Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864)
gp: K = bnfinit(y^22 - 8*y^21 + 36*y^20 - 80*y^19 - 80*y^18 + 1944*y^17 - 10302*y^16 + 26484*y^15 - 25065*y^14 + 88360*y^13 + 105286*y^12 + 116412*y^11 - 3684854*y^10 - 5338040*y^9 + 49064310*y^8 - 83237556*y^7 - 20546277*y^6 + 172274184*y^5 - 94548680*y^4 - 74599880*y^3 + 71987376*y^2 - 9019808*y - 3254864, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864)
\( x^{22} - 8 x^{21} + 36 x^{20} - 80 x^{19} - 80 x^{18} + 1944 x^{17} - 10302 x^{16} + 26484 x^{15} + \cdots - 3254864 \)
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: |
\(1368763750033950505738752000000000000000000000000000\)
\(\medspace = 2^{36}\cdot 3^{16}\cdot 5^{27}\cdot 199^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(211.05\) | 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: |
\(2\), \(3\), \(5\), \(199\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{30}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{6}a^{6}-\frac{4}{15}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{6}a^{2}-\frac{7}{15}$, $\frac{1}{30}a^{11}-\frac{1}{3}a^{8}+\frac{1}{6}a^{7}+\frac{1}{15}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{6}a^{3}+\frac{1}{3}a^{2}-\frac{2}{15}a-\frac{1}{3}$, $\frac{1}{30}a^{12}-\frac{1}{6}a^{8}-\frac{4}{15}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{6}a^{4}-\frac{7}{15}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{30}a^{13}-\frac{1}{6}a^{9}-\frac{4}{15}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{6}a^{5}-\frac{7}{15}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{150}a^{14}-\frac{1}{150}a^{13}+\frac{1}{150}a^{12}-\frac{1}{150}a^{11}+\frac{1}{150}a^{10}-\frac{23}{150}a^{9}+\frac{13}{150}a^{8}-\frac{53}{150}a^{7}+\frac{43}{150}a^{6}-\frac{11}{50}a^{5}-\frac{19}{150}a^{4}+\frac{3}{50}a^{3}+\frac{13}{75}a^{2}-\frac{6}{25}a-\frac{9}{25}$, $\frac{1}{150}a^{15}-\frac{1}{75}a^{10}-\frac{1}{15}a^{9}+\frac{1}{15}a^{8}+\frac{4}{15}a^{7}+\frac{2}{5}a^{6}-\frac{31}{75}a^{5}+\frac{4}{15}a^{4}-\frac{13}{30}a^{3}+\frac{4}{15}a^{2}+\frac{2}{5}a-\frac{17}{75}$, $\frac{1}{150}a^{16}-\frac{1}{75}a^{11}+\frac{1}{15}a^{9}-\frac{1}{15}a^{8}+\frac{1}{15}a^{7}+\frac{19}{75}a^{6}-\frac{4}{15}a^{5}+\frac{7}{30}a^{4}-\frac{1}{15}a^{3}+\frac{1}{15}a^{2}-\frac{17}{75}a+\frac{1}{15}$, $\frac{1}{300}a^{17}-\frac{1}{300}a^{16}-\frac{1}{300}a^{14}+\frac{1}{300}a^{13}+\frac{1}{150}a^{12}+\frac{1}{100}a^{11}-\frac{1}{300}a^{10}-\frac{47}{300}a^{9}-\frac{17}{75}a^{8}+\frac{47}{100}a^{7}-\frac{121}{300}a^{6}+\frac{23}{50}a^{5}-\frac{101}{300}a^{4}-\frac{13}{100}a^{3}+\frac{11}{30}a^{2}-\frac{2}{5}a-\frac{29}{75}$, $\frac{1}{900}a^{18}+\frac{1}{900}a^{17}+\frac{1}{450}a^{16}-\frac{1}{300}a^{15}+\frac{1}{900}a^{14}+\frac{1}{450}a^{13}+\frac{1}{100}a^{12}-\frac{1}{180}a^{11}-\frac{13}{900}a^{10}+\frac{13}{225}a^{9}-\frac{229}{900}a^{8}-\frac{37}{180}a^{7}-\frac{1}{25}a^{6}-\frac{67}{900}a^{5}-\frac{419}{900}a^{4}+\frac{4}{15}a^{3}+\frac{181}{450}a^{2}+\frac{79}{225}a+\frac{112}{225}$, $\frac{1}{10800}a^{19}+\frac{1}{675}a^{17}-\frac{1}{1350}a^{16}+\frac{1}{2700}a^{15}+\frac{1}{675}a^{14}+\frac{11}{5400}a^{13}+\frac{1}{675}a^{12}-\frac{137}{10800}a^{11}+\frac{7}{540}a^{10}-\frac{253}{5400}a^{9}-\frac{49}{2700}a^{8}-\frac{383}{5400}a^{7}-\frac{137}{540}a^{6}+\frac{109}{5400}a^{5}+\frac{611}{2700}a^{4}+\frac{2567}{10800}a^{3}+\frac{131}{2700}a^{2}+\frac{277}{900}a+\frac{743}{2700}$, $\frac{1}{21600}a^{20}-\frac{1}{2700}a^{18}+\frac{1}{5400}a^{17}+\frac{2}{675}a^{16}+\frac{1}{1350}a^{15}-\frac{19}{10800}a^{14}+\frac{1}{5400}a^{13}-\frac{281}{21600}a^{12}+\frac{1}{270}a^{11}+\frac{137}{10800}a^{10}-\frac{61}{1350}a^{9}-\frac{3539}{10800}a^{8}-\frac{233}{2700}a^{7}+\frac{3439}{10800}a^{6}+\frac{1211}{5400}a^{5}+\frac{3947}{21600}a^{4}+\frac{1}{108}a^{3}+\frac{57}{200}a^{2}+\frac{863}{5400}a+\frac{23}{90}$, $\frac{1}{10\!\cdots\!00}a^{21}+\frac{64\!\cdots\!21}{11\!\cdots\!40}a^{20}-\frac{12\!\cdots\!89}{29\!\cdots\!00}a^{19}-\frac{35\!\cdots\!17}{26\!\cdots\!00}a^{18}+\frac{30\!\cdots\!21}{29\!\cdots\!60}a^{17}+\frac{17\!\cdots\!43}{29\!\cdots\!60}a^{16}+\frac{27\!\cdots\!47}{17\!\cdots\!00}a^{15}-\frac{68\!\cdots\!67}{43\!\cdots\!00}a^{14}+\frac{54\!\cdots\!49}{35\!\cdots\!00}a^{13}-\frac{17\!\cdots\!93}{52\!\cdots\!00}a^{12}+\frac{98\!\cdots\!93}{70\!\cdots\!44}a^{11}+\frac{53\!\cdots\!19}{17\!\cdots\!60}a^{10}-\frac{22\!\cdots\!87}{52\!\cdots\!00}a^{9}+\frac{12\!\cdots\!13}{29\!\cdots\!00}a^{8}+\frac{15\!\cdots\!71}{58\!\cdots\!00}a^{7}-\frac{22\!\cdots\!87}{21\!\cdots\!00}a^{6}-\frac{39\!\cdots\!87}{35\!\cdots\!00}a^{5}-\frac{17\!\cdots\!43}{70\!\cdots\!44}a^{4}-\frac{22\!\cdots\!78}{16\!\cdots\!25}a^{3}+\frac{95\!\cdots\!87}{87\!\cdots\!00}a^{2}+\frac{19\!\cdots\!17}{43\!\cdots\!00}a-\frac{24\!\cdots\!29}{65\!\cdots\!00}$
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: | $13$ | 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: |
$\frac{11\!\cdots\!21}{10\!\cdots\!00}a^{21}-\frac{24\!\cdots\!41}{26\!\cdots\!00}a^{20}+\frac{14\!\cdots\!52}{33\!\cdots\!35}a^{19}-\frac{61\!\cdots\!81}{52\!\cdots\!60}a^{18}-\frac{32\!\cdots\!37}{13\!\cdots\!40}a^{17}+\frac{27\!\cdots\!73}{13\!\cdots\!00}a^{16}-\frac{65\!\cdots\!23}{52\!\cdots\!00}a^{15}+\frac{19\!\cdots\!31}{52\!\cdots\!60}a^{14}-\frac{11\!\cdots\!37}{21\!\cdots\!40}a^{13}+\frac{22\!\cdots\!57}{17\!\cdots\!20}a^{12}+\frac{43\!\cdots\!31}{17\!\cdots\!00}a^{11}+\frac{38\!\cdots\!69}{44\!\cdots\!00}a^{10}-\frac{41\!\cdots\!63}{10\!\cdots\!20}a^{9}-\frac{72\!\cdots\!99}{26\!\cdots\!80}a^{8}+\frac{56\!\cdots\!27}{10\!\cdots\!20}a^{7}-\frac{33\!\cdots\!49}{26\!\cdots\!00}a^{6}+\frac{72\!\cdots\!19}{10\!\cdots\!00}a^{5}+\frac{75\!\cdots\!07}{52\!\cdots\!60}a^{4}-\frac{11\!\cdots\!01}{52\!\cdots\!60}a^{3}+\frac{55\!\cdots\!07}{52\!\cdots\!60}a^{2}-\frac{11\!\cdots\!71}{16\!\cdots\!75}a-\frac{36\!\cdots\!07}{55\!\cdots\!25}$, $\frac{61\!\cdots\!81}{10\!\cdots\!00}a^{21}-\frac{74\!\cdots\!19}{17\!\cdots\!60}a^{20}+\frac{15\!\cdots\!49}{87\!\cdots\!00}a^{19}-\frac{84\!\cdots\!01}{26\!\cdots\!00}a^{18}-\frac{41\!\cdots\!61}{54\!\cdots\!75}a^{17}+\frac{47\!\cdots\!21}{43\!\cdots\!00}a^{16}-\frac{30\!\cdots\!23}{58\!\cdots\!00}a^{15}+\frac{65\!\cdots\!07}{58\!\cdots\!20}a^{14}-\frac{17\!\cdots\!71}{35\!\cdots\!00}a^{13}+\frac{24\!\cdots\!59}{52\!\cdots\!80}a^{12}+\frac{35\!\cdots\!97}{35\!\cdots\!20}a^{11}+\frac{63\!\cdots\!01}{43\!\cdots\!00}a^{10}-\frac{10\!\cdots\!07}{52\!\cdots\!00}a^{9}-\frac{21\!\cdots\!91}{43\!\cdots\!00}a^{8}+\frac{44\!\cdots\!41}{17\!\cdots\!00}a^{7}-\frac{27\!\cdots\!11}{97\!\cdots\!00}a^{6}-\frac{87\!\cdots\!73}{23\!\cdots\!80}a^{5}+\frac{65\!\cdots\!37}{87\!\cdots\!00}a^{4}+\frac{21\!\cdots\!06}{32\!\cdots\!05}a^{3}-\frac{38\!\cdots\!43}{87\!\cdots\!00}a^{2}+\frac{29\!\cdots\!03}{43\!\cdots\!40}a+\frac{13\!\cdots\!17}{65\!\cdots\!00}$, $\frac{41\!\cdots\!21}{10\!\cdots\!00}a^{21}-\frac{26\!\cdots\!33}{87\!\cdots\!00}a^{20}+\frac{74\!\cdots\!07}{58\!\cdots\!20}a^{19}-\frac{65\!\cdots\!53}{26\!\cdots\!00}a^{18}-\frac{99\!\cdots\!57}{21\!\cdots\!00}a^{17}+\frac{32\!\cdots\!93}{43\!\cdots\!00}a^{16}-\frac{21\!\cdots\!43}{58\!\cdots\!00}a^{15}+\frac{49\!\cdots\!03}{58\!\cdots\!20}a^{14}-\frac{18\!\cdots\!47}{35\!\cdots\!00}a^{13}+\frac{84\!\cdots\!53}{26\!\cdots\!00}a^{12}+\frac{20\!\cdots\!37}{35\!\cdots\!20}a^{11}+\frac{12\!\cdots\!63}{16\!\cdots\!00}a^{10}-\frac{15\!\cdots\!83}{10\!\cdots\!60}a^{9}-\frac{12\!\cdots\!99}{43\!\cdots\!00}a^{8}+\frac{31\!\cdots\!37}{17\!\cdots\!00}a^{7}-\frac{22\!\cdots\!59}{97\!\cdots\!00}a^{6}-\frac{24\!\cdots\!41}{11\!\cdots\!00}a^{5}+\frac{10\!\cdots\!39}{17\!\cdots\!60}a^{4}-\frac{20\!\cdots\!21}{32\!\cdots\!50}a^{3}-\frac{29\!\cdots\!23}{87\!\cdots\!00}a^{2}+\frac{74\!\cdots\!91}{73\!\cdots\!00}a+\frac{15\!\cdots\!69}{65\!\cdots\!00}$, $\frac{39\!\cdots\!97}{10\!\cdots\!00}a^{21}-\frac{23\!\cdots\!23}{87\!\cdots\!00}a^{20}+\frac{95\!\cdots\!63}{87\!\cdots\!00}a^{19}-\frac{51\!\cdots\!01}{26\!\cdots\!00}a^{18}-\frac{35\!\cdots\!03}{73\!\cdots\!90}a^{17}+\frac{99\!\cdots\!67}{14\!\cdots\!00}a^{16}-\frac{56\!\cdots\!57}{17\!\cdots\!00}a^{15}+\frac{11\!\cdots\!33}{17\!\cdots\!60}a^{14}-\frac{95\!\cdots\!63}{35\!\cdots\!00}a^{13}+\frac{79\!\cdots\!03}{26\!\cdots\!00}a^{12}+\frac{11\!\cdots\!13}{17\!\cdots\!00}a^{11}+\frac{17\!\cdots\!99}{16\!\cdots\!00}a^{10}-\frac{67\!\cdots\!19}{52\!\cdots\!00}a^{9}-\frac{46\!\cdots\!93}{14\!\cdots\!00}a^{8}+\frac{89\!\cdots\!23}{58\!\cdots\!00}a^{7}-\frac{14\!\cdots\!79}{87\!\cdots\!00}a^{6}-\frac{83\!\cdots\!23}{35\!\cdots\!00}a^{5}+\frac{37\!\cdots\!29}{87\!\cdots\!00}a^{4}+\frac{76\!\cdots\!63}{13\!\cdots\!00}a^{3}-\frac{68\!\cdots\!53}{29\!\cdots\!00}a^{2}+\frac{10\!\cdots\!17}{21\!\cdots\!00}a+\frac{84\!\cdots\!03}{65\!\cdots\!00}$, $\frac{25\!\cdots\!77}{10\!\cdots\!00}a^{21}-\frac{13\!\cdots\!67}{73\!\cdots\!00}a^{20}+\frac{70\!\cdots\!21}{87\!\cdots\!00}a^{19}-\frac{42\!\cdots\!21}{26\!\cdots\!00}a^{18}-\frac{11\!\cdots\!33}{43\!\cdots\!40}a^{17}+\frac{20\!\cdots\!61}{43\!\cdots\!00}a^{16}-\frac{13\!\cdots\!63}{58\!\cdots\!00}a^{15}+\frac{16\!\cdots\!33}{29\!\cdots\!00}a^{14}-\frac{13\!\cdots\!03}{35\!\cdots\!00}a^{13}+\frac{25\!\cdots\!79}{13\!\cdots\!00}a^{12}+\frac{64\!\cdots\!57}{19\!\cdots\!00}a^{11}+\frac{41\!\cdots\!93}{10\!\cdots\!50}a^{10}-\frac{18\!\cdots\!75}{21\!\cdots\!32}a^{9}-\frac{18\!\cdots\!81}{10\!\cdots\!50}a^{8}+\frac{20\!\cdots\!89}{17\!\cdots\!00}a^{7}-\frac{45\!\cdots\!23}{29\!\cdots\!00}a^{6}-\frac{14\!\cdots\!41}{11\!\cdots\!00}a^{5}+\frac{16\!\cdots\!51}{43\!\cdots\!00}a^{4}-\frac{35\!\cdots\!71}{65\!\cdots\!00}a^{3}-\frac{65\!\cdots\!53}{29\!\cdots\!00}a^{2}+\frac{38\!\cdots\!88}{54\!\cdots\!75}a+\frac{10\!\cdots\!97}{65\!\cdots\!00}$, $\frac{27\!\cdots\!73}{35\!\cdots\!00}a^{21}-\frac{41\!\cdots\!03}{58\!\cdots\!00}a^{20}+\frac{15\!\cdots\!29}{43\!\cdots\!00}a^{19}-\frac{88\!\cdots\!57}{87\!\cdots\!00}a^{18}+\frac{18\!\cdots\!33}{43\!\cdots\!00}a^{17}+\frac{65\!\cdots\!77}{43\!\cdots\!00}a^{16}-\frac{68\!\cdots\!71}{70\!\cdots\!44}a^{15}+\frac{68\!\cdots\!71}{21\!\cdots\!00}a^{14}-\frac{18\!\cdots\!89}{35\!\cdots\!00}a^{13}+\frac{21\!\cdots\!61}{17\!\cdots\!00}a^{12}-\frac{15\!\cdots\!19}{35\!\cdots\!20}a^{11}+\frac{11\!\cdots\!87}{87\!\cdots\!00}a^{10}-\frac{53\!\cdots\!67}{17\!\cdots\!00}a^{9}-\frac{89\!\cdots\!41}{87\!\cdots\!00}a^{8}+\frac{70\!\cdots\!51}{17\!\cdots\!00}a^{7}-\frac{47\!\cdots\!87}{43\!\cdots\!00}a^{6}+\frac{33\!\cdots\!39}{35\!\cdots\!00}a^{5}+\frac{66\!\cdots\!81}{17\!\cdots\!00}a^{4}-\frac{37\!\cdots\!41}{32\!\cdots\!00}a^{3}+\frac{52\!\cdots\!61}{87\!\cdots\!00}a^{2}-\frac{66\!\cdots\!87}{14\!\cdots\!00}a-\frac{45\!\cdots\!73}{18\!\cdots\!25}$, $\frac{13\!\cdots\!59}{21\!\cdots\!32}a^{21}-\frac{97\!\cdots\!51}{17\!\cdots\!60}a^{20}+\frac{24\!\cdots\!59}{97\!\cdots\!00}a^{19}-\frac{76\!\cdots\!07}{13\!\cdots\!00}a^{18}-\frac{51\!\cdots\!41}{10\!\cdots\!50}a^{17}+\frac{72\!\cdots\!66}{54\!\cdots\!75}a^{16}-\frac{63\!\cdots\!99}{87\!\cdots\!00}a^{15}+\frac{10\!\cdots\!97}{54\!\cdots\!75}a^{14}-\frac{12\!\cdots\!23}{65\!\cdots\!00}a^{13}+\frac{30\!\cdots\!01}{52\!\cdots\!80}a^{12}+\frac{64\!\cdots\!71}{16\!\cdots\!00}a^{11}-\frac{42\!\cdots\!19}{43\!\cdots\!00}a^{10}-\frac{68\!\cdots\!07}{26\!\cdots\!00}a^{9}-\frac{11\!\cdots\!01}{43\!\cdots\!00}a^{8}+\frac{32\!\cdots\!09}{87\!\cdots\!00}a^{7}-\frac{65\!\cdots\!56}{10\!\cdots\!35}a^{6}-\frac{28\!\cdots\!13}{17\!\cdots\!00}a^{5}+\frac{22\!\cdots\!99}{19\!\cdots\!40}a^{4}-\frac{17\!\cdots\!11}{26\!\cdots\!00}a^{3}-\frac{17\!\cdots\!17}{43\!\cdots\!00}a^{2}+\frac{52\!\cdots\!59}{10\!\cdots\!50}a-\frac{15\!\cdots\!57}{13\!\cdots\!20}$, $\frac{66\!\cdots\!47}{52\!\cdots\!00}a^{21}-\frac{89\!\cdots\!57}{87\!\cdots\!00}a^{20}+\frac{30\!\cdots\!77}{65\!\cdots\!80}a^{19}-\frac{14\!\cdots\!91}{13\!\cdots\!00}a^{18}-\frac{18\!\cdots\!51}{21\!\cdots\!00}a^{17}+\frac{53\!\cdots\!69}{21\!\cdots\!00}a^{16}-\frac{43\!\cdots\!61}{32\!\cdots\!00}a^{15}+\frac{12\!\cdots\!86}{36\!\cdots\!45}a^{14}-\frac{64\!\cdots\!69}{17\!\cdots\!00}a^{13}+\frac{30\!\cdots\!59}{26\!\cdots\!00}a^{12}+\frac{50\!\cdots\!33}{43\!\cdots\!00}a^{11}+\frac{18\!\cdots\!71}{14\!\cdots\!00}a^{10}-\frac{24\!\cdots\!31}{52\!\cdots\!80}a^{9}-\frac{26\!\cdots\!51}{43\!\cdots\!00}a^{8}+\frac{54\!\cdots\!93}{87\!\cdots\!00}a^{7}-\frac{20\!\cdots\!61}{18\!\cdots\!25}a^{6}-\frac{10\!\cdots\!43}{11\!\cdots\!40}a^{5}+\frac{38\!\cdots\!89}{17\!\cdots\!60}a^{4}-\frac{39\!\cdots\!47}{26\!\cdots\!00}a^{3}-\frac{30\!\cdots\!57}{43\!\cdots\!00}a^{2}+\frac{36\!\cdots\!03}{36\!\cdots\!45}a-\frac{17\!\cdots\!57}{65\!\cdots\!00}$, $\frac{18\!\cdots\!63}{26\!\cdots\!00}a^{21}-\frac{11\!\cdots\!59}{21\!\cdots\!00}a^{20}+\frac{19\!\cdots\!93}{87\!\cdots\!00}a^{19}-\frac{70\!\cdots\!18}{16\!\cdots\!25}a^{18}-\frac{15\!\cdots\!23}{18\!\cdots\!25}a^{17}+\frac{32\!\cdots\!79}{24\!\cdots\!00}a^{16}-\frac{28\!\cdots\!99}{43\!\cdots\!00}a^{15}+\frac{32\!\cdots\!33}{21\!\cdots\!00}a^{14}-\frac{75\!\cdots\!31}{87\!\cdots\!00}a^{13}+\frac{19\!\cdots\!31}{32\!\cdots\!50}a^{12}+\frac{32\!\cdots\!11}{29\!\cdots\!00}a^{11}+\frac{68\!\cdots\!79}{43\!\cdots\!40}a^{10}-\frac{33\!\cdots\!29}{13\!\cdots\!20}a^{9}-\frac{19\!\cdots\!01}{36\!\cdots\!50}a^{8}+\frac{57\!\cdots\!38}{18\!\cdots\!25}a^{7}-\frac{86\!\cdots\!03}{21\!\cdots\!00}a^{6}-\frac{13\!\cdots\!51}{35\!\cdots\!72}a^{5}+\frac{53\!\cdots\!76}{54\!\cdots\!75}a^{4}-\frac{11\!\cdots\!87}{26\!\cdots\!00}a^{3}-\frac{30\!\cdots\!16}{54\!\cdots\!75}a^{2}+\frac{10\!\cdots\!49}{73\!\cdots\!00}a+\frac{22\!\cdots\!61}{65\!\cdots\!00}$, $\frac{49\!\cdots\!13}{35\!\cdots\!72}a^{21}-\frac{17\!\cdots\!81}{17\!\cdots\!00}a^{20}+\frac{12\!\cdots\!23}{29\!\cdots\!00}a^{19}-\frac{16\!\cdots\!23}{21\!\cdots\!00}a^{18}-\frac{79\!\cdots\!57}{43\!\cdots\!00}a^{17}+\frac{57\!\cdots\!61}{21\!\cdots\!00}a^{16}-\frac{54\!\cdots\!03}{43\!\cdots\!00}a^{15}+\frac{23\!\cdots\!43}{87\!\cdots\!00}a^{14}-\frac{10\!\cdots\!57}{87\!\cdots\!00}a^{13}+\frac{20\!\cdots\!33}{17\!\cdots\!00}a^{12}+\frac{21\!\cdots\!61}{87\!\cdots\!00}a^{11}+\frac{31\!\cdots\!11}{87\!\cdots\!00}a^{10}-\frac{36\!\cdots\!93}{73\!\cdots\!00}a^{9}-\frac{10\!\cdots\!93}{87\!\cdots\!00}a^{8}+\frac{26\!\cdots\!43}{43\!\cdots\!40}a^{7}-\frac{11\!\cdots\!71}{17\!\cdots\!60}a^{6}-\frac{76\!\cdots\!17}{87\!\cdots\!00}a^{5}+\frac{60\!\cdots\!41}{35\!\cdots\!20}a^{4}+\frac{50\!\cdots\!89}{29\!\cdots\!00}a^{3}-\frac{13\!\cdots\!13}{14\!\cdots\!00}a^{2}+\frac{87\!\cdots\!99}{43\!\cdots\!00}a+\frac{11\!\cdots\!81}{21\!\cdots\!00}$, $\frac{22\!\cdots\!87}{10\!\cdots\!00}a^{21}-\frac{13\!\cdots\!53}{87\!\cdots\!00}a^{20}+\frac{55\!\cdots\!99}{87\!\cdots\!00}a^{19}-\frac{60\!\cdots\!39}{52\!\cdots\!80}a^{18}-\frac{44\!\cdots\!07}{16\!\cdots\!20}a^{17}+\frac{57\!\cdots\!17}{14\!\cdots\!00}a^{16}-\frac{13\!\cdots\!31}{70\!\cdots\!44}a^{15}+\frac{35\!\cdots\!99}{87\!\cdots\!00}a^{14}-\frac{62\!\cdots\!53}{35\!\cdots\!00}a^{13}+\frac{45\!\cdots\!61}{26\!\cdots\!00}a^{12}+\frac{65\!\cdots\!71}{17\!\cdots\!00}a^{11}+\frac{16\!\cdots\!23}{29\!\cdots\!60}a^{10}-\frac{38\!\cdots\!29}{52\!\cdots\!00}a^{9}-\frac{26\!\cdots\!89}{14\!\cdots\!00}a^{8}+\frac{52\!\cdots\!41}{58\!\cdots\!00}a^{7}-\frac{86\!\cdots\!21}{87\!\cdots\!00}a^{6}-\frac{46\!\cdots\!13}{35\!\cdots\!00}a^{5}+\frac{22\!\cdots\!51}{87\!\cdots\!00}a^{4}+\frac{91\!\cdots\!43}{32\!\cdots\!50}a^{3}-\frac{40\!\cdots\!11}{29\!\cdots\!00}a^{2}+\frac{63\!\cdots\!93}{21\!\cdots\!00}a+\frac{50\!\cdots\!07}{65\!\cdots\!00}$, $\frac{11\!\cdots\!47}{10\!\cdots\!00}a^{21}-\frac{20\!\cdots\!30}{24\!\cdots\!63}a^{20}+\frac{34\!\cdots\!39}{87\!\cdots\!00}a^{19}-\frac{24\!\cdots\!43}{26\!\cdots\!00}a^{18}-\frac{35\!\cdots\!52}{54\!\cdots\!75}a^{17}+\frac{89\!\cdots\!19}{43\!\cdots\!00}a^{16}-\frac{65\!\cdots\!69}{58\!\cdots\!00}a^{15}+\frac{29\!\cdots\!53}{97\!\cdots\!00}a^{14}-\frac{11\!\cdots\!37}{35\!\cdots\!00}a^{13}+\frac{13\!\cdots\!41}{13\!\cdots\!00}a^{12}+\frac{52\!\cdots\!57}{58\!\cdots\!00}a^{11}+\frac{24\!\cdots\!63}{21\!\cdots\!00}a^{10}-\frac{20\!\cdots\!13}{52\!\cdots\!00}a^{9}-\frac{10\!\cdots\!97}{21\!\cdots\!00}a^{8}+\frac{91\!\cdots\!63}{17\!\cdots\!00}a^{7}-\frac{28\!\cdots\!73}{29\!\cdots\!00}a^{6}-\frac{70\!\cdots\!73}{39\!\cdots\!00}a^{5}+\frac{77\!\cdots\!57}{43\!\cdots\!00}a^{4}-\frac{84\!\cdots\!87}{65\!\cdots\!00}a^{3}-\frac{15\!\cdots\!19}{29\!\cdots\!00}a^{2}+\frac{88\!\cdots\!98}{10\!\cdots\!35}a-\frac{14\!\cdots\!57}{65\!\cdots\!00}$, $\frac{12\!\cdots\!49}{10\!\cdots\!00}a^{21}-\frac{70\!\cdots\!53}{87\!\cdots\!00}a^{20}+\frac{28\!\cdots\!17}{87\!\cdots\!80}a^{19}-\frac{13\!\cdots\!29}{26\!\cdots\!00}a^{18}-\frac{18\!\cdots\!83}{10\!\cdots\!50}a^{17}+\frac{18\!\cdots\!47}{87\!\cdots\!80}a^{16}-\frac{55\!\cdots\!87}{58\!\cdots\!00}a^{15}+\frac{54\!\cdots\!23}{29\!\cdots\!00}a^{14}-\frac{11\!\cdots\!07}{35\!\cdots\!00}a^{13}+\frac{26\!\cdots\!01}{26\!\cdots\!00}a^{12}+\frac{45\!\cdots\!63}{17\!\cdots\!00}a^{11}+\frac{20\!\cdots\!49}{43\!\cdots\!00}a^{10}-\frac{20\!\cdots\!67}{52\!\cdots\!00}a^{9}-\frac{51\!\cdots\!41}{43\!\cdots\!00}a^{8}+\frac{76\!\cdots\!77}{17\!\cdots\!00}a^{7}-\frac{37\!\cdots\!71}{97\!\cdots\!00}a^{6}-\frac{98\!\cdots\!81}{11\!\cdots\!00}a^{5}+\frac{87\!\cdots\!39}{87\!\cdots\!00}a^{4}+\frac{93\!\cdots\!89}{26\!\cdots\!00}a^{3}-\frac{99\!\cdots\!87}{17\!\cdots\!60}a^{2}+\frac{59\!\cdots\!53}{10\!\cdots\!35}a+\frac{28\!\cdots\!13}{65\!\cdots\!10}$
| 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: | \( 396267155700000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 396267155700000000 \cdot 1}{2\cdot\sqrt{1368763750033950505738752000000000000000000000000000}}\cr\approx \mathstrut & 0.832554658791912 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864)
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 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864, 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 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864);
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 - 8*x^21 + 36*x^20 - 80*x^19 - 80*x^18 + 1944*x^17 - 10302*x^16 + 26484*x^15 - 25065*x^14 + 88360*x^13 + 105286*x^12 + 116412*x^11 - 3684854*x^10 - 5338040*x^9 + 49064310*x^8 - 83237556*x^7 - 20546277*x^6 + 172274184*x^5 - 94548680*x^4 - 74599880*x^3 + 71987376*x^2 - 9019808*x - 3254864);
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
$M_{11}\wr C_2$ (as 22T48):
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 125452800 |
| The 65 conjugacy class representatives for $M_{11}\wr C_2$ are not computed |
| Character table for $M_{11}\wr C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
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 24 sibling: | data not computed |
| Degree 44 sibling: | 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 | R | R | $22$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $22$ | $16{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $22$ | $16{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| Deg $16$ | $8$ | $2$ | $32$ | ||||
|
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.16.14.4 | $x^{16} - 12 x^{8} + 72$ | $8$ | $2$ | $14$ | 16T124 | $[\ ]_{8}^{8}$ | |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.10.13.1 | $x^{10} + 15 x^{4} + 5$ | $10$ | $1$ | $13$ | $D_5$ | $[3/2]_{2}$ | |
| 5.10.13.1 | $x^{10} + 15 x^{4} + 5$ | $10$ | $1$ | $13$ | $D_5$ | $[3/2]_{2}$ | |
|
\(199\)
| $\Q_{199}$ | $x + 196$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{199}$ | $x + 196$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{199}$ | $x + 196$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{199}$ | $x + 196$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 199.5.0.1 | $x^{5} + 3 x + 196$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 199.5.0.1 | $x^{5} + 3 x + 196$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 199.8.6.2 | $x^{8} - 152434 x^{4} - 7462570844$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |