!\cdots\!94}{76\!\cdots\!61}a^{11}+\frac{41\!\cdots\!20}{25\!\cdots\!87}a^{10}+\frac{21\!\cdots\!01}{36\!\cdots\!41}a^{9}-\frac{99\!\cdots\!95}{76\!\cdots\!61}a^{8}+\frac{59\!\cdots\!51}{11\!\cdots\!69}a^{7}+\frac{50\!\cdots\!72}{84\!\cdots\!29}a^{6}-\frac{94\!\cdots\!17}{84\!\cdots\!29}a^{5}+\frac{32\!\cdots\!15}{28\!\cdots\!43}a^{4}-\frac{41\!\cdots\!00}{93\!\cdots\!81}a^{3}-\frac{27\!\cdots\!51}{31\!\cdots\!27}a^{2}+\frac{17\!\cdots\!44}{31\!\cdots\!27}a-\frac{41\!\cdots\!75}{44\!\cdots\!61}$, $\frac{25\!\cdots\!21}{32\!\cdots\!31}a^{20}+\frac{16\!\cdots\!60}{32\!\cdots\!31}a^{19}-\frac{69\!\cdots\!60}{10\!\cdots\!77}a^{18}-\frac{12\!\cdots\!15}{32\!\cdots\!31}a^{17}+\frac{92\!\cdots\!66}{14\!\cdots\!97}a^{16}+\frac{45\!\cdots\!33}{84\!\cdots\!29}a^{15}-\frac{11\!\cdots\!72}{32\!\cdots\!31}a^{14}+\frac{17\!\cdots\!73}{32\!\cdots\!31}a^{13}+\frac{75\!\cdots\!21}{40\!\cdots\!51}a^{12}-\frac{12\!\cdots\!53}{32\!\cdots\!31}a^{11}-\frac{14\!\cdots\!32}{32\!\cdots\!31}a^{10}+\frac{28\!\cdots\!27}{10\!\cdots\!77}a^{9}-\frac{42\!\cdots\!64}{47\!\cdots\!33}a^{8}-\frac{33\!\cdots\!58}{11\!\cdots\!69}a^{7}+\frac{20\!\cdots\!43}{10\!\cdots\!77}a^{6}-\frac{13\!\cdots\!69}{17\!\cdots\!79}a^{5}+\frac{12\!\cdots\!89}{40\!\cdots\!51}a^{4}+\frac{95\!\cdots\!82}{36\!\cdots\!59}a^{3}+\frac{94\!\cdots\!77}{40\!\cdots\!51}a^{2}+\frac{18\!\cdots\!00}{58\!\cdots\!93}a-\frac{15\!\cdots\!02}{58\!\cdots\!93}$, $\frac{42\!\cdots\!16}{98\!\cdots\!93}a^{20}+\frac{78\!\cdots\!70}{32\!\cdots\!31}a^{19}-\frac{41\!\cdots\!31}{10\!\cdots\!77}a^{18}-\frac{26\!\cdots\!71}{98\!\cdots\!93}a^{17}+\frac{53\!\cdots\!76}{14\!\cdots\!97}a^{16}+\frac{13\!\cdots\!43}{36\!\cdots\!41}a^{15}-\frac{20\!\cdots\!37}{98\!\cdots\!93}a^{14}+\frac{71\!\cdots\!49}{32\!\cdots\!31}a^{13}+\frac{12\!\cdots\!38}{12\!\cdots\!53}a^{12}-\frac{17\!\cdots\!50}{98\!\cdots\!93}a^{11}-\frac{12\!\cdots\!56}{47\!\cdots\!33}a^{10}+\frac{42\!\cdots\!15}{32\!\cdots\!31}a^{9}-\frac{50\!\cdots\!78}{98\!\cdots\!93}a^{8}-\frac{16\!\cdots\!12}{11\!\cdots\!69}a^{7}+\frac{35\!\cdots\!00}{36\!\cdots\!59}a^{6}-\frac{17\!\cdots\!56}{36\!\cdots\!59}a^{5}+\frac{59\!\cdots\!26}{36\!\cdots\!59}a^{4}+\frac{23\!\cdots\!60}{36\!\cdots\!59}a^{3}+\frac{68\!\cdots\!54}{12\!\cdots\!53}a^{2}+\frac{30\!\cdots\!26}{40\!\cdots\!51}a-\frac{15\!\cdots\!51}{58\!\cdots\!93}$, $\frac{89\!\cdots\!82}{47\!\cdots\!33}a^{20}-\frac{65\!\cdots\!33}{32\!\cdots\!31}a^{19}-\frac{22\!\cdots\!99}{10\!\cdots\!77}a^{18}+\frac{74\!\cdots\!46}{47\!\cdots\!33}a^{17}+\frac{32\!\cdots\!06}{14\!\cdots\!97}a^{16}-\frac{86\!\cdots\!52}{84\!\cdots\!29}a^{15}-\frac{52\!\cdots\!36}{32\!\cdots\!31}a^{14}+\frac{72\!\cdots\!89}{32\!\cdots\!31}a^{13}+\frac{71\!\cdots\!13}{10\!\cdots\!77}a^{12}-\frac{45\!\cdots\!00}{32\!\cdots\!31}a^{11}-\frac{30\!\cdots\!09}{32\!\cdots\!31}a^{10}+\frac{24\!\cdots\!15}{36\!\cdots\!59}a^{9}-\frac{11\!\cdots\!94}{32\!\cdots\!31}a^{8}-\frac{98\!\cdots\!26}{11\!\cdots\!69}a^{7}+\frac{38\!\cdots\!17}{10\!\cdots\!77}a^{6}-\frac{20\!\cdots\!62}{40\!\cdots\!51}a^{5}+\frac{52\!\cdots\!67}{52\!\cdots\!37}a^{4}-\frac{31\!\cdots\!67}{36\!\cdots\!59}a^{3}-\frac{88\!\cdots\!95}{40\!\cdots\!51}a^{2}+\frac{80\!\cdots\!80}{40\!\cdots\!51}a-\frac{50\!\cdots\!16}{58\!\cdots\!93}$, $\frac{33\!\cdots\!90}{98\!\cdots\!93}a^{20}+\frac{50\!\cdots\!38}{47\!\cdots\!33}a^{19}-\frac{12\!\cdots\!31}{36\!\cdots\!59}a^{18}-\frac{15\!\cdots\!01}{98\!\cdots\!93}a^{17}+\frac{47\!\cdots\!38}{14\!\cdots\!97}a^{16}+\frac{64\!\cdots\!27}{25\!\cdots\!87}a^{15}-\frac{19\!\cdots\!19}{98\!\cdots\!93}a^{14}+\frac{53\!\cdots\!16}{32\!\cdots\!31}a^{13}+\frac{13\!\cdots\!96}{15\!\cdots\!11}a^{12}-\frac{15\!\cdots\!28}{98\!\cdots\!93}a^{11}-\frac{14\!\cdots\!89}{32\!\cdots\!31}a^{10}+\frac{36\!\cdots\!06}{32\!\cdots\!31}a^{9}-\frac{57\!\cdots\!33}{98\!\cdots\!93}a^{8}-\frac{21\!\cdots\!31}{11\!\cdots\!69}a^{7}+\frac{29\!\cdots\!77}{36\!\cdots\!59}a^{6}-\frac{13\!\cdots\!62}{36\!\cdots\!59}a^{5}+\frac{32\!\cdots\!10}{36\!\cdots\!59}a^{4}+\frac{34\!\cdots\!31}{36\!\cdots\!59}a^{3}+\frac{23\!\cdots\!21}{12\!\cdots\!53}a^{2}+\frac{74\!\cdots\!30}{40\!\cdots\!51}a-\frac{83\!\cdots\!01}{58\!\cdots\!93}$
| 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: | \( 2150750435.99 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 2150750435.99 \cdot 1}{2\cdot\sqrt{140982304436617562370493756640408961}}\cr\approx \mathstrut & 0.549295954684 \end{aligned}\] (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 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459)
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 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459, 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 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459);
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 - x^18 + 90*x^17 + 33*x^16 - 517*x^15 + 792*x^14 + 2043*x^13 - 5479*x^12 + 1746*x^11 + 30675*x^10 - 29471*x^9 - 28809*x^8 + 255240*x^7 - 240039*x^6 + 419256*x^5 + 41958*x^4 + 4131*x^3 + 176418*x^2 - 546750*x + 270459);
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
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 12 conjugacy class representatives for $D_{21}$ |
| Character table for $D_{21}$ |
Intermediate fields
| 3.1.12231.1, 7.1.3442951.1 |
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: | deg 42 |
| 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 | $21$ | ${\href{/padicField/7.2.0.1}{2} }^{10}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $21$ | ${\href{/padicField/13.2.0.1}{2} }^{10}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/23.2.0.1}{2} }^{10}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/37.7.0.1}{7} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/53.2.0.1}{2} }^{10}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ |
| 3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ | |
| 3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ | |
| 3.6.8.5 | $x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ | $3$ | $2$ | $8$ | $S_3$ | $[2]^{2}$ | |
|
\(151\)
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |