Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256)
gp: K = bnfinit(y^16 - 2*y^15 - 33*y^14 + 99*y^13 + 446*y^12 - 1762*y^11 - 2240*y^10 + 13297*y^9 - 3421*y^8 - 31124*y^7 + 29095*y^6 + 7636*y^5 - 16064*y^4 + 3272*y^3 + 1168*y^2 - 704*y + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256)
\( x^{16} - 2 x^{15} - 33 x^{14} + 99 x^{13} + 446 x^{12} - 1762 x^{11} - 2240 x^{10} + 13297 x^{9} + \cdots + 256 \)
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: |
\(121616372173473638916015625\)
\(\medspace = 5^{14}\cdot 109^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.69\) | 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: | $5^{7/8}109^{1/2}\approx 42.68860891456584$ | ||
| Ramified primes: |
\(5\), \(109\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{3}{16}a^{7}-\frac{1}{4}a^{6}-\frac{7}{16}a^{5}-\frac{3}{16}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{272}a^{14}-\frac{1}{34}a^{13}-\frac{7}{272}a^{12}+\frac{11}{272}a^{11}+\frac{3}{34}a^{10}-\frac{5}{34}a^{9}+\frac{11}{136}a^{8}+\frac{71}{272}a^{7}+\frac{123}{272}a^{6}+\frac{67}{136}a^{5}-\frac{3}{16}a^{4}-\frac{9}{34}a^{3}-\frac{8}{17}a^{2}+\frac{3}{17}a-\frac{7}{17}$, $\frac{1}{57\!\cdots\!36}a^{15}-\frac{45\!\cdots\!73}{28\!\cdots\!68}a^{14}-\frac{10\!\cdots\!49}{57\!\cdots\!36}a^{13}-\frac{27\!\cdots\!69}{57\!\cdots\!36}a^{12}-\frac{40\!\cdots\!85}{77\!\cdots\!64}a^{11}+\frac{29\!\cdots\!35}{26\!\cdots\!88}a^{10}-\frac{28\!\cdots\!37}{14\!\cdots\!84}a^{9}+\frac{17\!\cdots\!57}{13\!\cdots\!96}a^{8}+\frac{12\!\cdots\!95}{33\!\cdots\!08}a^{7}+\frac{59\!\cdots\!75}{14\!\cdots\!84}a^{6}+\frac{17\!\cdots\!35}{57\!\cdots\!36}a^{5}-\frac{11\!\cdots\!47}{65\!\cdots\!72}a^{4}+\frac{21\!\cdots\!69}{71\!\cdots\!92}a^{3}-\frac{11\!\cdots\!31}{71\!\cdots\!92}a^{2}+\frac{24\!\cdots\!21}{35\!\cdots\!96}a+\frac{29\!\cdots\!67}{89\!\cdots\!49}$
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: |
$\frac{185753217632647}{10\!\cdots\!16}a^{15}-\frac{150355489521985}{54\!\cdots\!08}a^{14}-\frac{62\!\cdots\!27}{10\!\cdots\!16}a^{13}+\frac{16\!\cdots\!25}{10\!\cdots\!16}a^{12}+\frac{11\!\cdots\!15}{14\!\cdots\!84}a^{11}-\frac{14\!\cdots\!73}{54\!\cdots\!08}a^{10}-\frac{13\!\cdots\!07}{27\!\cdots\!04}a^{9}+\frac{22\!\cdots\!67}{10\!\cdots\!16}a^{8}+\frac{19\!\cdots\!25}{10\!\cdots\!16}a^{7}-\frac{35\!\cdots\!87}{67\!\cdots\!76}a^{6}+\frac{19\!\cdots\!85}{63\!\cdots\!48}a^{5}+\frac{69\!\cdots\!89}{27\!\cdots\!04}a^{4}-\frac{32\!\cdots\!63}{16\!\cdots\!19}a^{3}-\frac{39\!\cdots\!59}{13\!\cdots\!52}a^{2}+\frac{46\!\cdots\!05}{67\!\cdots\!76}a-\frac{41078402337562}{99287015587307}$, $\frac{49\!\cdots\!11}{57\!\cdots\!36}a^{15}-\frac{41\!\cdots\!47}{28\!\cdots\!68}a^{14}-\frac{71\!\cdots\!51}{57\!\cdots\!36}a^{13}+\frac{30\!\cdots\!73}{57\!\cdots\!36}a^{12}-\frac{43\!\cdots\!83}{77\!\cdots\!64}a^{11}-\frac{21\!\cdots\!65}{26\!\cdots\!88}a^{10}+\frac{14\!\cdots\!80}{89\!\cdots\!49}a^{9}+\frac{83\!\cdots\!39}{13\!\cdots\!96}a^{8}-\frac{86\!\cdots\!19}{57\!\cdots\!36}a^{7}-\frac{19\!\cdots\!25}{14\!\cdots\!84}a^{6}+\frac{25\!\cdots\!17}{57\!\cdots\!36}a^{5}+\frac{24\!\cdots\!37}{13\!\cdots\!44}a^{4}-\frac{25\!\cdots\!09}{71\!\cdots\!92}a^{3}+\frac{76\!\cdots\!17}{71\!\cdots\!92}a^{2}+\frac{34\!\cdots\!87}{35\!\cdots\!96}a+\frac{55\!\cdots\!69}{89\!\cdots\!49}$, $\frac{96\!\cdots\!51}{16\!\cdots\!04}a^{15}-\frac{68\!\cdots\!77}{84\!\cdots\!52}a^{14}-\frac{32\!\cdots\!87}{16\!\cdots\!04}a^{13}+\frac{76\!\cdots\!73}{16\!\cdots\!04}a^{12}+\frac{63\!\cdots\!53}{22\!\cdots\!96}a^{11}-\frac{64\!\cdots\!29}{76\!\cdots\!32}a^{10}-\frac{72\!\cdots\!95}{42\!\cdots\!76}a^{9}+\frac{26\!\cdots\!43}{41\!\cdots\!44}a^{8}+\frac{26\!\cdots\!49}{16\!\cdots\!04}a^{7}-\frac{66\!\cdots\!11}{42\!\cdots\!76}a^{6}+\frac{12\!\cdots\!17}{16\!\cdots\!04}a^{5}+\frac{23\!\cdots\!37}{38\!\cdots\!16}a^{4}-\frac{13\!\cdots\!77}{42\!\cdots\!76}a^{3}-\frac{37\!\cdots\!31}{10\!\cdots\!94}a^{2}+\frac{13\!\cdots\!28}{52\!\cdots\!97}a-\frac{45\!\cdots\!13}{52\!\cdots\!97}$, $\frac{67\!\cdots\!49}{28\!\cdots\!68}a^{15}-\frac{28\!\cdots\!44}{89\!\cdots\!49}a^{14}-\frac{22\!\cdots\!83}{28\!\cdots\!68}a^{13}+\frac{51\!\cdots\!99}{28\!\cdots\!68}a^{12}+\frac{22\!\cdots\!35}{19\!\cdots\!16}a^{11}-\frac{21\!\cdots\!35}{65\!\cdots\!72}a^{10}-\frac{63\!\cdots\!59}{84\!\cdots\!52}a^{9}+\frac{18\!\cdots\!55}{69\!\cdots\!48}a^{8}+\frac{27\!\cdots\!15}{28\!\cdots\!68}a^{7}-\frac{94\!\cdots\!47}{14\!\cdots\!84}a^{6}+\frac{65\!\cdots\!73}{28\!\cdots\!68}a^{5}+\frac{21\!\cdots\!77}{65\!\cdots\!72}a^{4}-\frac{87\!\cdots\!17}{71\!\cdots\!92}a^{3}-\frac{55\!\cdots\!83}{17\!\cdots\!98}a^{2}+\frac{27\!\cdots\!34}{89\!\cdots\!49}a-\frac{49\!\cdots\!01}{89\!\cdots\!49}$, $\frac{11\!\cdots\!33}{52\!\cdots\!76}a^{15}-\frac{48\!\cdots\!49}{26\!\cdots\!88}a^{14}-\frac{23\!\cdots\!45}{30\!\cdots\!28}a^{13}+\frac{70\!\cdots\!43}{52\!\cdots\!76}a^{12}+\frac{85\!\cdots\!43}{70\!\cdots\!24}a^{11}-\frac{71\!\cdots\!95}{26\!\cdots\!88}a^{10}-\frac{11\!\cdots\!39}{13\!\cdots\!44}a^{9}+\frac{29\!\cdots\!93}{12\!\cdots\!36}a^{8}+\frac{12\!\cdots\!55}{52\!\cdots\!76}a^{7}-\frac{99\!\cdots\!65}{13\!\cdots\!44}a^{6}-\frac{32\!\cdots\!21}{52\!\cdots\!76}a^{5}+\frac{55\!\cdots\!61}{65\!\cdots\!72}a^{4}-\frac{41\!\cdots\!91}{16\!\cdots\!18}a^{3}-\frac{15\!\cdots\!93}{65\!\cdots\!72}a^{2}+\frac{22\!\cdots\!41}{32\!\cdots\!36}a+\frac{17\!\cdots\!66}{81\!\cdots\!59}$, $\frac{27\!\cdots\!63}{69\!\cdots\!48}a^{15}-\frac{21\!\cdots\!25}{34\!\cdots\!24}a^{14}-\frac{93\!\cdots\!79}{69\!\cdots\!48}a^{13}+\frac{23\!\cdots\!57}{69\!\cdots\!48}a^{12}+\frac{18\!\cdots\!05}{94\!\cdots\!52}a^{11}-\frac{19\!\cdots\!69}{31\!\cdots\!84}a^{10}-\frac{20\!\cdots\!09}{17\!\cdots\!12}a^{9}+\frac{32\!\cdots\!19}{69\!\cdots\!48}a^{8}+\frac{56\!\cdots\!09}{69\!\cdots\!48}a^{7}-\frac{20\!\cdots\!83}{17\!\cdots\!12}a^{6}+\frac{25\!\cdots\!45}{41\!\cdots\!44}a^{5}+\frac{85\!\cdots\!97}{15\!\cdots\!92}a^{4}-\frac{68\!\cdots\!03}{17\!\cdots\!12}a^{3}-\frac{86\!\cdots\!41}{87\!\cdots\!56}a^{2}+\frac{19\!\cdots\!67}{43\!\cdots\!78}a-\frac{23\!\cdots\!01}{12\!\cdots\!17}$, $\frac{42\!\cdots\!65}{71\!\cdots\!92}a^{15}-\frac{61\!\cdots\!53}{71\!\cdots\!92}a^{14}-\frac{35\!\cdots\!63}{17\!\cdots\!98}a^{13}+\frac{33\!\cdots\!73}{71\!\cdots\!92}a^{12}+\frac{55\!\cdots\!53}{19\!\cdots\!16}a^{11}-\frac{57\!\cdots\!81}{65\!\cdots\!72}a^{10}-\frac{12\!\cdots\!23}{71\!\cdots\!92}a^{9}+\frac{59\!\cdots\!59}{87\!\cdots\!56}a^{8}+\frac{11\!\cdots\!03}{71\!\cdots\!92}a^{7}-\frac{11\!\cdots\!89}{71\!\cdots\!92}a^{6}+\frac{13\!\cdots\!65}{17\!\cdots\!98}a^{5}+\frac{11\!\cdots\!59}{16\!\cdots\!18}a^{4}-\frac{14\!\cdots\!13}{35\!\cdots\!96}a^{3}-\frac{57\!\cdots\!03}{35\!\cdots\!96}a^{2}+\frac{22\!\cdots\!57}{17\!\cdots\!98}a-\frac{16\!\cdots\!65}{89\!\cdots\!49}$, $\frac{73\!\cdots\!53}{33\!\cdots\!08}a^{15}-\frac{47\!\cdots\!27}{28\!\cdots\!68}a^{14}-\frac{43\!\cdots\!77}{57\!\cdots\!36}a^{13}+\frac{72\!\cdots\!23}{57\!\cdots\!36}a^{12}+\frac{92\!\cdots\!61}{77\!\cdots\!64}a^{11}-\frac{66\!\cdots\!09}{26\!\cdots\!88}a^{10}-\frac{12\!\cdots\!49}{14\!\cdots\!84}a^{9}+\frac{29\!\cdots\!33}{13\!\cdots\!96}a^{8}+\frac{12\!\cdots\!27}{57\!\cdots\!36}a^{7}-\frac{22\!\cdots\!25}{35\!\cdots\!96}a^{6}-\frac{13\!\cdots\!21}{57\!\cdots\!36}a^{5}+\frac{38\!\cdots\!77}{76\!\cdots\!32}a^{4}-\frac{34\!\cdots\!07}{17\!\cdots\!98}a^{3}-\frac{59\!\cdots\!99}{71\!\cdots\!92}a^{2}+\frac{33\!\cdots\!07}{35\!\cdots\!96}a-\frac{26\!\cdots\!47}{89\!\cdots\!49}$, $\frac{94\!\cdots\!33}{57\!\cdots\!36}a^{15}-\frac{64\!\cdots\!75}{28\!\cdots\!68}a^{14}-\frac{31\!\cdots\!25}{57\!\cdots\!36}a^{13}+\frac{73\!\cdots\!63}{57\!\cdots\!36}a^{12}+\frac{62\!\cdots\!57}{77\!\cdots\!64}a^{11}-\frac{62\!\cdots\!21}{26\!\cdots\!88}a^{10}-\frac{42\!\cdots\!39}{84\!\cdots\!52}a^{9}+\frac{25\!\cdots\!53}{13\!\cdots\!96}a^{8}+\frac{30\!\cdots\!07}{57\!\cdots\!36}a^{7}-\frac{32\!\cdots\!27}{71\!\cdots\!92}a^{6}+\frac{11\!\cdots\!47}{57\!\cdots\!36}a^{5}+\frac{26\!\cdots\!69}{13\!\cdots\!44}a^{4}-\frac{41\!\cdots\!45}{35\!\cdots\!96}a^{3}-\frac{23\!\cdots\!81}{71\!\cdots\!92}a^{2}+\frac{27\!\cdots\!07}{35\!\cdots\!96}a-\frac{58\!\cdots\!11}{89\!\cdots\!49}$, $\frac{66\!\cdots\!29}{57\!\cdots\!36}a^{15}-\frac{14\!\cdots\!09}{28\!\cdots\!68}a^{14}-\frac{19\!\cdots\!45}{57\!\cdots\!36}a^{13}+\frac{11\!\cdots\!91}{57\!\cdots\!36}a^{12}+\frac{22\!\cdots\!99}{77\!\cdots\!64}a^{11}-\frac{86\!\cdots\!21}{26\!\cdots\!88}a^{10}+\frac{23\!\cdots\!03}{14\!\cdots\!84}a^{9}+\frac{32\!\cdots\!85}{13\!\cdots\!96}a^{8}-\frac{20\!\cdots\!97}{57\!\cdots\!36}a^{7}-\frac{55\!\cdots\!05}{14\!\cdots\!84}a^{6}+\frac{62\!\cdots\!19}{57\!\cdots\!36}a^{5}-\frac{11\!\cdots\!99}{32\!\cdots\!36}a^{4}-\frac{13\!\cdots\!89}{35\!\cdots\!96}a^{3}+\frac{13\!\cdots\!15}{71\!\cdots\!92}a^{2}-\frac{55\!\cdots\!11}{35\!\cdots\!96}a+\frac{13\!\cdots\!97}{89\!\cdots\!49}$, $\frac{12\!\cdots\!31}{57\!\cdots\!36}a^{15}-\frac{11\!\cdots\!61}{28\!\cdots\!68}a^{14}-\frac{24\!\cdots\!71}{33\!\cdots\!08}a^{13}+\frac{11\!\cdots\!33}{57\!\cdots\!36}a^{12}+\frac{77\!\cdots\!35}{77\!\cdots\!64}a^{11}-\frac{92\!\cdots\!79}{26\!\cdots\!88}a^{10}-\frac{77\!\cdots\!61}{14\!\cdots\!84}a^{9}+\frac{37\!\cdots\!43}{13\!\cdots\!96}a^{8}-\frac{16\!\cdots\!87}{57\!\cdots\!36}a^{7}-\frac{10\!\cdots\!19}{17\!\cdots\!98}a^{6}+\frac{27\!\cdots\!41}{57\!\cdots\!36}a^{5}+\frac{12\!\cdots\!13}{13\!\cdots\!44}a^{4}-\frac{31\!\cdots\!23}{17\!\cdots\!98}a^{3}+\frac{37\!\cdots\!45}{71\!\cdots\!92}a^{2}-\frac{53\!\cdots\!63}{35\!\cdots\!96}a-\frac{94\!\cdots\!93}{89\!\cdots\!49}$
| 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: | \( 39488908.4077 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 39488908.4077 \cdot 1}{2\cdot\sqrt{121616372173473638916015625}}\cr\approx \mathstrut & 0.714345888861 \end{aligned}\] (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 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256)
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 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256, 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 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256);
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 - 33*x^14 + 99*x^13 + 446*x^12 - 1762*x^11 - 2240*x^10 + 13297*x^9 - 3421*x^8 - 31124*x^7 + 29095*x^6 + 7636*x^5 - 16064*x^4 + 3272*x^3 + 1168*x^2 - 704*x + 256);
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
$C_4\wr C_2$ (as 16T42):
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 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{545}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{109}) \), 4.4.1485125.1 x2, 4.4.13625.1 x2, \(\Q(\sqrt{5}, \sqrt{109})\), 8.4.11027981328125.2, 8.4.11027981328125.1, 8.8.2205596265625.1 |
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 |
| Degree 8 siblings: | 8.4.11027981328125.1, 8.4.11027981328125.2 |
| Degree 16 sibling: | 16.0.10236206731207275390625.2 |
| Minimal sibling: | 8.4.11027981328125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
| 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
|
\(109\)
| 109.8.4.1 | $x^{8} + 28776 x^{7} + 310522274 x^{6} + 1489272514288 x^{5} + 2678510521605233 x^{4} + 178743008720712 x^{3} + 29612720181709536 x^{2} + 263388846138180416 x + 20054316486246464$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 109.8.4.1 | $x^{8} + 28776 x^{7} + 310522274 x^{6} + 1489272514288 x^{5} + 2678510521605233 x^{4} + 178743008720712 x^{3} + 29612720181709536 x^{2} + 263388846138180416 x + 20054316486246464$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |