Normalized defining polynomial
\( x^{18} + 468 x^{16} + 91260 x^{14} + 9596496 x^{12} + 588128112 x^{10} + 21172612032 x^{8} + 428157265536 x^{6} + 4337177495040 x^{4} + 16914992230656 x^{2} + 18296013072896 \)
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: | \(-6765429537581484920649996381709729642447896576=-\,2^{27}\cdot 3^{44}\cdot 13^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $351.66$ | 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: | $2, 3, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2808=2^{3}\cdot 3^{3}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2808}(1,·)$, $\chi_{2808}(1219,·)$, $\chi_{2808}(2785,·)$, $\chi_{2808}(529,·)$, $\chi_{2808}(2155,·)$, $\chi_{2808}(1465,·)$, $\chi_{2808}(283,·)$, $\chi_{2808}(979,·)$, $\chi_{2808}(2401,·)$, $\chi_{2808}(1819,·)$, $\chi_{2808}(913,·)$, $\chi_{2808}(937,·)$, $\chi_{2808}(43,·)$, $\chi_{2808}(883,·)$, $\chi_{2808}(2755,·)$, $\chi_{2808}(1849,·)$, $\chi_{2808}(1915,·)$, $\chi_{2808}(1873,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{104} a^{6}$, $\frac{1}{104} a^{7}$, $\frac{1}{208} a^{8}$, $\frac{1}{838864} a^{9} + \frac{9}{32264} a^{7} + \frac{351}{16132} a^{5} + \frac{1037}{8066} a^{3} - \frac{392}{4033} a$, $\frac{1}{1689472096} a^{10} + \frac{1645581}{844736048} a^{8} + \frac{561117}{422368024} a^{6} + \frac{1396455}{16244924} a^{4} - \frac{1754747}{8122462} a^{2} + \frac{292}{1007}$, $\frac{1}{1689472096} a^{11} + \frac{11}{64979696} a^{9} - \frac{134659}{52796003} a^{7} + \frac{542519}{16244924} a^{5} + \frac{852149}{4061231} a^{3} + \frac{453603}{4061231} a$, $\frac{1}{43926274496} a^{12} + \frac{559885}{844736048} a^{8} - \frac{1568537}{422368024} a^{6} + \frac{976113}{8122462} a^{4} + \frac{921288}{4061231} a^{2} - \frac{191}{1007}$, $\frac{1}{43926274496} a^{13} - \frac{7}{844736048} a^{9} + \frac{982513}{211184012} a^{7} + \frac{184611}{8122462} a^{5} + \frac{1990605}{8122462} a^{3} - \frac{599113}{4061231} a$, $\frac{1}{87852548992} a^{14} + \frac{81275}{52796003} a^{8} + \frac{605243}{422368024} a^{6} - \frac{417903}{16244924} a^{4} - \frac{36771}{427498} a^{2} + \frac{30}{1007}$, $\frac{1}{87852548992} a^{15} + \frac{363}{844736048} a^{9} - \frac{1233539}{422368024} a^{7} - \frac{936509}{8122462} a^{5} - \frac{13377}{213749} a^{3} - \frac{1979612}{4061231} a$, $\frac{1}{175705097984} a^{16} - \frac{1578485}{844736048} a^{8} - \frac{605769}{422368024} a^{6} + \frac{58096}{4061231} a^{4} + \frac{1441513}{8122462} a^{2} - \frac{261}{1007}$, $\frac{1}{175705097984} a^{17} + \frac{491}{844736048} a^{9} + \frac{344757}{105592006} a^{7} - \frac{1935687}{16244924} a^{5} - \frac{1897699}{8122462} a^{3} + \frac{1357138}{4061231} a$
Class group and class number
$C_{3}\times C_{35551278}$, which has order $106653834$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 54961806.57802202 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-26}) \), \(\Q(\zeta_{9})^+\), 6.0.7380232704.12, 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 | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | $18$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ | $18$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | $18$ |
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 | Data not computed | ||||||
| $3$ | 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 13 | Data not computed | ||||||