Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096)
gp: K = bnfinit(y^18 - 3*y^17 + 10*y^16 - 9*y^15 - 7*y^14 + 118*y^13 + 42*y^12 - 536*y^11 + 2032*y^10 - 3800*y^9 + 4088*y^8 + 1616*y^7 - 9680*y^6 + 16256*y^5 - 2176*y^4 - 33024*y^3 + 44288*y^2 - 22528*y + 4096, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096)
\( x^{18} - 3 x^{17} + 10 x^{16} - 9 x^{15} - 7 x^{14} + 118 x^{13} + 42 x^{12} - 536 x^{11} + 2032 x^{10} + \cdots + 4096 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-200930163501792205662554161152\)
\(\medspace = -\,2^{18}\cdot 3^{9}\cdot 7^{10}\cdot 13^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.46\) | 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: | $2^{3/2}3^{1/2}7^{5/6}13^{5/6}\approx 210.20358299362027$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}+\frac{3}{32}a^{8}+\frac{1}{16}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{9}-\frac{1}{32}a^{8}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{13}-\frac{1}{64}a^{11}-\frac{1}{64}a^{10}+\frac{1}{16}a^{9}-\frac{3}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{3}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{15}+\frac{1}{256}a^{14}+\frac{1}{128}a^{13}+\frac{3}{256}a^{12}-\frac{3}{256}a^{11}+\frac{3}{128}a^{10}-\frac{5}{128}a^{9}+\frac{1}{32}a^{8}+\frac{1}{32}a^{7}+\frac{5}{32}a^{6}-\frac{1}{32}a^{5}-\frac{3}{16}a^{4}+\frac{5}{16}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{123392}a^{16}-\frac{5}{123392}a^{15}-\frac{109}{30848}a^{14}-\frac{713}{123392}a^{13}-\frac{277}{123392}a^{12}-\frac{245}{15424}a^{11}-\frac{2343}{61696}a^{10}+\frac{2001}{30848}a^{9}+\frac{737}{15424}a^{8}-\frac{277}{15424}a^{7}+\frac{2393}{15424}a^{6}-\frac{26}{241}a^{5}-\frac{849}{7712}a^{4}+\frac{181}{3856}a^{3}-\frac{311}{1928}a^{2}+\frac{145}{964}a-\frac{93}{241}$, $\frac{1}{15\!\cdots\!28}a^{17}-\frac{57\!\cdots\!23}{15\!\cdots\!28}a^{16}-\frac{33\!\cdots\!33}{79\!\cdots\!64}a^{15}-\frac{11\!\cdots\!25}{15\!\cdots\!28}a^{14}+\frac{12\!\cdots\!49}{15\!\cdots\!28}a^{13}-\frac{11\!\cdots\!39}{79\!\cdots\!64}a^{12}-\frac{15\!\cdots\!27}{79\!\cdots\!64}a^{11}+\frac{12\!\cdots\!13}{24\!\cdots\!52}a^{10}-\frac{63\!\cdots\!69}{99\!\cdots\!08}a^{9}+\frac{11\!\cdots\!33}{19\!\cdots\!16}a^{8}-\frac{23\!\cdots\!89}{19\!\cdots\!16}a^{7}+\frac{20\!\cdots\!11}{99\!\cdots\!08}a^{6}-\frac{10\!\cdots\!45}{99\!\cdots\!08}a^{5}+\frac{20\!\cdots\!11}{24\!\cdots\!52}a^{4}-\frac{15\!\cdots\!77}{17\!\cdots\!68}a^{3}+\frac{23\!\cdots\!65}{62\!\cdots\!88}a^{2}+\frac{25\!\cdots\!53}{62\!\cdots\!88}a-\frac{76\!\cdots\!63}{15\!\cdots\!72}$
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{33791063290749}{3853273527406592} a^{17} + \frac{83163500008499}{3853273527406592} a^{16} - \frac{146255895426307}{1926636763703296} a^{15} + \frac{145556417080237}{3853273527406592} a^{14} + \frac{319135754802959}{3853273527406592} a^{13} - \frac{1906863789803913}{1926636763703296} a^{12} - \frac{1740131002464181}{1926636763703296} a^{11} + \frac{1018760688311745}{240829595462912} a^{10} - \frac{3735376769984855}{240829595462912} a^{9} + \frac{12000190567123727}{481659190925824} a^{8} - \frac{10690311906789047}{481659190925824} a^{7} - \frac{6344304316505175}{240829595462912} a^{6} + \frac{17048333286350749}{240829595462912} a^{5} - \frac{6245044816945993}{60207398865728} a^{4} - \frac{1135028298646069}{30103699432864} a^{3} + \frac{4071134319790741}{15051849716432} a^{2} - \frac{3625977850620357}{15051849716432} a + \frac{243921580809807}{3762962429108} \)
(order $6$)
| 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: |
$\frac{48\!\cdots\!01}{15\!\cdots\!28}a^{17}-\frac{95\!\cdots\!11}{15\!\cdots\!28}a^{16}+\frac{20\!\cdots\!87}{79\!\cdots\!64}a^{15}-\frac{78\!\cdots\!53}{15\!\cdots\!28}a^{14}-\frac{18\!\cdots\!27}{15\!\cdots\!28}a^{13}+\frac{27\!\cdots\!61}{79\!\cdots\!64}a^{12}+\frac{35\!\cdots\!89}{79\!\cdots\!64}a^{11}-\frac{11\!\cdots\!45}{12\!\cdots\!76}a^{10}+\frac{54\!\cdots\!67}{99\!\cdots\!08}a^{9}-\frac{13\!\cdots\!55}{19\!\cdots\!16}a^{8}+\frac{16\!\cdots\!07}{19\!\cdots\!16}a^{7}+\frac{90\!\cdots\!79}{99\!\cdots\!08}a^{6}-\frac{18\!\cdots\!21}{99\!\cdots\!08}a^{5}+\frac{10\!\cdots\!71}{24\!\cdots\!52}a^{4}+\frac{29\!\cdots\!75}{17\!\cdots\!68}a^{3}-\frac{39\!\cdots\!75}{62\!\cdots\!88}a^{2}+\frac{54\!\cdots\!09}{62\!\cdots\!88}a-\frac{85\!\cdots\!55}{15\!\cdots\!72}$, $\frac{91\!\cdots\!03}{79\!\cdots\!64}a^{17}-\frac{16\!\cdots\!01}{79\!\cdots\!64}a^{16}+\frac{33\!\cdots\!99}{39\!\cdots\!32}a^{15}+\frac{90\!\cdots\!33}{79\!\cdots\!64}a^{14}-\frac{83\!\cdots\!77}{79\!\cdots\!64}a^{13}+\frac{48\!\cdots\!69}{39\!\cdots\!32}a^{12}+\frac{81\!\cdots\!35}{39\!\cdots\!32}a^{11}-\frac{41\!\cdots\!19}{99\!\cdots\!08}a^{10}+\frac{86\!\cdots\!83}{49\!\cdots\!04}a^{9}-\frac{20\!\cdots\!61}{99\!\cdots\!08}a^{8}+\frac{13\!\cdots\!29}{99\!\cdots\!08}a^{7}+\frac{22\!\cdots\!95}{49\!\cdots\!04}a^{6}-\frac{31\!\cdots\!71}{49\!\cdots\!04}a^{5}+\frac{28\!\cdots\!07}{31\!\cdots\!44}a^{4}+\frac{10\!\cdots\!25}{89\!\cdots\!84}a^{3}-\frac{44\!\cdots\!09}{15\!\cdots\!72}a^{2}+\frac{35\!\cdots\!55}{31\!\cdots\!44}a-\frac{49\!\cdots\!43}{77\!\cdots\!86}$, $\frac{49\!\cdots\!61}{15\!\cdots\!28}a^{17}-\frac{67\!\cdots\!67}{15\!\cdots\!28}a^{16}+\frac{16\!\cdots\!11}{79\!\cdots\!64}a^{15}+\frac{19\!\cdots\!03}{15\!\cdots\!28}a^{14}-\frac{42\!\cdots\!39}{15\!\cdots\!28}a^{13}+\frac{25\!\cdots\!69}{79\!\cdots\!64}a^{12}+\frac{55\!\cdots\!61}{79\!\cdots\!64}a^{11}-\frac{90\!\cdots\!81}{99\!\cdots\!08}a^{10}+\frac{42\!\cdots\!35}{99\!\cdots\!08}a^{9}-\frac{71\!\cdots\!43}{19\!\cdots\!16}a^{8}+\frac{32\!\cdots\!95}{19\!\cdots\!16}a^{7}+\frac{13\!\cdots\!63}{99\!\cdots\!08}a^{6}-\frac{11\!\cdots\!01}{99\!\cdots\!08}a^{5}+\frac{41\!\cdots\!65}{24\!\cdots\!52}a^{4}+\frac{75\!\cdots\!15}{17\!\cdots\!68}a^{3}-\frac{39\!\cdots\!61}{62\!\cdots\!88}a^{2}+\frac{22\!\cdots\!77}{62\!\cdots\!88}a+\frac{31\!\cdots\!73}{15\!\cdots\!72}$, $\frac{94\!\cdots\!61}{49\!\cdots\!04}a^{17}-\frac{91\!\cdots\!41}{19\!\cdots\!16}a^{16}+\frac{32\!\cdots\!89}{19\!\cdots\!16}a^{15}-\frac{19\!\cdots\!05}{24\!\cdots\!52}a^{14}-\frac{37\!\cdots\!95}{19\!\cdots\!16}a^{13}+\frac{42\!\cdots\!41}{19\!\cdots\!16}a^{12}+\frac{49\!\cdots\!37}{24\!\cdots\!52}a^{11}-\frac{91\!\cdots\!77}{99\!\cdots\!08}a^{10}+\frac{16\!\cdots\!27}{49\!\cdots\!04}a^{9}-\frac{13\!\cdots\!91}{24\!\cdots\!52}a^{8}+\frac{11\!\cdots\!29}{24\!\cdots\!52}a^{7}+\frac{14\!\cdots\!59}{24\!\cdots\!52}a^{6}-\frac{59\!\cdots\!81}{38\!\cdots\!43}a^{5}+\frac{26\!\cdots\!45}{12\!\cdots\!76}a^{4}+\frac{77\!\cdots\!49}{89\!\cdots\!84}a^{3}-\frac{18\!\cdots\!15}{31\!\cdots\!44}a^{2}+\frac{75\!\cdots\!95}{15\!\cdots\!72}a-\frac{45\!\cdots\!18}{38\!\cdots\!43}$, $\frac{10\!\cdots\!93}{17\!\cdots\!08}a^{17}-\frac{22\!\cdots\!51}{17\!\cdots\!08}a^{16}+\frac{42\!\cdots\!23}{87\!\cdots\!04}a^{15}-\frac{22\!\cdots\!09}{17\!\cdots\!08}a^{14}-\frac{90\!\cdots\!15}{17\!\cdots\!08}a^{13}+\frac{57\!\cdots\!21}{87\!\cdots\!04}a^{12}+\frac{70\!\cdots\!89}{87\!\cdots\!04}a^{11}-\frac{13\!\cdots\!89}{54\!\cdots\!44}a^{10}+\frac{10\!\cdots\!47}{10\!\cdots\!88}a^{9}-\frac{31\!\cdots\!39}{21\!\cdots\!76}a^{8}+\frac{26\!\cdots\!55}{21\!\cdots\!76}a^{7}+\frac{20\!\cdots\!91}{10\!\cdots\!88}a^{6}-\frac{44\!\cdots\!25}{10\!\cdots\!88}a^{5}+\frac{16\!\cdots\!55}{27\!\cdots\!72}a^{4}+\frac{49\!\cdots\!37}{13\!\cdots\!36}a^{3}-\frac{11\!\cdots\!07}{68\!\cdots\!68}a^{2}+\frac{81\!\cdots\!73}{68\!\cdots\!68}a-\frac{44\!\cdots\!75}{17\!\cdots\!92}$, $\frac{34\!\cdots\!03}{87\!\cdots\!04}a^{17}-\frac{82\!\cdots\!05}{87\!\cdots\!04}a^{16}+\frac{14\!\cdots\!87}{43\!\cdots\!52}a^{15}-\frac{13\!\cdots\!35}{87\!\cdots\!04}a^{14}-\frac{31\!\cdots\!37}{87\!\cdots\!04}a^{13}+\frac{19\!\cdots\!09}{43\!\cdots\!52}a^{12}+\frac{18\!\cdots\!59}{43\!\cdots\!52}a^{11}-\frac{20\!\cdots\!01}{10\!\cdots\!88}a^{10}+\frac{37\!\cdots\!33}{54\!\cdots\!44}a^{9}-\frac{11\!\cdots\!73}{10\!\cdots\!88}a^{8}+\frac{10\!\cdots\!13}{10\!\cdots\!88}a^{7}+\frac{65\!\cdots\!79}{54\!\cdots\!44}a^{6}-\frac{16\!\cdots\!75}{54\!\cdots\!44}a^{5}+\frac{31\!\cdots\!19}{68\!\cdots\!68}a^{4}+\frac{12\!\cdots\!53}{68\!\cdots\!68}a^{3}-\frac{20\!\cdots\!23}{17\!\cdots\!92}a^{2}+\frac{35\!\cdots\!43}{34\!\cdots\!84}a-\frac{23\!\cdots\!63}{85\!\cdots\!46}$, $\frac{18\!\cdots\!81}{28\!\cdots\!88}a^{17}-\frac{21\!\cdots\!35}{14\!\cdots\!44}a^{16}+\frac{15\!\cdots\!07}{28\!\cdots\!88}a^{15}-\frac{75\!\cdots\!31}{28\!\cdots\!88}a^{14}-\frac{52\!\cdots\!53}{89\!\cdots\!84}a^{13}+\frac{20\!\cdots\!79}{28\!\cdots\!88}a^{12}+\frac{47\!\cdots\!95}{71\!\cdots\!72}a^{11}-\frac{42\!\cdots\!69}{14\!\cdots\!44}a^{10}+\frac{19\!\cdots\!89}{17\!\cdots\!68}a^{9}-\frac{31\!\cdots\!23}{17\!\cdots\!68}a^{8}+\frac{14\!\cdots\!89}{89\!\cdots\!84}a^{7}+\frac{69\!\cdots\!69}{35\!\cdots\!36}a^{6}-\frac{45\!\cdots\!47}{89\!\cdots\!84}a^{5}+\frac{13\!\cdots\!01}{17\!\cdots\!68}a^{4}+\frac{13\!\cdots\!97}{44\!\cdots\!92}a^{3}-\frac{86\!\cdots\!41}{44\!\cdots\!92}a^{2}+\frac{95\!\cdots\!16}{55\!\cdots\!49}a-\frac{25\!\cdots\!75}{55\!\cdots\!49}$, $\frac{75\!\cdots\!89}{22\!\cdots\!04}a^{17}-\frac{15\!\cdots\!19}{22\!\cdots\!04}a^{16}+\frac{29\!\cdots\!07}{11\!\cdots\!52}a^{15}-\frac{14\!\cdots\!09}{22\!\cdots\!04}a^{14}-\frac{67\!\cdots\!19}{22\!\cdots\!04}a^{13}+\frac{37\!\cdots\!17}{11\!\cdots\!52}a^{12}+\frac{53\!\cdots\!81}{11\!\cdots\!52}a^{11}-\frac{11\!\cdots\!85}{89\!\cdots\!84}a^{10}+\frac{70\!\cdots\!45}{14\!\cdots\!44}a^{9}-\frac{19\!\cdots\!15}{28\!\cdots\!88}a^{8}+\frac{14\!\cdots\!51}{28\!\cdots\!88}a^{7}+\frac{15\!\cdots\!07}{14\!\cdots\!44}a^{6}-\frac{28\!\cdots\!29}{14\!\cdots\!44}a^{5}+\frac{99\!\cdots\!39}{35\!\cdots\!36}a^{4}+\frac{45\!\cdots\!75}{17\!\cdots\!68}a^{3}-\frac{71\!\cdots\!47}{89\!\cdots\!84}a^{2}+\frac{45\!\cdots\!85}{89\!\cdots\!84}a-\frac{23\!\cdots\!23}{22\!\cdots\!96}$
| 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: | \( 297602831.7716659 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 297602831.7716659 \cdot 1}{6\cdot\sqrt{200930163501792205662554161152}}\cr\approx \mathstrut & 1.68881491303713 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096)
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^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096, 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^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096);
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^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096);
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
$C_3\times S_3^2$ (as 18T46):
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 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.728.1, 6.0.14309568.1, 6.0.223587.2 |
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
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.898666574987777490026496.5 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.5.1 | $x^{6} + 21$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 13.6.5.3 | $x^{6} + 39$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |