Normalized defining polynomial
\( x^{18} - 225 x^{15} + 21511 x^{12} - 1110330 x^{9} + 32240575 x^{6} - 489793320 x^{3} + 3004685307 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-28228240504985523195322963100012436711747=-\,3^{27}\cdot 37^{6}\cdot 1063^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $176.71$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 37, 1063$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{15337577509076523} a^{15} + \frac{1643145514301869}{5112525836358841} a^{12} + \frac{2625424173800479}{15337577509076523} a^{9} + \frac{1617222725841669}{5112525836358841} a^{6} + \frac{7361298174471697}{15337577509076523} a^{3} - \frac{791684727669486}{5112525836358841}$, $\frac{1}{22132124345597422689} a^{16} + \frac{990673552146262830}{2459124927288602521} a^{13} - \frac{2344023934714907540}{22132124345597422689} a^{10} + \frac{50380183176606067}{189163455945277117} a^{7} + \frac{2169959726954261440}{22132124345597422689} a^{4} - \frac{66726730781888095}{2459124927288602521} a$, $\frac{1}{10645551810232360313409} a^{17} + \frac{170670293535059836779}{1182839090025817812601} a^{14} - \frac{4118919152215835527694}{10645551810232360313409} a^{11} - \frac{1334819827983760669}{6999047869975253329} a^{8} + \frac{644001565749279519421}{10645551810232360313409} a^{5} + \frac{366342887435219887534}{1182839090025817812601} a^{2}$
Class group and class number
$C_{3}\times C_{3}\times C_{6}\times C_{18}$, which has order $972$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{4357560858863}{22132124345597422689} a^{16} - \frac{95700818581364}{2459124927288602521} a^{13} + \frac{70916952875455346}{22132124345597422689} a^{10} - \frac{25797618217993284}{189163455945277117} a^{7} + \frac{65385242226787739567}{22132124345597422689} a^{4} - \frac{61658237602357818640}{2459124927288602521} a \) (order $18$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 122998830744.33968 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3^2\times S_3$ (as 18T17):
| A solvable group of order 54 |
| The 27 conjugacy class representatives for $C_3^2\times S_3$ |
| Character table for $C_3^2\times S_3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), Deg 6, \(\Q(\zeta_{9})\), 6.0.41767044147.3, 6.0.30448175183163.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
| 3.6.9.10 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
| $37$ | $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 1063 | Data not computed | ||||||