Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 324*x^14 - 1208*x^13 + 38634*x^12 + 291024*x^11 - 1435904*x^10 - 22872192*x^9 - 52823469*x^8 + 457836672*x^7 + 3556633824*x^6 + 10251984768*x^5 + 10991206666*x^4 - 10633046448*x^3 - 41850562356*x^2 - 41586585624*x - 13976251839)
gp: K = bnfinit(y^16 - 324*y^14 - 1208*y^13 + 38634*y^12 + 291024*y^11 - 1435904*y^10 - 22872192*y^9 - 52823469*y^8 + 457836672*y^7 + 3556633824*y^6 + 10251984768*y^5 + 10991206666*y^4 - 10633046448*y^3 - 41850562356*y^2 - 41586585624*y - 13976251839, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 324*x^14 - 1208*x^13 + 38634*x^12 + 291024*x^11 - 1435904*x^10 - 22872192*x^9 - 52823469*x^8 + 457836672*x^7 + 3556633824*x^6 + 10251984768*x^5 + 10991206666*x^4 - 10633046448*x^3 - 41850562356*x^2 - 41586585624*x - 13976251839);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 324*x^14 - 1208*x^13 + 38634*x^12 + 291024*x^11 - 1435904*x^10 - 22872192*x^9 - 52823469*x^8 + 457836672*x^7 + 3556633824*x^6 + 10251984768*x^5 + 10991206666*x^4 - 10633046448*x^3 - 41850562356*x^2 - 41586585624*x - 13976251839)
\( x^{16} - 324 x^{14} - 1208 x^{13} + 38634 x^{12} + 291024 x^{11} - 1435904 x^{10} - 22872192 x^{9} + \cdots - 13976251839 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(305159650169696945720591486976000000000000\)
\(\medspace = 2^{24}\cdot 3^{16}\cdot 5^{12}\cdot 29^{4}\cdot 31^{2}\cdot 50461^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(391.55\) | 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: |
\(2\), \(3\), \(5\), \(29\), \(31\), \(50461\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{40}a^{8}-\frac{1}{20}a^{6}-\frac{1}{10}a^{5}-\frac{1}{8}a^{4}+\frac{1}{10}a^{3}-\frac{1}{4}a^{2}+\frac{3}{10}a+\frac{9}{40}$, $\frac{1}{40}a^{9}-\frac{1}{20}a^{7}-\frac{1}{10}a^{6}-\frac{1}{8}a^{5}+\frac{1}{10}a^{4}-\frac{1}{4}a^{3}+\frac{3}{10}a^{2}+\frac{9}{40}a$, $\frac{1}{40}a^{10}-\frac{1}{10}a^{7}+\frac{1}{40}a^{6}+\frac{3}{20}a^{5}+\frac{1}{4}a^{3}+\frac{9}{40}a^{2}-\frac{3}{20}a+\frac{1}{5}$, $\frac{1}{40}a^{11}+\frac{1}{40}a^{7}-\frac{1}{20}a^{6}+\frac{1}{10}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{3}{20}a^{2}-\frac{1}{10}a-\frac{1}{10}$, $\frac{1}{80}a^{12}-\frac{1}{80}a^{10}-\frac{1}{10}a^{7}-\frac{1}{16}a^{6}+\frac{1}{10}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{13}{80}a^{2}+\frac{23}{80}$, $\frac{1}{720}a^{13}-\frac{1}{80}a^{11}-\frac{1}{360}a^{10}+\frac{1}{120}a^{9}-\frac{17}{720}a^{7}-\frac{13}{120}a^{6}-\frac{11}{120}a^{5}-\frac{2}{15}a^{4}-\frac{9}{80}a^{3}+\frac{37}{120}a^{2}+\frac{37}{720}a-\frac{1}{60}$, $\frac{1}{12960}a^{14}-\frac{1}{4320}a^{13}-\frac{1}{240}a^{12}-\frac{47}{12960}a^{11}-\frac{7}{864}a^{10}-\frac{1}{240}a^{9}-\frac{25}{2592}a^{8}-\frac{91}{1440}a^{7}-\frac{11}{864}a^{6}+\frac{239}{2160}a^{5}-\frac{353}{1440}a^{4}+\frac{1931}{4320}a^{3}-\frac{2191}{6480}a^{2}+\frac{979}{4320}a+\frac{649}{1440}$, $\frac{1}{16\!\cdots\!80}a^{15}+\frac{10\!\cdots\!07}{27\!\cdots\!80}a^{14}+\frac{14\!\cdots\!31}{18\!\cdots\!20}a^{13}+\frac{39\!\cdots\!27}{16\!\cdots\!80}a^{12}-\frac{97\!\cdots\!69}{92\!\cdots\!96}a^{11}-\frac{10\!\cdots\!79}{18\!\cdots\!20}a^{10}-\frac{38\!\cdots\!67}{16\!\cdots\!80}a^{9}+\frac{34\!\cdots\!09}{34\!\cdots\!61}a^{8}-\frac{69\!\cdots\!49}{13\!\cdots\!40}a^{7}-\frac{65\!\cdots\!53}{55\!\cdots\!60}a^{6}-\frac{44\!\cdots\!09}{61\!\cdots\!40}a^{5}+\frac{78\!\cdots\!15}{69\!\cdots\!22}a^{4}-\frac{47\!\cdots\!69}{16\!\cdots\!80}a^{3}+\frac{66\!\cdots\!59}{18\!\cdots\!20}a^{2}-\frac{33\!\cdots\!89}{30\!\cdots\!20}a+\frac{42\!\cdots\!79}{61\!\cdots\!40}$
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: | $15$ | 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: |
$\frac{21\!\cdots\!72}{52\!\cdots\!15}a^{15}-\frac{18\!\cdots\!52}{17\!\cdots\!05}a^{14}-\frac{76\!\cdots\!48}{57\!\cdots\!35}a^{13}-\frac{77\!\cdots\!06}{52\!\cdots\!15}a^{12}+\frac{32\!\cdots\!76}{19\!\cdots\!45}a^{11}+\frac{44\!\cdots\!48}{57\!\cdots\!35}a^{10}-\frac{43\!\cdots\!76}{52\!\cdots\!15}a^{9}-\frac{12\!\cdots\!93}{17\!\cdots\!05}a^{8}-\frac{18\!\cdots\!72}{17\!\cdots\!05}a^{7}+\frac{34\!\cdots\!96}{17\!\cdots\!05}a^{6}+\frac{17\!\cdots\!76}{19\!\cdots\!45}a^{5}+\frac{27\!\cdots\!74}{17\!\cdots\!05}a^{4}-\frac{67\!\cdots\!64}{52\!\cdots\!15}a^{3}-\frac{24\!\cdots\!52}{57\!\cdots\!35}a^{2}-\frac{74\!\cdots\!36}{14\!\cdots\!27}a-\frac{38\!\cdots\!53}{19\!\cdots\!45}$, $\frac{52\!\cdots\!41}{15\!\cdots\!60}a^{15}-\frac{44\!\cdots\!57}{23\!\cdots\!40}a^{14}-\frac{76\!\cdots\!01}{77\!\cdots\!58}a^{13}+\frac{56\!\cdots\!50}{38\!\cdots\!29}a^{12}+\frac{11\!\cdots\!97}{92\!\cdots\!96}a^{11}+\frac{23\!\cdots\!23}{77\!\cdots\!80}a^{10}-\frac{50\!\cdots\!47}{77\!\cdots\!80}a^{9}-\frac{48\!\cdots\!89}{11\!\cdots\!70}a^{8}+\frac{26\!\cdots\!19}{57\!\cdots\!80}a^{7}+\frac{98\!\cdots\!29}{77\!\cdots\!80}a^{6}+\frac{19\!\cdots\!29}{38\!\cdots\!90}a^{5}+\frac{16\!\cdots\!36}{21\!\cdots\!05}a^{4}-\frac{31\!\cdots\!59}{12\!\cdots\!30}a^{3}-\frac{12\!\cdots\!23}{57\!\cdots\!35}a^{2}-\frac{19\!\cdots\!19}{77\!\cdots\!80}a-\frac{11\!\cdots\!59}{12\!\cdots\!30}$, $\frac{52\!\cdots\!41}{15\!\cdots\!60}a^{15}-\frac{44\!\cdots\!57}{23\!\cdots\!40}a^{14}-\frac{76\!\cdots\!01}{77\!\cdots\!58}a^{13}+\frac{56\!\cdots\!50}{38\!\cdots\!29}a^{12}+\frac{11\!\cdots\!97}{92\!\cdots\!96}a^{11}+\frac{23\!\cdots\!23}{77\!\cdots\!80}a^{10}-\frac{50\!\cdots\!47}{77\!\cdots\!80}a^{9}-\frac{48\!\cdots\!89}{11\!\cdots\!70}a^{8}+\frac{26\!\cdots\!19}{57\!\cdots\!80}a^{7}+\frac{98\!\cdots\!29}{77\!\cdots\!80}a^{6}+\frac{19\!\cdots\!29}{38\!\cdots\!90}a^{5}+\frac{16\!\cdots\!36}{21\!\cdots\!05}a^{4}-\frac{31\!\cdots\!59}{12\!\cdots\!30}a^{3}-\frac{12\!\cdots\!23}{57\!\cdots\!35}a^{2}-\frac{19\!\cdots\!19}{77\!\cdots\!80}a-\frac{11\!\cdots\!89}{12\!\cdots\!30}$, $\frac{58\!\cdots\!17}{41\!\cdots\!20}a^{15}-\frac{10\!\cdots\!11}{13\!\cdots\!44}a^{14}-\frac{37\!\cdots\!71}{92\!\cdots\!60}a^{13}+\frac{25\!\cdots\!37}{41\!\cdots\!20}a^{12}+\frac{93\!\cdots\!33}{18\!\cdots\!92}a^{11}+\frac{14\!\cdots\!27}{11\!\cdots\!70}a^{10}-\frac{55\!\cdots\!57}{20\!\cdots\!60}a^{9}-\frac{11\!\cdots\!09}{69\!\cdots\!20}a^{8}+\frac{57\!\cdots\!03}{27\!\cdots\!80}a^{7}+\frac{36\!\cdots\!91}{69\!\cdots\!20}a^{6}+\frac{77\!\cdots\!42}{38\!\cdots\!29}a^{5}+\frac{50\!\cdots\!26}{17\!\cdots\!05}a^{4}-\frac{91\!\cdots\!13}{83\!\cdots\!40}a^{3}-\frac{49\!\cdots\!17}{57\!\cdots\!35}a^{2}-\frac{29\!\cdots\!31}{30\!\cdots\!20}a-\frac{53\!\cdots\!37}{15\!\cdots\!60}$, $\frac{36\!\cdots\!69}{10\!\cdots\!30}a^{15}-\frac{54\!\cdots\!47}{27\!\cdots\!80}a^{14}-\frac{94\!\cdots\!07}{92\!\cdots\!60}a^{13}+\frac{12\!\cdots\!27}{83\!\cdots\!40}a^{12}+\frac{11\!\cdots\!41}{92\!\cdots\!60}a^{11}+\frac{14\!\cdots\!83}{46\!\cdots\!80}a^{10}-\frac{13\!\cdots\!47}{20\!\cdots\!60}a^{9}-\frac{11\!\cdots\!37}{27\!\cdots\!80}a^{8}+\frac{28\!\cdots\!03}{55\!\cdots\!76}a^{7}+\frac{18\!\cdots\!11}{13\!\cdots\!40}a^{6}+\frac{21\!\cdots\!25}{42\!\cdots\!81}a^{5}+\frac{20\!\cdots\!03}{27\!\cdots\!80}a^{4}-\frac{22\!\cdots\!91}{83\!\cdots\!40}a^{3}-\frac{22\!\cdots\!37}{10\!\cdots\!40}a^{2}-\frac{15\!\cdots\!19}{61\!\cdots\!64}a-\frac{13\!\cdots\!49}{15\!\cdots\!60}$, $\frac{45\!\cdots\!74}{17\!\cdots\!05}a^{15}-\frac{16\!\cdots\!81}{18\!\cdots\!20}a^{14}-\frac{18\!\cdots\!29}{22\!\cdots\!20}a^{13}-\frac{11\!\cdots\!23}{27\!\cdots\!88}a^{12}+\frac{19\!\cdots\!71}{18\!\cdots\!20}a^{11}+\frac{28\!\cdots\!97}{68\!\cdots\!60}a^{10}-\frac{14\!\cdots\!73}{27\!\cdots\!80}a^{9}-\frac{26\!\cdots\!73}{61\!\cdots\!40}a^{8}+\frac{96\!\cdots\!79}{18\!\cdots\!20}a^{7}+\frac{21\!\cdots\!63}{18\!\cdots\!20}a^{6}+\frac{16\!\cdots\!71}{30\!\cdots\!20}a^{5}+\frac{16\!\cdots\!93}{18\!\cdots\!20}a^{4}-\frac{68\!\cdots\!51}{55\!\cdots\!60}a^{3}-\frac{13\!\cdots\!17}{57\!\cdots\!35}a^{2}-\frac{17\!\cdots\!43}{61\!\cdots\!40}a-\frac{73\!\cdots\!53}{68\!\cdots\!60}$, $\frac{20\!\cdots\!81}{20\!\cdots\!60}a^{15}-\frac{32\!\cdots\!27}{55\!\cdots\!60}a^{14}-\frac{51\!\cdots\!83}{18\!\cdots\!20}a^{13}+\frac{42\!\cdots\!77}{83\!\cdots\!40}a^{12}+\frac{12\!\cdots\!17}{36\!\cdots\!84}a^{11}+\frac{13\!\cdots\!81}{18\!\cdots\!20}a^{10}-\frac{15\!\cdots\!29}{83\!\cdots\!40}a^{9}-\frac{61\!\cdots\!29}{55\!\cdots\!60}a^{8}+\frac{86\!\cdots\!79}{55\!\cdots\!60}a^{7}+\frac{19\!\cdots\!19}{55\!\cdots\!60}a^{6}+\frac{13\!\cdots\!31}{10\!\cdots\!40}a^{5}+\frac{10\!\cdots\!23}{55\!\cdots\!60}a^{4}-\frac{12\!\cdots\!91}{16\!\cdots\!80}a^{3}-\frac{17\!\cdots\!39}{30\!\cdots\!20}a^{2}-\frac{38\!\cdots\!81}{61\!\cdots\!40}a-\frac{13\!\cdots\!67}{61\!\cdots\!40}$, $\frac{16\!\cdots\!57}{55\!\cdots\!60}a^{15}-\frac{78\!\cdots\!71}{18\!\cdots\!20}a^{14}-\frac{96\!\cdots\!31}{10\!\cdots\!40}a^{13}-\frac{23\!\cdots\!39}{11\!\cdots\!52}a^{12}+\frac{21\!\cdots\!93}{18\!\cdots\!20}a^{11}+\frac{87\!\cdots\!03}{12\!\cdots\!30}a^{10}-\frac{57\!\cdots\!97}{11\!\cdots\!52}a^{9}-\frac{36\!\cdots\!61}{61\!\cdots\!40}a^{8}-\frac{12\!\cdots\!87}{18\!\cdots\!20}a^{7}+\frac{33\!\cdots\!63}{23\!\cdots\!40}a^{6}+\frac{50\!\cdots\!39}{61\!\cdots\!40}a^{5}+\frac{32\!\cdots\!61}{18\!\cdots\!20}a^{4}+\frac{15\!\cdots\!19}{27\!\cdots\!80}a^{3}-\frac{73\!\cdots\!13}{18\!\cdots\!20}a^{2}-\frac{38\!\cdots\!79}{61\!\cdots\!40}a-\frac{30\!\cdots\!19}{11\!\cdots\!60}$, $\frac{18\!\cdots\!57}{46\!\cdots\!48}a^{15}-\frac{19\!\cdots\!87}{12\!\cdots\!28}a^{14}-\frac{16\!\cdots\!05}{13\!\cdots\!92}a^{13}+\frac{22\!\cdots\!87}{23\!\cdots\!40}a^{12}+\frac{31\!\cdots\!09}{20\!\cdots\!80}a^{11}+\frac{11\!\cdots\!11}{20\!\cdots\!80}a^{10}-\frac{71\!\cdots\!33}{92\!\cdots\!60}a^{9}-\frac{36\!\cdots\!01}{61\!\cdots\!40}a^{8}+\frac{33\!\cdots\!99}{12\!\cdots\!28}a^{7}+\frac{10\!\cdots\!01}{61\!\cdots\!40}a^{6}+\frac{82\!\cdots\!87}{11\!\cdots\!60}a^{5}+\frac{70\!\cdots\!07}{61\!\cdots\!40}a^{4}-\frac{45\!\cdots\!87}{18\!\cdots\!20}a^{3}-\frac{16\!\cdots\!81}{51\!\cdots\!20}a^{2}-\frac{76\!\cdots\!11}{20\!\cdots\!80}a-\frac{31\!\cdots\!23}{22\!\cdots\!20}$, $\frac{34\!\cdots\!87}{33\!\cdots\!56}a^{15}-\frac{31\!\cdots\!29}{55\!\cdots\!60}a^{14}-\frac{13\!\cdots\!89}{46\!\cdots\!80}a^{13}+\frac{73\!\cdots\!89}{16\!\cdots\!80}a^{12}+\frac{22\!\cdots\!81}{61\!\cdots\!40}a^{11}+\frac{84\!\cdots\!97}{92\!\cdots\!60}a^{10}-\frac{66\!\cdots\!11}{33\!\cdots\!56}a^{9}-\frac{68\!\cdots\!33}{55\!\cdots\!60}a^{8}+\frac{84\!\cdots\!97}{55\!\cdots\!60}a^{7}+\frac{10\!\cdots\!91}{27\!\cdots\!80}a^{6}+\frac{18\!\cdots\!31}{12\!\cdots\!28}a^{5}+\frac{12\!\cdots\!19}{55\!\cdots\!60}a^{4}-\frac{33\!\cdots\!03}{41\!\cdots\!20}a^{3}-\frac{23\!\cdots\!79}{36\!\cdots\!84}a^{2}-\frac{14\!\cdots\!63}{20\!\cdots\!80}a-\frac{39\!\cdots\!47}{15\!\cdots\!60}$, $\frac{99\!\cdots\!87}{83\!\cdots\!40}a^{15}-\frac{19\!\cdots\!11}{27\!\cdots\!80}a^{14}-\frac{31\!\cdots\!67}{92\!\cdots\!60}a^{13}+\frac{24\!\cdots\!43}{41\!\cdots\!20}a^{12}+\frac{19\!\cdots\!13}{46\!\cdots\!80}a^{11}+\frac{90\!\cdots\!33}{92\!\cdots\!60}a^{10}-\frac{18\!\cdots\!79}{83\!\cdots\!40}a^{9}-\frac{38\!\cdots\!61}{27\!\cdots\!80}a^{8}+\frac{24\!\cdots\!07}{13\!\cdots\!40}a^{7}+\frac{12\!\cdots\!91}{27\!\cdots\!80}a^{6}+\frac{51\!\cdots\!27}{30\!\cdots\!20}a^{5}+\frac{67\!\cdots\!19}{27\!\cdots\!80}a^{4}-\frac{75\!\cdots\!49}{83\!\cdots\!40}a^{3}-\frac{33\!\cdots\!91}{46\!\cdots\!80}a^{2}-\frac{31\!\cdots\!57}{38\!\cdots\!90}a-\frac{89\!\cdots\!79}{30\!\cdots\!20}$, $\frac{29\!\cdots\!11}{41\!\cdots\!20}a^{15}-\frac{10\!\cdots\!77}{27\!\cdots\!80}a^{14}-\frac{19\!\cdots\!19}{92\!\cdots\!60}a^{13}+\frac{11\!\cdots\!39}{41\!\cdots\!20}a^{12}+\frac{49\!\cdots\!99}{18\!\cdots\!92}a^{11}+\frac{59\!\cdots\!91}{92\!\cdots\!60}a^{10}-\frac{11\!\cdots\!01}{83\!\cdots\!64}a^{9}-\frac{24\!\cdots\!89}{27\!\cdots\!80}a^{8}+\frac{71\!\cdots\!85}{55\!\cdots\!76}a^{7}+\frac{15\!\cdots\!01}{55\!\cdots\!76}a^{6}+\frac{15\!\cdots\!99}{15\!\cdots\!60}a^{5}+\frac{34\!\cdots\!87}{27\!\cdots\!80}a^{4}-\frac{76\!\cdots\!27}{83\!\cdots\!40}a^{3}-\frac{19\!\cdots\!21}{46\!\cdots\!80}a^{2}-\frac{24\!\cdots\!89}{61\!\cdots\!64}a-\frac{37\!\cdots\!11}{30\!\cdots\!20}$, $\frac{93\!\cdots\!23}{41\!\cdots\!20}a^{15}-\frac{65\!\cdots\!31}{55\!\cdots\!60}a^{14}-\frac{24\!\cdots\!35}{36\!\cdots\!84}a^{13}+\frac{65\!\cdots\!07}{83\!\cdots\!40}a^{12}+\frac{15\!\cdots\!57}{18\!\cdots\!20}a^{11}+\frac{40\!\cdots\!93}{18\!\cdots\!20}a^{10}-\frac{36\!\cdots\!73}{83\!\cdots\!40}a^{9}-\frac{15\!\cdots\!69}{55\!\cdots\!60}a^{8}+\frac{16\!\cdots\!99}{55\!\cdots\!60}a^{7}+\frac{47\!\cdots\!43}{55\!\cdots\!60}a^{6}+\frac{10\!\cdots\!61}{30\!\cdots\!20}a^{5}+\frac{28\!\cdots\!43}{55\!\cdots\!60}a^{4}-\frac{22\!\cdots\!63}{16\!\cdots\!80}a^{3}-\frac{13\!\cdots\!57}{92\!\cdots\!60}a^{2}-\frac{10\!\cdots\!21}{61\!\cdots\!40}a-\frac{41\!\cdots\!79}{61\!\cdots\!40}$, $\frac{14\!\cdots\!91}{41\!\cdots\!32}a^{15}-\frac{61\!\cdots\!49}{55\!\cdots\!76}a^{14}-\frac{19\!\cdots\!77}{18\!\cdots\!92}a^{13}-\frac{54\!\cdots\!99}{83\!\cdots\!40}a^{12}+\frac{12\!\cdots\!01}{92\!\cdots\!60}a^{11}+\frac{64\!\cdots\!63}{11\!\cdots\!70}a^{10}-\frac{69\!\cdots\!43}{10\!\cdots\!83}a^{9}-\frac{15\!\cdots\!77}{27\!\cdots\!80}a^{8}+\frac{65\!\cdots\!31}{27\!\cdots\!80}a^{7}+\frac{53\!\cdots\!99}{34\!\cdots\!10}a^{6}+\frac{54\!\cdots\!07}{77\!\cdots\!58}a^{5}+\frac{66\!\cdots\!67}{55\!\cdots\!76}a^{4}-\frac{11\!\cdots\!79}{83\!\cdots\!40}a^{3}-\frac{29\!\cdots\!41}{92\!\cdots\!60}a^{2}-\frac{24\!\cdots\!13}{61\!\cdots\!64}a-\frac{56\!\cdots\!11}{38\!\cdots\!90}$, $\frac{31\!\cdots\!81}{16\!\cdots\!80}a^{15}-\frac{59\!\cdots\!77}{55\!\cdots\!60}a^{14}-\frac{12\!\cdots\!53}{23\!\cdots\!74}a^{13}+\frac{13\!\cdots\!19}{16\!\cdots\!80}a^{12}+\frac{12\!\cdots\!29}{18\!\cdots\!20}a^{11}+\frac{19\!\cdots\!79}{11\!\cdots\!70}a^{10}-\frac{61\!\cdots\!37}{16\!\cdots\!80}a^{9}-\frac{12\!\cdots\!69}{55\!\cdots\!60}a^{8}+\frac{15\!\cdots\!99}{55\!\cdots\!60}a^{7}+\frac{99\!\cdots\!85}{13\!\cdots\!44}a^{6}+\frac{17\!\cdots\!37}{61\!\cdots\!40}a^{5}+\frac{22\!\cdots\!31}{55\!\cdots\!60}a^{4}-\frac{78\!\cdots\!49}{52\!\cdots\!15}a^{3}-\frac{22\!\cdots\!09}{18\!\cdots\!20}a^{2}-\frac{82\!\cdots\!23}{61\!\cdots\!40}a-\frac{14\!\cdots\!39}{30\!\cdots\!20}$
| 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: | \( 5717295893090000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 5717295893090000 \cdot 1}{2\cdot\sqrt{305159650169696945720591486976000000000000}}\cr\approx \mathstrut & 0.339138464940143 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 324*x^14 - 1208*x^13 + 38634*x^12 + 291024*x^11 - 1435904*x^10 - 22872192*x^9 - 52823469*x^8 + 457836672*x^7 + 3556633824*x^6 + 10251984768*x^5 + 10991206666*x^4 - 10633046448*x^3 - 41850562356*x^2 - 41586585624*x - 13976251839)
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^16 - 324*x^14 - 1208*x^13 + 38634*x^12 + 291024*x^11 - 1435904*x^10 - 22872192*x^9 - 52823469*x^8 + 457836672*x^7 + 3556633824*x^6 + 10251984768*x^5 + 10991206666*x^4 - 10633046448*x^3 - 41850562356*x^2 - 41586585624*x - 13976251839, 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^16 - 324*x^14 - 1208*x^13 + 38634*x^12 + 291024*x^11 - 1435904*x^10 - 22872192*x^9 - 52823469*x^8 + 457836672*x^7 + 3556633824*x^6 + 10251984768*x^5 + 10991206666*x^4 - 10633046448*x^3 - 41850562356*x^2 - 41586585624*x - 13976251839);
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^16 - 324*x^14 - 1208*x^13 + 38634*x^12 + 291024*x^11 - 1435904*x^10 - 22872192*x^9 - 52823469*x^8 + 457836672*x^7 + 3556633824*x^6 + 10251984768*x^5 + 10991206666*x^4 - 10633046448*x^3 - 41850562356*x^2 - 41586585624*x - 13976251839);
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
$(C_2^2:S_4)^2.C_6$ (as 16T1853):
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 55296 |
| The 59 conjugacy class representatives for $(C_2^2:S_4)^2.C_6$ |
| Character table for $(C_2^2:S_4)^2.C_6$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1 |
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
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | 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 | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{5}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | R | R | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $24$ | |||
|
\(3\)
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.12.16.17 | $x^{12} - 24 x^{11} + 306 x^{10} - 2580 x^{9} + 22869 x^{8} - 7236 x^{7} + 99846 x^{6} - 4536 x^{5} + 179658 x^{4} + 3996 x^{3} + 191808 x^{2} + 110889$ | $3$ | $4$ | $16$ | 12T73 | $[2, 2, 2]^{4}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.12.10.2 | $x^{12} - 20 x^{6} - 100$ | $6$ | $2$ | $10$ | $C_6\times S_3$ | $[\ ]_{6}^{6}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.6.3.2 | $x^{6} + 1682 x^{2} - 658503$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 29.6.0.1 | $x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(50461\)
| $\Q_{50461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{50461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |