Normalized defining polynomial
\( x^{18} - 6 x^{17} - 10 x^{16} + 97 x^{15} + 2 x^{14} - 560 x^{13} + 222 x^{12} + 1460 x^{11} - 630 x^{10} - 1983 x^{9} + 630 x^{8} + 1460 x^{7} - 222 x^{6} - 560 x^{5} - 2 x^{4} + 97 x^{3} + 10 x^{2} - 6 x - 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(17706519352529288000000000=2^{12}\cdot 5^{9}\cdot 19^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.27$ | 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: | $2, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{175} a^{16} - \frac{2}{175} a^{15} - \frac{17}{175} a^{14} + \frac{27}{175} a^{13} - \frac{82}{175} a^{12} + \frac{2}{25} a^{11} + \frac{3}{25} a^{10} - \frac{17}{175} a^{9} + \frac{23}{175} a^{8} + \frac{17}{175} a^{7} + \frac{3}{25} a^{6} - \frac{2}{25} a^{5} - \frac{82}{175} a^{4} - \frac{27}{175} a^{3} - \frac{17}{175} a^{2} + \frac{2}{175} a + \frac{1}{175}$, $\frac{1}{100975} a^{17} - \frac{6}{20195} a^{16} + \frac{7634}{100975} a^{15} + \frac{4983}{100975} a^{14} + \frac{45432}{100975} a^{13} - \frac{1017}{2885} a^{12} + \frac{6342}{14425} a^{11} + \frac{92}{4039} a^{10} + \frac{28989}{100975} a^{9} + \frac{32763}{100975} a^{8} - \frac{233}{577} a^{7} - \frac{1851}{14425} a^{6} - \frac{5706}{20195} a^{5} + \frac{21764}{100975} a^{4} - \frac{19771}{100975} a^{3} + \frac{45313}{100975} a^{2} + \frac{9376}{20195} a - \frac{5849}{14425}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 4327937.33071 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
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
| \(\Q(\sqrt{5}) \), 3.3.7220.1 x3, 3.3.361.1, 6.6.260642000.1, 6.6.722000.1 x2, 6.6.16290125.1, 9.9.376367048000.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 sibling: | 6.6.722000.1 |
| Degree 9 sibling: | 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 | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$ |
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 | Data not computed | ||||||
| $5$ | 5.6.3.1 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} - 10 x^{4} + 25 x^{2} - 500$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $19$ | 19.9.6.2 | $x^{9} + 228 x^{6} + 16967 x^{3} + 438976$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 19.9.6.2 | $x^{9} + 228 x^{6} + 16967 x^{3} + 438976$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |