Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888)
gp: K = bnfinit(y^16 - 5*y^15 - 826*y^14 + 3919*y^13 + 163389*y^12 - 1327426*y^11 - 5307893*y^10 + 66746609*y^9 + 35529210*y^8 - 1329168051*y^7 + 432751103*y^6 + 12242227394*y^5 - 4542410360*y^4 - 48328678280*y^3 + 7339560560*y^2 + 60542675904*y + 12486085888, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888)
\( x^{16} - 5 x^{15} - 826 x^{14} + 3919 x^{13} + 163389 x^{12} - 1327426 x^{11} - 5307893 x^{10} + \cdots + 12486085888 \)
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: |
\(21767174858780577201904716993499183222663790161\)
\(\medspace = 13^{14}\cdot 157^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(787.25\) | 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: | $13^{7/8}157^{7/8}\approx 787.2499271616497$ | ||
| Ramified primes: |
\(13\), \(157\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{56}a^{12}+\frac{1}{28}a^{11}-\frac{1}{14}a^{10}-\frac{1}{56}a^{9}-\frac{1}{28}a^{8}-\frac{1}{14}a^{7}-\frac{13}{56}a^{6}-\frac{1}{4}a^{5}+\frac{1}{14}a^{4}-\frac{3}{56}a^{3}-\frac{1}{28}a^{2}-\frac{1}{2}a+\frac{3}{7}$, $\frac{1}{1120}a^{13}+\frac{1}{560}a^{12}+\frac{1}{112}a^{11}-\frac{71}{1120}a^{10}-\frac{1}{70}a^{9}-\frac{3}{56}a^{8}-\frac{41}{1120}a^{7}+\frac{19}{80}a^{6}-\frac{15}{112}a^{5}-\frac{37}{224}a^{4}-\frac{39}{280}a^{3}-\frac{3}{20}a^{2}-\frac{11}{140}a-\frac{2}{5}$, $\frac{1}{15680}a^{14}-\frac{1}{7840}a^{13}-\frac{19}{7840}a^{12}+\frac{649}{15680}a^{11}-\frac{313}{3920}a^{10}-\frac{129}{3920}a^{9}+\frac{1399}{15680}a^{8}+\frac{59}{1568}a^{7}-\frac{1187}{7840}a^{6}+\frac{475}{3136}a^{5}+\frac{123}{1960}a^{4}-\frac{61}{245}a^{3}+\frac{83}{1960}a^{2}+\frac{207}{490}a-\frac{22}{245}$, $\frac{1}{66\!\cdots\!60}a^{15}-\frac{55\!\cdots\!95}{19\!\cdots\!16}a^{14}-\frac{17\!\cdots\!67}{47\!\cdots\!40}a^{13}+\frac{57\!\cdots\!71}{66\!\cdots\!60}a^{12}-\frac{33\!\cdots\!19}{66\!\cdots\!60}a^{11}-\frac{65\!\cdots\!17}{33\!\cdots\!80}a^{10}-\frac{56\!\cdots\!97}{66\!\cdots\!60}a^{9}+\frac{51\!\cdots\!17}{13\!\cdots\!40}a^{8}+\frac{39\!\cdots\!21}{33\!\cdots\!80}a^{7}+\frac{20\!\cdots\!03}{95\!\cdots\!80}a^{6}-\frac{49\!\cdots\!81}{66\!\cdots\!60}a^{5}-\frac{53\!\cdots\!83}{33\!\cdots\!80}a^{4}+\frac{80\!\cdots\!77}{41\!\cdots\!60}a^{3}+\frac{21\!\cdots\!91}{23\!\cdots\!52}a^{2}+\frac{19\!\cdots\!53}{41\!\cdots\!60}a+\frac{17\!\cdots\!98}{10\!\cdots\!65}$
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
$C_{4}$, 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: |
$\frac{84\!\cdots\!01}{12\!\cdots\!80}a^{15}+\frac{10\!\cdots\!21}{64\!\cdots\!40}a^{14}-\frac{17\!\cdots\!13}{32\!\cdots\!20}a^{13}-\frac{36\!\cdots\!87}{25\!\cdots\!76}a^{12}+\frac{34\!\cdots\!21}{32\!\cdots\!20}a^{11}-\frac{62\!\cdots\!89}{64\!\cdots\!40}a^{10}-\frac{76\!\cdots\!49}{12\!\cdots\!80}a^{9}+\frac{47\!\cdots\!53}{64\!\cdots\!40}a^{8}+\frac{23\!\cdots\!89}{16\!\cdots\!60}a^{7}-\frac{15\!\cdots\!77}{12\!\cdots\!80}a^{6}-\frac{28\!\cdots\!57}{16\!\cdots\!60}a^{5}+\frac{17\!\cdots\!11}{64\!\cdots\!40}a^{4}+\frac{34\!\cdots\!37}{40\!\cdots\!90}a^{3}+\frac{13\!\cdots\!43}{80\!\cdots\!80}a^{2}-\frac{96\!\cdots\!89}{80\!\cdots\!80}a-\frac{11\!\cdots\!11}{20\!\cdots\!95}$, $\frac{25\!\cdots\!83}{10\!\cdots\!92}a^{15}-\frac{57\!\cdots\!21}{74\!\cdots\!80}a^{14}-\frac{59\!\cdots\!27}{74\!\cdots\!80}a^{13}+\frac{64\!\cdots\!47}{10\!\cdots\!92}a^{12}-\frac{24\!\cdots\!83}{51\!\cdots\!60}a^{11}-\frac{61\!\cdots\!53}{51\!\cdots\!60}a^{10}+\frac{61\!\cdots\!77}{10\!\cdots\!92}a^{9}+\frac{29\!\cdots\!41}{74\!\cdots\!80}a^{8}-\frac{13\!\cdots\!67}{51\!\cdots\!60}a^{7}-\frac{64\!\cdots\!41}{14\!\cdots\!56}a^{6}+\frac{38\!\cdots\!35}{10\!\cdots\!92}a^{5}+\frac{75\!\cdots\!37}{51\!\cdots\!60}a^{4}-\frac{11\!\cdots\!53}{64\!\cdots\!20}a^{3}-\frac{11\!\cdots\!39}{92\!\cdots\!60}a^{2}+\frac{15\!\cdots\!07}{64\!\cdots\!20}a+\frac{16\!\cdots\!78}{32\!\cdots\!31}$, $\frac{56\!\cdots\!87}{10\!\cdots\!20}a^{15}+\frac{26\!\cdots\!59}{74\!\cdots\!28}a^{14}-\frac{66\!\cdots\!63}{14\!\cdots\!56}a^{13}-\frac{40\!\cdots\!81}{10\!\cdots\!20}a^{12}+\frac{44\!\cdots\!71}{51\!\cdots\!60}a^{11}-\frac{30\!\cdots\!23}{12\!\cdots\!40}a^{10}-\frac{84\!\cdots\!83}{20\!\cdots\!84}a^{9}+\frac{11\!\cdots\!43}{92\!\cdots\!60}a^{8}+\frac{43\!\cdots\!19}{51\!\cdots\!60}a^{7}-\frac{32\!\cdots\!01}{14\!\cdots\!60}a^{6}-\frac{45\!\cdots\!81}{51\!\cdots\!60}a^{5}+\frac{36\!\cdots\!97}{25\!\cdots\!80}a^{4}+\frac{50\!\cdots\!81}{12\!\cdots\!40}a^{3}-\frac{57\!\cdots\!41}{18\!\cdots\!40}a^{2}-\frac{22\!\cdots\!51}{32\!\cdots\!10}a-\frac{19\!\cdots\!71}{16\!\cdots\!55}$, $\frac{75\!\cdots\!53}{36\!\cdots\!60}a^{15}+\frac{43\!\cdots\!33}{18\!\cdots\!80}a^{14}-\frac{38\!\cdots\!47}{22\!\cdots\!60}a^{13}-\frac{85\!\cdots\!67}{36\!\cdots\!60}a^{12}+\frac{29\!\cdots\!53}{90\!\cdots\!40}a^{11}-\frac{13\!\cdots\!39}{18\!\cdots\!80}a^{10}-\frac{59\!\cdots\!01}{36\!\cdots\!60}a^{9}+\frac{74\!\cdots\!09}{18\!\cdots\!80}a^{8}+\frac{33\!\cdots\!73}{90\!\cdots\!40}a^{7}-\frac{24\!\cdots\!77}{36\!\cdots\!60}a^{6}-\frac{18\!\cdots\!21}{45\!\cdots\!20}a^{5}+\frac{65\!\cdots\!93}{18\!\cdots\!80}a^{4}+\frac{86\!\cdots\!91}{45\!\cdots\!20}a^{3}-\frac{41\!\cdots\!79}{11\!\cdots\!30}a^{2}-\frac{66\!\cdots\!49}{22\!\cdots\!60}a-\frac{32\!\cdots\!81}{56\!\cdots\!65}$, $\frac{17\!\cdots\!21}{66\!\cdots\!56}a^{15}-\frac{93\!\cdots\!43}{95\!\cdots\!80}a^{14}-\frac{21\!\cdots\!39}{95\!\cdots\!08}a^{13}+\frac{31\!\cdots\!59}{41\!\cdots\!60}a^{12}+\frac{29\!\cdots\!11}{66\!\cdots\!60}a^{11}-\frac{50\!\cdots\!29}{16\!\cdots\!40}a^{10}-\frac{52\!\cdots\!31}{33\!\cdots\!80}a^{9}+\frac{14\!\cdots\!03}{95\!\cdots\!80}a^{8}+\frac{58\!\cdots\!91}{33\!\cdots\!80}a^{7}-\frac{72\!\cdots\!11}{23\!\cdots\!20}a^{6}+\frac{48\!\cdots\!13}{13\!\cdots\!12}a^{5}+\frac{22\!\cdots\!57}{83\!\cdots\!20}a^{4}-\frac{71\!\cdots\!23}{83\!\cdots\!20}a^{3}-\frac{12\!\cdots\!83}{11\!\cdots\!60}a^{2}+\frac{30\!\cdots\!23}{10\!\cdots\!65}a+\frac{12\!\cdots\!11}{10\!\cdots\!65}$, $\frac{25\!\cdots\!01}{83\!\cdots\!20}a^{15}-\frac{74\!\cdots\!11}{95\!\cdots\!80}a^{14}-\frac{38\!\cdots\!24}{14\!\cdots\!95}a^{13}+\frac{38\!\cdots\!53}{66\!\cdots\!56}a^{12}+\frac{69\!\cdots\!19}{13\!\cdots\!12}a^{11}-\frac{94\!\cdots\!63}{33\!\cdots\!80}a^{10}-\frac{39\!\cdots\!13}{16\!\cdots\!40}a^{9}+\frac{14\!\cdots\!83}{95\!\cdots\!80}a^{8}+\frac{80\!\cdots\!57}{16\!\cdots\!40}a^{7}-\frac{14\!\cdots\!49}{47\!\cdots\!40}a^{6}-\frac{40\!\cdots\!83}{66\!\cdots\!60}a^{5}+\frac{78\!\cdots\!37}{33\!\cdots\!80}a^{4}+\frac{37\!\cdots\!17}{83\!\cdots\!20}a^{3}-\frac{50\!\cdots\!27}{11\!\cdots\!60}a^{2}-\frac{35\!\cdots\!73}{41\!\cdots\!60}a-\frac{17\!\cdots\!58}{10\!\cdots\!65}$, $\frac{65\!\cdots\!07}{66\!\cdots\!60}a^{15}+\frac{14\!\cdots\!29}{59\!\cdots\!80}a^{14}-\frac{76\!\cdots\!15}{95\!\cdots\!08}a^{13}-\frac{14\!\cdots\!73}{66\!\cdots\!60}a^{12}+\frac{51\!\cdots\!77}{33\!\cdots\!80}a^{11}-\frac{61\!\cdots\!61}{41\!\cdots\!60}a^{10}-\frac{54\!\cdots\!39}{66\!\cdots\!60}a^{9}+\frac{24\!\cdots\!31}{23\!\cdots\!20}a^{8}+\frac{62\!\cdots\!31}{33\!\cdots\!80}a^{7}-\frac{11\!\cdots\!29}{95\!\cdots\!80}a^{6}-\frac{66\!\cdots\!71}{33\!\cdots\!80}a^{5}-\frac{17\!\cdots\!89}{33\!\cdots\!28}a^{4}+\frac{28\!\cdots\!37}{41\!\cdots\!66}a^{3}+\frac{33\!\cdots\!93}{59\!\cdots\!80}a^{2}-\frac{13\!\cdots\!57}{10\!\cdots\!65}a-\frac{61\!\cdots\!79}{10\!\cdots\!65}$, $\frac{13\!\cdots\!79}{33\!\cdots\!80}a^{15}-\frac{61\!\cdots\!47}{47\!\cdots\!40}a^{14}-\frac{15\!\cdots\!39}{47\!\cdots\!40}a^{13}+\frac{32\!\cdots\!23}{33\!\cdots\!80}a^{12}+\frac{21\!\cdots\!03}{33\!\cdots\!80}a^{11}-\frac{13\!\cdots\!89}{33\!\cdots\!80}a^{10}-\frac{18\!\cdots\!79}{66\!\cdots\!56}a^{9}+\frac{10\!\cdots\!63}{47\!\cdots\!40}a^{8}+\frac{16\!\cdots\!37}{33\!\cdots\!80}a^{7}-\frac{20\!\cdots\!17}{47\!\cdots\!40}a^{6}-\frac{18\!\cdots\!19}{33\!\cdots\!80}a^{5}+\frac{25\!\cdots\!01}{66\!\cdots\!56}a^{4}+\frac{39\!\cdots\!47}{83\!\cdots\!20}a^{3}-\frac{32\!\cdots\!57}{29\!\cdots\!90}a^{2}-\frac{65\!\cdots\!49}{41\!\cdots\!60}a-\frac{30\!\cdots\!94}{10\!\cdots\!65}$, $\frac{10\!\cdots\!27}{83\!\cdots\!20}a^{15}-\frac{32\!\cdots\!21}{47\!\cdots\!40}a^{14}-\frac{10\!\cdots\!27}{95\!\cdots\!08}a^{13}+\frac{44\!\cdots\!31}{83\!\cdots\!20}a^{12}+\frac{14\!\cdots\!79}{66\!\cdots\!56}a^{11}-\frac{59\!\cdots\!43}{33\!\cdots\!80}a^{10}-\frac{13\!\cdots\!39}{20\!\cdots\!30}a^{9}+\frac{42\!\cdots\!13}{47\!\cdots\!40}a^{8}+\frac{88\!\cdots\!81}{33\!\cdots\!80}a^{7}-\frac{20\!\cdots\!33}{11\!\cdots\!60}a^{6}+\frac{32\!\cdots\!97}{33\!\cdots\!80}a^{5}+\frac{52\!\cdots\!59}{33\!\cdots\!80}a^{4}-\frac{79\!\cdots\!59}{83\!\cdots\!20}a^{3}-\frac{18\!\cdots\!19}{29\!\cdots\!90}a^{2}+\frac{96\!\cdots\!67}{41\!\cdots\!60}a+\frac{77\!\cdots\!31}{10\!\cdots\!65}$, $\frac{32\!\cdots\!71}{58\!\cdots\!60}a^{15}-\frac{21\!\cdots\!09}{12\!\cdots\!40}a^{14}-\frac{15\!\cdots\!79}{16\!\cdots\!96}a^{13}+\frac{87\!\cdots\!11}{58\!\cdots\!60}a^{12}+\frac{30\!\cdots\!87}{58\!\cdots\!60}a^{11}-\frac{16\!\cdots\!73}{58\!\cdots\!60}a^{10}+\frac{92\!\cdots\!13}{58\!\cdots\!60}a^{9}+\frac{12\!\cdots\!87}{84\!\cdots\!80}a^{8}-\frac{67\!\cdots\!79}{58\!\cdots\!60}a^{7}-\frac{30\!\cdots\!37}{84\!\cdots\!80}a^{6}+\frac{10\!\cdots\!29}{58\!\cdots\!60}a^{5}+\frac{46\!\cdots\!41}{11\!\cdots\!72}a^{4}+\frac{12\!\cdots\!54}{36\!\cdots\!71}a^{3}-\frac{13\!\cdots\!67}{10\!\cdots\!60}a^{2}-\frac{13\!\cdots\!13}{73\!\cdots\!20}a-\frac{53\!\cdots\!89}{18\!\cdots\!55}$, $\frac{58\!\cdots\!59}{47\!\cdots\!04}a^{15}+\frac{76\!\cdots\!79}{95\!\cdots\!80}a^{14}-\frac{24\!\cdots\!67}{23\!\cdots\!20}a^{13}-\frac{60\!\cdots\!63}{68\!\cdots\!20}a^{12}+\frac{26\!\cdots\!33}{13\!\cdots\!40}a^{11}-\frac{25\!\cdots\!89}{47\!\cdots\!40}a^{10}-\frac{16\!\cdots\!19}{17\!\cdots\!80}a^{9}+\frac{27\!\cdots\!81}{95\!\cdots\!80}a^{8}+\frac{25\!\cdots\!25}{12\!\cdots\!62}a^{7}-\frac{23\!\cdots\!13}{47\!\cdots\!40}a^{6}-\frac{42\!\cdots\!91}{19\!\cdots\!16}a^{5}+\frac{13\!\cdots\!03}{47\!\cdots\!40}a^{4}+\frac{17\!\cdots\!69}{17\!\cdots\!80}a^{3}-\frac{50\!\cdots\!73}{11\!\cdots\!60}a^{2}-\frac{12\!\cdots\!57}{85\!\cdots\!40}a-\frac{41\!\cdots\!18}{14\!\cdots\!95}$, $\frac{93\!\cdots\!09}{15\!\cdots\!80}a^{15}-\frac{30\!\cdots\!19}{55\!\cdots\!60}a^{14}-\frac{78\!\cdots\!61}{15\!\cdots\!96}a^{13}+\frac{69\!\cdots\!81}{15\!\cdots\!80}a^{12}+\frac{73\!\cdots\!63}{77\!\cdots\!40}a^{11}-\frac{18\!\cdots\!63}{15\!\cdots\!08}a^{10}-\frac{16\!\cdots\!69}{15\!\cdots\!80}a^{9}+\frac{15\!\cdots\!59}{27\!\cdots\!80}a^{8}-\frac{79\!\cdots\!55}{97\!\cdots\!38}a^{7}-\frac{63\!\cdots\!85}{63\!\cdots\!84}a^{6}+\frac{15\!\cdots\!83}{77\!\cdots\!40}a^{5}+\frac{62\!\cdots\!69}{77\!\cdots\!40}a^{4}-\frac{29\!\cdots\!59}{19\!\cdots\!60}a^{3}-\frac{34\!\cdots\!01}{13\!\cdots\!40}a^{2}+\frac{70\!\cdots\!51}{19\!\cdots\!76}a+\frac{61\!\cdots\!08}{48\!\cdots\!19}$, $\frac{40\!\cdots\!97}{91\!\cdots\!20}a^{15}+\frac{13\!\cdots\!89}{13\!\cdots\!60}a^{14}-\frac{34\!\cdots\!59}{93\!\cdots\!40}a^{13}-\frac{15\!\cdots\!17}{91\!\cdots\!20}a^{12}+\frac{65\!\cdots\!69}{91\!\cdots\!20}a^{11}-\frac{99\!\cdots\!07}{45\!\cdots\!60}a^{10}-\frac{31\!\cdots\!37}{91\!\cdots\!20}a^{9}+\frac{15\!\cdots\!19}{13\!\cdots\!60}a^{8}+\frac{33\!\cdots\!27}{45\!\cdots\!60}a^{7}-\frac{10\!\cdots\!81}{53\!\cdots\!08}a^{6}-\frac{73\!\cdots\!37}{91\!\cdots\!20}a^{5}+\frac{11\!\cdots\!19}{91\!\cdots\!72}a^{4}+\frac{44\!\cdots\!89}{11\!\cdots\!40}a^{3}-\frac{32\!\cdots\!43}{16\!\cdots\!20}a^{2}-\frac{32\!\cdots\!57}{57\!\cdots\!20}a-\frac{32\!\cdots\!31}{28\!\cdots\!21}$, $\frac{92\!\cdots\!63}{33\!\cdots\!80}a^{15}+\frac{30\!\cdots\!59}{11\!\cdots\!60}a^{14}-\frac{10\!\cdots\!53}{47\!\cdots\!40}a^{13}-\frac{87\!\cdots\!79}{33\!\cdots\!80}a^{12}+\frac{18\!\cdots\!67}{41\!\cdots\!60}a^{11}-\frac{36\!\cdots\!33}{33\!\cdots\!80}a^{10}-\frac{14\!\cdots\!31}{66\!\cdots\!56}a^{9}+\frac{71\!\cdots\!29}{11\!\cdots\!60}a^{8}+\frac{14\!\cdots\!59}{33\!\cdots\!80}a^{7}-\frac{48\!\cdots\!39}{47\!\cdots\!40}a^{6}-\frac{48\!\cdots\!79}{10\!\cdots\!65}a^{5}+\frac{38\!\cdots\!89}{66\!\cdots\!56}a^{4}+\frac{86\!\cdots\!67}{41\!\cdots\!60}a^{3}-\frac{54\!\cdots\!83}{59\!\cdots\!80}a^{2}-\frac{11\!\cdots\!73}{41\!\cdots\!60}a-\frac{47\!\cdots\!23}{10\!\cdots\!65}$, $\frac{11\!\cdots\!37}{14\!\cdots\!80}a^{15}+\frac{12\!\cdots\!49}{14\!\cdots\!60}a^{14}-\frac{64\!\cdots\!53}{10\!\cdots\!32}a^{13}-\frac{12\!\cdots\!59}{14\!\cdots\!80}a^{12}+\frac{88\!\cdots\!83}{71\!\cdots\!24}a^{11}-\frac{19\!\cdots\!01}{71\!\cdots\!40}a^{10}-\frac{88\!\cdots\!21}{14\!\cdots\!80}a^{9}+\frac{15\!\cdots\!31}{10\!\cdots\!20}a^{8}+\frac{12\!\cdots\!89}{89\!\cdots\!30}a^{7}-\frac{48\!\cdots\!23}{20\!\cdots\!40}a^{6}-\frac{70\!\cdots\!96}{44\!\cdots\!15}a^{5}+\frac{76\!\cdots\!63}{71\!\cdots\!40}a^{4}+\frac{31\!\cdots\!58}{44\!\cdots\!15}a^{3}-\frac{58\!\cdots\!31}{12\!\cdots\!90}a^{2}-\frac{87\!\cdots\!81}{89\!\cdots\!30}a-\frac{92\!\cdots\!01}{44\!\cdots\!15}$
| 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: | \( 2472425742860000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 2472425742860000000 \cdot 4}{2\cdot\sqrt{21767174858780577201904716993499183222663790161}}\cr\approx \mathstrut & 2.19650476674310 \end{aligned}\] (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 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888)
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 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888, 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 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888);
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 - 826*x^14 + 3919*x^13 + 163389*x^12 - 1327426*x^11 - 5307893*x^10 + 66746609*x^9 + 35529210*x^8 - 1329168051*x^7 + 432751103*x^6 + 12242227394*x^5 - 4542410360*x^4 - 48328678280*x^3 + 7339560560*x^2 + 60542675904*x + 12486085888);
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 | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.8.7.1 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 13.8.7.1 | $x^{8} + 52$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(157\)
| 157.8.7.2 | $x^{8} + 157$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 157.8.7.2 | $x^{8} + 157$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |