Normalized defining polynomial
\( x^{18} - 9 x^{17} + 39 x^{16} - 98 x^{15} + 153 x^{14} - 129 x^{13} - 29 x^{12} + 459 x^{11} - 918 x^{10} + \cdots + 27 \)
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: | \(-2727747884710191986159616\) \(\medspace = -\,2^{12}\cdot 3^{24}\cdot 11^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.78\) | 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: | $2^{2/3}3^{4/3}11^{1/2}\approx 22.779525808351707$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | 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 over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{6}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{4}{9}a^{6}-\frac{4}{9}a^{5}-\frac{4}{9}a^{4}-\frac{4}{9}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{8}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{9}-\frac{2}{9}a^{3}$, $\frac{1}{27}a^{16}+\frac{1}{27}a^{15}+\frac{1}{9}a^{12}+\frac{4}{27}a^{10}+\frac{4}{27}a^{9}-\frac{1}{9}a^{8}+\frac{2}{9}a^{6}+\frac{1}{3}a^{5}+\frac{4}{27}a^{4}-\frac{5}{27}a^{3}+\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{12\!\cdots\!91}a^{17}+\frac{19\!\cdots\!86}{12\!\cdots\!91}a^{16}+\frac{61\!\cdots\!36}{12\!\cdots\!91}a^{15}+\frac{18\!\cdots\!86}{41\!\cdots\!97}a^{14}-\frac{94\!\cdots\!95}{41\!\cdots\!97}a^{13}+\frac{39\!\cdots\!27}{41\!\cdots\!97}a^{12}-\frac{19\!\cdots\!31}{12\!\cdots\!91}a^{11}+\frac{18\!\cdots\!24}{12\!\cdots\!91}a^{10}-\frac{10\!\cdots\!28}{12\!\cdots\!91}a^{9}-\frac{62\!\cdots\!27}{41\!\cdots\!97}a^{8}+\frac{20\!\cdots\!31}{41\!\cdots\!97}a^{7}-\frac{45\!\cdots\!59}{41\!\cdots\!97}a^{6}+\frac{54\!\cdots\!51}{12\!\cdots\!91}a^{5}-\frac{43\!\cdots\!53}{12\!\cdots\!91}a^{4}+\frac{58\!\cdots\!68}{12\!\cdots\!91}a^{3}+\frac{13\!\cdots\!52}{46\!\cdots\!33}a^{2}+\frac{54\!\cdots\!68}{46\!\cdots\!33}a+\frac{16\!\cdots\!41}{46\!\cdots\!33}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{217656374579678}{25\!\cdots\!93}a^{17}-\frac{18\!\cdots\!57}{23\!\cdots\!37}a^{16}+\frac{84\!\cdots\!24}{23\!\cdots\!37}a^{15}-\frac{76\!\cdots\!28}{77\!\cdots\!79}a^{14}+\frac{13\!\cdots\!30}{77\!\cdots\!79}a^{13}-\frac{45\!\cdots\!16}{25\!\cdots\!93}a^{12}+\frac{25\!\cdots\!72}{77\!\cdots\!79}a^{11}+\frac{92\!\cdots\!36}{23\!\cdots\!37}a^{10}-\frac{22\!\cdots\!88}{23\!\cdots\!37}a^{9}+\frac{56\!\cdots\!60}{856142941704031}a^{8}+\frac{88\!\cdots\!52}{77\!\cdots\!79}a^{7}-\frac{34\!\cdots\!36}{25\!\cdots\!93}a^{6}-\frac{56\!\cdots\!48}{77\!\cdots\!79}a^{5}+\frac{20\!\cdots\!37}{23\!\cdots\!37}a^{4}+\frac{36\!\cdots\!82}{23\!\cdots\!37}a^{3}+\frac{63\!\cdots\!38}{77\!\cdots\!79}a^{2}+\frac{18\!\cdots\!06}{856142941704031}a+\frac{16\!\cdots\!47}{856142941704031}$, $\frac{11\!\cdots\!30}{69\!\cdots\!11}a^{17}-\frac{10\!\cdots\!52}{69\!\cdots\!11}a^{16}+\frac{47\!\cdots\!68}{69\!\cdots\!11}a^{15}-\frac{42\!\cdots\!76}{23\!\cdots\!37}a^{14}+\frac{74\!\cdots\!22}{23\!\cdots\!37}a^{13}-\frac{76\!\cdots\!70}{23\!\cdots\!37}a^{12}+\frac{43\!\cdots\!64}{69\!\cdots\!11}a^{11}+\frac{51\!\cdots\!00}{69\!\cdots\!11}a^{10}-\frac{12\!\cdots\!08}{69\!\cdots\!11}a^{9}+\frac{28\!\cdots\!61}{23\!\cdots\!37}a^{8}+\frac{50\!\cdots\!80}{23\!\cdots\!37}a^{7}-\frac{58\!\cdots\!44}{23\!\cdots\!37}a^{6}-\frac{96\!\cdots\!96}{69\!\cdots\!11}a^{5}+\frac{11\!\cdots\!59}{69\!\cdots\!11}a^{4}+\frac{20\!\cdots\!34}{69\!\cdots\!11}a^{3}+\frac{11\!\cdots\!90}{77\!\cdots\!79}a^{2}+\frac{35\!\cdots\!58}{856142941704031}a+\frac{76\!\cdots\!99}{25\!\cdots\!93}$, $\frac{10\!\cdots\!36}{13\!\cdots\!99}a^{17}-\frac{10\!\cdots\!28}{13\!\cdots\!99}a^{16}+\frac{50\!\cdots\!25}{13\!\cdots\!99}a^{15}-\frac{14\!\cdots\!97}{13\!\cdots\!99}a^{14}+\frac{27\!\cdots\!73}{13\!\cdots\!99}a^{13}-\frac{33\!\cdots\!23}{13\!\cdots\!99}a^{12}+\frac{18\!\cdots\!17}{13\!\cdots\!99}a^{11}+\frac{14\!\cdots\!77}{46\!\cdots\!33}a^{10}-\frac{15\!\cdots\!44}{15\!\cdots\!11}a^{9}+\frac{16\!\cdots\!83}{15\!\cdots\!11}a^{8}+\frac{83\!\cdots\!99}{13\!\cdots\!99}a^{7}-\frac{21\!\cdots\!97}{13\!\cdots\!99}a^{6}-\frac{85\!\cdots\!32}{46\!\cdots\!33}a^{5}+\frac{12\!\cdots\!32}{13\!\cdots\!99}a^{4}+\frac{14\!\cdots\!87}{13\!\cdots\!99}a^{3}+\frac{25\!\cdots\!18}{13\!\cdots\!99}a^{2}-\frac{20\!\cdots\!48}{46\!\cdots\!33}a-\frac{85\!\cdots\!00}{15\!\cdots\!11}$, $\frac{36\!\cdots\!48}{12\!\cdots\!91}a^{17}-\frac{30\!\cdots\!13}{12\!\cdots\!91}a^{16}+\frac{12\!\cdots\!17}{12\!\cdots\!91}a^{15}-\frac{84\!\cdots\!13}{41\!\cdots\!97}a^{14}+\frac{79\!\cdots\!40}{41\!\cdots\!97}a^{13}+\frac{54\!\cdots\!21}{41\!\cdots\!97}a^{12}-\frac{91\!\cdots\!40}{12\!\cdots\!91}a^{11}+\frac{21\!\cdots\!25}{12\!\cdots\!91}a^{10}-\frac{24\!\cdots\!79}{12\!\cdots\!91}a^{9}-\frac{73\!\cdots\!34}{41\!\cdots\!97}a^{8}+\frac{32\!\cdots\!74}{41\!\cdots\!97}a^{7}-\frac{67\!\cdots\!59}{41\!\cdots\!97}a^{6}-\frac{11\!\cdots\!01}{12\!\cdots\!91}a^{5}+\frac{38\!\cdots\!56}{12\!\cdots\!91}a^{4}+\frac{11\!\cdots\!95}{12\!\cdots\!91}a^{3}+\frac{28\!\cdots\!06}{46\!\cdots\!33}a^{2}+\frac{82\!\cdots\!11}{46\!\cdots\!33}a+\frac{99\!\cdots\!36}{46\!\cdots\!33}$, $\frac{10\!\cdots\!54}{12\!\cdots\!91}a^{17}-\frac{10\!\cdots\!18}{12\!\cdots\!91}a^{16}+\frac{51\!\cdots\!49}{12\!\cdots\!91}a^{15}-\frac{49\!\cdots\!63}{41\!\cdots\!97}a^{14}+\frac{90\!\cdots\!78}{41\!\cdots\!97}a^{13}-\frac{98\!\cdots\!21}{41\!\cdots\!97}a^{12}+\frac{66\!\cdots\!89}{12\!\cdots\!91}a^{11}+\frac{61\!\cdots\!60}{12\!\cdots\!91}a^{10}-\frac{16\!\cdots\!92}{12\!\cdots\!91}a^{9}+\frac{48\!\cdots\!44}{41\!\cdots\!97}a^{8}+\frac{48\!\cdots\!29}{41\!\cdots\!97}a^{7}-\frac{10\!\cdots\!21}{41\!\cdots\!97}a^{6}-\frac{15\!\cdots\!08}{12\!\cdots\!91}a^{5}+\frac{27\!\cdots\!66}{12\!\cdots\!91}a^{4}+\frac{12\!\cdots\!79}{12\!\cdots\!91}a^{3}-\frac{37\!\cdots\!34}{46\!\cdots\!33}a^{2}-\frac{26\!\cdots\!79}{46\!\cdots\!33}a-\frac{62\!\cdots\!87}{46\!\cdots\!33}$, $\frac{57\!\cdots\!29}{46\!\cdots\!33}a^{17}-\frac{15\!\cdots\!10}{13\!\cdots\!99}a^{16}+\frac{74\!\cdots\!86}{15\!\cdots\!11}a^{15}-\frac{55\!\cdots\!83}{46\!\cdots\!33}a^{14}+\frac{25\!\cdots\!66}{13\!\cdots\!99}a^{13}-\frac{19\!\cdots\!35}{13\!\cdots\!99}a^{12}-\frac{95\!\cdots\!39}{13\!\cdots\!99}a^{11}+\frac{83\!\cdots\!79}{13\!\cdots\!99}a^{10}-\frac{15\!\cdots\!47}{13\!\cdots\!99}a^{9}+\frac{40\!\cdots\!89}{13\!\cdots\!99}a^{8}+\frac{29\!\cdots\!62}{13\!\cdots\!99}a^{7}-\frac{15\!\cdots\!13}{13\!\cdots\!99}a^{6}-\frac{28\!\cdots\!36}{13\!\cdots\!99}a^{5}+\frac{33\!\cdots\!48}{46\!\cdots\!33}a^{4}+\frac{42\!\cdots\!73}{13\!\cdots\!99}a^{3}+\frac{32\!\cdots\!64}{13\!\cdots\!99}a^{2}+\frac{13\!\cdots\!29}{15\!\cdots\!11}a+\frac{21\!\cdots\!41}{15\!\cdots\!11}$, $\frac{30\!\cdots\!96}{41\!\cdots\!97}a^{17}-\frac{27\!\cdots\!53}{41\!\cdots\!97}a^{16}+\frac{46\!\cdots\!69}{15\!\cdots\!11}a^{15}-\frac{11\!\cdots\!47}{13\!\cdots\!99}a^{14}+\frac{22\!\cdots\!80}{13\!\cdots\!99}a^{13}-\frac{32\!\cdots\!04}{15\!\cdots\!11}a^{12}+\frac{68\!\cdots\!70}{41\!\cdots\!97}a^{11}+\frac{70\!\cdots\!65}{41\!\cdots\!97}a^{10}-\frac{98\!\cdots\!84}{13\!\cdots\!99}a^{9}+\frac{10\!\cdots\!07}{13\!\cdots\!99}a^{8}+\frac{10\!\cdots\!30}{13\!\cdots\!99}a^{7}-\frac{51\!\cdots\!60}{46\!\cdots\!33}a^{6}+\frac{13\!\cdots\!95}{41\!\cdots\!97}a^{5}-\frac{45\!\cdots\!72}{41\!\cdots\!97}a^{4}+\frac{12\!\cdots\!76}{13\!\cdots\!99}a^{3}+\frac{32\!\cdots\!28}{13\!\cdots\!99}a^{2}+\frac{26\!\cdots\!73}{15\!\cdots\!11}a+\frac{78\!\cdots\!73}{15\!\cdots\!11}$, $\frac{15\!\cdots\!91}{12\!\cdots\!91}a^{17}-\frac{14\!\cdots\!35}{12\!\cdots\!91}a^{16}+\frac{65\!\cdots\!72}{12\!\cdots\!91}a^{15}-\frac{56\!\cdots\!86}{41\!\cdots\!97}a^{14}+\frac{94\!\cdots\!91}{41\!\cdots\!97}a^{13}-\frac{91\!\cdots\!86}{41\!\cdots\!97}a^{12}+\frac{26\!\cdots\!38}{12\!\cdots\!91}a^{11}+\frac{71\!\cdots\!86}{12\!\cdots\!91}a^{10}-\frac{16\!\cdots\!16}{12\!\cdots\!91}a^{9}+\frac{29\!\cdots\!83}{41\!\cdots\!97}a^{8}+\frac{73\!\cdots\!34}{41\!\cdots\!97}a^{7}-\frac{58\!\cdots\!28}{41\!\cdots\!97}a^{6}-\frac{17\!\cdots\!72}{12\!\cdots\!91}a^{5}+\frac{87\!\cdots\!71}{12\!\cdots\!91}a^{4}+\frac{33\!\cdots\!58}{12\!\cdots\!91}a^{3}+\frac{29\!\cdots\!83}{13\!\cdots\!99}a^{2}+\frac{37\!\cdots\!82}{46\!\cdots\!33}a+\frac{55\!\cdots\!72}{46\!\cdots\!33}$ | 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: | \( 110286.485801 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 110286.485801 \cdot 3}{2\cdot\sqrt{2727747884710191986159616}}\cr\approx \mathstrut & 1.52872813649 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 18T3):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3 \times C_3$ |
Character table for $S_3 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{-11}) \), 3.1.3564.2 x3, \(\Q(\zeta_{9})^+\), 6.0.139723056.2, 6.0.1724976.2 x2, 6.0.8732691.1, 9.3.45270270144.7 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 sibling: | 6.0.1724976.2 |
Degree 9 sibling: | 9.3.45270270144.7 |
Minimal sibling: | 6.0.1724976.2 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(11\) | 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 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |