Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916)
gp: K = bnfinit(y^22 - 3*y^21 + 45*y^20 - 12*y^19 + 1263*y^18 - 891*y^17 + 11013*y^16 + 12294*y^15 + 42993*y^14 + 18555*y^13 + 42921*y^12 + 71280*y^11 + 36558*y^10 + 25902*y^9 + 32238*y^8 + 31986*y^7 + 15876*y^6 + 6156*y^5 + 13644*y^4 + 11880*y^3 + 972*y^2 + 1944*y + 2916, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916)
\( x^{22} - 3 x^{21} + 45 x^{20} - 12 x^{19} + 1263 x^{18} - 891 x^{17} + 11013 x^{16} + 12294 x^{15} + \cdots + 2916 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-11914309055197948986020479547159042393930184375104869730372026368\)
\(\medspace = -\,2^{24}\cdot 3^{29}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(817.61\) | 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: |
\(2\), \(3\), \(7\), \(23\), \(137\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{18}a^{10}-\frac{1}{2}a^{8}-\frac{1}{3}a^{7}-\frac{1}{6}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{18}a^{11}-\frac{1}{6}a^{9}-\frac{1}{3}a^{8}-\frac{1}{6}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{18}a^{12}+\frac{1}{3}a^{8}+\frac{1}{6}a^{6}-\frac{1}{3}a^{3}$, $\frac{1}{18}a^{13}+\frac{1}{6}a^{7}-\frac{1}{3}a^{4}$, $\frac{1}{18}a^{14}+\frac{1}{6}a^{8}-\frac{1}{3}a^{5}$, $\frac{1}{18}a^{15}-\frac{1}{6}a^{9}-\frac{1}{3}a^{6}$, $\frac{1}{18}a^{16}-\frac{1}{2}a^{8}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{36}a^{17}-\frac{1}{36}a^{16}-\frac{1}{36}a^{14}-\frac{1}{36}a^{12}-\frac{1}{36}a^{11}-\frac{1}{6}a^{9}-\frac{1}{2}a^{7}-\frac{1}{3}a^{6}-\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{108}a^{18}-\frac{1}{36}a^{16}-\frac{1}{36}a^{15}-\frac{1}{36}a^{14}-\frac{1}{36}a^{13}-\frac{1}{36}a^{11}-\frac{1}{18}a^{9}+\frac{1}{6}a^{7}+\frac{1}{6}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{108}a^{19}-\frac{1}{36}a^{15}-\frac{1}{36}a^{11}-\frac{1}{6}a^{9}-\frac{1}{3}a^{8}-\frac{1}{6}a^{7}+\frac{1}{3}a^{6}+\frac{1}{6}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{729000}a^{20}+\frac{539}{121500}a^{19}-\frac{1}{250}a^{18}+\frac{16}{1215}a^{17}+\frac{6589}{243000}a^{16}-\frac{146}{10125}a^{15}-\frac{71}{30375}a^{14}-\frac{233}{10125}a^{13}+\frac{2173}{81000}a^{12}+\frac{59}{60750}a^{11}-\frac{179}{13500}a^{10}-\frac{833}{6750}a^{9}+\frac{584}{3375}a^{8}-\frac{253}{40500}a^{7}+\frac{19}{3375}a^{6}-\frac{34}{2025}a^{5}-\frac{251}{675}a^{4}-\frac{478}{1125}a^{3}-\frac{2699}{20250}a^{2}-\frac{1169}{6750}a+\frac{361}{2250}$, $\frac{1}{30\!\cdots\!00}a^{21}+\frac{62\!\cdots\!59}{10\!\cdots\!00}a^{20}-\frac{58\!\cdots\!03}{16\!\cdots\!00}a^{19}+\frac{89\!\cdots\!27}{50\!\cdots\!00}a^{18}-\frac{55\!\cdots\!61}{10\!\cdots\!00}a^{17}-\frac{41\!\cdots\!53}{11\!\cdots\!00}a^{16}-\frac{84\!\cdots\!19}{10\!\cdots\!00}a^{15}-\frac{11\!\cdots\!43}{84\!\cdots\!50}a^{14}-\frac{73\!\cdots\!29}{33\!\cdots\!00}a^{13}+\frac{21\!\cdots\!03}{10\!\cdots\!00}a^{12}-\frac{34\!\cdots\!79}{16\!\cdots\!00}a^{11}-\frac{34\!\cdots\!63}{56\!\cdots\!00}a^{10}+\frac{79\!\cdots\!37}{14\!\cdots\!75}a^{9}-\frac{35\!\cdots\!59}{16\!\cdots\!00}a^{8}+\frac{11\!\cdots\!37}{62\!\cdots\!00}a^{7}-\frac{73\!\cdots\!44}{42\!\cdots\!25}a^{6}+\frac{41\!\cdots\!82}{20\!\cdots\!85}a^{5}+\frac{22\!\cdots\!92}{46\!\cdots\!25}a^{4}-\frac{58\!\cdots\!67}{64\!\cdots\!50}a^{3}-\frac{55\!\cdots\!19}{14\!\cdots\!75}a^{2}+\frac{81\!\cdots\!36}{46\!\cdots\!25}a+\frac{92\!\cdots\!41}{31\!\cdots\!50}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
\( -\frac{45857543861397661588583459}{404115560235275960661290458524} a^{21} - \frac{2181086504872815458901569}{44901728915030662295698939836} a^{20} - \frac{178216142039565906970027051}{44901728915030662295698939836} a^{19} - \frac{539925034001650159653401131}{33676296686272996721774204877} a^{18} - \frac{18838203925820066566983220577}{134705186745091986887096819508} a^{17} - \frac{17489914613439699792040572251}{44901728915030662295698939836} a^{16} - \frac{125478777649349894314594768633}{134705186745091986887096819508} a^{15} - \frac{42130229910541202487660866179}{7483621485838443715949823306} a^{14} - \frac{437757791793002786648961750035}{44901728915030662295698939836} a^{13} - \frac{2585127815797723748584080269551}{134705186745091986887096819508} a^{12} - \frac{485052396037237426537672653287}{44901728915030662295698939836} a^{11} - \frac{84166308867181713814939181060}{3741810742919221857974911653} a^{10} - \frac{181554584272409390007286070047}{7483621485838443715949823306} a^{9} - \frac{192874228667814523305394214015}{11225432228757665573924734959} a^{8} - \frac{40848039178800285815335173547}{7483621485838443715949823306} a^{7} - \frac{148928184816721105621690756067}{22450864457515331147849469918} a^{6} - \frac{71083581891760346712654921431}{7483621485838443715949823306} a^{5} - \frac{655827849591219147265103117}{138585583071082291036107839} a^{4} - \frac{1433494370743172721762961097}{863494786827512736455748843} a^{3} - \frac{3014364559620487801239175585}{1247270247639740619324970551} a^{2} - \frac{1097996282465898389484983687}{415756749213246873108323517} a + \frac{30621804865368512430518822}{415756749213246873108323517} \)
(order $6$)
| 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: |
$\frac{11\!\cdots\!23}{15\!\cdots\!00}a^{21}-\frac{39\!\cdots\!59}{16\!\cdots\!85}a^{20}+\frac{28\!\cdots\!33}{84\!\cdots\!50}a^{19}-\frac{74\!\cdots\!71}{50\!\cdots\!00}a^{18}+\frac{23\!\cdots\!11}{25\!\cdots\!50}a^{17}-\frac{69\!\cdots\!29}{84\!\cdots\!25}a^{16}+\frac{41\!\cdots\!89}{50\!\cdots\!00}a^{15}+\frac{22\!\cdots\!68}{31\!\cdots\!75}a^{14}+\frac{25\!\cdots\!01}{84\!\cdots\!50}a^{13}+\frac{45\!\cdots\!29}{10\!\cdots\!00}a^{12}+\frac{10\!\cdots\!16}{42\!\cdots\!25}a^{11}+\frac{97\!\cdots\!21}{28\!\cdots\!75}a^{10}+\frac{44\!\cdots\!74}{28\!\cdots\!75}a^{9}+\frac{72\!\cdots\!14}{16\!\cdots\!85}a^{8}+\frac{30\!\cdots\!21}{56\!\cdots\!50}a^{7}+\frac{12\!\cdots\!89}{84\!\cdots\!50}a^{6}+\frac{21\!\cdots\!71}{28\!\cdots\!75}a^{5}+\frac{10\!\cdots\!34}{15\!\cdots\!75}a^{4}+\frac{86\!\cdots\!77}{32\!\cdots\!25}a^{3}+\frac{22\!\cdots\!97}{51\!\cdots\!25}a^{2}-\frac{60\!\cdots\!43}{17\!\cdots\!75}a+\frac{71\!\cdots\!91}{15\!\cdots\!75}$, $\frac{39\!\cdots\!84}{42\!\cdots\!25}a^{21}-\frac{76\!\cdots\!69}{56\!\cdots\!00}a^{20}+\frac{60\!\cdots\!13}{18\!\cdots\!50}a^{19}+\frac{10\!\cdots\!33}{14\!\cdots\!75}a^{18}+\frac{51\!\cdots\!79}{56\!\cdots\!00}a^{17}+\frac{18\!\cdots\!52}{15\!\cdots\!75}a^{16}+\frac{69\!\cdots\!63}{28\!\cdots\!50}a^{15}+\frac{68\!\cdots\!07}{18\!\cdots\!00}a^{14}+\frac{55\!\cdots\!81}{93\!\cdots\!50}a^{13}+\frac{20\!\cdots\!83}{14\!\cdots\!75}a^{12}+\frac{85\!\cdots\!89}{18\!\cdots\!00}a^{11}+\frac{13\!\cdots\!08}{51\!\cdots\!25}a^{10}+\frac{38\!\cdots\!67}{31\!\cdots\!50}a^{9}+\frac{66\!\cdots\!47}{46\!\cdots\!25}a^{8}+\frac{35\!\cdots\!69}{17\!\cdots\!75}a^{7}+\frac{55\!\cdots\!51}{46\!\cdots\!25}a^{6}+\frac{21\!\cdots\!01}{20\!\cdots\!50}a^{5}+\frac{11\!\cdots\!43}{17\!\cdots\!75}a^{4}+\frac{61\!\cdots\!39}{71\!\cdots\!25}a^{3}+\frac{39\!\cdots\!39}{15\!\cdots\!75}a^{2}+\frac{12\!\cdots\!28}{17\!\cdots\!75}a+\frac{34\!\cdots\!78}{17\!\cdots\!75}$, $\frac{55\!\cdots\!09}{15\!\cdots\!00}a^{21}-\frac{64\!\cdots\!79}{56\!\cdots\!00}a^{20}+\frac{27\!\cdots\!97}{16\!\cdots\!50}a^{19}-\frac{68\!\cdots\!23}{12\!\cdots\!75}a^{18}+\frac{55\!\cdots\!94}{12\!\cdots\!75}a^{17}-\frac{16\!\cdots\!86}{42\!\cdots\!25}a^{16}+\frac{85\!\cdots\!47}{25\!\cdots\!50}a^{15}+\frac{21\!\cdots\!61}{56\!\cdots\!00}a^{14}+\frac{16\!\cdots\!53}{16\!\cdots\!00}a^{13}-\frac{90\!\cdots\!98}{12\!\cdots\!75}a^{12}-\frac{24\!\cdots\!59}{16\!\cdots\!00}a^{11}-\frac{17\!\cdots\!94}{14\!\cdots\!75}a^{10}-\frac{30\!\cdots\!01}{14\!\cdots\!75}a^{9}-\frac{71\!\cdots\!89}{84\!\cdots\!50}a^{8}-\frac{10\!\cdots\!01}{14\!\cdots\!75}a^{7}-\frac{23\!\cdots\!31}{42\!\cdots\!25}a^{6}+\frac{24\!\cdots\!57}{56\!\cdots\!50}a^{5}-\frac{76\!\cdots\!12}{17\!\cdots\!75}a^{4}-\frac{19\!\cdots\!59}{64\!\cdots\!25}a^{3}+\frac{13\!\cdots\!97}{46\!\cdots\!25}a^{2}-\frac{11\!\cdots\!43}{15\!\cdots\!75}a-\frac{15\!\cdots\!18}{15\!\cdots\!75}$, $\frac{16\!\cdots\!03}{37\!\cdots\!25}a^{21}-\frac{59\!\cdots\!47}{56\!\cdots\!00}a^{20}+\frac{15\!\cdots\!73}{84\!\cdots\!25}a^{19}+\frac{82\!\cdots\!61}{12\!\cdots\!75}a^{18}+\frac{25\!\cdots\!43}{50\!\cdots\!00}a^{17}-\frac{73\!\cdots\!21}{84\!\cdots\!50}a^{16}+\frac{48\!\cdots\!23}{12\!\cdots\!75}a^{15}+\frac{47\!\cdots\!73}{56\!\cdots\!00}a^{14}+\frac{12\!\cdots\!27}{84\!\cdots\!50}a^{13}+\frac{33\!\cdots\!97}{25\!\cdots\!50}a^{12}+\frac{97\!\cdots\!63}{16\!\cdots\!00}a^{11}+\frac{55\!\cdots\!08}{14\!\cdots\!75}a^{10}+\frac{50\!\cdots\!89}{28\!\cdots\!50}a^{9}-\frac{60\!\cdots\!26}{42\!\cdots\!25}a^{8}+\frac{36\!\cdots\!82}{14\!\cdots\!75}a^{7}+\frac{79\!\cdots\!92}{42\!\cdots\!25}a^{6}-\frac{66\!\cdots\!49}{56\!\cdots\!50}a^{5}+\frac{35\!\cdots\!27}{51\!\cdots\!25}a^{4}+\frac{71\!\cdots\!63}{64\!\cdots\!25}a^{3}-\frac{99\!\cdots\!29}{46\!\cdots\!25}a^{2}+\frac{86\!\cdots\!76}{15\!\cdots\!75}a+\frac{34\!\cdots\!26}{15\!\cdots\!75}$, $\frac{70\!\cdots\!99}{75\!\cdots\!50}a^{21}-\frac{31\!\cdots\!27}{42\!\cdots\!25}a^{20}+\frac{30\!\cdots\!97}{84\!\cdots\!50}a^{19}+\frac{39\!\cdots\!63}{50\!\cdots\!00}a^{18}+\frac{29\!\cdots\!61}{25\!\cdots\!50}a^{17}+\frac{28\!\cdots\!59}{16\!\cdots\!00}a^{16}+\frac{46\!\cdots\!77}{50\!\cdots\!00}a^{15}+\frac{18\!\cdots\!59}{56\!\cdots\!00}a^{14}+\frac{23\!\cdots\!67}{33\!\cdots\!00}a^{13}+\frac{27\!\cdots\!01}{25\!\cdots\!50}a^{12}+\frac{56\!\cdots\!53}{67\!\cdots\!40}a^{11}+\frac{37\!\cdots\!23}{28\!\cdots\!50}a^{10}+\frac{20\!\cdots\!21}{14\!\cdots\!75}a^{9}+\frac{48\!\cdots\!87}{42\!\cdots\!25}a^{8}+\frac{48\!\cdots\!26}{14\!\cdots\!75}a^{7}+\frac{33\!\cdots\!73}{84\!\cdots\!50}a^{6}+\frac{29\!\cdots\!31}{56\!\cdots\!50}a^{5}+\frac{75\!\cdots\!64}{15\!\cdots\!75}a^{4}+\frac{49\!\cdots\!28}{32\!\cdots\!25}a^{3}+\frac{12\!\cdots\!73}{93\!\cdots\!25}a^{2}+\frac{41\!\cdots\!58}{31\!\cdots\!75}a+\frac{12\!\cdots\!02}{15\!\cdots\!75}$, $\frac{60\!\cdots\!01}{30\!\cdots\!00}a^{21}-\frac{70\!\cdots\!31}{11\!\cdots\!00}a^{20}+\frac{14\!\cdots\!94}{16\!\cdots\!85}a^{19}-\frac{74\!\cdots\!72}{25\!\cdots\!75}a^{18}+\frac{60\!\cdots\!16}{25\!\cdots\!75}a^{17}-\frac{37\!\cdots\!33}{16\!\cdots\!50}a^{16}+\frac{46\!\cdots\!04}{25\!\cdots\!75}a^{15}+\frac{23\!\cdots\!29}{11\!\cdots\!00}a^{14}+\frac{17\!\cdots\!17}{33\!\cdots\!00}a^{13}-\frac{19\!\cdots\!69}{50\!\cdots\!50}a^{12}-\frac{26\!\cdots\!01}{33\!\cdots\!00}a^{11}-\frac{19\!\cdots\!66}{28\!\cdots\!75}a^{10}-\frac{33\!\cdots\!14}{28\!\cdots\!75}a^{9}-\frac{39\!\cdots\!98}{84\!\cdots\!25}a^{8}-\frac{11\!\cdots\!14}{28\!\cdots\!75}a^{7}-\frac{25\!\cdots\!09}{84\!\cdots\!25}a^{6}+\frac{22\!\cdots\!23}{11\!\cdots\!90}a^{5}-\frac{25\!\cdots\!04}{10\!\cdots\!25}a^{4}-\frac{21\!\cdots\!51}{12\!\cdots\!45}a^{3}+\frac{14\!\cdots\!58}{93\!\cdots\!25}a^{2}-\frac{13\!\cdots\!77}{31\!\cdots\!75}a-\frac{17\!\cdots\!52}{31\!\cdots\!75}$, $\frac{22\!\cdots\!71}{15\!\cdots\!00}a^{21}-\frac{19\!\cdots\!51}{56\!\cdots\!00}a^{20}+\frac{53\!\cdots\!84}{84\!\cdots\!25}a^{19}+\frac{65\!\cdots\!01}{25\!\cdots\!50}a^{18}+\frac{46\!\cdots\!47}{25\!\cdots\!50}a^{17}-\frac{76\!\cdots\!93}{84\!\cdots\!50}a^{16}+\frac{19\!\cdots\!84}{12\!\cdots\!75}a^{15}+\frac{15\!\cdots\!59}{56\!\cdots\!00}a^{14}+\frac{12\!\cdots\!57}{16\!\cdots\!00}a^{13}+\frac{16\!\cdots\!51}{25\!\cdots\!50}a^{12}+\frac{11\!\cdots\!29}{16\!\cdots\!00}a^{11}+\frac{16\!\cdots\!14}{14\!\cdots\!75}a^{10}+\frac{27\!\cdots\!37}{28\!\cdots\!50}a^{9}+\frac{39\!\cdots\!09}{84\!\cdots\!50}a^{8}+\frac{41\!\cdots\!06}{14\!\cdots\!75}a^{7}+\frac{36\!\cdots\!97}{84\!\cdots\!50}a^{6}+\frac{20\!\cdots\!33}{56\!\cdots\!50}a^{5}+\frac{22\!\cdots\!22}{17\!\cdots\!75}a^{4}+\frac{68\!\cdots\!54}{64\!\cdots\!25}a^{3}+\frac{82\!\cdots\!68}{46\!\cdots\!25}a^{2}+\frac{12\!\cdots\!08}{15\!\cdots\!75}a-\frac{47\!\cdots\!17}{15\!\cdots\!75}$, $\frac{32\!\cdots\!57}{15\!\cdots\!00}a^{21}-\frac{27\!\cdots\!17}{56\!\cdots\!00}a^{20}+\frac{30\!\cdots\!87}{33\!\cdots\!00}a^{19}+\frac{98\!\cdots\!67}{25\!\cdots\!50}a^{18}+\frac{13\!\cdots\!73}{50\!\cdots\!00}a^{17}-\frac{17\!\cdots\!37}{16\!\cdots\!00}a^{16}+\frac{11\!\cdots\!37}{50\!\cdots\!00}a^{15}+\frac{58\!\cdots\!82}{14\!\cdots\!75}a^{14}+\frac{18\!\cdots\!19}{16\!\cdots\!00}a^{13}+\frac{46\!\cdots\!59}{50\!\cdots\!00}a^{12}+\frac{17\!\cdots\!43}{16\!\cdots\!00}a^{11}+\frac{24\!\cdots\!88}{14\!\cdots\!75}a^{10}+\frac{20\!\cdots\!27}{14\!\cdots\!75}a^{9}+\frac{57\!\cdots\!53}{84\!\cdots\!50}a^{8}+\frac{12\!\cdots\!79}{28\!\cdots\!50}a^{7}+\frac{26\!\cdots\!37}{42\!\cdots\!25}a^{6}+\frac{30\!\cdots\!11}{56\!\cdots\!50}a^{5}+\frac{33\!\cdots\!99}{17\!\cdots\!75}a^{4}+\frac{99\!\cdots\!68}{64\!\cdots\!25}a^{3}+\frac{11\!\cdots\!06}{46\!\cdots\!25}a^{2}+\frac{18\!\cdots\!36}{15\!\cdots\!75}a-\frac{61\!\cdots\!39}{15\!\cdots\!75}$, $\frac{54\!\cdots\!79}{30\!\cdots\!00}a^{21}-\frac{23\!\cdots\!49}{33\!\cdots\!00}a^{20}+\frac{36\!\cdots\!44}{42\!\cdots\!25}a^{19}-\frac{23\!\cdots\!39}{25\!\cdots\!50}a^{18}+\frac{23\!\cdots\!31}{10\!\cdots\!00}a^{17}-\frac{11\!\cdots\!73}{33\!\cdots\!00}a^{16}+\frac{11\!\cdots\!93}{50\!\cdots\!00}a^{15}+\frac{47\!\cdots\!69}{14\!\cdots\!75}a^{14}+\frac{49\!\cdots\!99}{67\!\cdots\!00}a^{13}-\frac{27\!\cdots\!69}{10\!\cdots\!00}a^{12}+\frac{83\!\cdots\!82}{84\!\cdots\!25}a^{11}+\frac{25\!\cdots\!19}{56\!\cdots\!00}a^{10}+\frac{36\!\cdots\!44}{14\!\cdots\!75}a^{9}+\frac{42\!\cdots\!47}{16\!\cdots\!00}a^{8}+\frac{20\!\cdots\!81}{56\!\cdots\!00}a^{7}+\frac{11\!\cdots\!06}{42\!\cdots\!25}a^{6}+\frac{30\!\cdots\!29}{56\!\cdots\!50}a^{5}+\frac{10\!\cdots\!86}{15\!\cdots\!75}a^{4}+\frac{12\!\cdots\!49}{64\!\cdots\!50}a^{3}+\frac{55\!\cdots\!23}{93\!\cdots\!25}a^{2}-\frac{10\!\cdots\!32}{31\!\cdots\!75}a+\frac{19\!\cdots\!01}{31\!\cdots\!50}$, $\frac{15\!\cdots\!99}{42\!\cdots\!25}a^{21}-\frac{25\!\cdots\!87}{18\!\cdots\!00}a^{20}+\frac{16\!\cdots\!37}{93\!\cdots\!50}a^{19}-\frac{10\!\cdots\!41}{56\!\cdots\!00}a^{18}+\frac{13\!\cdots\!47}{28\!\cdots\!50}a^{17}-\frac{67\!\cdots\!57}{93\!\cdots\!50}a^{16}+\frac{50\!\cdots\!47}{11\!\cdots\!00}a^{15}+\frac{36\!\cdots\!21}{62\!\cdots\!00}a^{14}+\frac{26\!\cdots\!91}{18\!\cdots\!00}a^{13}-\frac{30\!\cdots\!87}{56\!\cdots\!00}a^{12}+\frac{32\!\cdots\!57}{18\!\cdots\!00}a^{11}+\frac{19\!\cdots\!01}{15\!\cdots\!75}a^{10}-\frac{33\!\cdots\!71}{15\!\cdots\!75}a^{9}+\frac{10\!\cdots\!61}{93\!\cdots\!50}a^{8}+\frac{34\!\cdots\!34}{15\!\cdots\!75}a^{7}+\frac{67\!\cdots\!79}{93\!\cdots\!50}a^{6}-\frac{82\!\cdots\!49}{12\!\cdots\!10}a^{5}+\frac{31\!\cdots\!89}{51\!\cdots\!25}a^{4}+\frac{18\!\cdots\!93}{35\!\cdots\!25}a^{3}-\frac{42\!\cdots\!91}{51\!\cdots\!25}a^{2}+\frac{20\!\cdots\!37}{51\!\cdots\!25}a+\frac{14\!\cdots\!86}{17\!\cdots\!75}$
| 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: | \( 1462413343710000000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 1462413343710000000000000 \cdot 1}{6\cdot\sqrt{11914309055197948986020479547159042393930184375104869730372026368}}\cr\approx \mathstrut & 1.34543590076382 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916)
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 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916, 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 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916);
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 - 3*x^21 + 45*x^20 - 12*x^19 + 1263*x^18 - 891*x^17 + 11013*x^16 + 12294*x^15 + 42993*x^14 + 18555*x^13 + 42921*x^12 + 71280*x^11 + 36558*x^10 + 25902*x^9 + 32238*x^8 + 31986*x^7 + 15876*x^6 + 6156*x^5 + 13644*x^4 + 11880*x^3 + 972*x^2 + 1944*x + 2916);
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
$C_2\times A_{11}$ (as 22T46):
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 39916800 |
| The 62 conjugacy class representatives for $C_2\times A_{11}$ are not computed |
| Character table for $C_2\times A_{11}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.7.63019333158425674204677255696384.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | R | $22$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | $22$ | ${\href{/padicField/19.5.0.1}{5} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.8.12.13 | $x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$ | $4$ | $2$ | $12$ | $D_4$ | $[2, 2]^{2}$ | |
| 2.12.12.28 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 130 x^{7} + 159 x^{6} + 132 x^{5} + 10 x^{4} - 100 x^{3} - 53 x^{2} + 22 x + 19$ | $6$ | $2$ | $12$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| Deg $18$ | $18$ | $1$ | $27$ | ||||
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(23\)
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.14.0.1 | $x^{14} + x^{8} + 5 x^{7} + 16 x^{6} + x^{5} + 18 x^{4} + 19 x^{3} + x^{2} + 22 x + 5$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(137\)
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 137.10.8.1 | $x^{10} + 655 x^{9} + 171625 x^{8} + 22488770 x^{7} + 1474044185 x^{6} + 38714410755 x^{5} + 4422222290 x^{4} + 225901280 x^{3} + 3082903315 x^{2} + 201591447850 x + 5280771081809$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 137.10.8.1 | $x^{10} + 655 x^{9} + 171625 x^{8} + 22488770 x^{7} + 1474044185 x^{6} + 38714410755 x^{5} + 4422222290 x^{4} + 225901280 x^{3} + 3082903315 x^{2} + 201591447850 x + 5280771081809$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |