Normalized defining polynomial
\( x^{18} - 9 x^{17} - 63 x^{16} + 708 x^{15} + 1962 x^{14} - 26838 x^{13} - 31584 x^{12} + \cdots + 9800651699 \)
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: |
\(-32139998655544385099809577787619644058971\)
\(\medspace = -\,3^{44}\cdot 7^{12}\cdot 11^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(177.99\) | 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: | $3^{22/9}7^{2/3}11^{1/2}\approx 177.98829233603956$ | ||
| Ramified primes: |
\(3\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2079=3^{3}\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2079}(1,·)$, $\chi_{2079}(1222,·)$, $\chi_{2079}(1033,·)$, $\chi_{2079}(529,·)$, $\chi_{2079}(340,·)$, $\chi_{2079}(1948,·)$, $\chi_{2079}(1759,·)$, $\chi_{2079}(1891,·)$, $\chi_{2079}(1255,·)$, $\chi_{2079}(1066,·)$, $\chi_{2079}(1387,·)$, $\chi_{2079}(1198,·)$, $\chi_{2079}(562,·)$, $\chi_{2079}(373,·)$, $\chi_{2079}(694,·)$, $\chi_{2079}(505,·)$, $\chi_{2079}(1915,·)$, $\chi_{2079}(1726,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{7}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-\frac{3}{7}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{8}-\frac{2}{7}a^{5}-\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{9}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{133}a^{10}-\frac{8}{133}a^{9}+\frac{4}{133}a^{8}+\frac{1}{19}a^{7}+\frac{1}{19}a^{6}-\frac{2}{19}a^{5}+\frac{1}{7}a^{4}+\frac{27}{133}a^{3}-\frac{37}{133}a^{2}+\frac{32}{133}a+\frac{5}{19}$, $\frac{1}{133}a^{11}-\frac{3}{133}a^{9}+\frac{1}{133}a^{8}+\frac{6}{133}a^{7}+\frac{4}{133}a^{6}+\frac{59}{133}a^{5}+\frac{46}{133}a^{4}-\frac{7}{19}a^{3}-\frac{55}{133}a^{2}-\frac{13}{133}a-\frac{5}{133}$, $\frac{1}{931}a^{12}+\frac{1}{931}a^{11}-\frac{2}{931}a^{10}+\frac{9}{931}a^{9}-\frac{27}{931}a^{8}-\frac{40}{931}a^{7}+\frac{13}{931}a^{6}+\frac{34}{931}a^{5}-\frac{383}{931}a^{4}+\frac{208}{931}a^{3}-\frac{295}{931}a^{2}-\frac{233}{931}a-\frac{27}{931}$, $\frac{1}{931}a^{13}-\frac{3}{931}a^{11}-\frac{3}{931}a^{10}-\frac{3}{49}a^{9}+\frac{64}{931}a^{8}-\frac{45}{931}a^{7}+\frac{8}{133}a^{6}+\frac{45}{931}a^{5}-\frac{207}{931}a^{4}-\frac{349}{931}a^{3}+\frac{181}{931}a^{2}+\frac{290}{931}a+\frac{69}{931}$, $\frac{1}{931}a^{14}-\frac{2}{133}a^{9}-\frac{1}{133}a^{8}-\frac{22}{931}a^{7}-\frac{1}{133}a^{6}-\frac{46}{133}a^{5}+\frac{2}{19}a^{4}+\frac{16}{133}a^{3}+\frac{8}{133}a-\frac{137}{931}$, $\frac{1}{931}a^{15}+\frac{2}{133}a^{9}+\frac{34}{931}a^{8}-\frac{6}{133}a^{7}+\frac{6}{133}a^{6}-\frac{52}{133}a^{5}-\frac{60}{133}a^{4}+\frac{54}{133}a^{3}-\frac{47}{133}a^{2}+\frac{45}{931}a+\frac{51}{133}$, $\frac{1}{16488941}a^{16}+\frac{1125}{16488941}a^{15}+\frac{4190}{16488941}a^{14}+\frac{410}{16488941}a^{13}-\frac{1237}{2355563}a^{12}-\frac{2234}{867839}a^{11}-\frac{35789}{16488941}a^{10}+\frac{858461}{16488941}a^{9}-\frac{387367}{16488941}a^{8}-\frac{760279}{16488941}a^{7}-\frac{5268}{123977}a^{6}-\frac{5748731}{16488941}a^{5}+\frac{267524}{16488941}a^{4}-\frac{3562940}{16488941}a^{3}+\frac{1373455}{16488941}a^{2}-\frac{6976082}{16488941}a-\frac{7629852}{16488941}$, $\frac{1}{11\!\cdots\!49}a^{17}+\frac{23\!\cdots\!05}{11\!\cdots\!49}a^{16}+\frac{12\!\cdots\!50}{11\!\cdots\!49}a^{15}+\frac{16\!\cdots\!53}{15\!\cdots\!07}a^{14}+\frac{75\!\cdots\!00}{58\!\cdots\!71}a^{13}-\frac{40\!\cdots\!68}{11\!\cdots\!49}a^{12}+\frac{12\!\cdots\!26}{11\!\cdots\!49}a^{11}+\frac{23\!\cdots\!81}{11\!\cdots\!49}a^{10}+\frac{19\!\cdots\!33}{11\!\cdots\!49}a^{9}-\frac{14\!\cdots\!86}{11\!\cdots\!49}a^{8}+\frac{15\!\cdots\!19}{11\!\cdots\!49}a^{7}+\frac{66\!\cdots\!39}{11\!\cdots\!49}a^{6}+\frac{25\!\cdots\!95}{11\!\cdots\!49}a^{5}-\frac{46\!\cdots\!13}{11\!\cdots\!49}a^{4}+\frac{27\!\cdots\!85}{58\!\cdots\!71}a^{3}-\frac{51\!\cdots\!61}{11\!\cdots\!49}a^{2}+\frac{53\!\cdots\!30}{11\!\cdots\!49}a-\frac{71\!\cdots\!02}{11\!\cdots\!49}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{3}\times C_{277677}$, which has order $833031$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: |
$\frac{32\!\cdots\!78}{59\!\cdots\!61}a^{17}-\frac{12\!\cdots\!28}{41\!\cdots\!27}a^{16}-\frac{20\!\cdots\!56}{41\!\cdots\!27}a^{15}+\frac{12\!\cdots\!29}{59\!\cdots\!61}a^{14}+\frac{15\!\cdots\!85}{59\!\cdots\!61}a^{13}-\frac{29\!\cdots\!75}{41\!\cdots\!27}a^{12}-\frac{35\!\cdots\!47}{41\!\cdots\!27}a^{11}+\frac{53\!\cdots\!91}{41\!\cdots\!27}a^{10}+\frac{39\!\cdots\!25}{22\!\cdots\!33}a^{9}-\frac{45\!\cdots\!16}{41\!\cdots\!27}a^{8}-\frac{97\!\cdots\!81}{41\!\cdots\!27}a^{7}-\frac{20\!\cdots\!23}{41\!\cdots\!27}a^{6}+\frac{79\!\cdots\!91}{41\!\cdots\!27}a^{5}+\frac{89\!\cdots\!64}{41\!\cdots\!27}a^{4}-\frac{29\!\cdots\!19}{41\!\cdots\!27}a^{3}-\frac{10\!\cdots\!37}{47\!\cdots\!43}a^{2}-\frac{88\!\cdots\!43}{47\!\cdots\!43}a+\frac{19\!\cdots\!75}{41\!\cdots\!27}$, $\frac{97\!\cdots\!20}{59\!\cdots\!61}a^{17}-\frac{68\!\cdots\!53}{41\!\cdots\!27}a^{16}-\frac{37\!\cdots\!14}{41\!\cdots\!27}a^{15}+\frac{81\!\cdots\!90}{59\!\cdots\!61}a^{14}+\frac{99\!\cdots\!68}{59\!\cdots\!61}a^{13}-\frac{22\!\cdots\!76}{41\!\cdots\!27}a^{12}+\frac{87\!\cdots\!50}{41\!\cdots\!27}a^{11}+\frac{49\!\cdots\!11}{41\!\cdots\!27}a^{10}-\frac{29\!\cdots\!73}{22\!\cdots\!33}a^{9}-\frac{67\!\cdots\!18}{41\!\cdots\!27}a^{8}+\frac{10\!\cdots\!50}{41\!\cdots\!27}a^{7}+\frac{55\!\cdots\!78}{41\!\cdots\!27}a^{6}-\frac{93\!\cdots\!78}{41\!\cdots\!27}a^{5}-\frac{28\!\cdots\!64}{41\!\cdots\!27}a^{4}+\frac{42\!\cdots\!45}{41\!\cdots\!27}a^{3}+\frac{90\!\cdots\!56}{47\!\cdots\!43}a^{2}+\frac{21\!\cdots\!78}{47\!\cdots\!43}a-\frac{82\!\cdots\!37}{41\!\cdots\!27}$, $\frac{35\!\cdots\!28}{11\!\cdots\!49}a^{17}-\frac{72\!\cdots\!59}{15\!\cdots\!07}a^{16}-\frac{16\!\cdots\!50}{11\!\cdots\!49}a^{15}+\frac{42\!\cdots\!71}{11\!\cdots\!49}a^{14}+\frac{42\!\cdots\!69}{15\!\cdots\!07}a^{13}-\frac{16\!\cdots\!51}{11\!\cdots\!49}a^{12}-\frac{27\!\cdots\!81}{11\!\cdots\!49}a^{11}+\frac{37\!\cdots\!99}{11\!\cdots\!49}a^{10}+\frac{14\!\cdots\!93}{11\!\cdots\!49}a^{9}-\frac{50\!\cdots\!61}{11\!\cdots\!49}a^{8}-\frac{48\!\cdots\!63}{11\!\cdots\!49}a^{7}+\frac{32\!\cdots\!90}{11\!\cdots\!49}a^{6}+\frac{75\!\cdots\!48}{11\!\cdots\!49}a^{5}-\frac{14\!\cdots\!64}{11\!\cdots\!49}a^{4}-\frac{40\!\cdots\!80}{11\!\cdots\!49}a^{3}-\frac{11\!\cdots\!27}{11\!\cdots\!49}a^{2}-\frac{23\!\cdots\!74}{15\!\cdots\!07}a-\frac{19\!\cdots\!07}{58\!\cdots\!71}$, $\frac{36\!\cdots\!28}{11\!\cdots\!49}a^{17}-\frac{25\!\cdots\!05}{83\!\cdots\!53}a^{16}-\frac{19\!\cdots\!26}{11\!\cdots\!49}a^{15}+\frac{24\!\cdots\!76}{11\!\cdots\!49}a^{14}+\frac{56\!\cdots\!46}{15\!\cdots\!07}a^{13}-\frac{89\!\cdots\!43}{11\!\cdots\!49}a^{12}+\frac{18\!\cdots\!92}{11\!\cdots\!49}a^{11}+\frac{18\!\cdots\!09}{11\!\cdots\!49}a^{10}-\frac{20\!\cdots\!50}{11\!\cdots\!49}a^{9}-\frac{32\!\cdots\!32}{15\!\cdots\!07}a^{8}+\frac{61\!\cdots\!42}{15\!\cdots\!07}a^{7}+\frac{16\!\cdots\!94}{11\!\cdots\!49}a^{6}-\frac{36\!\cdots\!21}{11\!\cdots\!49}a^{5}-\frac{91\!\cdots\!67}{11\!\cdots\!49}a^{4}+\frac{15\!\cdots\!65}{11\!\cdots\!49}a^{3}+\frac{36\!\cdots\!54}{11\!\cdots\!49}a^{2}+\frac{65\!\cdots\!16}{11\!\cdots\!49}a-\frac{30\!\cdots\!77}{15\!\cdots\!07}$, $\frac{57\!\cdots\!96}{22\!\cdots\!01}a^{17}-\frac{36\!\cdots\!61}{11\!\cdots\!49}a^{16}-\frac{13\!\cdots\!08}{11\!\cdots\!49}a^{15}+\frac{43\!\cdots\!88}{15\!\cdots\!07}a^{14}+\frac{31\!\cdots\!60}{15\!\cdots\!07}a^{13}-\frac{12\!\cdots\!81}{11\!\cdots\!49}a^{12}+\frac{32\!\cdots\!72}{11\!\cdots\!49}a^{11}+\frac{29\!\cdots\!52}{11\!\cdots\!49}a^{10}-\frac{14\!\cdots\!39}{11\!\cdots\!49}a^{9}-\frac{42\!\cdots\!44}{11\!\cdots\!49}a^{8}+\frac{21\!\cdots\!54}{11\!\cdots\!49}a^{7}+\frac{34\!\cdots\!36}{11\!\cdots\!49}a^{6}-\frac{13\!\cdots\!43}{11\!\cdots\!49}a^{5}-\frac{13\!\cdots\!63}{11\!\cdots\!49}a^{4}+\frac{84\!\cdots\!44}{11\!\cdots\!49}a^{3}-\frac{42\!\cdots\!67}{58\!\cdots\!71}a^{2}-\frac{23\!\cdots\!32}{58\!\cdots\!71}a-\frac{19\!\cdots\!21}{11\!\cdots\!49}$, $\frac{77\!\cdots\!50}{11\!\cdots\!49}a^{17}-\frac{12\!\cdots\!92}{15\!\cdots\!07}a^{16}-\frac{38\!\cdots\!64}{11\!\cdots\!49}a^{15}+\frac{97\!\cdots\!00}{15\!\cdots\!07}a^{14}+\frac{34\!\cdots\!62}{83\!\cdots\!53}a^{13}-\frac{25\!\cdots\!50}{11\!\cdots\!49}a^{12}+\frac{17\!\cdots\!86}{11\!\cdots\!49}a^{11}+\frac{50\!\cdots\!03}{11\!\cdots\!49}a^{10}-\frac{33\!\cdots\!55}{58\!\cdots\!71}a^{9}-\frac{56\!\cdots\!59}{11\!\cdots\!49}a^{8}+\frac{81\!\cdots\!06}{11\!\cdots\!49}a^{7}+\frac{31\!\cdots\!79}{11\!\cdots\!49}a^{6}-\frac{28\!\cdots\!85}{11\!\cdots\!49}a^{5}-\frac{70\!\cdots\!69}{11\!\cdots\!49}a^{4}-\frac{12\!\cdots\!12}{11\!\cdots\!49}a^{3}+\frac{23\!\cdots\!81}{11\!\cdots\!49}a^{2}-\frac{83\!\cdots\!55}{11\!\cdots\!49}a+\frac{23\!\cdots\!00}{11\!\cdots\!49}$, $\frac{10\!\cdots\!90}{11\!\cdots\!49}a^{17}-\frac{91\!\cdots\!06}{15\!\cdots\!07}a^{16}-\frac{12\!\cdots\!96}{11\!\cdots\!49}a^{15}+\frac{76\!\cdots\!71}{11\!\cdots\!49}a^{14}+\frac{70\!\cdots\!77}{15\!\cdots\!07}a^{13}-\frac{33\!\cdots\!81}{11\!\cdots\!49}a^{12}-\frac{10\!\cdots\!27}{11\!\cdots\!49}a^{11}+\frac{74\!\cdots\!33}{11\!\cdots\!49}a^{10}+\frac{11\!\cdots\!39}{11\!\cdots\!49}a^{9}-\frac{13\!\cdots\!00}{15\!\cdots\!07}a^{8}-\frac{88\!\cdots\!19}{15\!\cdots\!07}a^{7}+\frac{68\!\cdots\!21}{11\!\cdots\!49}a^{6}-\frac{11\!\cdots\!19}{11\!\cdots\!49}a^{5}-\frac{20\!\cdots\!45}{11\!\cdots\!49}a^{4}+\frac{18\!\cdots\!81}{11\!\cdots\!49}a^{3}-\frac{15\!\cdots\!50}{11\!\cdots\!49}a^{2}-\frac{10\!\cdots\!73}{11\!\cdots\!49}a+\frac{44\!\cdots\!36}{15\!\cdots\!07}$, $\frac{52\!\cdots\!22}{11\!\cdots\!49}a^{17}-\frac{39\!\cdots\!30}{11\!\cdots\!49}a^{16}-\frac{17\!\cdots\!94}{58\!\cdots\!71}a^{15}+\frac{31\!\cdots\!28}{11\!\cdots\!49}a^{14}+\frac{13\!\cdots\!36}{15\!\cdots\!07}a^{13}-\frac{11\!\cdots\!56}{11\!\cdots\!49}a^{12}-\frac{10\!\cdots\!94}{11\!\cdots\!49}a^{11}+\frac{25\!\cdots\!58}{11\!\cdots\!49}a^{10}-\frac{95\!\cdots\!74}{11\!\cdots\!49}a^{9}-\frac{34\!\cdots\!18}{11\!\cdots\!49}a^{8}+\frac{46\!\cdots\!94}{11\!\cdots\!49}a^{7}+\frac{29\!\cdots\!85}{11\!\cdots\!49}a^{6}-\frac{27\!\cdots\!79}{55\!\cdots\!51}a^{5}-\frac{16\!\cdots\!10}{11\!\cdots\!49}a^{4}+\frac{31\!\cdots\!49}{11\!\cdots\!49}a^{3}+\frac{50\!\cdots\!00}{11\!\cdots\!49}a^{2}+\frac{18\!\cdots\!73}{11\!\cdots\!49}a+\frac{30\!\cdots\!91}{58\!\cdots\!71}$
| 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: | \( 10392888.21418944 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 10392888.21418944 \cdot 833031}{2\cdot\sqrt{32139998655544385099809577787619644058971}}\cr\approx \mathstrut & 0.368521908652069 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{9})^+\), 6.0.8732691.1, 9.9.3691950281939241.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | R | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | R | R | $18$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
|
\(7\)
| 7.18.12.2 | $x^{18} - 14 x^{15} + 441 x^{12} + 3773 x^{9} - 91238 x^{6} + 201684 x^{3} + 1058841$ | $3$ | $6$ | $12$ | $C_{18}$ | $[\ ]_{3}^{6}$ |
|
\(11\)
| 11.18.9.2 | $x^{18} + 99 x^{16} + 4356 x^{14} + 111804 x^{12} + 18 x^{11} + 1844782 x^{10} - 3744 x^{9} + 20287674 x^{8} - 20196 x^{7} + 148892436 x^{6} + 1280664 x^{5} + 702432669 x^{4} + 3521970 x^{3} + 1922907025 x^{2} - 17918856 x + 2360533311$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |