Normalized defining polynomial
\( x^{18} + 133 x^{16} + 5852 x^{14} + 104937 x^{12} + 902139 x^{10} + 4121537 x^{8} + 10383443 x^{6} + 14115822 x^{4} + 9410548 x^{2} + 2352637 \)
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: | \(-18893592163670789529965168685732855808=-\,2^{18}\cdot 7^{15}\cdot 19^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $117.74$ | 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: | $2, 7, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(532=2^{2}\cdot 7\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{532}(1,·)$, $\chi_{532}(197,·)$, $\chi_{532}(457,·)$, $\chi_{532}(75,·)$, $\chi_{532}(335,·)$, $\chi_{532}(531,·)$, $\chi_{532}(277,·)$, $\chi_{532}(121,·)$, $\chi_{532}(411,·)$, $\chi_{532}(31,·)$, $\chi_{532}(227,·)$, $\chi_{532}(103,·)$, $\chi_{532}(429,·)$, $\chi_{532}(27,·)$, $\chi_{532}(305,·)$, $\chi_{532}(501,·)$, $\chi_{532}(505,·)$, $\chi_{532}(255,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{133} a^{6}$, $\frac{1}{133} a^{7}$, $\frac{1}{133} a^{8}$, $\frac{1}{133} a^{9}$, $\frac{1}{931} a^{10} - \frac{3}{7} a^{4}$, $\frac{1}{931} a^{11} - \frac{3}{7} a^{5}$, $\frac{1}{2971752} a^{12} + \frac{11}{22344} a^{10} - \frac{1}{7448} a^{8} - \frac{23}{7448} a^{6} - \frac{59}{168} a^{4} - \frac{11}{56} a^{2} - \frac{47}{168}$, $\frac{1}{2971752} a^{13} + \frac{11}{22344} a^{11} - \frac{1}{7448} a^{9} - \frac{23}{7448} a^{7} - \frac{59}{168} a^{5} - \frac{11}{56} a^{3} - \frac{47}{168} a$, $\frac{1}{2971752} a^{14} + \frac{1}{2793} a^{10} - \frac{2}{931} a^{8} + \frac{1}{798} a^{6} + \frac{1}{42} a^{4} + \frac{2}{21} a^{2} + \frac{7}{24}$, $\frac{1}{2971752} a^{15} + \frac{1}{2793} a^{11} - \frac{2}{931} a^{9} + \frac{1}{798} a^{7} + \frac{1}{42} a^{5} + \frac{2}{21} a^{3} + \frac{7}{24} a$, $\frac{1}{20802264} a^{16} + \frac{1}{19551} a^{10} + \frac{1}{294} a^{8} + \frac{1}{5586} a^{6} - \frac{5}{147} a^{4} - \frac{65}{168} a^{2} + \frac{2}{21}$, $\frac{1}{20802264} a^{17} + \frac{1}{19551} a^{11} + \frac{1}{294} a^{9} + \frac{1}{5586} a^{7} - \frac{5}{147} a^{5} - \frac{65}{168} a^{3} + \frac{2}{21} a$
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{868}$, which has order $222208$ (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: | \( 833965.2438558349 \) (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
| \(\Q(\sqrt{-133}) \), 3.3.361.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.17689.1, 6.0.54355325248.1, 6.0.2663410937152.2, 6.0.7377869632.2, 6.0.2663410937152.1, 9.9.5534900853769.1 |
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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 | $x^{6} - 2 x^{4} + x^{2} - 3$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| $7$ | 7.6.5.2 | $x^{6} - 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.6.5.2 | $x^{6} - 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.5.2 | $x^{6} - 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 19 | Data not computed | ||||||