Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408)
gp: K = bnfinit(y^16 - 3*y^15 - 47*y^14 + 80*y^13 + 1033*y^12 - 1001*y^11 - 15701*y^10 - 212*y^9 + 181451*y^8 + 353057*y^7 - 570157*y^6 - 3438310*y^5 - 6547056*y^4 - 7290744*y^3 - 5355776*y^2 - 2001824*y + 25408, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408)
\( x^{16} - 3 x^{15} - 47 x^{14} + 80 x^{13} + 1033 x^{12} - 1001 x^{11} - 15701 x^{10} - 212 x^{9} + \cdots + 25408 \)
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: |
\(800737337114777368288115578449\)
\(\medspace = 3^{8}\cdot 73^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(73.96\) | 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: | $3^{1/2}73^{7/8}\approx 73.95510022503493$ | ||
| Ramified primes: |
\(3\), \(73\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{5}+\frac{1}{8}a^{3}+\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{16}a^{10}+\frac{1}{16}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{5}{16}a^{2}+\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{32}a^{11}-\frac{1}{16}a^{9}-\frac{1}{32}a^{7}-\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{32}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{8}+\frac{1}{8}a^{5}-\frac{7}{32}a^{4}+\frac{1}{8}a^{3}+\frac{5}{16}a^{2}-\frac{3}{8}a+\frac{1}{4}$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{9}-\frac{1}{8}a^{6}-\frac{7}{32}a^{5}-\frac{1}{8}a^{4}-\frac{3}{16}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{512}a^{14}-\frac{1}{512}a^{13}-\frac{5}{512}a^{12}-\frac{3}{256}a^{11}-\frac{15}{512}a^{10}-\frac{15}{512}a^{9}+\frac{25}{512}a^{8}-\frac{11}{256}a^{7}+\frac{27}{512}a^{6}-\frac{113}{512}a^{5}-\frac{75}{512}a^{4}-\frac{1}{64}a^{3}-\frac{37}{128}a^{2}-\frac{1}{8}a-\frac{3}{32}$, $\frac{1}{44\!\cdots\!56}a^{15}-\frac{30\!\cdots\!03}{44\!\cdots\!56}a^{14}-\frac{60\!\cdots\!03}{44\!\cdots\!56}a^{13}-\frac{13\!\cdots\!71}{11\!\cdots\!64}a^{12}+\frac{16\!\cdots\!41}{44\!\cdots\!56}a^{11}-\frac{43\!\cdots\!41}{44\!\cdots\!56}a^{10}-\frac{17\!\cdots\!81}{44\!\cdots\!56}a^{9}+\frac{12\!\cdots\!49}{55\!\cdots\!32}a^{8}-\frac{50\!\cdots\!97}{44\!\cdots\!56}a^{7}+\frac{51\!\cdots\!65}{44\!\cdots\!56}a^{6}+\frac{20\!\cdots\!39}{44\!\cdots\!56}a^{5}+\frac{37\!\cdots\!87}{22\!\cdots\!28}a^{4}-\frac{17\!\cdots\!05}{11\!\cdots\!64}a^{3}-\frac{29\!\cdots\!83}{55\!\cdots\!32}a^{2}-\frac{20\!\cdots\!75}{27\!\cdots\!16}a-\frac{30\!\cdots\!73}{13\!\cdots\!08}$
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: |
$\frac{17\!\cdots\!33}{20\!\cdots\!72}a^{15}-\frac{12\!\cdots\!29}{20\!\cdots\!72}a^{14}-\frac{44\!\cdots\!41}{20\!\cdots\!72}a^{13}+\frac{19\!\cdots\!97}{10\!\cdots\!36}a^{12}+\frac{62\!\cdots\!93}{20\!\cdots\!72}a^{11}-\frac{62\!\cdots\!11}{20\!\cdots\!72}a^{10}-\frac{84\!\cdots\!95}{20\!\cdots\!72}a^{9}+\frac{32\!\cdots\!17}{10\!\cdots\!36}a^{8}+\frac{14\!\cdots\!87}{20\!\cdots\!72}a^{7}-\frac{24\!\cdots\!09}{20\!\cdots\!72}a^{6}-\frac{11\!\cdots\!11}{20\!\cdots\!72}a^{5}-\frac{36\!\cdots\!67}{62\!\cdots\!96}a^{4}-\frac{40\!\cdots\!29}{50\!\cdots\!68}a^{3}+\frac{40\!\cdots\!27}{62\!\cdots\!96}a^{2}+\frac{70\!\cdots\!81}{12\!\cdots\!92}a-\frac{31\!\cdots\!39}{15\!\cdots\!24}$, $\frac{14\!\cdots\!33}{22\!\cdots\!28}a^{15}-\frac{64\!\cdots\!61}{27\!\cdots\!16}a^{14}-\frac{35\!\cdots\!91}{11\!\cdots\!64}a^{13}+\frac{16\!\cdots\!89}{22\!\cdots\!28}a^{12}+\frac{15\!\cdots\!79}{22\!\cdots\!28}a^{11}-\frac{13\!\cdots\!11}{11\!\cdots\!64}a^{10}-\frac{11\!\cdots\!47}{11\!\cdots\!64}a^{9}+\frac{17\!\cdots\!15}{22\!\cdots\!28}a^{8}+\frac{28\!\cdots\!85}{22\!\cdots\!28}a^{7}+\frac{16\!\cdots\!25}{11\!\cdots\!64}a^{6}-\frac{33\!\cdots\!37}{55\!\cdots\!32}a^{5}-\frac{45\!\cdots\!47}{22\!\cdots\!28}a^{4}-\frac{14\!\cdots\!47}{55\!\cdots\!32}a^{3}-\frac{10\!\cdots\!37}{55\!\cdots\!32}a^{2}-\frac{21\!\cdots\!99}{13\!\cdots\!08}a-\frac{10\!\cdots\!19}{13\!\cdots\!08}$, $\frac{58\!\cdots\!45}{22\!\cdots\!28}a^{15}-\frac{11\!\cdots\!89}{55\!\cdots\!32}a^{14}-\frac{59\!\cdots\!93}{11\!\cdots\!64}a^{13}+\frac{14\!\cdots\!01}{22\!\cdots\!28}a^{12}+\frac{13\!\cdots\!63}{22\!\cdots\!28}a^{11}-\frac{11\!\cdots\!25}{11\!\cdots\!64}a^{10}-\frac{89\!\cdots\!81}{11\!\cdots\!64}a^{9}+\frac{25\!\cdots\!39}{22\!\cdots\!28}a^{8}+\frac{39\!\cdots\!73}{22\!\cdots\!28}a^{7}-\frac{69\!\cdots\!93}{11\!\cdots\!64}a^{6}-\frac{53\!\cdots\!07}{27\!\cdots\!16}a^{5}-\frac{10\!\cdots\!19}{22\!\cdots\!28}a^{4}+\frac{23\!\cdots\!01}{55\!\cdots\!32}a^{3}+\frac{40\!\cdots\!95}{55\!\cdots\!32}a^{2}+\frac{10\!\cdots\!65}{13\!\cdots\!08}a+\frac{53\!\cdots\!41}{13\!\cdots\!08}$, $\frac{30\!\cdots\!89}{11\!\cdots\!64}a^{15}-\frac{15\!\cdots\!27}{11\!\cdots\!64}a^{14}-\frac{10\!\cdots\!19}{11\!\cdots\!64}a^{13}+\frac{11\!\cdots\!41}{27\!\cdots\!16}a^{12}+\frac{20\!\cdots\!65}{11\!\cdots\!64}a^{11}-\frac{75\!\cdots\!57}{11\!\cdots\!64}a^{10}-\frac{31\!\cdots\!69}{11\!\cdots\!64}a^{9}+\frac{82\!\cdots\!93}{13\!\cdots\!08}a^{8}+\frac{40\!\cdots\!91}{11\!\cdots\!64}a^{7}+\frac{20\!\cdots\!65}{11\!\cdots\!64}a^{6}-\frac{21\!\cdots\!29}{11\!\cdots\!64}a^{5}-\frac{29\!\cdots\!33}{55\!\cdots\!32}a^{4}-\frac{19\!\cdots\!07}{27\!\cdots\!16}a^{3}-\frac{82\!\cdots\!03}{13\!\cdots\!08}a^{2}-\frac{19\!\cdots\!23}{69\!\cdots\!04}a-\frac{15\!\cdots\!91}{34\!\cdots\!52}$, $\frac{61\!\cdots\!07}{44\!\cdots\!56}a^{15}-\frac{25\!\cdots\!57}{44\!\cdots\!56}a^{14}-\frac{25\!\cdots\!73}{44\!\cdots\!56}a^{13}+\frac{19\!\cdots\!41}{11\!\cdots\!64}a^{12}+\frac{53\!\cdots\!39}{44\!\cdots\!56}a^{11}-\frac{11\!\cdots\!87}{44\!\cdots\!56}a^{10}-\frac{80\!\cdots\!71}{44\!\cdots\!56}a^{9}+\frac{24\!\cdots\!73}{13\!\cdots\!08}a^{8}+\frac{98\!\cdots\!77}{44\!\cdots\!56}a^{7}+\frac{11\!\cdots\!87}{44\!\cdots\!56}a^{6}-\frac{41\!\cdots\!79}{44\!\cdots\!56}a^{5}-\frac{80\!\cdots\!95}{22\!\cdots\!28}a^{4}-\frac{64\!\cdots\!19}{11\!\cdots\!64}a^{3}-\frac{32\!\cdots\!57}{55\!\cdots\!32}a^{2}-\frac{10\!\cdots\!81}{27\!\cdots\!16}a-\frac{13\!\cdots\!15}{13\!\cdots\!08}$, $\frac{61\!\cdots\!19}{22\!\cdots\!28}a^{15}+\frac{60\!\cdots\!35}{11\!\cdots\!64}a^{14}-\frac{12\!\cdots\!19}{69\!\cdots\!04}a^{13}-\frac{93\!\cdots\!95}{22\!\cdots\!28}a^{12}+\frac{10\!\cdots\!29}{22\!\cdots\!28}a^{11}+\frac{63\!\cdots\!29}{55\!\cdots\!32}a^{10}-\frac{41\!\cdots\!61}{55\!\cdots\!32}a^{9}-\frac{49\!\cdots\!57}{22\!\cdots\!28}a^{8}+\frac{17\!\cdots\!67}{22\!\cdots\!28}a^{7}+\frac{10\!\cdots\!03}{27\!\cdots\!16}a^{6}+\frac{29\!\cdots\!51}{11\!\cdots\!64}a^{5}-\frac{53\!\cdots\!27}{22\!\cdots\!28}a^{4}-\frac{30\!\cdots\!73}{55\!\cdots\!32}a^{3}-\frac{29\!\cdots\!37}{55\!\cdots\!32}a^{2}-\frac{26\!\cdots\!61}{13\!\cdots\!08}a+\frac{37\!\cdots\!13}{13\!\cdots\!08}$, $\frac{22\!\cdots\!83}{22\!\cdots\!28}a^{15}-\frac{12\!\cdots\!71}{22\!\cdots\!28}a^{14}-\frac{68\!\cdots\!43}{22\!\cdots\!28}a^{13}+\frac{17\!\cdots\!13}{11\!\cdots\!64}a^{12}+\frac{13\!\cdots\!23}{22\!\cdots\!28}a^{11}-\frac{54\!\cdots\!85}{22\!\cdots\!28}a^{10}-\frac{19\!\cdots\!61}{22\!\cdots\!28}a^{9}+\frac{23\!\cdots\!45}{11\!\cdots\!64}a^{8}+\frac{26\!\cdots\!29}{22\!\cdots\!28}a^{7}+\frac{13\!\cdots\!85}{22\!\cdots\!28}a^{6}-\frac{13\!\cdots\!97}{22\!\cdots\!28}a^{5}-\frac{10\!\cdots\!27}{55\!\cdots\!32}a^{4}-\frac{14\!\cdots\!71}{55\!\cdots\!32}a^{3}-\frac{34\!\cdots\!09}{13\!\cdots\!08}a^{2}-\frac{19\!\cdots\!09}{13\!\cdots\!08}a-\frac{12\!\cdots\!97}{34\!\cdots\!52}$, $\frac{80\!\cdots\!65}{44\!\cdots\!56}a^{15}-\frac{47\!\cdots\!17}{44\!\cdots\!56}a^{14}-\frac{24\!\cdots\!69}{44\!\cdots\!56}a^{13}+\frac{71\!\cdots\!85}{22\!\cdots\!28}a^{12}+\frac{45\!\cdots\!89}{44\!\cdots\!56}a^{11}-\frac{22\!\cdots\!19}{44\!\cdots\!56}a^{10}-\frac{69\!\cdots\!19}{44\!\cdots\!56}a^{9}+\frac{10\!\cdots\!45}{22\!\cdots\!28}a^{8}+\frac{97\!\cdots\!11}{44\!\cdots\!56}a^{7}-\frac{12\!\cdots\!45}{44\!\cdots\!56}a^{6}-\frac{61\!\cdots\!75}{44\!\cdots\!56}a^{5}-\frac{18\!\cdots\!01}{69\!\cdots\!04}a^{4}-\frac{22\!\cdots\!85}{11\!\cdots\!64}a^{3}-\frac{10\!\cdots\!33}{13\!\cdots\!08}a^{2}+\frac{25\!\cdots\!53}{27\!\cdots\!16}a+\frac{40\!\cdots\!17}{34\!\cdots\!52}$, $\frac{31\!\cdots\!53}{55\!\cdots\!32}a^{15}-\frac{73\!\cdots\!83}{22\!\cdots\!28}a^{14}-\frac{40\!\cdots\!73}{22\!\cdots\!28}a^{13}+\frac{22\!\cdots\!43}{22\!\cdots\!28}a^{12}+\frac{37\!\cdots\!47}{11\!\cdots\!64}a^{11}-\frac{35\!\cdots\!03}{22\!\cdots\!28}a^{10}-\frac{11\!\cdots\!11}{22\!\cdots\!28}a^{9}+\frac{33\!\cdots\!85}{22\!\cdots\!28}a^{8}+\frac{75\!\cdots\!87}{11\!\cdots\!64}a^{7}-\frac{37\!\cdots\!45}{22\!\cdots\!28}a^{6}-\frac{88\!\cdots\!33}{22\!\cdots\!28}a^{5}-\frac{18\!\cdots\!43}{22\!\cdots\!28}a^{4}-\frac{25\!\cdots\!19}{27\!\cdots\!16}a^{3}-\frac{48\!\cdots\!37}{55\!\cdots\!32}a^{2}-\frac{19\!\cdots\!13}{34\!\cdots\!52}a-\frac{72\!\cdots\!95}{13\!\cdots\!08}$, $\frac{25\!\cdots\!39}{27\!\cdots\!16}a^{15}-\frac{50\!\cdots\!81}{11\!\cdots\!64}a^{14}-\frac{38\!\cdots\!51}{11\!\cdots\!64}a^{13}+\frac{15\!\cdots\!21}{11\!\cdots\!64}a^{12}+\frac{37\!\cdots\!77}{55\!\cdots\!32}a^{11}-\frac{25\!\cdots\!01}{11\!\cdots\!64}a^{10}-\frac{11\!\cdots\!49}{11\!\cdots\!64}a^{9}+\frac{22\!\cdots\!15}{11\!\cdots\!64}a^{8}+\frac{72\!\cdots\!05}{55\!\cdots\!32}a^{7}+\frac{77\!\cdots\!57}{11\!\cdots\!64}a^{6}-\frac{75\!\cdots\!95}{11\!\cdots\!64}a^{5}-\frac{20\!\cdots\!85}{11\!\cdots\!64}a^{4}-\frac{16\!\cdots\!27}{69\!\cdots\!04}a^{3}-\frac{52\!\cdots\!47}{27\!\cdots\!16}a^{2}-\frac{32\!\cdots\!52}{43\!\cdots\!69}a+\frac{10\!\cdots\!11}{69\!\cdots\!04}$, $\frac{51\!\cdots\!93}{22\!\cdots\!28}a^{15}-\frac{21\!\cdots\!83}{22\!\cdots\!28}a^{14}-\frac{22\!\cdots\!31}{22\!\cdots\!28}a^{13}+\frac{17\!\cdots\!15}{55\!\cdots\!32}a^{12}+\frac{45\!\cdots\!17}{22\!\cdots\!28}a^{11}-\frac{11\!\cdots\!01}{22\!\cdots\!28}a^{10}-\frac{68\!\cdots\!97}{22\!\cdots\!28}a^{9}+\frac{55\!\cdots\!79}{13\!\cdots\!08}a^{8}+\frac{84\!\cdots\!03}{22\!\cdots\!28}a^{7}+\frac{74\!\cdots\!25}{22\!\cdots\!28}a^{6}-\frac{39\!\cdots\!81}{22\!\cdots\!28}a^{5}-\frac{63\!\cdots\!37}{11\!\cdots\!64}a^{4}-\frac{43\!\cdots\!37}{55\!\cdots\!32}a^{3}-\frac{18\!\cdots\!51}{27\!\cdots\!16}a^{2}-\frac{48\!\cdots\!99}{13\!\cdots\!08}a+\frac{30\!\cdots\!99}{69\!\cdots\!04}$
| 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: | \( 9033509529.05 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 9033509529.05 \cdot 1}{2\cdot\sqrt{800737337114777368288115578449}}\cr\approx \mathstrut & 2.01391395084 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408)
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 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408, 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 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408);
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 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408);
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
$C_2^2:C_8$ (as 16T24):
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 $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{73}) \), 4.4.389017.1, 4.2.15987.1, 4.2.1167051.1, 8.4.99426586671873.1, 8.8.894839280046857.1, 8.4.1362008036601.1 |
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 16 sibling: | 16.0.9885646137219473682569328129.1 |
| Minimal sibling: | 16.0.9885646137219473682569328129.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(73\)
| 73.16.14.1 | $x^{16} + 560 x^{15} + 137240 x^{14} + 19227600 x^{13} + 1684816700 x^{12} + 94599694000 x^{11} + 3327837457000 x^{10} + 67300032450000 x^{9} + 609674268043896 x^{8} + 336500162290880 x^{7} + 83195946420160 x^{6} + 11826359641600 x^{5} + 1175201013000 x^{4} + 6895769020000 x^{3} + 239006660174000 x^{2} + 4775207729180000 x + 41740387870087204$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |