Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024)
gp: K = bnfinit(y^16 - 2*y^15 - 521*y^14 + 786*y^13 + 55734*y^12 - 567292*y^11 + 1341198*y^10 + 149101604*y^9 - 132927939*y^8 - 14337715994*y^7 - 50939514293*y^6 + 995895836266*y^5 + 14997315506204*y^4 + 4919298712952*y^3 - 796637993795424*y^2 - 1453431519915008*y + 10858129511809024, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024)
\( x^{16} - 2 x^{15} - 521 x^{14} + 786 x^{13} + 55734 x^{12} - 567292 x^{11} + 1341198 x^{10} + \cdots + 10\!\cdots\!24 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3815787068614420330917907539305897270844547529\)
\(\medspace = 23^{12}\cdot 89^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(706.07\) | 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: | $23^{3/4}89^{15/16}\approx 706.0720305549709$ | ||
| Ramified primes: |
\(23\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{89}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{16}a^{7}+\frac{3}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{128}a^{8}-\frac{1}{32}a^{7}+\frac{1}{64}a^{6}-\frac{1}{16}a^{5}+\frac{1}{128}a^{4}-\frac{1}{32}a^{3}-\frac{1}{32}a^{2}+\frac{1}{8}a$, $\frac{1}{128}a^{9}+\frac{1}{64}a^{7}+\frac{1}{128}a^{5}-\frac{1}{32}a^{3}$, $\frac{1}{512}a^{10}+\frac{1}{512}a^{9}+\frac{5}{256}a^{7}-\frac{3}{512}a^{6}+\frac{17}{512}a^{5}-\frac{3}{256}a^{4}-\frac{31}{128}a^{3}-\frac{15}{64}a^{2}+\frac{7}{16}a$, $\frac{1}{1024}a^{11}+\frac{3}{1024}a^{9}-\frac{1}{512}a^{8}+\frac{11}{1024}a^{7}-\frac{1}{256}a^{6}-\frac{51}{1024}a^{5}+\frac{31}{512}a^{4}+\frac{49}{256}a^{3}+\frac{25}{128}a^{2}-\frac{13}{32}a$, $\frac{1}{1024}a^{12}-\frac{1}{1024}a^{10}+\frac{1}{512}a^{9}+\frac{3}{1024}a^{8}+\frac{1}{256}a^{7}+\frac{9}{1024}a^{6}+\frac{1}{512}a^{5}+\frac{5}{256}a^{4}+\frac{31}{128}a^{3}+\frac{7}{32}a^{2}-\frac{1}{2}a$, $\frac{1}{90112}a^{13}+\frac{17}{90112}a^{12}+\frac{35}{90112}a^{11}+\frac{25}{90112}a^{10}+\frac{241}{90112}a^{9}+\frac{103}{90112}a^{8}-\frac{1863}{90112}a^{7}-\frac{893}{90112}a^{6}+\frac{1429}{45056}a^{5}-\frac{1341}{22528}a^{4}+\frac{2529}{11264}a^{3}+\frac{127}{2816}a^{2}-\frac{43}{88}a+\frac{3}{11}$, $\frac{1}{7929856}a^{14}-\frac{17}{3964928}a^{13}+\frac{21}{61952}a^{12}-\frac{7}{22528}a^{11}-\frac{13}{32768}a^{10}-\frac{197}{180224}a^{9}-\frac{5123}{1982464}a^{8}+\frac{25581}{991232}a^{7}+\frac{174417}{7929856}a^{6}+\frac{119799}{3964928}a^{5}-\frac{97271}{1982464}a^{4}+\frac{8457}{991232}a^{3}-\frac{4157}{247808}a^{2}-\frac{577}{7744}a-\frac{37}{121}$, $\frac{1}{64\!\cdots\!72}a^{15}+\frac{25\!\cdots\!09}{64\!\cdots\!72}a^{14}-\frac{11\!\cdots\!31}{32\!\cdots\!36}a^{13}-\frac{37\!\cdots\!45}{25\!\cdots\!12}a^{12}+\frac{14\!\cdots\!41}{29\!\cdots\!76}a^{11}-\frac{72\!\cdots\!07}{29\!\cdots\!76}a^{10}+\frac{16\!\cdots\!45}{50\!\cdots\!24}a^{9}-\frac{43\!\cdots\!43}{16\!\cdots\!68}a^{8}-\frac{11\!\cdots\!47}{64\!\cdots\!72}a^{7}+\frac{11\!\cdots\!45}{64\!\cdots\!72}a^{6}-\frac{42\!\cdots\!07}{29\!\cdots\!76}a^{5}-\frac{56\!\cdots\!41}{14\!\cdots\!88}a^{4}-\frac{54\!\cdots\!91}{73\!\cdots\!44}a^{3}+\frac{28\!\cdots\!01}{20\!\cdots\!96}a^{2}-\frac{46\!\cdots\!03}{63\!\cdots\!28}a+\frac{10\!\cdots\!61}{49\!\cdots\!76}$
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: | $11$ | 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: |
$\frac{55\!\cdots\!11}{79\!\cdots\!76}a^{15}-\frac{12\!\cdots\!75}{15\!\cdots\!52}a^{14}-\frac{23\!\cdots\!01}{79\!\cdots\!76}a^{13}+\frac{19\!\cdots\!99}{62\!\cdots\!92}a^{12}+\frac{33\!\cdots\!81}{32\!\cdots\!28}a^{11}-\frac{35\!\cdots\!03}{72\!\cdots\!16}a^{10}+\frac{21\!\cdots\!55}{39\!\cdots\!88}a^{9}+\frac{22\!\cdots\!53}{39\!\cdots\!88}a^{8}-\frac{46\!\cdots\!85}{79\!\cdots\!76}a^{7}-\frac{74\!\cdots\!67}{15\!\cdots\!52}a^{6}+\frac{37\!\cdots\!47}{79\!\cdots\!76}a^{5}+\frac{25\!\cdots\!61}{39\!\cdots\!88}a^{4}+\frac{95\!\cdots\!65}{19\!\cdots\!44}a^{3}-\frac{16\!\cdots\!59}{45\!\cdots\!76}a^{2}-\frac{28\!\cdots\!53}{14\!\cdots\!68}a+\frac{14\!\cdots\!37}{19\!\cdots\!81}$, $\frac{71\!\cdots\!31}{10\!\cdots\!28}a^{15}-\frac{77\!\cdots\!33}{10\!\cdots\!28}a^{14}-\frac{15\!\cdots\!49}{50\!\cdots\!64}a^{13}+\frac{12\!\cdots\!91}{39\!\cdots\!88}a^{12}+\frac{42\!\cdots\!13}{42\!\cdots\!84}a^{11}-\frac{22\!\cdots\!21}{46\!\cdots\!24}a^{10}+\frac{42\!\cdots\!29}{79\!\cdots\!76}a^{9}+\frac{14\!\cdots\!75}{25\!\cdots\!32}a^{8}-\frac{59\!\cdots\!77}{10\!\cdots\!28}a^{7}-\frac{47\!\cdots\!25}{10\!\cdots\!28}a^{6}+\frac{24\!\cdots\!53}{50\!\cdots\!64}a^{5}+\frac{16\!\cdots\!79}{25\!\cdots\!32}a^{4}+\frac{61\!\cdots\!69}{12\!\cdots\!16}a^{3}-\frac{10\!\cdots\!83}{28\!\cdots\!64}a^{2}-\frac{18\!\cdots\!11}{90\!\cdots\!52}a+\frac{61\!\cdots\!31}{77\!\cdots\!24}$, $\frac{13\!\cdots\!37}{10\!\cdots\!28}a^{15}-\frac{14\!\cdots\!99}{10\!\cdots\!28}a^{14}-\frac{27\!\cdots\!43}{50\!\cdots\!64}a^{13}+\frac{46\!\cdots\!75}{79\!\cdots\!76}a^{12}+\frac{73\!\cdots\!27}{42\!\cdots\!84}a^{11}-\frac{40\!\cdots\!27}{46\!\cdots\!24}a^{10}+\frac{38\!\cdots\!11}{39\!\cdots\!88}a^{9}+\frac{25\!\cdots\!41}{25\!\cdots\!32}a^{8}-\frac{10\!\cdots\!31}{10\!\cdots\!28}a^{7}-\frac{86\!\cdots\!55}{10\!\cdots\!28}a^{6}+\frac{39\!\cdots\!39}{50\!\cdots\!64}a^{5}+\frac{29\!\cdots\!65}{25\!\cdots\!32}a^{4}+\frac{11\!\cdots\!79}{12\!\cdots\!16}a^{3}-\frac{19\!\cdots\!17}{28\!\cdots\!64}a^{2}-\frac{32\!\cdots\!85}{90\!\cdots\!52}a+\frac{10\!\cdots\!81}{77\!\cdots\!24}$, $\frac{90\!\cdots\!35}{35\!\cdots\!88}a^{15}-\frac{10\!\cdots\!25}{35\!\cdots\!88}a^{14}-\frac{23\!\cdots\!33}{17\!\cdots\!44}a^{13}+\frac{24\!\cdots\!15}{13\!\cdots\!48}a^{12}+\frac{21\!\cdots\!41}{14\!\cdots\!64}a^{11}-\frac{68\!\cdots\!49}{16\!\cdots\!04}a^{10}+\frac{63\!\cdots\!61}{27\!\cdots\!96}a^{9}+\frac{51\!\cdots\!79}{88\!\cdots\!72}a^{8}-\frac{23\!\cdots\!59}{32\!\cdots\!08}a^{7}-\frac{13\!\cdots\!57}{35\!\cdots\!88}a^{6}+\frac{14\!\cdots\!13}{17\!\cdots\!44}a^{5}+\frac{10\!\cdots\!51}{88\!\cdots\!72}a^{4}-\frac{18\!\cdots\!19}{44\!\cdots\!36}a^{3}+\frac{21\!\cdots\!03}{11\!\cdots\!84}a^{2}+\frac{26\!\cdots\!83}{34\!\cdots\!12}a-\frac{44\!\cdots\!71}{27\!\cdots\!54}$, $\frac{16\!\cdots\!79}{71\!\cdots\!76}a^{15}-\frac{18\!\cdots\!29}{71\!\cdots\!76}a^{14}-\frac{43\!\cdots\!85}{35\!\cdots\!88}a^{13}+\frac{85\!\cdots\!17}{55\!\cdots\!92}a^{12}+\frac{36\!\cdots\!57}{29\!\cdots\!28}a^{11}-\frac{12\!\cdots\!09}{32\!\cdots\!08}a^{10}+\frac{11\!\cdots\!69}{55\!\cdots\!92}a^{9}+\frac{89\!\cdots\!07}{17\!\cdots\!44}a^{8}-\frac{43\!\cdots\!25}{71\!\cdots\!76}a^{7}-\frac{24\!\cdots\!21}{71\!\cdots\!76}a^{6}+\frac{23\!\cdots\!73}{35\!\cdots\!88}a^{5}+\frac{21\!\cdots\!27}{17\!\cdots\!44}a^{4}-\frac{28\!\cdots\!67}{88\!\cdots\!72}a^{3}+\frac{54\!\cdots\!59}{22\!\cdots\!68}a^{2}+\frac{38\!\cdots\!23}{69\!\cdots\!24}a-\frac{56\!\cdots\!85}{54\!\cdots\!08}$, $\frac{10\!\cdots\!77}{71\!\cdots\!76}a^{15}-\frac{16\!\cdots\!43}{71\!\cdots\!76}a^{14}-\frac{16\!\cdots\!29}{32\!\cdots\!08}a^{13}+\frac{77\!\cdots\!99}{86\!\cdots\!28}a^{12}-\frac{88\!\cdots\!77}{29\!\cdots\!28}a^{11}-\frac{24\!\cdots\!11}{32\!\cdots\!08}a^{10}+\frac{16\!\cdots\!51}{11\!\cdots\!84}a^{9}+\frac{32\!\cdots\!05}{17\!\cdots\!44}a^{8}-\frac{67\!\cdots\!23}{71\!\cdots\!76}a^{7}-\frac{36\!\cdots\!69}{64\!\cdots\!16}a^{6}+\frac{55\!\cdots\!51}{35\!\cdots\!88}a^{5}+\frac{23\!\cdots\!81}{17\!\cdots\!44}a^{4}+\frac{25\!\cdots\!43}{88\!\cdots\!72}a^{3}-\frac{14\!\cdots\!43}{22\!\cdots\!68}a^{2}-\frac{88\!\cdots\!83}{69\!\cdots\!24}a+\frac{46\!\cdots\!45}{54\!\cdots\!08}$, $\frac{38\!\cdots\!35}{35\!\cdots\!88}a^{15}-\frac{62\!\cdots\!17}{35\!\cdots\!88}a^{14}-\frac{63\!\cdots\!17}{17\!\cdots\!44}a^{13}+\frac{37\!\cdots\!99}{55\!\cdots\!92}a^{12}-\frac{40\!\cdots\!79}{14\!\cdots\!64}a^{11}-\frac{85\!\cdots\!09}{16\!\cdots\!04}a^{10}+\frac{50\!\cdots\!47}{43\!\cdots\!64}a^{9}+\frac{32\!\cdots\!47}{88\!\cdots\!72}a^{8}-\frac{23\!\cdots\!85}{35\!\cdots\!88}a^{7}-\frac{13\!\cdots\!53}{35\!\cdots\!88}a^{6}+\frac{21\!\cdots\!41}{17\!\cdots\!44}a^{5}+\frac{87\!\cdots\!95}{88\!\cdots\!72}a^{4}+\frac{63\!\cdots\!77}{44\!\cdots\!36}a^{3}-\frac{53\!\cdots\!57}{11\!\cdots\!84}a^{2}-\frac{21\!\cdots\!05}{34\!\cdots\!12}a+\frac{14\!\cdots\!17}{27\!\cdots\!54}$, $\frac{10\!\cdots\!71}{16\!\cdots\!68}a^{15}+\frac{12\!\cdots\!75}{16\!\cdots\!68}a^{14}-\frac{18\!\cdots\!25}{80\!\cdots\!84}a^{13}-\frac{67\!\cdots\!27}{25\!\cdots\!12}a^{12}-\frac{78\!\cdots\!37}{73\!\cdots\!44}a^{11}-\frac{27\!\cdots\!93}{73\!\cdots\!44}a^{10}-\frac{11\!\cdots\!73}{25\!\cdots\!12}a^{9}+\frac{14\!\cdots\!95}{40\!\cdots\!92}a^{8}+\frac{64\!\cdots\!59}{16\!\cdots\!68}a^{7}-\frac{58\!\cdots\!37}{16\!\cdots\!68}a^{6}-\frac{67\!\cdots\!39}{80\!\cdots\!84}a^{5}-\frac{20\!\cdots\!97}{40\!\cdots\!92}a^{4}+\frac{52\!\cdots\!97}{20\!\cdots\!96}a^{3}+\frac{18\!\cdots\!17}{45\!\cdots\!84}a^{2}+\frac{40\!\cdots\!51}{13\!\cdots\!92}a-\frac{62\!\cdots\!55}{12\!\cdots\!94}$, $\frac{23\!\cdots\!93}{32\!\cdots\!36}a^{15}+\frac{27\!\cdots\!37}{32\!\cdots\!36}a^{14}-\frac{60\!\cdots\!15}{16\!\cdots\!68}a^{13}-\frac{15\!\cdots\!87}{25\!\cdots\!12}a^{12}+\frac{55\!\cdots\!93}{14\!\cdots\!88}a^{11}-\frac{41\!\cdots\!27}{14\!\cdots\!88}a^{10}+\frac{24\!\cdots\!39}{50\!\cdots\!24}a^{9}+\frac{78\!\cdots\!03}{73\!\cdots\!44}a^{8}+\frac{80\!\cdots\!73}{32\!\cdots\!36}a^{7}-\frac{30\!\cdots\!99}{32\!\cdots\!36}a^{6}-\frac{10\!\cdots\!73}{16\!\cdots\!68}a^{5}+\frac{40\!\cdots\!45}{80\!\cdots\!84}a^{4}+\frac{49\!\cdots\!19}{40\!\cdots\!92}a^{3}+\frac{43\!\cdots\!61}{10\!\cdots\!48}a^{2}-\frac{13\!\cdots\!71}{31\!\cdots\!64}a-\frac{60\!\cdots\!67}{24\!\cdots\!88}$, $\frac{22\!\cdots\!57}{32\!\cdots\!36}a^{15}-\frac{24\!\cdots\!19}{32\!\cdots\!36}a^{14}-\frac{47\!\cdots\!71}{16\!\cdots\!68}a^{13}+\frac{40\!\cdots\!67}{12\!\cdots\!56}a^{12}+\frac{13\!\cdots\!55}{13\!\cdots\!08}a^{11}-\frac{71\!\cdots\!19}{14\!\cdots\!88}a^{10}+\frac{66\!\cdots\!19}{12\!\cdots\!56}a^{9}+\frac{45\!\cdots\!93}{80\!\cdots\!84}a^{8}-\frac{19\!\cdots\!31}{32\!\cdots\!36}a^{7}-\frac{14\!\cdots\!63}{32\!\cdots\!36}a^{6}+\frac{96\!\cdots\!43}{16\!\cdots\!68}a^{5}+\frac{51\!\cdots\!69}{80\!\cdots\!84}a^{4}+\frac{18\!\cdots\!19}{40\!\cdots\!92}a^{3}-\frac{35\!\cdots\!85}{91\!\cdots\!68}a^{2}-\frac{58\!\cdots\!73}{28\!\cdots\!24}a+\frac{20\!\cdots\!53}{24\!\cdots\!88}$, $\frac{10\!\cdots\!55}{16\!\cdots\!68}a^{15}+\frac{19\!\cdots\!91}{16\!\cdots\!68}a^{14}-\frac{75\!\cdots\!21}{80\!\cdots\!84}a^{13}-\frac{17\!\cdots\!97}{12\!\cdots\!56}a^{12}+\frac{54\!\cdots\!63}{73\!\cdots\!44}a^{11}-\frac{15\!\cdots\!45}{73\!\cdots\!44}a^{10}-\frac{88\!\cdots\!07}{25\!\cdots\!12}a^{9}+\frac{97\!\cdots\!03}{40\!\cdots\!92}a^{8}+\frac{65\!\cdots\!03}{16\!\cdots\!68}a^{7}-\frac{14\!\cdots\!89}{16\!\cdots\!68}a^{6}-\frac{41\!\cdots\!27}{80\!\cdots\!84}a^{5}-\frac{16\!\cdots\!33}{40\!\cdots\!92}a^{4}+\frac{23\!\cdots\!53}{20\!\cdots\!96}a^{3}+\frac{13\!\cdots\!63}{50\!\cdots\!24}a^{2}+\frac{45\!\cdots\!71}{15\!\cdots\!32}a-\frac{42\!\cdots\!87}{12\!\cdots\!94}$
| 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: | \( 5313448120150000000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 5313448120150000000000 \cdot 1}{2\cdot\sqrt{3815787068614420330917907539305897270844547529}}\cr\approx \mathstrut & 17159.8632152005 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024)
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 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024, 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 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024);
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 - 2*x^15 - 521*x^14 + 786*x^13 + 55734*x^12 - 567292*x^11 + 1341198*x^10 + 149101604*x^9 - 132927939*x^8 - 14337715994*x^7 - 50939514293*x^6 + 995895836266*x^5 + 14997315506204*x^4 + 4919298712952*x^3 - 796637993795424*x^2 - 1453431519915008*x + 10858129511809024);
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
$\OD_{32}$ (as 16T22):
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 20 conjugacy class representatives for $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.12377740988499730889.1 |
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
| Galois closure: | deg 32 |
| 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 | ${\href{/padicField/2.1.0.1}{1} }^{16}$ | $16$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{8}$ | $16$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | $16$ | R | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.16.12.3 | $x^{16} - 1311 x^{12} + 563385 x^{8} - 73050668 x^{4} + 34980125$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |
|
\(89\)
| 89.16.15.1 | $x^{16} + 979$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |