Normalized defining polynomial
\( x^{18} - 2 x^{17} + 4 x^{16} + 3 x^{15} - 26 x^{14} + 99 x^{13} - 59 x^{12} - 417 x^{11} + 864 x^{10} + 18 x^{9} - 2591 x^{8} + 3082 x^{7} + 1768 x^{6} - 5352 x^{5} + 3039 x^{4} + 2938 x^{3} - 2001 x^{2} - 524 x + 263 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5799380344807175895894517717=13^{3}\cdot 1129^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.87$ | 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: | $13, 1129$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{33} a^{16} + \frac{5}{11} a^{15} + \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{8}{33} a^{12} + \frac{2}{33} a^{11} + \frac{13}{33} a^{10} + \frac{3}{11} a^{9} - \frac{10}{33} a^{8} - \frac{5}{33} a^{7} - \frac{5}{33} a^{6} - \frac{7}{33} a^{5} - \frac{2}{33} a^{4} - \frac{5}{11} a^{3} + \frac{5}{33} a^{2} - \frac{16}{33} a + \frac{5}{33}$, $\frac{1}{271220894658448865965288006047} a^{17} + \frac{2094161222323297921675978310}{271220894658448865965288006047} a^{16} + \frac{15443866824285871634804237953}{90406964886149621988429335349} a^{15} - \frac{41250886378899614195494399933}{90406964886149621988429335349} a^{14} + \frac{3068801257982223373903787363}{24656444968949896905935273277} a^{13} - \frac{30232001091332471854245208304}{271220894658448865965288006047} a^{12} + \frac{123481238020994433902052808697}{271220894658448865965288006047} a^{11} - \frac{3226303832425691021972037065}{24656444968949896905935273277} a^{10} + \frac{35782421273044166360513697911}{271220894658448865965288006047} a^{9} + \frac{117564289632128577210834577151}{271220894658448865965288006047} a^{8} + \frac{30001348281217511616579752936}{90406964886149621988429335349} a^{7} - \frac{11915870263193832950387122784}{271220894658448865965288006047} a^{6} + \frac{79856159101252481464865839658}{271220894658448865965288006047} a^{5} + \frac{1823973787651922921123061902}{7330294450228347728791567731} a^{4} + \frac{48416766180832690980219838475}{271220894658448865965288006047} a^{3} - \frac{2144964455272726135875832691}{8218814989649965635311757759} a^{2} + \frac{2931445811796138991457094119}{8218814989649965635311757759} a - \frac{113536375581805458725723546987}{271220894658448865965288006047}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $11$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 9741468.33714 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^2:D_9$ (as 18T67):
| A solvable group of order 144 |
| The 18 conjugacy class representatives for $C_2\times C_2^2:D_9$ |
| Character table for $C_2\times C_2^2:D_9$ |
Intermediate fields
| 3.3.1129.1, 6.2.16570333.1, 9.9.1624709678881.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 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 | $18$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 1129 | Data not computed | ||||||