Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504)
gp: K = bnfinit(y^16 - 4*y^15 - 29*y^14 + 2382*y^13 - 82573*y^12 - 427336*y^11 + 11394403*y^10 + 27090344*y^9 - 189686251*y^8 - 287852338*y^7 - 2171051651*y^6 - 22553825868*y^5 - 971088042492*y^4 - 3371533703552*y^3 - 5254806287296*y^2 + 16552670883328*y - 7727214997504, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504)
\( x^{16} - 4 x^{15} - 29 x^{14} + 2382 x^{13} - 82573 x^{12} - 427336 x^{11} + 11394403 x^{10} + \cdots - 7727214997504 \)
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: |
\(7017513031942338923028700587492388019023549177\)
\(\medspace = 31^{12}\cdot 73^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(733.48\) | 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: | $31^{3/4}73^{15/16}\approx 733.4769184806004$ | ||
| Ramified primes: |
\(31\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{73}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}+\frac{1}{32}a^{9}-\frac{1}{16}a^{8}+\frac{1}{8}a^{6}+\frac{3}{32}a^{5}+\frac{5}{32}a^{4}+\frac{15}{32}a^{3}-\frac{3}{16}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{9}-\frac{1}{16}a^{8}+\frac{1}{8}a^{7}+\frac{7}{32}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{9}{32}a^{3}-\frac{3}{16}a^{2}$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{10}-\frac{1}{16}a^{9}+\frac{7}{32}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{1}{32}a^{4}+\frac{5}{16}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{128}a^{14}-\frac{1}{128}a^{12}-\frac{1}{64}a^{11}+\frac{7}{128}a^{10}-\frac{1}{32}a^{9}+\frac{3}{128}a^{8}+\frac{1}{32}a^{7}-\frac{7}{128}a^{6}-\frac{9}{64}a^{5}+\frac{25}{128}a^{4}+\frac{1}{8}a^{3}-\frac{11}{32}a^{2}+\frac{3}{8}a$, $\frac{1}{32\!\cdots\!16}a^{15}+\frac{72\!\cdots\!67}{10\!\cdots\!88}a^{14}-\frac{37\!\cdots\!73}{32\!\cdots\!16}a^{13}+\frac{17\!\cdots\!53}{16\!\cdots\!08}a^{12}-\frac{33\!\cdots\!57}{32\!\cdots\!16}a^{11}+\frac{42\!\cdots\!41}{81\!\cdots\!04}a^{10}+\frac{45\!\cdots\!63}{32\!\cdots\!16}a^{9}-\frac{35\!\cdots\!47}{81\!\cdots\!04}a^{8}+\frac{53\!\cdots\!21}{32\!\cdots\!16}a^{7}-\frac{16\!\cdots\!07}{16\!\cdots\!08}a^{6}+\frac{58\!\cdots\!65}{32\!\cdots\!16}a^{5}-\frac{17\!\cdots\!19}{40\!\cdots\!52}a^{4}+\frac{11\!\cdots\!21}{81\!\cdots\!04}a^{3}+\frac{11\!\cdots\!97}{20\!\cdots\!76}a^{2}+\frac{10\!\cdots\!27}{31\!\cdots\!59}a+\frac{11\!\cdots\!72}{31\!\cdots\!59}$
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: |
$\frac{76\!\cdots\!17}{50\!\cdots\!44}a^{15}-\frac{14\!\cdots\!39}{25\!\cdots\!72}a^{14}-\frac{27\!\cdots\!65}{25\!\cdots\!72}a^{13}+\frac{17\!\cdots\!91}{50\!\cdots\!44}a^{12}-\frac{76\!\cdots\!01}{63\!\cdots\!18}a^{11}-\frac{19\!\cdots\!15}{25\!\cdots\!72}a^{10}+\frac{10\!\cdots\!31}{50\!\cdots\!44}a^{9}+\frac{26\!\cdots\!03}{25\!\cdots\!72}a^{8}-\frac{19\!\cdots\!01}{25\!\cdots\!72}a^{7}-\frac{40\!\cdots\!79}{50\!\cdots\!44}a^{6}-\frac{12\!\cdots\!73}{63\!\cdots\!18}a^{5}+\frac{50\!\cdots\!25}{25\!\cdots\!72}a^{4}+\frac{27\!\cdots\!22}{31\!\cdots\!59}a^{3}+\frac{38\!\cdots\!62}{31\!\cdots\!59}a^{2}-\frac{12\!\cdots\!84}{31\!\cdots\!59}a+\frac{26\!\cdots\!31}{31\!\cdots\!59}$, $\frac{11\!\cdots\!73}{25\!\cdots\!72}a^{15}-\frac{30\!\cdots\!73}{12\!\cdots\!36}a^{14}-\frac{15\!\cdots\!09}{12\!\cdots\!36}a^{13}+\frac{27\!\cdots\!27}{25\!\cdots\!72}a^{12}-\frac{11\!\cdots\!84}{31\!\cdots\!59}a^{11}-\frac{17\!\cdots\!25}{12\!\cdots\!36}a^{10}+\frac{13\!\cdots\!51}{25\!\cdots\!72}a^{9}+\frac{58\!\cdots\!45}{12\!\cdots\!36}a^{8}-\frac{15\!\cdots\!09}{12\!\cdots\!36}a^{7}-\frac{43\!\cdots\!91}{25\!\cdots\!72}a^{6}-\frac{17\!\cdots\!52}{31\!\cdots\!59}a^{5}-\frac{72\!\cdots\!01}{12\!\cdots\!36}a^{4}-\frac{12\!\cdots\!08}{31\!\cdots\!59}a^{3}-\frac{21\!\cdots\!72}{31\!\cdots\!59}a^{2}+\frac{57\!\cdots\!16}{31\!\cdots\!59}a+\frac{58\!\cdots\!69}{31\!\cdots\!59}$, $\frac{14\!\cdots\!85}{63\!\cdots\!18}a^{15}-\frac{26\!\cdots\!59}{25\!\cdots\!72}a^{14}-\frac{19\!\cdots\!84}{31\!\cdots\!59}a^{13}+\frac{14\!\cdots\!79}{25\!\cdots\!72}a^{12}-\frac{24\!\cdots\!95}{12\!\cdots\!36}a^{11}-\frac{21\!\cdots\!05}{25\!\cdots\!72}a^{10}+\frac{86\!\cdots\!57}{31\!\cdots\!59}a^{9}+\frac{70\!\cdots\!71}{25\!\cdots\!72}a^{8}-\frac{38\!\cdots\!63}{63\!\cdots\!18}a^{7}+\frac{17\!\cdots\!97}{25\!\cdots\!72}a^{6}-\frac{18\!\cdots\!75}{12\!\cdots\!36}a^{5}-\frac{26\!\cdots\!99}{25\!\cdots\!72}a^{4}-\frac{62\!\cdots\!98}{31\!\cdots\!59}a^{3}-\frac{10\!\cdots\!46}{31\!\cdots\!59}a^{2}+\frac{28\!\cdots\!92}{31\!\cdots\!59}a+\frac{32\!\cdots\!31}{31\!\cdots\!59}$, $\frac{11\!\cdots\!47}{50\!\cdots\!44}a^{15}-\frac{22\!\cdots\!73}{25\!\cdots\!72}a^{14}-\frac{42\!\cdots\!45}{25\!\cdots\!72}a^{13}+\frac{26\!\cdots\!27}{50\!\cdots\!44}a^{12}-\frac{47\!\cdots\!53}{25\!\cdots\!72}a^{11}-\frac{15\!\cdots\!01}{12\!\cdots\!36}a^{10}+\frac{16\!\cdots\!73}{50\!\cdots\!44}a^{9}+\frac{41\!\cdots\!25}{25\!\cdots\!72}a^{8}-\frac{30\!\cdots\!21}{25\!\cdots\!72}a^{7}-\frac{63\!\cdots\!55}{50\!\cdots\!44}a^{6}-\frac{77\!\cdots\!27}{25\!\cdots\!72}a^{5}+\frac{19\!\cdots\!19}{63\!\cdots\!18}a^{4}+\frac{42\!\cdots\!30}{31\!\cdots\!59}a^{3}+\frac{11\!\cdots\!91}{63\!\cdots\!18}a^{2}-\frac{18\!\cdots\!52}{31\!\cdots\!59}a+\frac{88\!\cdots\!13}{31\!\cdots\!59}$, $\frac{88\!\cdots\!09}{10\!\cdots\!88}a^{15}-\frac{10\!\cdots\!07}{10\!\cdots\!88}a^{14}+\frac{10\!\cdots\!11}{10\!\cdots\!88}a^{13}+\frac{24\!\cdots\!91}{10\!\cdots\!88}a^{12}-\frac{91\!\cdots\!21}{10\!\cdots\!88}a^{11}+\frac{20\!\cdots\!93}{10\!\cdots\!88}a^{10}+\frac{12\!\cdots\!51}{10\!\cdots\!88}a^{9}-\frac{74\!\cdots\!09}{10\!\cdots\!88}a^{8}-\frac{21\!\cdots\!55}{10\!\cdots\!88}a^{7}+\frac{36\!\cdots\!61}{10\!\cdots\!88}a^{6}-\frac{14\!\cdots\!39}{10\!\cdots\!88}a^{5}-\frac{53\!\cdots\!69}{10\!\cdots\!88}a^{4}-\frac{20\!\cdots\!61}{31\!\cdots\!59}a^{3}+\frac{25\!\cdots\!91}{25\!\cdots\!72}a^{2}-\frac{41\!\cdots\!99}{31\!\cdots\!59}a+\frac{15\!\cdots\!23}{31\!\cdots\!59}$, $\frac{34\!\cdots\!65}{50\!\cdots\!44}a^{15}-\frac{25\!\cdots\!57}{63\!\cdots\!18}a^{14}-\frac{55\!\cdots\!99}{31\!\cdots\!59}a^{13}+\frac{42\!\cdots\!71}{25\!\cdots\!72}a^{12}-\frac{30\!\cdots\!47}{50\!\cdots\!44}a^{11}-\frac{10\!\cdots\!47}{50\!\cdots\!44}a^{10}+\frac{43\!\cdots\!27}{50\!\cdots\!44}a^{9}+\frac{75\!\cdots\!11}{12\!\cdots\!36}a^{8}-\frac{60\!\cdots\!17}{31\!\cdots\!59}a^{7}-\frac{15\!\cdots\!85}{25\!\cdots\!72}a^{6}-\frac{41\!\cdots\!53}{50\!\cdots\!44}a^{5}-\frac{59\!\cdots\!61}{50\!\cdots\!44}a^{4}-\frac{20\!\cdots\!32}{31\!\cdots\!59}a^{3}-\frac{13\!\cdots\!15}{12\!\cdots\!36}a^{2}+\frac{91\!\cdots\!14}{31\!\cdots\!59}a+\frac{91\!\cdots\!89}{31\!\cdots\!59}$, $\frac{10\!\cdots\!99}{10\!\cdots\!88}a^{15}+\frac{29\!\cdots\!05}{10\!\cdots\!88}a^{14}-\frac{12\!\cdots\!97}{10\!\cdots\!88}a^{13}+\frac{62\!\cdots\!67}{10\!\cdots\!88}a^{12}-\frac{12\!\cdots\!47}{10\!\cdots\!88}a^{11}-\frac{22\!\cdots\!45}{10\!\cdots\!88}a^{10}-\frac{31\!\cdots\!31}{10\!\cdots\!88}a^{9}+\frac{56\!\cdots\!59}{10\!\cdots\!88}a^{8}+\frac{18\!\cdots\!49}{10\!\cdots\!88}a^{7}+\frac{21\!\cdots\!01}{10\!\cdots\!88}a^{6}+\frac{16\!\cdots\!27}{10\!\cdots\!88}a^{5}+\frac{23\!\cdots\!29}{10\!\cdots\!88}a^{4}+\frac{22\!\cdots\!65}{31\!\cdots\!59}a^{3}+\frac{25\!\cdots\!97}{25\!\cdots\!72}a^{2}-\frac{10\!\cdots\!05}{31\!\cdots\!59}a+\frac{51\!\cdots\!31}{31\!\cdots\!59}$, $\frac{87\!\cdots\!77}{25\!\cdots\!72}a^{15}+\frac{14\!\cdots\!79}{63\!\cdots\!18}a^{14}+\frac{29\!\cdots\!79}{10\!\cdots\!88}a^{13}+\frac{10\!\cdots\!13}{10\!\cdots\!88}a^{12}-\frac{38\!\cdots\!59}{25\!\cdots\!72}a^{11}-\frac{28\!\cdots\!39}{10\!\cdots\!88}a^{10}-\frac{45\!\cdots\!27}{10\!\cdots\!88}a^{9}+\frac{36\!\cdots\!27}{50\!\cdots\!44}a^{8}+\frac{33\!\cdots\!73}{10\!\cdots\!88}a^{7}+\frac{30\!\cdots\!91}{10\!\cdots\!88}a^{6}+\frac{20\!\cdots\!19}{63\!\cdots\!18}a^{5}+\frac{31\!\cdots\!67}{10\!\cdots\!88}a^{4}+\frac{90\!\cdots\!15}{10\!\cdots\!88}a^{3}+\frac{44\!\cdots\!43}{50\!\cdots\!44}a^{2}-\frac{47\!\cdots\!09}{12\!\cdots\!36}a+\frac{57\!\cdots\!41}{31\!\cdots\!59}$, $\frac{30\!\cdots\!69}{32\!\cdots\!16}a^{15}-\frac{15\!\cdots\!11}{50\!\cdots\!44}a^{14}-\frac{94\!\cdots\!21}{32\!\cdots\!16}a^{13}+\frac{35\!\cdots\!45}{16\!\cdots\!08}a^{12}-\frac{24\!\cdots\!01}{32\!\cdots\!16}a^{11}-\frac{36\!\cdots\!31}{81\!\cdots\!04}a^{10}+\frac{33\!\cdots\!23}{32\!\cdots\!16}a^{9}+\frac{26\!\cdots\!45}{81\!\cdots\!04}a^{8}-\frac{48\!\cdots\!87}{32\!\cdots\!16}a^{7}-\frac{61\!\cdots\!67}{16\!\cdots\!08}a^{6}-\frac{73\!\cdots\!83}{32\!\cdots\!16}a^{5}-\frac{90\!\cdots\!95}{40\!\cdots\!52}a^{4}-\frac{74\!\cdots\!11}{81\!\cdots\!04}a^{3}-\frac{77\!\cdots\!59}{20\!\cdots\!76}a^{2}-\frac{49\!\cdots\!45}{63\!\cdots\!18}a+\frac{30\!\cdots\!85}{31\!\cdots\!59}$, $\frac{10\!\cdots\!71}{32\!\cdots\!16}a^{15}-\frac{80\!\cdots\!17}{20\!\cdots\!76}a^{14}+\frac{67\!\cdots\!45}{32\!\cdots\!16}a^{13}+\frac{10\!\cdots\!59}{16\!\cdots\!08}a^{12}-\frac{10\!\cdots\!19}{32\!\cdots\!16}a^{11}+\frac{89\!\cdots\!95}{81\!\cdots\!04}a^{10}+\frac{96\!\cdots\!97}{32\!\cdots\!16}a^{9}-\frac{11\!\cdots\!93}{81\!\cdots\!04}a^{8}+\frac{14\!\cdots\!91}{32\!\cdots\!16}a^{7}-\frac{71\!\cdots\!81}{16\!\cdots\!08}a^{6}+\frac{87\!\cdots\!27}{32\!\cdots\!16}a^{5}-\frac{11\!\cdots\!67}{40\!\cdots\!52}a^{4}-\frac{84\!\cdots\!41}{81\!\cdots\!04}a^{3}-\frac{65\!\cdots\!49}{20\!\cdots\!76}a^{2}+\frac{96\!\cdots\!73}{12\!\cdots\!36}a-\frac{10\!\cdots\!22}{31\!\cdots\!59}$, $\frac{14\!\cdots\!39}{32\!\cdots\!16}a^{15}-\frac{71\!\cdots\!77}{50\!\cdots\!44}a^{14}-\frac{44\!\cdots\!43}{32\!\cdots\!16}a^{13}+\frac{16\!\cdots\!07}{16\!\cdots\!08}a^{12}-\frac{11\!\cdots\!39}{32\!\cdots\!16}a^{11}-\frac{17\!\cdots\!97}{81\!\cdots\!04}a^{10}+\frac{15\!\cdots\!17}{32\!\cdots\!16}a^{9}+\frac{12\!\cdots\!79}{81\!\cdots\!04}a^{8}-\frac{22\!\cdots\!01}{32\!\cdots\!16}a^{7}-\frac{28\!\cdots\!45}{16\!\cdots\!08}a^{6}-\frac{34\!\cdots\!81}{32\!\cdots\!16}a^{5}-\frac{43\!\cdots\!85}{40\!\cdots\!52}a^{4}-\frac{34\!\cdots\!65}{81\!\cdots\!04}a^{3}-\frac{36\!\cdots\!69}{20\!\cdots\!76}a^{2}-\frac{46\!\cdots\!67}{12\!\cdots\!36}a+\frac{14\!\cdots\!41}{31\!\cdots\!59}$
| 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: | \( 416510986026000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 416510986026000000 \cdot 1}{2\cdot\sqrt{7017513031942338923028700587492388019023549177}}\cr\approx \mathstrut & 0.991891777625763 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504)
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 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504, 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 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504);
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 - 4*x^15 - 29*x^14 + 2382*x^13 - 82573*x^12 - 427336*x^11 + 11394403*x^10 + 27090344*x^9 - 189686251*x^8 - 287852338*x^7 - 2171051651*x^6 - 22553825868*x^5 - 971088042492*x^4 - 3371533703552*x^3 - 5254806287296*x^2 + 16552670883328*x - 7727214997504);
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
| \(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.10202504527754980537.1 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | $16$ | R | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | $16$ | $16$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.16.12.3 | $x^{16} - 1488 x^{12} + 738048 x^{8} - 109154224 x^{4} + 24935067$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |
|
\(73\)
| 73.16.15.1 | $x^{16} + 657$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |