Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776)
gp: K = bnfinit(y^16 - 5*y^15 - 297*y^14 + 1060*y^13 + 35573*y^12 - 76475*y^11 - 2207799*y^10 + 1841560*y^9 + 75196675*y^8 + 21744135*y^7 - 1351490139*y^6 - 1541936250*y^5 + 10995877008*y^4 + 17851139640*y^3 - 27965699712*y^2 - 36379432800*y + 35867779776, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776)
\( x^{16} - 5 x^{15} - 297 x^{14} + 1060 x^{13} + 35573 x^{12} - 76475 x^{11} - 2207799 x^{10} + \cdots + 35867779776 \)
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: |
\(16225116300501260546649351778570556640625\)
\(\medspace = 5^{14}\cdot 149^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(325.94\) | 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: | $5^{7/8}149^{7/8}\approx 325.939307617564$ | ||
| Ramified primes: |
\(5\), \(149\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{6}-\frac{1}{6}a^{5}-\frac{1}{4}a^{4}-\frac{5}{12}a^{2}+\frac{1}{3}a$, $\frac{1}{12}a^{7}-\frac{1}{12}a^{5}+\frac{1}{12}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{72}a^{8}-\frac{1}{72}a^{7}-\frac{1}{36}a^{6}-\frac{5}{24}a^{5}-\frac{1}{36}a^{4}-\frac{7}{72}a^{3}-\frac{23}{72}a^{2}+\frac{1}{12}a-\frac{1}{2}$, $\frac{1}{144}a^{9}-\frac{1}{144}a^{8}-\frac{1}{72}a^{7}-\frac{1}{48}a^{6}+\frac{5}{72}a^{5}+\frac{29}{144}a^{4}-\frac{23}{144}a^{3}-\frac{1}{8}a^{2}+\frac{1}{12}a$, $\frac{1}{432}a^{10}-\frac{1}{144}a^{8}-\frac{17}{432}a^{7}-\frac{17}{432}a^{6}+\frac{11}{48}a^{5}+\frac{1}{72}a^{4}+\frac{19}{432}a^{3}-\frac{29}{72}a^{2}-\frac{5}{12}a$, $\frac{1}{432}a^{11}-\frac{1}{216}a^{8}-\frac{5}{432}a^{7}-\frac{1}{24}a^{6}+\frac{5}{24}a^{5}+\frac{35}{216}a^{4}+\frac{11}{48}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{2592}a^{12}-\frac{1}{1296}a^{11}-\frac{1}{1296}a^{9}-\frac{1}{2592}a^{8}-\frac{11}{648}a^{7}+\frac{1}{48}a^{6}-\frac{289}{1296}a^{5}+\frac{391}{2592}a^{4}-\frac{17}{144}a^{3}+\frac{1}{12}a^{2}+\frac{1}{12}a-\frac{1}{2}$, $\frac{1}{23328}a^{13}+\frac{1}{5832}a^{12}+\frac{1}{1296}a^{11}+\frac{1}{1458}a^{10}+\frac{77}{23328}a^{9}-\frac{19}{11664}a^{8}-\frac{11}{1296}a^{7}+\frac{395}{11664}a^{6}-\frac{4391}{23328}a^{5}-\frac{1}{432}a^{4}+\frac{29}{432}a^{3}-\frac{5}{18}a^{2}-\frac{2}{9}a-\frac{1}{2}$, $\frac{1}{139968}a^{14}+\frac{1}{139968}a^{13}-\frac{7}{46656}a^{12}+\frac{35}{69984}a^{11}+\frac{29}{139968}a^{10}-\frac{53}{139968}a^{9}-\frac{109}{46656}a^{8}+\frac{17}{69984}a^{7}-\frac{3845}{139968}a^{6}-\frac{3169}{46656}a^{5}+\frac{635}{5184}a^{4}-\frac{41}{216}a^{3}+\frac{23}{432}a^{2}+\frac{4}{9}a-\frac{1}{4}$, $\frac{1}{32\!\cdots\!08}a^{15}-\frac{51\!\cdots\!99}{32\!\cdots\!08}a^{14}-\frac{15\!\cdots\!05}{10\!\cdots\!36}a^{13}+\frac{99\!\cdots\!73}{16\!\cdots\!04}a^{12}+\frac{18\!\cdots\!29}{32\!\cdots\!08}a^{11}+\frac{19\!\cdots\!31}{32\!\cdots\!08}a^{10}-\frac{25\!\cdots\!99}{10\!\cdots\!36}a^{9}-\frac{49\!\cdots\!55}{16\!\cdots\!04}a^{8}-\frac{14\!\cdots\!07}{75\!\cdots\!56}a^{7}-\frac{35\!\cdots\!37}{10\!\cdots\!36}a^{6}-\frac{29\!\cdots\!67}{11\!\cdots\!04}a^{5}+\frac{19\!\cdots\!53}{83\!\cdots\!16}a^{4}+\frac{15\!\cdots\!93}{99\!\cdots\!92}a^{3}-\frac{28\!\cdots\!32}{10\!\cdots\!77}a^{2}+\frac{38\!\cdots\!13}{92\!\cdots\!24}a-\frac{27\!\cdots\!01}{12\!\cdots\!17}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{4}$, which has order $4$ (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: |
$\frac{30\!\cdots\!33}{44\!\cdots\!96}a^{15}-\frac{14\!\cdots\!87}{22\!\cdots\!48}a^{14}-\frac{13\!\cdots\!73}{74\!\cdots\!16}a^{13}+\frac{66\!\cdots\!07}{44\!\cdots\!96}a^{12}+\frac{80\!\cdots\!83}{44\!\cdots\!96}a^{11}-\frac{72\!\cdots\!59}{55\!\cdots\!12}a^{10}-\frac{71\!\cdots\!33}{74\!\cdots\!16}a^{9}+\frac{23\!\cdots\!05}{44\!\cdots\!96}a^{8}+\frac{29\!\cdots\!31}{10\!\cdots\!72}a^{7}-\frac{39\!\cdots\!65}{37\!\cdots\!08}a^{6}-\frac{96\!\cdots\!75}{20\!\cdots\!56}a^{5}+\frac{17\!\cdots\!57}{18\!\cdots\!72}a^{4}+\frac{48\!\cdots\!25}{13\!\cdots\!04}a^{3}-\frac{12\!\cdots\!19}{45\!\cdots\!68}a^{2}-\frac{15\!\cdots\!35}{21\!\cdots\!98}a+\frac{81\!\cdots\!55}{14\!\cdots\!32}$, $\frac{56\!\cdots\!55}{77\!\cdots\!44}a^{15}-\frac{50\!\cdots\!97}{77\!\cdots\!44}a^{14}-\frac{49\!\cdots\!93}{25\!\cdots\!48}a^{13}+\frac{30\!\cdots\!11}{19\!\cdots\!36}a^{12}+\frac{15\!\cdots\!67}{77\!\cdots\!44}a^{11}-\frac{10\!\cdots\!87}{77\!\cdots\!44}a^{10}-\frac{28\!\cdots\!43}{25\!\cdots\!48}a^{9}+\frac{12\!\cdots\!17}{19\!\cdots\!36}a^{8}+\frac{26\!\cdots\!29}{77\!\cdots\!44}a^{7}-\frac{35\!\cdots\!23}{25\!\cdots\!48}a^{6}-\frac{16\!\cdots\!83}{28\!\cdots\!72}a^{5}+\frac{23\!\cdots\!13}{15\!\cdots\!04}a^{4}+\frac{53\!\cdots\!67}{11\!\cdots\!28}a^{3}-\frac{12\!\cdots\!79}{19\!\cdots\!88}a^{2}-\frac{66\!\cdots\!32}{61\!\cdots\!37}a+\frac{13\!\cdots\!53}{12\!\cdots\!74}$, $\frac{55\!\cdots\!29}{77\!\cdots\!44}a^{15}-\frac{38\!\cdots\!67}{19\!\cdots\!36}a^{14}-\frac{29\!\cdots\!05}{12\!\cdots\!24}a^{13}+\frac{25\!\cdots\!93}{77\!\cdots\!44}a^{12}+\frac{22\!\cdots\!23}{77\!\cdots\!44}a^{11}-\frac{20\!\cdots\!97}{38\!\cdots\!72}a^{10}-\frac{24\!\cdots\!47}{12\!\cdots\!24}a^{9}-\frac{13\!\cdots\!05}{77\!\cdots\!44}a^{8}+\frac{51\!\cdots\!57}{77\!\cdots\!44}a^{7}+\frac{16\!\cdots\!07}{12\!\cdots\!24}a^{6}-\frac{94\!\cdots\!17}{79\!\cdots\!52}a^{5}-\frac{10\!\cdots\!29}{31\!\cdots\!08}a^{4}+\frac{97\!\cdots\!95}{11\!\cdots\!28}a^{3}+\frac{23\!\cdots\!37}{79\!\cdots\!52}a^{2}-\frac{71\!\cdots\!25}{22\!\cdots\!32}a-\frac{12\!\cdots\!63}{24\!\cdots\!48}$, $\frac{13\!\cdots\!15}{32\!\cdots\!08}a^{15}-\frac{13\!\cdots\!33}{32\!\cdots\!08}a^{14}-\frac{11\!\cdots\!99}{10\!\cdots\!36}a^{13}+\frac{15\!\cdots\!53}{16\!\cdots\!04}a^{12}+\frac{33\!\cdots\!39}{32\!\cdots\!08}a^{11}-\frac{27\!\cdots\!23}{32\!\cdots\!08}a^{10}-\frac{56\!\cdots\!53}{10\!\cdots\!36}a^{9}+\frac{57\!\cdots\!49}{16\!\cdots\!04}a^{8}+\frac{11\!\cdots\!91}{75\!\cdots\!56}a^{7}-\frac{76\!\cdots\!03}{10\!\cdots\!36}a^{6}-\frac{27\!\cdots\!49}{11\!\cdots\!04}a^{5}+\frac{68\!\cdots\!39}{11\!\cdots\!88}a^{4}+\frac{16\!\cdots\!91}{99\!\cdots\!92}a^{3}-\frac{38\!\cdots\!13}{20\!\cdots\!54}a^{2}-\frac{30\!\cdots\!07}{10\!\cdots\!36}a+\frac{33\!\cdots\!04}{12\!\cdots\!17}$, $\frac{18\!\cdots\!99}{10\!\cdots\!36}a^{15}-\frac{18\!\cdots\!99}{10\!\cdots\!36}a^{14}-\frac{15\!\cdots\!49}{35\!\cdots\!12}a^{13}+\frac{53\!\cdots\!39}{13\!\cdots\!92}a^{12}+\frac{47\!\cdots\!19}{10\!\cdots\!36}a^{11}-\frac{37\!\cdots\!29}{10\!\cdots\!36}a^{10}-\frac{81\!\cdots\!51}{35\!\cdots\!12}a^{9}+\frac{38\!\cdots\!51}{26\!\cdots\!84}a^{8}+\frac{16\!\cdots\!51}{25\!\cdots\!52}a^{7}-\frac{10\!\cdots\!97}{35\!\cdots\!12}a^{6}-\frac{12\!\cdots\!81}{11\!\cdots\!04}a^{5}+\frac{16\!\cdots\!55}{66\!\cdots\!28}a^{4}+\frac{26\!\cdots\!83}{33\!\cdots\!64}a^{3}-\frac{40\!\cdots\!79}{55\!\cdots\!44}a^{2}-\frac{14\!\cdots\!41}{92\!\cdots\!24}a+\frac{65\!\cdots\!45}{51\!\cdots\!68}$, $\frac{84\!\cdots\!89}{14\!\cdots\!88}a^{15}-\frac{78\!\cdots\!81}{17\!\cdots\!56}a^{14}-\frac{27\!\cdots\!79}{17\!\cdots\!56}a^{13}+\frac{60\!\cdots\!61}{59\!\cdots\!52}a^{12}+\frac{15\!\cdots\!93}{89\!\cdots\!28}a^{11}-\frac{16\!\cdots\!65}{17\!\cdots\!56}a^{10}-\frac{17\!\cdots\!37}{17\!\cdots\!56}a^{9}+\frac{22\!\cdots\!11}{59\!\cdots\!52}a^{8}+\frac{66\!\cdots\!37}{20\!\cdots\!96}a^{7}-\frac{13\!\cdots\!87}{17\!\cdots\!56}a^{6}-\frac{33\!\cdots\!89}{59\!\cdots\!52}a^{5}+\frac{13\!\cdots\!21}{22\!\cdots\!76}a^{4}+\frac{24\!\cdots\!51}{55\!\cdots\!44}a^{3}-\frac{10\!\cdots\!75}{55\!\cdots\!44}a^{2}-\frac{47\!\cdots\!07}{46\!\cdots\!12}a+\frac{37\!\cdots\!23}{51\!\cdots\!68}$, $\frac{16\!\cdots\!87}{77\!\cdots\!44}a^{15}-\frac{15\!\cdots\!57}{77\!\cdots\!44}a^{14}-\frac{14\!\cdots\!77}{25\!\cdots\!48}a^{13}+\frac{43\!\cdots\!75}{96\!\cdots\!68}a^{12}+\frac{44\!\cdots\!39}{77\!\cdots\!44}a^{11}-\frac{30\!\cdots\!83}{77\!\cdots\!44}a^{10}-\frac{79\!\cdots\!19}{25\!\cdots\!48}a^{9}+\frac{19\!\cdots\!39}{12\!\cdots\!71}a^{8}+\frac{71\!\cdots\!45}{77\!\cdots\!44}a^{7}-\frac{85\!\cdots\!47}{25\!\cdots\!48}a^{6}-\frac{43\!\cdots\!99}{28\!\cdots\!72}a^{5}+\frac{45\!\cdots\!49}{15\!\cdots\!04}a^{4}+\frac{13\!\cdots\!71}{11\!\cdots\!28}a^{3}-\frac{17\!\cdots\!33}{19\!\cdots\!88}a^{2}-\frac{44\!\cdots\!25}{18\!\cdots\!11}a+\frac{23\!\cdots\!37}{12\!\cdots\!74}$, $\frac{49\!\cdots\!72}{25\!\cdots\!11}a^{15}-\frac{13\!\cdots\!09}{80\!\cdots\!52}a^{14}-\frac{34\!\cdots\!89}{67\!\cdots\!96}a^{13}+\frac{32\!\cdots\!37}{80\!\cdots\!52}a^{12}+\frac{22\!\cdots\!99}{40\!\cdots\!76}a^{11}-\frac{28\!\cdots\!97}{80\!\cdots\!52}a^{10}-\frac{40\!\cdots\!63}{13\!\cdots\!92}a^{9}+\frac{11\!\cdots\!57}{80\!\cdots\!52}a^{8}+\frac{86\!\cdots\!81}{93\!\cdots\!32}a^{7}-\frac{78\!\cdots\!35}{26\!\cdots\!84}a^{6}-\frac{11\!\cdots\!75}{74\!\cdots\!44}a^{5}+\frac{85\!\cdots\!39}{33\!\cdots\!64}a^{4}+\frac{29\!\cdots\!25}{24\!\cdots\!48}a^{3}-\frac{65\!\cdots\!15}{83\!\cdots\!16}a^{2}-\frac{32\!\cdots\!89}{12\!\cdots\!17}a+\frac{49\!\cdots\!95}{25\!\cdots\!34}$, $\frac{15\!\cdots\!23}{32\!\cdots\!08}a^{15}-\frac{14\!\cdots\!19}{32\!\cdots\!08}a^{14}-\frac{13\!\cdots\!85}{10\!\cdots\!36}a^{13}+\frac{85\!\cdots\!81}{80\!\cdots\!52}a^{12}+\frac{41\!\cdots\!15}{32\!\cdots\!08}a^{11}-\frac{29\!\cdots\!17}{32\!\cdots\!08}a^{10}-\frac{72\!\cdots\!79}{10\!\cdots\!36}a^{9}+\frac{77\!\cdots\!89}{20\!\cdots\!88}a^{8}+\frac{14\!\cdots\!95}{75\!\cdots\!56}a^{7}-\frac{82\!\cdots\!85}{10\!\cdots\!36}a^{6}-\frac{38\!\cdots\!47}{11\!\cdots\!04}a^{5}+\frac{44\!\cdots\!35}{66\!\cdots\!28}a^{4}+\frac{24\!\cdots\!57}{99\!\cdots\!92}a^{3}-\frac{33\!\cdots\!35}{16\!\cdots\!32}a^{2}-\frac{15\!\cdots\!51}{30\!\cdots\!08}a+\frac{20\!\cdots\!69}{51\!\cdots\!68}$, $\frac{18\!\cdots\!75}{52\!\cdots\!48}a^{15}-\frac{52\!\cdots\!07}{15\!\cdots\!44}a^{14}-\frac{14\!\cdots\!43}{15\!\cdots\!44}a^{13}+\frac{20\!\cdots\!31}{26\!\cdots\!24}a^{12}+\frac{14\!\cdots\!81}{15\!\cdots\!44}a^{11}-\frac{10\!\cdots\!21}{15\!\cdots\!44}a^{10}-\frac{78\!\cdots\!09}{15\!\cdots\!44}a^{9}+\frac{73\!\cdots\!51}{26\!\cdots\!24}a^{8}+\frac{54\!\cdots\!97}{36\!\cdots\!08}a^{7}-\frac{88\!\cdots\!23}{15\!\cdots\!44}a^{6}-\frac{42\!\cdots\!01}{17\!\cdots\!16}a^{5}+\frac{11\!\cdots\!25}{24\!\cdots\!28}a^{4}+\frac{29\!\cdots\!29}{16\!\cdots\!52}a^{3}-\frac{21\!\cdots\!68}{15\!\cdots\!33}a^{2}-\frac{49\!\cdots\!45}{13\!\cdots\!96}a+\frac{32\!\cdots\!23}{11\!\cdots\!58}$, $\frac{34\!\cdots\!51}{32\!\cdots\!08}a^{15}-\frac{31\!\cdots\!71}{32\!\cdots\!08}a^{14}-\frac{29\!\cdots\!89}{10\!\cdots\!36}a^{13}+\frac{23\!\cdots\!99}{10\!\cdots\!44}a^{12}+\frac{89\!\cdots\!51}{32\!\cdots\!08}a^{11}-\frac{64\!\cdots\!57}{32\!\cdots\!08}a^{10}-\frac{15\!\cdots\!03}{10\!\cdots\!36}a^{9}+\frac{66\!\cdots\!67}{80\!\cdots\!52}a^{8}+\frac{33\!\cdots\!55}{75\!\cdots\!56}a^{7}-\frac{17\!\cdots\!85}{10\!\cdots\!36}a^{6}-\frac{86\!\cdots\!31}{11\!\cdots\!04}a^{5}+\frac{96\!\cdots\!61}{66\!\cdots\!28}a^{4}+\frac{54\!\cdots\!69}{99\!\cdots\!92}a^{3}-\frac{71\!\cdots\!33}{16\!\cdots\!32}a^{2}-\frac{10\!\cdots\!89}{92\!\cdots\!24}a+\frac{45\!\cdots\!39}{51\!\cdots\!68}$, $\frac{78\!\cdots\!07}{40\!\cdots\!76}a^{15}-\frac{15\!\cdots\!13}{80\!\cdots\!52}a^{14}-\frac{65\!\cdots\!41}{13\!\cdots\!92}a^{13}+\frac{22\!\cdots\!01}{50\!\cdots\!22}a^{12}+\frac{97\!\cdots\!81}{20\!\cdots\!88}a^{11}-\frac{31\!\cdots\!41}{80\!\cdots\!52}a^{10}-\frac{33\!\cdots\!79}{13\!\cdots\!92}a^{9}+\frac{65\!\cdots\!93}{40\!\cdots\!76}a^{8}+\frac{66\!\cdots\!03}{93\!\cdots\!32}a^{7}-\frac{88\!\cdots\!57}{26\!\cdots\!84}a^{6}-\frac{10\!\cdots\!09}{93\!\cdots\!93}a^{5}+\frac{49\!\cdots\!41}{16\!\cdots\!32}a^{4}+\frac{10\!\cdots\!21}{12\!\cdots\!24}a^{3}-\frac{39\!\cdots\!13}{41\!\cdots\!08}a^{2}-\frac{12\!\cdots\!29}{77\!\cdots\!02}a+\frac{20\!\cdots\!81}{12\!\cdots\!17}$, $\frac{23\!\cdots\!63}{19\!\cdots\!68}a^{15}-\frac{81\!\cdots\!55}{78\!\cdots\!72}a^{14}-\frac{80\!\cdots\!19}{26\!\cdots\!24}a^{13}+\frac{18\!\cdots\!03}{78\!\cdots\!72}a^{12}+\frac{12\!\cdots\!97}{39\!\cdots\!36}a^{11}-\frac{16\!\cdots\!31}{78\!\cdots\!72}a^{10}-\frac{43\!\cdots\!37}{26\!\cdots\!24}a^{9}+\frac{65\!\cdots\!73}{78\!\cdots\!72}a^{8}+\frac{44\!\cdots\!29}{90\!\cdots\!52}a^{7}-\frac{42\!\cdots\!23}{26\!\cdots\!24}a^{6}-\frac{20\!\cdots\!27}{26\!\cdots\!24}a^{5}+\frac{13\!\cdots\!13}{96\!\cdots\!12}a^{4}+\frac{22\!\cdots\!19}{40\!\cdots\!88}a^{3}-\frac{33\!\cdots\!05}{80\!\cdots\!76}a^{2}-\frac{22\!\cdots\!27}{20\!\cdots\!44}a+\frac{20\!\cdots\!53}{22\!\cdots\!16}$, $\frac{10\!\cdots\!59}{80\!\cdots\!52}a^{15}-\frac{13\!\cdots\!97}{10\!\cdots\!44}a^{14}-\frac{84\!\cdots\!19}{26\!\cdots\!84}a^{13}+\frac{64\!\cdots\!17}{20\!\cdots\!88}a^{12}+\frac{24\!\cdots\!79}{80\!\cdots\!52}a^{11}-\frac{11\!\cdots\!19}{40\!\cdots\!76}a^{10}-\frac{39\!\cdots\!73}{26\!\cdots\!84}a^{9}+\frac{46\!\cdots\!41}{40\!\cdots\!76}a^{8}+\frac{75\!\cdots\!93}{18\!\cdots\!64}a^{7}-\frac{30\!\cdots\!69}{13\!\cdots\!92}a^{6}-\frac{61\!\cdots\!37}{99\!\cdots\!92}a^{5}+\frac{16\!\cdots\!37}{83\!\cdots\!16}a^{4}+\frac{21\!\cdots\!63}{49\!\cdots\!96}a^{3}-\frac{47\!\cdots\!77}{83\!\cdots\!16}a^{2}-\frac{32\!\cdots\!41}{46\!\cdots\!12}a+\frac{88\!\cdots\!07}{12\!\cdots\!17}$, $\frac{13\!\cdots\!79}{21\!\cdots\!56}a^{15}-\frac{10\!\cdots\!25}{21\!\cdots\!56}a^{14}-\frac{12\!\cdots\!55}{73\!\cdots\!52}a^{13}+\frac{12\!\cdots\!05}{10\!\cdots\!28}a^{12}+\frac{40\!\cdots\!15}{21\!\cdots\!56}a^{11}-\frac{21\!\cdots\!35}{21\!\cdots\!56}a^{10}-\frac{77\!\cdots\!45}{73\!\cdots\!52}a^{9}+\frac{45\!\cdots\!45}{10\!\cdots\!28}a^{8}+\frac{17\!\cdots\!95}{51\!\cdots\!92}a^{7}-\frac{61\!\cdots\!95}{73\!\cdots\!52}a^{6}-\frac{47\!\cdots\!69}{81\!\cdots\!28}a^{5}+\frac{17\!\cdots\!65}{22\!\cdots\!48}a^{4}+\frac{31\!\cdots\!85}{67\!\cdots\!44}a^{3}-\frac{15\!\cdots\!05}{56\!\cdots\!12}a^{2}-\frac{22\!\cdots\!35}{20\!\cdots\!56}a+\frac{14\!\cdots\!05}{17\!\cdots\!13}$
| 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: | \( 5709703146950000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 5709703146950000 \cdot 4}{2\cdot\sqrt{16225116300501260546649351778570556640625}}\cr\approx \mathstrut & 5.87529322729923 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776)
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 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776, 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 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776);
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 - 5*x^15 - 297*x^14 + 1060*x^13 + 35573*x^12 - 76475*x^11 - 2207799*x^10 + 1841560*x^9 + 75196675*x^8 + 21744135*x^7 - 1351490139*x^6 - 1541936250*x^5 + 10995877008*x^4 + 17851139640*x^3 - 27965699712*x^2 - 36379432800*x + 35867779776);
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
$\OD_{16}:C_2$ (as 16T36):
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 32 |
| The 11 conjugacy class representatives for $\OD_{16}:C_2$ |
| Character table for $\OD_{16}:C_2$ |
Intermediate fields
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(149\)
| 149.8.7.1 | $x^{8} + 596$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 149.8.7.1 | $x^{8} + 596$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |