Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300)
gp: K = bnfinit(y^22 - 9*y^21 + 41*y^20 - 140*y^19 + 383*y^18 - 720*y^17 + 530*y^16 + 1506*y^15 - 7373*y^14 + 22033*y^13 - 51771*y^12 + 76979*y^11 - 19458*y^10 - 188822*y^9 + 431163*y^8 - 422621*y^7 + 84439*y^6 + 244344*y^5 - 239399*y^4 + 37800*y^3 + 53515*y^2 - 17850*y - 6300, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300)
\( x^{22} - 9 x^{21} + 41 x^{20} - 140 x^{19} + 383 x^{18} - 720 x^{17} + 530 x^{16} + 1506 x^{15} + \cdots - 6300 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(113420524920318543882697355574770703125\)
\(\medspace = 5^{8}\cdot 7^{15}\cdot 11^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(53.67\) | 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: | $5^{1/2}7^{3/4}11^{11/12}\approx 86.67991835373819$ | ||
| Ramified primes: |
\(5\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{14}a^{17}+\frac{1}{14}a^{16}-\frac{2}{7}a^{15}-\frac{3}{14}a^{14}-\frac{1}{2}a^{13}+\frac{1}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}+\frac{2}{7}a^{8}-\frac{1}{14}a^{7}+\frac{3}{14}a^{6}+\frac{5}{14}a^{5}-\frac{1}{14}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{18}+\frac{1}{7}a^{16}-\frac{3}{7}a^{15}-\frac{2}{7}a^{14}+\frac{1}{7}a^{13}-\frac{1}{2}a^{12}-\frac{2}{7}a^{11}+\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{5}{14}a^{8}+\frac{2}{7}a^{7}-\frac{5}{14}a^{6}+\frac{1}{14}a^{5}-\frac{3}{7}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{14}a^{19}-\frac{1}{14}a^{16}-\frac{3}{14}a^{15}-\frac{3}{7}a^{14}-\frac{1}{14}a^{12}-\frac{1}{7}a^{11}-\frac{3}{7}a^{10}-\frac{5}{14}a^{9}-\frac{2}{7}a^{8}-\frac{3}{14}a^{7}+\frac{1}{7}a^{6}+\frac{5}{14}a^{5}-\frac{5}{14}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2030}a^{20}-\frac{1}{70}a^{19}+\frac{1}{2030}a^{18}-\frac{3}{406}a^{17}+\frac{9}{290}a^{16}-\frac{80}{203}a^{15}+\frac{38}{203}a^{14}+\frac{193}{1015}a^{13}+\frac{447}{2030}a^{12}+\frac{59}{1015}a^{11}+\frac{199}{2030}a^{10}+\frac{999}{2030}a^{9}-\frac{499}{1015}a^{8}-\frac{747}{2030}a^{7}-\frac{597}{2030}a^{6}+\frac{202}{1015}a^{5}-\frac{123}{1015}a^{4}-\frac{4}{145}a^{3}+\frac{53}{290}a^{2}-\frac{25}{58}a+\frac{5}{29}$, $\frac{1}{10\!\cdots\!90}a^{21}-\frac{25\!\cdots\!18}{16\!\cdots\!65}a^{20}+\frac{31\!\cdots\!33}{20\!\cdots\!10}a^{19}+\frac{86\!\cdots\!84}{71\!\cdots\!85}a^{18}-\frac{12\!\cdots\!31}{50\!\cdots\!95}a^{17}-\frac{18\!\cdots\!96}{23\!\cdots\!95}a^{16}-\frac{95\!\cdots\!97}{20\!\cdots\!38}a^{15}+\frac{30\!\cdots\!23}{23\!\cdots\!95}a^{14}+\frac{20\!\cdots\!63}{10\!\cdots\!90}a^{13}+\frac{30\!\cdots\!56}{10\!\cdots\!19}a^{12}-\frac{29\!\cdots\!91}{33\!\cdots\!30}a^{11}-\frac{16\!\cdots\!51}{50\!\cdots\!95}a^{10}+\frac{30\!\cdots\!13}{23\!\cdots\!95}a^{9}-\frac{19\!\cdots\!89}{69\!\cdots\!22}a^{8}+\frac{16\!\cdots\!57}{33\!\cdots\!30}a^{7}+\frac{43\!\cdots\!17}{10\!\cdots\!90}a^{6}-\frac{12\!\cdots\!47}{10\!\cdots\!90}a^{5}-\frac{51\!\cdots\!29}{33\!\cdots\!30}a^{4}+\frac{44\!\cdots\!27}{14\!\cdots\!78}a^{3}-\frac{10\!\cdots\!39}{47\!\cdots\!90}a^{2}+\frac{66\!\cdots\!31}{14\!\cdots\!17}a+\frac{14\!\cdots\!08}{47\!\cdots\!39}$
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: |
$\frac{41\!\cdots\!88}{16\!\cdots\!65}a^{21}-\frac{33\!\cdots\!42}{16\!\cdots\!65}a^{20}+\frac{38\!\cdots\!93}{47\!\cdots\!90}a^{19}-\frac{12\!\cdots\!95}{47\!\cdots\!39}a^{18}+\frac{21\!\cdots\!83}{33\!\cdots\!30}a^{17}-\frac{45\!\cdots\!04}{47\!\cdots\!39}a^{16}-\frac{91\!\cdots\!07}{66\!\cdots\!46}a^{15}+\frac{20\!\cdots\!93}{47\!\cdots\!90}a^{14}-\frac{46\!\cdots\!43}{33\!\cdots\!30}a^{13}+\frac{12\!\cdots\!83}{33\!\cdots\!30}a^{12}-\frac{13\!\cdots\!13}{16\!\cdots\!65}a^{11}+\frac{13\!\cdots\!32}{16\!\cdots\!65}a^{10}+\frac{46\!\cdots\!71}{47\!\cdots\!90}a^{9}-\frac{73\!\cdots\!26}{16\!\cdots\!65}a^{8}+\frac{95\!\cdots\!84}{16\!\cdots\!65}a^{7}-\frac{55\!\cdots\!01}{33\!\cdots\!30}a^{6}-\frac{64\!\cdots\!73}{16\!\cdots\!65}a^{5}+\frac{68\!\cdots\!52}{16\!\cdots\!65}a^{4}-\frac{78\!\cdots\!13}{34\!\cdots\!85}a^{3}-\frac{71\!\cdots\!20}{47\!\cdots\!39}a^{2}+\frac{39\!\cdots\!59}{95\!\cdots\!78}a+\frac{81\!\cdots\!73}{47\!\cdots\!39}$, $\frac{18\!\cdots\!13}{10\!\cdots\!90}a^{21}-\frac{55\!\cdots\!07}{33\!\cdots\!30}a^{20}+\frac{10\!\cdots\!87}{14\!\cdots\!70}a^{19}-\frac{15\!\cdots\!91}{71\!\cdots\!85}a^{18}+\frac{58\!\cdots\!49}{10\!\cdots\!90}a^{17}-\frac{23\!\cdots\!81}{23\!\cdots\!95}a^{16}+\frac{11\!\cdots\!33}{10\!\cdots\!19}a^{15}+\frac{85\!\cdots\!74}{23\!\cdots\!95}a^{14}-\frac{12\!\cdots\!13}{10\!\cdots\!90}a^{13}+\frac{34\!\cdots\!01}{10\!\cdots\!90}a^{12}-\frac{51\!\cdots\!25}{66\!\cdots\!46}a^{11}+\frac{88\!\cdots\!41}{10\!\cdots\!90}a^{10}+\frac{29\!\cdots\!25}{47\!\cdots\!39}a^{9}-\frac{18\!\cdots\!67}{50\!\cdots\!95}a^{8}+\frac{18\!\cdots\!49}{33\!\cdots\!30}a^{7}-\frac{52\!\cdots\!63}{20\!\cdots\!38}a^{6}-\frac{19\!\cdots\!69}{10\!\cdots\!90}a^{5}+\frac{52\!\cdots\!71}{16\!\cdots\!65}a^{4}-\frac{19\!\cdots\!57}{20\!\cdots\!10}a^{3}-\frac{13\!\cdots\!42}{23\!\cdots\!95}a^{2}+\frac{79\!\cdots\!65}{28\!\cdots\!34}a+\frac{32\!\cdots\!21}{47\!\cdots\!39}$, $\frac{32\!\cdots\!61}{28\!\cdots\!34}a^{21}-\frac{40\!\cdots\!51}{47\!\cdots\!90}a^{20}+\frac{46\!\cdots\!27}{14\!\cdots\!70}a^{19}-\frac{71\!\cdots\!19}{71\!\cdots\!85}a^{18}+\frac{34\!\cdots\!13}{14\!\cdots\!17}a^{17}-\frac{14\!\cdots\!53}{47\!\cdots\!90}a^{16}-\frac{35\!\cdots\!12}{14\!\cdots\!17}a^{15}+\frac{19\!\cdots\!17}{95\!\cdots\!78}a^{14}-\frac{40\!\cdots\!84}{71\!\cdots\!85}a^{13}+\frac{20\!\cdots\!59}{14\!\cdots\!70}a^{12}-\frac{19\!\cdots\!89}{68\!\cdots\!70}a^{11}+\frac{30\!\cdots\!03}{14\!\cdots\!70}a^{10}+\frac{14\!\cdots\!33}{23\!\cdots\!95}a^{9}-\frac{13\!\cdots\!53}{71\!\cdots\!85}a^{8}+\frac{46\!\cdots\!41}{23\!\cdots\!95}a^{7}+\frac{17\!\cdots\!13}{71\!\cdots\!85}a^{6}-\frac{25\!\cdots\!83}{10\!\cdots\!55}a^{5}+\frac{93\!\cdots\!71}{47\!\cdots\!90}a^{4}+\frac{51\!\cdots\!49}{20\!\cdots\!10}a^{3}-\frac{69\!\cdots\!43}{68\!\cdots\!70}a^{2}+\frac{11\!\cdots\!95}{40\!\cdots\!62}a+\frac{88\!\cdots\!87}{68\!\cdots\!77}$, $\frac{17\!\cdots\!57}{10\!\cdots\!90}a^{21}-\frac{23\!\cdots\!82}{16\!\cdots\!65}a^{20}+\frac{41\!\cdots\!02}{71\!\cdots\!85}a^{19}-\frac{28\!\cdots\!47}{14\!\cdots\!70}a^{18}+\frac{26\!\cdots\!78}{50\!\cdots\!95}a^{17}-\frac{21\!\cdots\!21}{23\!\cdots\!95}a^{16}+\frac{91\!\cdots\!65}{20\!\cdots\!38}a^{15}+\frac{53\!\cdots\!36}{23\!\cdots\!95}a^{14}-\frac{99\!\cdots\!30}{10\!\cdots\!19}a^{13}+\frac{30\!\cdots\!23}{10\!\cdots\!90}a^{12}-\frac{22\!\cdots\!33}{33\!\cdots\!30}a^{11}+\frac{44\!\cdots\!31}{50\!\cdots\!95}a^{10}-\frac{48\!\cdots\!43}{16\!\cdots\!10}a^{9}-\frac{11\!\cdots\!56}{50\!\cdots\!95}a^{8}+\frac{81\!\cdots\!09}{16\!\cdots\!65}a^{7}-\frac{49\!\cdots\!29}{10\!\cdots\!90}a^{6}+\frac{91\!\cdots\!16}{50\!\cdots\!95}a^{5}+\frac{55\!\cdots\!09}{33\!\cdots\!30}a^{4}-\frac{26\!\cdots\!83}{10\!\cdots\!55}a^{3}+\frac{56\!\cdots\!51}{47\!\cdots\!90}a^{2}+\frac{40\!\cdots\!11}{28\!\cdots\!34}a-\frac{13\!\cdots\!50}{47\!\cdots\!39}$, $\frac{61\!\cdots\!52}{16\!\cdots\!65}a^{21}-\frac{72\!\cdots\!09}{33\!\cdots\!30}a^{20}+\frac{15\!\cdots\!39}{23\!\cdots\!95}a^{19}-\frac{82\!\cdots\!29}{47\!\cdots\!90}a^{18}+\frac{55\!\cdots\!96}{16\!\cdots\!65}a^{17}+\frac{13\!\cdots\!04}{23\!\cdots\!95}a^{16}-\frac{64\!\cdots\!31}{33\!\cdots\!73}a^{15}+\frac{21\!\cdots\!47}{47\!\cdots\!90}a^{14}-\frac{29\!\cdots\!90}{33\!\cdots\!73}a^{13}+\frac{77\!\cdots\!21}{33\!\cdots\!30}a^{12}-\frac{49\!\cdots\!64}{16\!\cdots\!65}a^{11}-\frac{18\!\cdots\!11}{33\!\cdots\!30}a^{10}+\frac{58\!\cdots\!14}{23\!\cdots\!95}a^{9}-\frac{44\!\cdots\!97}{16\!\cdots\!65}a^{8}-\frac{50\!\cdots\!17}{33\!\cdots\!30}a^{7}+\frac{20\!\cdots\!67}{33\!\cdots\!30}a^{6}-\frac{13\!\cdots\!21}{33\!\cdots\!30}a^{5}-\frac{70\!\cdots\!11}{33\!\cdots\!30}a^{4}+\frac{23\!\cdots\!43}{68\!\cdots\!70}a^{3}-\frac{12\!\cdots\!59}{47\!\cdots\!90}a^{2}-\frac{76\!\cdots\!91}{95\!\cdots\!78}a-\frac{28\!\cdots\!27}{47\!\cdots\!39}$, $\frac{47\!\cdots\!11}{33\!\cdots\!30}a^{21}-\frac{18\!\cdots\!68}{16\!\cdots\!65}a^{20}+\frac{20\!\cdots\!07}{47\!\cdots\!90}a^{19}-\frac{65\!\cdots\!11}{47\!\cdots\!90}a^{18}+\frac{11\!\cdots\!53}{33\!\cdots\!30}a^{17}-\frac{24\!\cdots\!53}{47\!\cdots\!90}a^{16}-\frac{26\!\cdots\!11}{66\!\cdots\!46}a^{15}+\frac{10\!\cdots\!43}{47\!\cdots\!90}a^{14}-\frac{86\!\cdots\!91}{11\!\cdots\!37}a^{13}+\frac{69\!\cdots\!69}{33\!\cdots\!30}a^{12}-\frac{14\!\cdots\!47}{33\!\cdots\!30}a^{11}+\frac{74\!\cdots\!83}{16\!\cdots\!65}a^{10}+\frac{14\!\cdots\!63}{34\!\cdots\!85}a^{9}-\frac{36\!\cdots\!18}{16\!\cdots\!65}a^{8}+\frac{51\!\cdots\!36}{16\!\cdots\!65}a^{7}-\frac{47\!\cdots\!87}{33\!\cdots\!30}a^{6}-\frac{47\!\cdots\!59}{33\!\cdots\!30}a^{5}+\frac{36\!\cdots\!03}{16\!\cdots\!65}a^{4}-\frac{38\!\cdots\!93}{68\!\cdots\!70}a^{3}-\frac{29\!\cdots\!21}{47\!\cdots\!90}a^{2}+\frac{27\!\cdots\!99}{95\!\cdots\!78}a+\frac{50\!\cdots\!62}{47\!\cdots\!39}$, $\frac{13\!\cdots\!97}{14\!\cdots\!70}a^{21}-\frac{15\!\cdots\!63}{47\!\cdots\!90}a^{20}-\frac{38\!\cdots\!59}{14\!\cdots\!70}a^{19}+\frac{20\!\cdots\!12}{71\!\cdots\!85}a^{18}-\frac{10\!\cdots\!32}{71\!\cdots\!85}a^{17}+\frac{20\!\cdots\!21}{34\!\cdots\!85}a^{16}-\frac{39\!\cdots\!03}{28\!\cdots\!34}a^{15}+\frac{17\!\cdots\!91}{16\!\cdots\!10}a^{14}+\frac{19\!\cdots\!43}{14\!\cdots\!70}a^{13}-\frac{45\!\cdots\!33}{71\!\cdots\!85}a^{12}+\frac{26\!\cdots\!39}{95\!\cdots\!78}a^{11}-\frac{12\!\cdots\!61}{14\!\cdots\!70}a^{10}+\frac{65\!\cdots\!91}{47\!\cdots\!39}a^{9}+\frac{47\!\cdots\!97}{71\!\cdots\!85}a^{8}-\frac{93\!\cdots\!27}{23\!\cdots\!95}a^{7}+\frac{18\!\cdots\!91}{28\!\cdots\!34}a^{6}-\frac{50\!\cdots\!81}{14\!\cdots\!70}a^{5}-\frac{41\!\cdots\!71}{23\!\cdots\!95}a^{4}+\frac{34\!\cdots\!82}{10\!\cdots\!55}a^{3}-\frac{80\!\cdots\!51}{68\!\cdots\!70}a^{2}-\frac{75\!\cdots\!29}{20\!\cdots\!31}a+\frac{14\!\cdots\!43}{68\!\cdots\!77}$, $\frac{15\!\cdots\!87}{10\!\cdots\!90}a^{21}-\frac{80\!\cdots\!31}{66\!\cdots\!46}a^{20}+\frac{14\!\cdots\!09}{28\!\cdots\!34}a^{19}-\frac{22\!\cdots\!41}{14\!\cdots\!70}a^{18}+\frac{20\!\cdots\!43}{50\!\cdots\!95}a^{17}-\frac{15\!\cdots\!03}{23\!\cdots\!95}a^{16}+\frac{22\!\cdots\!49}{20\!\cdots\!38}a^{15}+\frac{38\!\cdots\!43}{16\!\cdots\!10}a^{14}-\frac{85\!\cdots\!43}{10\!\cdots\!90}a^{13}+\frac{24\!\cdots\!97}{10\!\cdots\!90}a^{12}-\frac{17\!\cdots\!11}{33\!\cdots\!30}a^{11}+\frac{11\!\cdots\!57}{20\!\cdots\!38}a^{10}+\frac{76\!\cdots\!68}{23\!\cdots\!95}a^{9}-\frac{23\!\cdots\!93}{10\!\cdots\!90}a^{8}+\frac{12\!\cdots\!33}{33\!\cdots\!73}a^{7}-\frac{11\!\cdots\!44}{50\!\cdots\!95}a^{6}-\frac{81\!\cdots\!80}{10\!\cdots\!19}a^{5}+\frac{75\!\cdots\!45}{33\!\cdots\!73}a^{4}-\frac{11\!\cdots\!42}{10\!\cdots\!55}a^{3}-\frac{54\!\cdots\!66}{23\!\cdots\!95}a^{2}+\frac{68\!\cdots\!97}{28\!\cdots\!34}a+\frac{11\!\cdots\!37}{47\!\cdots\!39}$, $\frac{15\!\cdots\!64}{10\!\cdots\!19}a^{21}-\frac{58\!\cdots\!43}{16\!\cdots\!65}a^{20}-\frac{70\!\cdots\!12}{71\!\cdots\!85}a^{19}+\frac{85\!\cdots\!73}{20\!\cdots\!10}a^{18}-\frac{20\!\cdots\!62}{10\!\cdots\!19}a^{17}+\frac{19\!\cdots\!18}{23\!\cdots\!95}a^{16}-\frac{13\!\cdots\!55}{10\!\cdots\!19}a^{15}-\frac{34\!\cdots\!59}{47\!\cdots\!39}a^{14}+\frac{14\!\cdots\!41}{50\!\cdots\!95}a^{13}-\frac{76\!\cdots\!21}{10\!\cdots\!90}a^{12}+\frac{63\!\cdots\!86}{16\!\cdots\!65}a^{11}-\frac{52\!\cdots\!36}{50\!\cdots\!95}a^{10}+\frac{22\!\cdots\!39}{23\!\cdots\!95}a^{9}+\frac{16\!\cdots\!39}{10\!\cdots\!90}a^{8}-\frac{83\!\cdots\!59}{16\!\cdots\!65}a^{7}+\frac{43\!\cdots\!11}{10\!\cdots\!90}a^{6}+\frac{66\!\cdots\!23}{10\!\cdots\!90}a^{5}-\frac{60\!\cdots\!62}{16\!\cdots\!65}a^{4}+\frac{16\!\cdots\!76}{10\!\cdots\!55}a^{3}+\frac{33\!\cdots\!27}{47\!\cdots\!90}a^{2}-\frac{14\!\cdots\!29}{28\!\cdots\!34}a-\frac{64\!\cdots\!72}{47\!\cdots\!39}$, $\frac{41\!\cdots\!89}{71\!\cdots\!85}a^{21}-\frac{25\!\cdots\!30}{47\!\cdots\!39}a^{20}+\frac{33\!\cdots\!81}{14\!\cdots\!17}a^{19}-\frac{54\!\cdots\!24}{71\!\cdots\!85}a^{18}+\frac{28\!\cdots\!99}{14\!\cdots\!70}a^{17}-\frac{16\!\cdots\!33}{47\!\cdots\!90}a^{16}+\frac{18\!\cdots\!07}{14\!\cdots\!17}a^{15}+\frac{52\!\cdots\!71}{47\!\cdots\!90}a^{14}-\frac{84\!\cdots\!31}{20\!\cdots\!10}a^{13}+\frac{82\!\cdots\!94}{71\!\cdots\!85}a^{12}-\frac{63\!\cdots\!42}{23\!\cdots\!95}a^{11}+\frac{49\!\cdots\!84}{14\!\cdots\!17}a^{10}+\frac{24\!\cdots\!84}{23\!\cdots\!95}a^{9}-\frac{89\!\cdots\!41}{71\!\cdots\!85}a^{8}+\frac{19\!\cdots\!93}{95\!\cdots\!78}a^{7}-\frac{18\!\cdots\!27}{14\!\cdots\!70}a^{6}-\frac{16\!\cdots\!25}{28\!\cdots\!34}a^{5}+\frac{12\!\cdots\!43}{95\!\cdots\!78}a^{4}-\frac{51\!\cdots\!91}{10\!\cdots\!55}a^{3}-\frac{16\!\cdots\!03}{68\!\cdots\!70}a^{2}+\frac{28\!\cdots\!07}{20\!\cdots\!31}a+\frac{23\!\cdots\!57}{68\!\cdots\!77}$, $\frac{84\!\cdots\!13}{33\!\cdots\!30}a^{21}-\frac{38\!\cdots\!64}{16\!\cdots\!65}a^{20}+\frac{35\!\cdots\!44}{34\!\cdots\!85}a^{19}-\frac{85\!\cdots\!14}{23\!\cdots\!95}a^{18}+\frac{32\!\cdots\!79}{33\!\cdots\!30}a^{17}-\frac{88\!\cdots\!19}{47\!\cdots\!90}a^{16}+\frac{94\!\cdots\!87}{66\!\cdots\!46}a^{15}+\frac{25\!\cdots\!57}{68\!\cdots\!70}a^{14}-\frac{12\!\cdots\!87}{66\!\cdots\!46}a^{13}+\frac{93\!\cdots\!46}{16\!\cdots\!65}a^{12}-\frac{44\!\cdots\!71}{33\!\cdots\!30}a^{11}+\frac{33\!\cdots\!19}{16\!\cdots\!65}a^{10}-\frac{27\!\cdots\!09}{47\!\cdots\!90}a^{9}-\frac{15\!\cdots\!23}{33\!\cdots\!30}a^{8}+\frac{36\!\cdots\!11}{33\!\cdots\!30}a^{7}-\frac{35\!\cdots\!81}{33\!\cdots\!30}a^{6}+\frac{47\!\cdots\!79}{16\!\cdots\!65}a^{5}+\frac{80\!\cdots\!69}{16\!\cdots\!65}a^{4}-\frac{17\!\cdots\!82}{34\!\cdots\!85}a^{3}+\frac{58\!\cdots\!47}{47\!\cdots\!90}a^{2}+\frac{29\!\cdots\!14}{47\!\cdots\!39}a-\frac{12\!\cdots\!95}{47\!\cdots\!39}$, $\frac{93\!\cdots\!83}{46\!\cdots\!70}a^{21}-\frac{24\!\cdots\!93}{15\!\cdots\!90}a^{20}+\frac{15\!\cdots\!28}{23\!\cdots\!35}a^{19}-\frac{10\!\cdots\!57}{46\!\cdots\!70}a^{18}+\frac{13\!\cdots\!22}{23\!\cdots\!35}a^{17}-\frac{75\!\cdots\!31}{77\!\cdots\!45}a^{16}+\frac{44\!\cdots\!81}{92\!\cdots\!14}a^{15}+\frac{37\!\cdots\!81}{15\!\cdots\!90}a^{14}-\frac{25\!\cdots\!08}{23\!\cdots\!35}a^{13}+\frac{10\!\cdots\!93}{31\!\cdots\!66}a^{12}-\frac{16\!\cdots\!19}{22\!\cdots\!70}a^{11}+\frac{45\!\cdots\!09}{46\!\cdots\!70}a^{10}-\frac{43\!\cdots\!99}{15\!\cdots\!90}a^{9}-\frac{23\!\cdots\!63}{92\!\cdots\!14}a^{8}+\frac{80\!\cdots\!31}{15\!\cdots\!90}a^{7}-\frac{25\!\cdots\!39}{46\!\cdots\!70}a^{6}+\frac{56\!\cdots\!72}{23\!\cdots\!35}a^{5}+\frac{94\!\cdots\!24}{77\!\cdots\!45}a^{4}-\frac{33\!\cdots\!15}{13\!\cdots\!02}a^{3}+\frac{31\!\cdots\!13}{22\!\cdots\!70}a^{2}-\frac{10\!\cdots\!35}{13\!\cdots\!02}a-\frac{40\!\cdots\!25}{22\!\cdots\!67}$, $\frac{37\!\cdots\!73}{50\!\cdots\!95}a^{21}-\frac{44\!\cdots\!69}{66\!\cdots\!46}a^{20}+\frac{11\!\cdots\!43}{40\!\cdots\!62}a^{19}-\frac{19\!\cdots\!29}{20\!\cdots\!10}a^{18}+\frac{12\!\cdots\!59}{50\!\cdots\!95}a^{17}-\frac{10\!\cdots\!94}{23\!\cdots\!95}a^{16}+\frac{12\!\cdots\!19}{10\!\cdots\!19}a^{15}+\frac{22\!\cdots\!39}{16\!\cdots\!10}a^{14}-\frac{50\!\cdots\!49}{10\!\cdots\!90}a^{13}+\frac{14\!\cdots\!81}{10\!\cdots\!90}a^{12}-\frac{54\!\cdots\!79}{16\!\cdots\!65}a^{11}+\frac{81\!\cdots\!61}{20\!\cdots\!38}a^{10}+\frac{75\!\cdots\!63}{47\!\cdots\!90}a^{9}-\frac{75\!\cdots\!02}{50\!\cdots\!95}a^{8}+\frac{81\!\cdots\!19}{33\!\cdots\!73}a^{7}-\frac{73\!\cdots\!67}{50\!\cdots\!95}a^{6}-\frac{12\!\cdots\!23}{20\!\cdots\!38}a^{5}+\frac{98\!\cdots\!53}{66\!\cdots\!46}a^{4}-\frac{57\!\cdots\!46}{10\!\cdots\!55}a^{3}-\frac{55\!\cdots\!28}{23\!\cdots\!95}a^{2}+\frac{41\!\cdots\!73}{28\!\cdots\!34}a+\frac{17\!\cdots\!68}{47\!\cdots\!39}$
| 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: | \( 118924316463 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 118924316463 \cdot 1}{2\cdot\sqrt{113420524920318543882697355574770703125}}\cr\approx \mathstrut & 0.867988083117009 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300)
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^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300, 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^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300);
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^22 - 9*x^21 + 41*x^20 - 140*x^19 + 383*x^18 - 720*x^17 + 530*x^16 + 1506*x^15 - 7373*x^14 + 22033*x^13 - 51771*x^12 + 76979*x^11 - 19458*x^10 - 188822*x^9 + 431163*x^8 - 422621*x^7 + 84439*x^6 + 244344*x^5 - 239399*x^4 + 37800*x^3 + 53515*x^2 - 17850*x - 6300);
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
$\PGL(2,11)$ (as 22T14):
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 1320 |
| The 13 conjugacy class representatives for $\PGL(2,11)$ |
| Character table for $\PGL(2,11)$ |
Intermediate fields
| \(\Q(\sqrt{77}) \) |
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 12 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Minimal sibling: | 12.0.179896172317964748078125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | R | R | ${\href{/padicField/13.5.0.1}{5} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.5.0.1}{5} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{11}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{11}$ | ${\href{/padicField/47.2.0.1}{2} }^{11}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.12.6.2 | $x^{12} + 25 x^{8} - 500 x^{6} + 625 x^{4} + 31250$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.4.3.2 | $x^{4} + 21$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.3.2 | $x^{4} + 21$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.8.6.2 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(11\)
| 11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 11.6.5.2 | $x^{6} + 11$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
| 11.12.11.1 | $x^{12} + 11$ | $12$ | $1$ | $11$ | $D_{12}$ | $[\ ]_{12}^{2}$ |