Normalized defining polynomial
\( x^{18} - 216 x^{16} + 20151 x^{14} - 1038600 x^{12} + 31811859 x^{10} - 83486 x^{9} + \cdots + 334789624707 \)
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: |
\(-3079394418562350896973143551074069022507008\)
\(\medspace = -\,2^{27}\cdot 3^{44}\cdot 13^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(229.33\) | 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: | $2^{3/2}3^{22/9}13^{2/3}\approx 229.3349766307829$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2808=2^{3}\cdot 3^{3}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2808}(1,·)$, $\chi_{2808}(211,·)$, $\chi_{2808}(2785,·)$, $\chi_{2808}(2635,·)$, $\chi_{2808}(913,·)$, $\chi_{2808}(1147,·)$, $\chi_{2808}(1699,·)$, $\chi_{2808}(1849,·)$, $\chi_{2808}(2401,·)$, $\chi_{2808}(2083,·)$, $\chi_{2808}(529,·)$, $\chi_{2808}(937,·)$, $\chi_{2808}(235,·)$, $\chi_{2808}(2107,·)$, $\chi_{2808}(1171,·)$, $\chi_{2808}(1465,·)$, $\chi_{2808}(763,·)$, $\chi_{2808}(1873,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13}a^{6}+\frac{6}{13}a^{4}-\frac{1}{13}a^{2}-\frac{5}{13}$, $\frac{1}{13}a^{7}+\frac{6}{13}a^{5}-\frac{1}{13}a^{3}-\frac{5}{13}a$, $\frac{1}{13}a^{8}+\frac{2}{13}a^{4}+\frac{1}{13}a^{2}+\frac{4}{13}$, $\frac{1}{13}a^{9}+\frac{2}{13}a^{5}+\frac{1}{13}a^{3}+\frac{4}{13}a$, $\frac{1}{689}a^{10}+\frac{23}{689}a^{9}-\frac{1}{689}a^{8}-\frac{16}{689}a^{7}+\frac{184}{689}a^{5}+\frac{5}{53}a^{4}-\frac{12}{53}a^{3}+\frac{304}{689}a^{2}+\frac{133}{689}a+\frac{175}{689}$, $\frac{1}{689}a^{11}+\frac{7}{689}a^{8}-\frac{3}{689}a^{7}+\frac{25}{689}a^{6}+\frac{179}{689}a^{5}+\frac{151}{689}a^{4}-\frac{30}{689}a^{3}+\frac{190}{689}a^{2}-\frac{287}{689}a+\frac{215}{689}$, $\frac{1}{170183}a^{12}-\frac{1}{170183}a^{10}-\frac{5}{689}a^{9}-\frac{2228}{170183}a^{8}-\frac{17}{689}a^{7}+\frac{6062}{170183}a^{6}-\frac{60}{689}a^{5}-\frac{69843}{170183}a^{4}+\frac{181}{689}a^{3}+\frac{78114}{170183}a^{2}-\frac{23}{53}a+\frac{47631}{170183}$, $\frac{1}{170183}a^{13}-\frac{1}{170183}a^{11}-\frac{5}{170183}a^{9}-\frac{22}{689}a^{8}-\frac{607}{170183}a^{7}-\frac{7}{689}a^{6}+\frac{13396}{170183}a^{5}+\frac{135}{689}a^{4}+\frac{16364}{170183}a^{3}-\frac{210}{689}a^{2}+\frac{41703}{170183}a-\frac{79}{689}$, $\frac{1}{170183}a^{14}-\frac{6}{170183}a^{10}+\frac{2}{53}a^{9}-\frac{2835}{170183}a^{8}-\frac{24}{689}a^{7}+\frac{6367}{170183}a^{6}+\frac{181}{689}a^{5}+\frac{38158}{170183}a^{4}+\frac{24}{689}a^{3}-\frac{37275}{170183}a^{2}-\frac{166}{689}a-\frac{57097}{170183}$, $\frac{1}{170183}a^{15}-\frac{6}{170183}a^{11}-\frac{6540}{170183}a^{9}+\frac{2}{689}a^{8}+\frac{4391}{170183}a^{7}+\frac{22}{689}a^{6}+\frac{47791}{170183}a^{5}+\frac{136}{689}a^{4}+\frac{22005}{170183}a^{3}-\frac{332}{689}a^{2}+\frac{18238}{170183}a-\frac{310}{689}$, $\frac{1}{170183}a^{16}+\frac{123}{170183}a^{10}+\frac{10}{689}a^{9}-\frac{2555}{170183}a^{8}+\frac{18}{689}a^{7}+\frac{5617}{170183}a^{6}-\frac{132}{689}a^{5}-\frac{68296}{170183}a^{4}+\frac{252}{689}a^{3}+\frac{53190}{170183}a^{2}-\frac{50}{689}a+\frac{25942}{170183}$, $\frac{1}{16\!\cdots\!47}a^{17}+\frac{20\!\cdots\!74}{16\!\cdots\!47}a^{16}-\frac{33\!\cdots\!26}{16\!\cdots\!47}a^{15}-\frac{26\!\cdots\!48}{16\!\cdots\!47}a^{14}+\frac{87\!\cdots\!67}{16\!\cdots\!47}a^{13}+\frac{99\!\cdots\!27}{16\!\cdots\!47}a^{12}-\frac{31\!\cdots\!71}{12\!\cdots\!19}a^{11}-\frac{85\!\cdots\!74}{16\!\cdots\!47}a^{10}-\frac{39\!\cdots\!30}{12\!\cdots\!19}a^{9}-\frac{25\!\cdots\!75}{16\!\cdots\!47}a^{8}-\frac{13\!\cdots\!32}{16\!\cdots\!47}a^{7}-\frac{14\!\cdots\!74}{16\!\cdots\!47}a^{6}+\frac{53\!\cdots\!81}{16\!\cdots\!47}a^{5}+\frac{37\!\cdots\!51}{16\!\cdots\!47}a^{4}-\frac{61\!\cdots\!19}{16\!\cdots\!47}a^{3}+\frac{76\!\cdots\!68}{16\!\cdots\!47}a^{2}-\frac{15\!\cdots\!07}{16\!\cdots\!47}a+\frac{92\!\cdots\!37}{16\!\cdots\!47}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{3}\times C_{934911}$, which has order $2804733$ (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: |
$\frac{43\!\cdots\!36}{28\!\cdots\!47}a^{17}-\frac{23\!\cdots\!56}{47\!\cdots\!43}a^{16}-\frac{12\!\cdots\!06}{36\!\cdots\!11}a^{15}+\frac{18\!\cdots\!16}{47\!\cdots\!43}a^{14}+\frac{12\!\cdots\!78}{36\!\cdots\!11}a^{13}-\frac{15\!\cdots\!40}{24\!\cdots\!97}a^{12}-\frac{49\!\cdots\!76}{28\!\cdots\!47}a^{11}+\frac{21\!\cdots\!64}{47\!\cdots\!43}a^{10}+\frac{15\!\cdots\!86}{28\!\cdots\!47}a^{9}-\frac{85\!\cdots\!16}{47\!\cdots\!43}a^{8}-\frac{28\!\cdots\!04}{28\!\cdots\!47}a^{7}+\frac{18\!\cdots\!71}{47\!\cdots\!43}a^{6}+\frac{28\!\cdots\!88}{28\!\cdots\!47}a^{5}-\frac{24\!\cdots\!56}{47\!\cdots\!43}a^{4}-\frac{12\!\cdots\!22}{36\!\cdots\!11}a^{3}+\frac{16\!\cdots\!51}{47\!\cdots\!43}a^{2}-\frac{11\!\cdots\!24}{36\!\cdots\!11}a-\frac{30\!\cdots\!92}{47\!\cdots\!43}$, $\frac{68\!\cdots\!76}{28\!\cdots\!47}a^{17}+\frac{45\!\cdots\!44}{47\!\cdots\!43}a^{16}-\frac{10\!\cdots\!74}{19\!\cdots\!69}a^{15}-\frac{27\!\cdots\!16}{47\!\cdots\!43}a^{14}+\frac{18\!\cdots\!46}{36\!\cdots\!11}a^{13}+\frac{42\!\cdots\!75}{47\!\cdots\!43}a^{12}-\frac{75\!\cdots\!76}{28\!\cdots\!47}a^{11}-\frac{30\!\cdots\!80}{47\!\cdots\!43}a^{10}+\frac{23\!\cdots\!98}{28\!\cdots\!47}a^{9}+\frac{11\!\cdots\!98}{47\!\cdots\!43}a^{8}-\frac{42\!\cdots\!56}{28\!\cdots\!47}a^{7}-\frac{26\!\cdots\!63}{47\!\cdots\!43}a^{6}+\frac{41\!\cdots\!00}{28\!\cdots\!47}a^{5}+\frac{16\!\cdots\!73}{24\!\cdots\!97}a^{4}-\frac{24\!\cdots\!06}{36\!\cdots\!11}a^{3}-\frac{19\!\cdots\!83}{47\!\cdots\!43}a^{2}+\frac{19\!\cdots\!00}{36\!\cdots\!11}a+\frac{78\!\cdots\!41}{47\!\cdots\!43}$, $\frac{23\!\cdots\!94}{16\!\cdots\!47}a^{17}+\frac{94\!\cdots\!52}{16\!\cdots\!47}a^{16}-\frac{48\!\cdots\!74}{16\!\cdots\!47}a^{15}-\frac{82\!\cdots\!43}{30\!\cdots\!99}a^{14}+\frac{41\!\cdots\!76}{16\!\cdots\!47}a^{13}+\frac{58\!\cdots\!27}{16\!\cdots\!47}a^{12}-\frac{18\!\cdots\!32}{16\!\cdots\!47}a^{11}-\frac{35\!\cdots\!58}{16\!\cdots\!47}a^{10}+\frac{42\!\cdots\!58}{16\!\cdots\!47}a^{9}+\frac{80\!\cdots\!74}{12\!\cdots\!19}a^{8}-\frac{39\!\cdots\!10}{16\!\cdots\!47}a^{7}-\frac{13\!\cdots\!69}{16\!\cdots\!47}a^{6}-\frac{30\!\cdots\!00}{16\!\cdots\!47}a^{5}+\frac{15\!\cdots\!94}{12\!\cdots\!19}a^{4}+\frac{18\!\cdots\!80}{16\!\cdots\!47}a^{3}+\frac{78\!\cdots\!59}{16\!\cdots\!47}a^{2}+\frac{62\!\cdots\!92}{12\!\cdots\!19}a-\frac{70\!\cdots\!47}{16\!\cdots\!47}$, $\frac{42\!\cdots\!24}{16\!\cdots\!47}a^{17}-\frac{89\!\cdots\!75}{16\!\cdots\!47}a^{16}-\frac{90\!\cdots\!42}{16\!\cdots\!47}a^{15}+\frac{20\!\cdots\!35}{16\!\cdots\!47}a^{14}+\frac{81\!\cdots\!16}{16\!\cdots\!47}a^{13}-\frac{19\!\cdots\!22}{16\!\cdots\!47}a^{12}-\frac{39\!\cdots\!28}{16\!\cdots\!47}a^{11}+\frac{95\!\cdots\!83}{16\!\cdots\!47}a^{10}+\frac{57\!\cdots\!46}{84\!\cdots\!13}a^{9}-\frac{25\!\cdots\!93}{16\!\cdots\!47}a^{8}-\frac{16\!\cdots\!62}{16\!\cdots\!47}a^{7}+\frac{34\!\cdots\!31}{16\!\cdots\!47}a^{6}+\frac{12\!\cdots\!20}{16\!\cdots\!47}a^{5}-\frac{19\!\cdots\!94}{16\!\cdots\!47}a^{4}-\frac{73\!\cdots\!90}{30\!\cdots\!99}a^{3}+\frac{65\!\cdots\!31}{16\!\cdots\!47}a^{2}-\frac{14\!\cdots\!82}{12\!\cdots\!19}a+\frac{94\!\cdots\!35}{16\!\cdots\!47}$, $\frac{40\!\cdots\!82}{16\!\cdots\!47}a^{17}-\frac{27\!\cdots\!01}{16\!\cdots\!47}a^{16}-\frac{78\!\cdots\!52}{16\!\cdots\!47}a^{15}+\frac{58\!\cdots\!33}{16\!\cdots\!47}a^{14}+\frac{57\!\cdots\!22}{16\!\cdots\!47}a^{13}-\frac{53\!\cdots\!34}{16\!\cdots\!47}a^{12}-\frac{16\!\cdots\!90}{16\!\cdots\!47}a^{11}+\frac{27\!\cdots\!48}{16\!\cdots\!47}a^{10}-\frac{13\!\cdots\!26}{16\!\cdots\!47}a^{9}-\frac{84\!\cdots\!51}{16\!\cdots\!47}a^{8}+\frac{18\!\cdots\!48}{16\!\cdots\!47}a^{7}+\frac{15\!\cdots\!80}{16\!\cdots\!47}a^{6}-\frac{36\!\cdots\!48}{16\!\cdots\!47}a^{5}-\frac{88\!\cdots\!13}{84\!\cdots\!13}a^{4}+\frac{19\!\cdots\!76}{16\!\cdots\!47}a^{3}+\frac{47\!\cdots\!19}{84\!\cdots\!13}a^{2}-\frac{51\!\cdots\!96}{12\!\cdots\!19}a+\frac{73\!\cdots\!65}{16\!\cdots\!47}$, $\frac{49\!\cdots\!56}{16\!\cdots\!47}a^{17}+\frac{33\!\cdots\!15}{16\!\cdots\!47}a^{16}-\frac{11\!\cdots\!44}{16\!\cdots\!47}a^{15}-\frac{37\!\cdots\!60}{65\!\cdots\!01}a^{14}+\frac{10\!\cdots\!72}{16\!\cdots\!47}a^{13}+\frac{10\!\cdots\!63}{16\!\cdots\!47}a^{12}-\frac{60\!\cdots\!26}{16\!\cdots\!47}a^{11}-\frac{67\!\cdots\!28}{16\!\cdots\!47}a^{10}+\frac{19\!\cdots\!70}{16\!\cdots\!47}a^{9}+\frac{25\!\cdots\!93}{16\!\cdots\!47}a^{8}-\frac{39\!\cdots\!90}{16\!\cdots\!47}a^{7}-\frac{59\!\cdots\!18}{16\!\cdots\!47}a^{6}+\frac{22\!\cdots\!40}{84\!\cdots\!13}a^{5}+\frac{76\!\cdots\!88}{16\!\cdots\!47}a^{4}-\frac{18\!\cdots\!38}{16\!\cdots\!47}a^{3}-\frac{43\!\cdots\!65}{16\!\cdots\!47}a^{2}-\frac{19\!\cdots\!80}{12\!\cdots\!19}a-\frac{86\!\cdots\!07}{16\!\cdots\!47}$, $\frac{29\!\cdots\!84}{16\!\cdots\!47}a^{17}+\frac{14\!\cdots\!57}{12\!\cdots\!19}a^{16}-\frac{33\!\cdots\!90}{84\!\cdots\!13}a^{15}-\frac{42\!\cdots\!23}{16\!\cdots\!47}a^{14}+\frac{58\!\cdots\!00}{16\!\cdots\!47}a^{13}+\frac{39\!\cdots\!49}{16\!\cdots\!47}a^{12}-\frac{16\!\cdots\!10}{84\!\cdots\!13}a^{11}-\frac{21\!\cdots\!43}{16\!\cdots\!47}a^{10}+\frac{94\!\cdots\!36}{16\!\cdots\!47}a^{9}+\frac{66\!\cdots\!09}{16\!\cdots\!47}a^{8}-\frac{17\!\cdots\!52}{16\!\cdots\!47}a^{7}-\frac{12\!\cdots\!81}{16\!\cdots\!47}a^{6}+\frac{18\!\cdots\!92}{16\!\cdots\!47}a^{5}+\frac{13\!\cdots\!49}{16\!\cdots\!47}a^{4}-\frac{92\!\cdots\!96}{16\!\cdots\!47}a^{3}-\frac{81\!\cdots\!34}{16\!\cdots\!47}a^{2}+\frac{18\!\cdots\!10}{12\!\cdots\!19}a+\frac{30\!\cdots\!49}{16\!\cdots\!47}$, $\frac{79\!\cdots\!50}{16\!\cdots\!47}a^{17}+\frac{96\!\cdots\!60}{16\!\cdots\!47}a^{16}-\frac{12\!\cdots\!20}{16\!\cdots\!47}a^{15}-\frac{89\!\cdots\!77}{84\!\cdots\!13}a^{14}+\frac{74\!\cdots\!98}{16\!\cdots\!47}a^{13}+\frac{12\!\cdots\!75}{16\!\cdots\!47}a^{12}-\frac{21\!\cdots\!84}{16\!\cdots\!47}a^{11}-\frac{41\!\cdots\!26}{16\!\cdots\!47}a^{10}+\frac{30\!\cdots\!88}{16\!\cdots\!47}a^{9}+\frac{56\!\cdots\!41}{16\!\cdots\!47}a^{8}-\frac{32\!\cdots\!38}{16\!\cdots\!47}a^{7}+\frac{37\!\cdots\!64}{16\!\cdots\!47}a^{6}+\frac{48\!\cdots\!84}{16\!\cdots\!47}a^{5}-\frac{38\!\cdots\!30}{30\!\cdots\!99}a^{4}-\frac{33\!\cdots\!98}{16\!\cdots\!47}a^{3}+\frac{17\!\cdots\!90}{16\!\cdots\!47}a^{2}-\frac{72\!\cdots\!40}{12\!\cdots\!19}a-\frac{36\!\cdots\!07}{84\!\cdots\!13}$
| 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: | \( 54961806.57802202 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 54961806.57802202 \cdot 2804733}{2\cdot\sqrt{3079394418562350896973143551074069022507008}}\cr\approx \mathstrut & 0.670360913649410 \end{aligned}\] (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
| \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{9})^+\), 6.0.3359232.1, 9.9.151470380950257681.2 |
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 | R | R | $18$ | $18$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.1.0.1}{1} }^{18}$ | $18$ | $18$ | $18$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.27.117 | $x^{18} + 16 x^{17} + 178 x^{16} + 2960 x^{15} + 45360 x^{14} + 447008 x^{13} + 3255456 x^{12} + 17891904 x^{11} + 60260960 x^{10} + 85138048 x^{9} - 288700480 x^{8} - 3555798272 x^{7} - 16235478272 x^{6} - 53744921088 x^{5} - 137665523200 x^{4} - 262308385792 x^{3} - 401534975744 x^{2} - 426755266560 x - 200836357632$ | $2$ | $9$ | $27$ | $C_{18}$ | $[3]^{9}$ |
|
\(3\)
| 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 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
|
\(13\)
| 13.18.12.2 | $x^{18} - 21970 x^{9} + 314171 x^{6} - 4084223 x^{3} + 9653618$ | $3$ | $6$ | $12$ | $C_{18}$ | $[\ ]_{3}^{6}$ |