Normalized defining polynomial
\( x^{18} + 474 x^{16} + 59013 x^{14} + 2045784 x^{12} + 19771488 x^{10} + 82905444 x^{8} + 170304408 x^{6} + 173056689 x^{4} + 79872318 x^{2} + 13312053 \)
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: | \(-58239839403585271065831634525715076154762985472=-\,2^{18}\cdot 3^{27}\cdot 79^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $396.34$ | 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: | $2, 3, 79$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2844=2^{2}\cdot 3^{2}\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2844}(1,·)$, $\chi_{2844}(1477,·)$, $\chi_{2844}(2315,·)$, $\chi_{2844}(529,·)$, $\chi_{2844}(1367,·)$, $\chi_{2844}(949,·)$, $\chi_{2844}(2843,·)$, $\chi_{2844}(947,·)$, $\chi_{2844}(419,·)$, $\chi_{2844}(1895,·)$, $\chi_{2844}(1129,·)$, $\chi_{2844}(2663,·)$, $\chi_{2844}(2077,·)$, $\chi_{2844}(1715,·)$, $\chi_{2844}(181,·)$, $\chi_{2844}(1897,·)$, $\chi_{2844}(2425,·)$, $\chi_{2844}(767,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{237} a^{6}$, $\frac{1}{237} a^{7}$, $\frac{1}{237} a^{8}$, $\frac{1}{237} a^{9}$, $\frac{1}{237} a^{10}$, $\frac{1}{237} a^{11}$, $\frac{1}{954873} a^{12} + \frac{1}{1343} a^{10} + \frac{4}{4029} a^{8} + \frac{1}{17} a^{4} - \frac{3}{17} a^{2} - \frac{4}{17}$, $\frac{1}{954873} a^{13} + \frac{1}{1343} a^{11} + \frac{4}{4029} a^{9} + \frac{1}{17} a^{5} - \frac{3}{17} a^{3} - \frac{4}{17} a$, $\frac{1}{133682220} a^{14} + \frac{4}{11140185} a^{12} - \frac{29}{20145} a^{10} + \frac{11}{8295} a^{8} + \frac{106}{141015} a^{6} + \frac{26}{85} a^{4} - \frac{171}{595} a^{2} + \frac{29}{140}$, $\frac{1}{133682220} a^{15} + \frac{4}{11140185} a^{13} - \frac{29}{20145} a^{11} + \frac{11}{8295} a^{9} + \frac{106}{141015} a^{7} + \frac{26}{85} a^{5} - \frac{171}{595} a^{3} + \frac{29}{140} a$, $\frac{1}{6991609516088400} a^{16} - \frac{3334819}{6991609516088400} a^{14} + \frac{2409671}{38842275089380} a^{12} - \frac{1486343799}{2458371841100} a^{10} + \frac{15040718177}{7375115523300} a^{8} - \frac{311013977}{1843778880825} a^{6} + \frac{225719227}{3111863090} a^{4} + \frac{47929377841}{124474523600} a^{2} + \frac{33880883409}{124474523600}$, $\frac{1}{6991609516088400} a^{17} - \frac{3334819}{6991609516088400} a^{15} + \frac{2409671}{38842275089380} a^{13} - \frac{1486343799}{2458371841100} a^{11} + \frac{15040718177}{7375115523300} a^{9} - \frac{311013977}{1843778880825} a^{7} + \frac{225719227}{3111863090} a^{5} + \frac{47929377841}{124474523600} a^{3} + \frac{33880883409}{124474523600} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{756}\times C_{14364}$, which has order $521240832$ (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: | \( 33738401.24471492 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_6$ (as 18T2):
| An abelian group of order 18 |
| The 18 conjugacy class representatives for $C_6 \times C_3$ |
| Character table for $C_6 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-237}) \), \(\Q(\zeta_{9})^+\), 3.3.6241.1, 3.3.505521.1, 3.3.505521.2, 6.0.621087144768.9, 6.0.5317153457472.1, 6.0.3876204870497088.1, 6.0.3876204870497088.2, 9.9.129186640449535761.1 |
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.6.5 | $x^{6} - 2 x^{4} + x^{2} - 3$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.6.6.5 | $x^{6} - 2 x^{4} + x^{2} - 3$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.5 | $x^{6} - 2 x^{4} + x^{2} - 3$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 3 | Data not computed | ||||||
| 79 | Data not computed | ||||||