Normalized defining polynomial
\( x^{18} - 7 x^{17} + 31 x^{16} - 98 x^{15} + 331 x^{14} - 957 x^{13} + 2693 x^{12} - 6227 x^{11} + 15226 x^{10} - 30674 x^{9} + 66392 x^{8} - 111391 x^{7} + 216508 x^{6} + \cdots + 713641 \)
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: | \(-680129765110548404696137774571\) \(\medspace = -\,11^{9}\cdot 19^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.43\) | 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: | $11^{1/2}19^{8/9}\approx 45.432455522435575$ | ||
Ramified primes: | \(11\), \(19\) | 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: | \(209=11\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(131,·)$, $\chi_{209}(197,·)$, $\chi_{209}(199,·)$, $\chi_{209}(23,·)$, $\chi_{209}(142,·)$, $\chi_{209}(144,·)$, $\chi_{209}(87,·)$, $\chi_{209}(153,·)$, $\chi_{209}(111,·)$, $\chi_{209}(100,·)$, $\chi_{209}(43,·)$, $\chi_{209}(45,·)$, $\chi_{209}(175,·)$, $\chi_{209}(177,·)$, $\chi_{209}(54,·)$, $\chi_{209}(120,·)$, $\chi_{209}(188,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{256}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37}a^{16}-\frac{12}{37}a^{15}+\frac{8}{37}a^{14}+\frac{7}{37}a^{13}+\frac{2}{37}a^{12}+\frac{6}{37}a^{11}+\frac{18}{37}a^{10}-\frac{7}{37}a^{9}+\frac{3}{37}a^{8}+\frac{10}{37}a^{7}+\frac{11}{37}a^{6}-\frac{18}{37}a^{5}+\frac{12}{37}a^{4}-\frac{8}{37}a^{3}+\frac{4}{37}a^{2}+\frac{14}{37}a-\frac{14}{37}$, $\frac{1}{28\!\cdots\!63}a^{17}-\frac{23\!\cdots\!74}{28\!\cdots\!63}a^{16}-\frac{12\!\cdots\!37}{28\!\cdots\!63}a^{15}+\frac{82\!\cdots\!81}{28\!\cdots\!63}a^{14}+\frac{11\!\cdots\!91}{28\!\cdots\!63}a^{13}-\frac{56\!\cdots\!75}{28\!\cdots\!63}a^{12}-\frac{12\!\cdots\!99}{28\!\cdots\!63}a^{11}-\frac{10\!\cdots\!65}{28\!\cdots\!63}a^{10}-\frac{96\!\cdots\!88}{28\!\cdots\!63}a^{9}-\frac{59\!\cdots\!07}{28\!\cdots\!63}a^{8}+\frac{99\!\cdots\!83}{28\!\cdots\!63}a^{7}+\frac{76\!\cdots\!56}{28\!\cdots\!63}a^{6}-\frac{96\!\cdots\!68}{28\!\cdots\!63}a^{5}+\frac{72\!\cdots\!59}{28\!\cdots\!63}a^{4}+\frac{48\!\cdots\!78}{28\!\cdots\!63}a^{3}+\frac{10\!\cdots\!38}{28\!\cdots\!63}a^{2}-\frac{11\!\cdots\!41}{28\!\cdots\!63}a-\frac{12\!\cdots\!02}{28\!\cdots\!63}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{542}$, which has order $1084$ (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: | $\frac{20\!\cdots\!38}{28\!\cdots\!63}a^{17}-\frac{54\!\cdots\!26}{76\!\cdots\!99}a^{16}+\frac{96\!\cdots\!46}{28\!\cdots\!63}a^{15}-\frac{32\!\cdots\!91}{28\!\cdots\!63}a^{14}+\frac{99\!\cdots\!15}{28\!\cdots\!63}a^{13}-\frac{30\!\cdots\!90}{28\!\cdots\!63}a^{12}+\frac{81\!\cdots\!21}{28\!\cdots\!63}a^{11}-\frac{19\!\cdots\!09}{28\!\cdots\!63}a^{10}+\frac{42\!\cdots\!66}{28\!\cdots\!63}a^{9}-\frac{91\!\cdots\!56}{28\!\cdots\!63}a^{8}+\frac{17\!\cdots\!73}{28\!\cdots\!63}a^{7}-\frac{31\!\cdots\!29}{28\!\cdots\!63}a^{6}+\frac{47\!\cdots\!24}{28\!\cdots\!63}a^{5}-\frac{75\!\cdots\!40}{28\!\cdots\!63}a^{4}+\frac{91\!\cdots\!81}{28\!\cdots\!63}a^{3}-\frac{11\!\cdots\!01}{28\!\cdots\!63}a^{2}+\frac{84\!\cdots\!24}{28\!\cdots\!63}a-\frac{92\!\cdots\!13}{28\!\cdots\!63}$, $\frac{39\!\cdots\!52}{28\!\cdots\!63}a^{17}-\frac{68\!\cdots\!54}{28\!\cdots\!63}a^{16}+\frac{45\!\cdots\!10}{28\!\cdots\!63}a^{15}-\frac{20\!\cdots\!65}{28\!\cdots\!63}a^{14}+\frac{69\!\cdots\!19}{28\!\cdots\!63}a^{13}-\frac{20\!\cdots\!28}{28\!\cdots\!63}a^{12}+\frac{54\!\cdots\!33}{28\!\cdots\!63}a^{11}-\frac{14\!\cdots\!65}{28\!\cdots\!63}a^{10}+\frac{32\!\cdots\!62}{28\!\cdots\!63}a^{9}-\frac{71\!\cdots\!93}{28\!\cdots\!63}a^{8}+\frac{13\!\cdots\!15}{28\!\cdots\!63}a^{7}-\frac{25\!\cdots\!15}{28\!\cdots\!63}a^{6}+\frac{37\!\cdots\!08}{28\!\cdots\!63}a^{5}-\frac{63\!\cdots\!50}{28\!\cdots\!63}a^{4}+\frac{66\!\cdots\!43}{28\!\cdots\!63}a^{3}-\frac{10\!\cdots\!47}{28\!\cdots\!63}a^{2}+\frac{57\!\cdots\!13}{28\!\cdots\!63}a-\frac{93\!\cdots\!84}{28\!\cdots\!63}$, $\frac{42\!\cdots\!42}{28\!\cdots\!63}a^{17}-\frac{28\!\cdots\!87}{28\!\cdots\!63}a^{16}+\frac{12\!\cdots\!26}{28\!\cdots\!63}a^{15}-\frac{38\!\cdots\!40}{28\!\cdots\!63}a^{14}+\frac{13\!\cdots\!56}{28\!\cdots\!63}a^{13}-\frac{37\!\cdots\!55}{28\!\cdots\!63}a^{12}+\frac{10\!\cdots\!40}{28\!\cdots\!63}a^{11}-\frac{24\!\cdots\!59}{28\!\cdots\!63}a^{10}+\frac{61\!\cdots\!40}{28\!\cdots\!63}a^{9}-\frac{12\!\cdots\!72}{28\!\cdots\!63}a^{8}+\frac{27\!\cdots\!60}{28\!\cdots\!63}a^{7}-\frac{44\!\cdots\!42}{28\!\cdots\!63}a^{6}+\frac{92\!\cdots\!95}{28\!\cdots\!63}a^{5}-\frac{11\!\cdots\!41}{28\!\cdots\!63}a^{4}+\frac{24\!\cdots\!39}{28\!\cdots\!63}a^{3}-\frac{20\!\cdots\!76}{28\!\cdots\!63}a^{2}+\frac{29\!\cdots\!12}{28\!\cdots\!63}a-\frac{42\!\cdots\!30}{28\!\cdots\!63}$, $\frac{12\!\cdots\!28}{28\!\cdots\!63}a^{17}-\frac{12\!\cdots\!35}{28\!\cdots\!63}a^{16}+\frac{63\!\cdots\!74}{28\!\cdots\!63}a^{15}-\frac{23\!\cdots\!48}{28\!\cdots\!63}a^{14}+\frac{76\!\cdots\!78}{28\!\cdots\!63}a^{13}-\frac{24\!\cdots\!23}{28\!\cdots\!63}a^{12}+\frac{69\!\cdots\!90}{28\!\cdots\!63}a^{11}-\frac{18\!\cdots\!55}{28\!\cdots\!63}a^{10}+\frac{43\!\cdots\!68}{28\!\cdots\!63}a^{9}-\frac{10\!\cdots\!64}{28\!\cdots\!63}a^{8}+\frac{20\!\cdots\!00}{28\!\cdots\!63}a^{7}-\frac{45\!\cdots\!60}{28\!\cdots\!63}a^{6}+\frac{76\!\cdots\!65}{28\!\cdots\!63}a^{5}-\frac{16\!\cdots\!91}{28\!\cdots\!63}a^{4}+\frac{21\!\cdots\!55}{28\!\cdots\!63}a^{3}-\frac{37\!\cdots\!67}{28\!\cdots\!63}a^{2}+\frac{28\!\cdots\!79}{28\!\cdots\!63}a-\frac{85\!\cdots\!01}{28\!\cdots\!63}$, $\frac{13\!\cdots\!56}{28\!\cdots\!63}a^{17}-\frac{10\!\cdots\!81}{28\!\cdots\!63}a^{16}+\frac{54\!\cdots\!66}{28\!\cdots\!63}a^{15}-\frac{21\!\cdots\!93}{28\!\cdots\!63}a^{14}+\frac{73\!\cdots\!95}{28\!\cdots\!63}a^{13}-\frac{20\!\cdots\!82}{28\!\cdots\!63}a^{12}+\frac{53\!\cdots\!09}{28\!\cdots\!63}a^{11}-\frac{12\!\cdots\!19}{28\!\cdots\!63}a^{10}+\frac{30\!\cdots\!76}{28\!\cdots\!63}a^{9}-\frac{57\!\cdots\!93}{28\!\cdots\!63}a^{8}+\frac{11\!\cdots\!75}{28\!\cdots\!63}a^{7}-\frac{17\!\cdots\!17}{28\!\cdots\!63}a^{6}+\frac{29\!\cdots\!68}{28\!\cdots\!63}a^{5}-\frac{29\!\cdots\!85}{28\!\cdots\!63}a^{4}+\frac{45\!\cdots\!85}{28\!\cdots\!63}a^{3}-\frac{18\!\cdots\!37}{28\!\cdots\!63}a^{2}+\frac{32\!\cdots\!43}{28\!\cdots\!63}a+\frac{46\!\cdots\!78}{28\!\cdots\!63}$, $\frac{45\!\cdots\!76}{28\!\cdots\!63}a^{17}-\frac{30\!\cdots\!58}{28\!\cdots\!63}a^{16}+\frac{11\!\cdots\!56}{28\!\cdots\!63}a^{15}-\frac{29\!\cdots\!61}{28\!\cdots\!63}a^{14}+\frac{89\!\cdots\!81}{28\!\cdots\!63}a^{13}-\frac{25\!\cdots\!72}{28\!\cdots\!63}a^{12}+\frac{67\!\cdots\!73}{28\!\cdots\!63}a^{11}-\frac{12\!\cdots\!69}{28\!\cdots\!63}a^{10}+\frac{27\!\cdots\!74}{28\!\cdots\!63}a^{9}-\frac{47\!\cdots\!53}{28\!\cdots\!63}a^{8}+\frac{10\!\cdots\!13}{28\!\cdots\!63}a^{7}-\frac{11\!\cdots\!34}{28\!\cdots\!63}a^{6}+\frac{22\!\cdots\!93}{28\!\cdots\!63}a^{5}-\frac{12\!\cdots\!15}{28\!\cdots\!63}a^{4}+\frac{35\!\cdots\!58}{28\!\cdots\!63}a^{3}+\frac{98\!\cdots\!06}{28\!\cdots\!63}a^{2}+\frac{64\!\cdots\!92}{76\!\cdots\!99}a+\frac{55\!\cdots\!56}{28\!\cdots\!63}$, $\frac{18\!\cdots\!44}{28\!\cdots\!63}a^{17}-\frac{82\!\cdots\!53}{28\!\cdots\!63}a^{16}+\frac{14\!\cdots\!66}{28\!\cdots\!63}a^{15}+\frac{13\!\cdots\!35}{28\!\cdots\!63}a^{14}-\frac{12\!\cdots\!37}{28\!\cdots\!63}a^{13}+\frac{12\!\cdots\!22}{28\!\cdots\!63}a^{12}-\frac{57\!\cdots\!15}{28\!\cdots\!63}a^{11}+\frac{28\!\cdots\!22}{28\!\cdots\!63}a^{10}-\frac{53\!\cdots\!71}{28\!\cdots\!63}a^{9}+\frac{16\!\cdots\!99}{28\!\cdots\!63}a^{8}-\frac{27\!\cdots\!57}{28\!\cdots\!63}a^{7}+\frac{62\!\cdots\!13}{28\!\cdots\!63}a^{6}-\frac{73\!\cdots\!10}{28\!\cdots\!63}a^{5}+\frac{16\!\cdots\!60}{28\!\cdots\!63}a^{4}-\frac{94\!\cdots\!94}{28\!\cdots\!63}a^{3}+\frac{19\!\cdots\!25}{28\!\cdots\!63}a^{2}-\frac{36\!\cdots\!77}{28\!\cdots\!63}a-\frac{30\!\cdots\!90}{28\!\cdots\!63}$, $\frac{15\!\cdots\!26}{28\!\cdots\!63}a^{17}-\frac{10\!\cdots\!70}{28\!\cdots\!63}a^{16}+\frac{34\!\cdots\!48}{28\!\cdots\!63}a^{15}-\frac{72\!\cdots\!61}{28\!\cdots\!63}a^{14}+\frac{21\!\cdots\!35}{28\!\cdots\!63}a^{13}-\frac{64\!\cdots\!34}{28\!\cdots\!63}a^{12}+\frac{15\!\cdots\!67}{28\!\cdots\!63}a^{11}-\frac{27\!\cdots\!02}{28\!\cdots\!63}a^{10}+\frac{56\!\cdots\!31}{28\!\cdots\!63}a^{9}-\frac{91\!\cdots\!31}{28\!\cdots\!63}a^{8}+\frac{18\!\cdots\!61}{28\!\cdots\!63}a^{7}-\frac{17\!\cdots\!96}{28\!\cdots\!63}a^{6}+\frac{34\!\cdots\!37}{28\!\cdots\!63}a^{5}+\frac{22\!\cdots\!42}{28\!\cdots\!63}a^{4}+\frac{48\!\cdots\!15}{28\!\cdots\!63}a^{3}+\frac{14\!\cdots\!17}{28\!\cdots\!63}a^{2}+\frac{15\!\cdots\!93}{28\!\cdots\!63}a+\frac{52\!\cdots\!05}{28\!\cdots\!63}$ (assuming GRH) | 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: | \( 22305.8950792 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22305.8950792 \cdot 1084}{2\cdot\sqrt{680129765110548404696137774571}}\cr\approx \mathstrut & 0.223739089291 \end{aligned}\] (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
\(\Q(\sqrt{-11}) \), 3.3.361.1, 6.0.173457251.1, \(\Q(\zeta_{19})^+\) |
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 | $18$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | R | $18$ | $18$ | R | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.1.0.1}{1} }^{18}$ | $18$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(19\) | 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |