Normalized defining polynomial
\( x^{18} - 18 x^{16} - 24 x^{15} + 135 x^{14} + 360 x^{13} - 952 x^{12} - 2160 x^{11} + 6159 x^{10} + 7376 x^{9} - 23706 x^{8} - 17784 x^{7} + 41945 x^{6} + 30024 x^{5} - 13692 x^{4} - 9984 x^{3} - 29520 x^{2} - 42624 x - 11072 \)
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: | \(129865547912188440278888339956826112=2^{23}\cdot 3^{18}\cdot 43^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $89.28$ | 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: | $2, 3, 43$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{3}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{7} - \frac{1}{16} a^{5} - \frac{3}{32} a^{4} - \frac{1}{32} a^{3} - \frac{3}{16} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{128} a^{9} - \frac{1}{128} a^{7} + \frac{1}{64} a^{6} - \frac{5}{128} a^{5} + \frac{1}{32} a^{4} + \frac{9}{128} a^{3} + \frac{1}{64} a^{2} - \frac{9}{32} a + \frac{3}{16}$, $\frac{1}{128} a^{10} - \frac{1}{128} a^{8} + \frac{1}{64} a^{7} + \frac{3}{128} a^{6} + \frac{1}{32} a^{5} - \frac{7}{128} a^{4} + \frac{1}{64} a^{3} + \frac{1}{32} a^{2} + \frac{3}{16} a - \frac{1}{4}$, $\frac{1}{512} a^{11} + \frac{1}{512} a^{10} + \frac{1}{512} a^{9} + \frac{1}{512} a^{8} + \frac{11}{512} a^{7} + \frac{3}{512} a^{6} + \frac{3}{512} a^{5} + \frac{51}{512} a^{4} + \frac{1}{16} a^{3} - \frac{5}{64} a^{2} - \frac{15}{32} a + \frac{11}{32}$, $\frac{1}{2048} a^{12} + \frac{1}{512} a^{10} + \frac{1}{512} a^{9} - \frac{13}{1024} a^{8} + \frac{15}{512} a^{7} + \frac{5}{512} a^{6} + \frac{11}{512} a^{5} - \frac{63}{2048} a^{4} - \frac{55}{512} a^{3} - \frac{7}{128} a^{2} - \frac{33}{128} a - \frac{13}{128}$, $\frac{1}{2048} a^{13} + \frac{1}{1024} a^{9} - \frac{1}{256} a^{8} + \frac{1}{256} a^{7} - \frac{1}{64} a^{6} - \frac{107}{2048} a^{5} + \frac{19}{256} a^{4} + \frac{7}{128} a^{3} + \frac{29}{128} a^{2} + \frac{23}{128} a + \frac{1}{32}$, $\frac{1}{4096} a^{14} - \frac{1}{4096} a^{13} - \frac{7}{2048} a^{10} + \frac{3}{2048} a^{9} + \frac{1}{128} a^{8} - \frac{11}{512} a^{7} + \frac{37}{4096} a^{6} + \frac{115}{4096} a^{5} + \frac{49}{512} a^{4} - \frac{3}{256} a^{3} - \frac{1}{4} a^{2} - \frac{47}{256} a + \frac{21}{64}$, $\frac{1}{65536} a^{15} - \frac{15}{65536} a^{13} - \frac{1}{32768} a^{12} - \frac{19}{32768} a^{11} - \frac{13}{8192} a^{10} + \frac{85}{32768} a^{9} - \frac{59}{16384} a^{8} - \frac{2019}{65536} a^{7} + \frac{117}{8192} a^{6} - \frac{1675}{65536} a^{5} - \frac{2561}{32768} a^{4} - \frac{315}{8192} a^{3} + \frac{429}{4096} a^{2} + \frac{1053}{4096} a + \frac{359}{2048}$, $\frac{1}{65536} a^{16} + \frac{1}{65536} a^{14} + \frac{7}{32768} a^{13} - \frac{3}{32768} a^{12} + \frac{3}{8192} a^{11} + \frac{101}{32768} a^{10} + \frac{45}{16384} a^{9} - \frac{419}{65536} a^{8} + \frac{133}{8192} a^{7} - \frac{1083}{65536} a^{6} - \frac{409}{32768} a^{5} - \frac{151}{8192} a^{4} - \frac{219}{4096} a^{3} + \frac{157}{4096} a^{2} + \frac{143}{2048} a + \frac{45}{128}$, $\frac{1}{131072} a^{17} - \frac{1}{131072} a^{16} - \frac{1}{131072} a^{15} + \frac{13}{131072} a^{14} + \frac{5}{65536} a^{13} - \frac{15}{65536} a^{12} - \frac{1}{65536} a^{11} - \frac{163}{65536} a^{10} - \frac{427}{131072} a^{9} - \frac{733}{131072} a^{8} + \frac{2403}{131072} a^{7} + \frac{2489}{131072} a^{6} - \frac{359}{8192} a^{5} + \frac{1779}{32768} a^{4} - \frac{925}{8192} a^{3} - \frac{793}{8192} a^{2} + \frac{81}{1024} a + \frac{449}{2048}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 414714038879 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1119744 |
| The 267 conjugacy class representatives for t18n926 are not computed |
| Character table for t18n926 is not computed |
Intermediate fields
| 3.3.1849.1, 6.4.27350408.1 |
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 | R | R | $18$ | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ | $18$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ | $18$ | ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ | R | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $18$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.3.3 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.6.4 | $x^{4} - 2 x^{2} + 20$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
| 2.4.10.7 | $x^{4} - 2 x^{2} + 3$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
| $3$ | 3.9.9.6 | $x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ |
| 3.9.9.2 | $x^{9} + 18 x^{3} + 27 x + 27$ | $3$ | $3$ | $9$ | $C_3^2 : S_3 $ | $[3/2, 3/2]_{2}^{3}$ | |
| 43 | Data not computed | ||||||