Normalized defining polynomial
\( x^{18} - x^{17} - 6 x^{16} + 19 x^{15} - 28 x^{14} - 10 x^{13} + 117 x^{12} - 297 x^{11} + 458 x^{10} - 535 x^{9} + 458 x^{8} - 297 x^{7} + 117 x^{6} - 10 x^{5} - 28 x^{4} + 19 x^{3} - 6 x^{2} - x + 1 \)
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: | \(105410344139098495736057=7^{12}\cdot 13^{5}\cdot 29^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.01$ | 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: | $7, 13, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{3}{13} a^{13} + \frac{3}{13} a^{12} - \frac{1}{13} a^{9} + \frac{6}{13} a^{8} + \frac{3}{13} a^{7} + \frac{6}{13} a^{6} - \frac{1}{13} a^{5} + \frac{3}{13} a^{2} + \frac{3}{13} a + \frac{1}{13}$, $\frac{1}{169} a^{15} + \frac{4}{169} a^{14} - \frac{59}{169} a^{13} - \frac{75}{169} a^{12} - \frac{3}{13} a^{11} + \frac{12}{169} a^{10} + \frac{18}{169} a^{9} - \frac{30}{169} a^{8} + \frac{9}{169} a^{7} - \frac{73}{169} a^{6} + \frac{25}{169} a^{5} - \frac{3}{13} a^{4} - \frac{36}{169} a^{3} - \frac{59}{169} a^{2} - \frac{22}{169} a - \frac{12}{169}$, $\frac{1}{169} a^{16} + \frac{3}{169} a^{14} + \frac{57}{169} a^{13} - \frac{12}{169} a^{12} - \frac{1}{169} a^{11} - \frac{30}{169} a^{10} - \frac{11}{169} a^{9} - \frac{79}{169} a^{8} - \frac{44}{169} a^{7} - \frac{60}{169} a^{6} - \frac{48}{169} a^{5} - \frac{49}{169} a^{4} - \frac{84}{169} a^{3} - \frac{59}{169} a^{2} - \frac{28}{169} a - \frac{43}{169}$, $\frac{1}{169} a^{17} + \frac{6}{169} a^{14} + \frac{48}{169} a^{13} - \frac{62}{169} a^{12} - \frac{82}{169} a^{11} - \frac{47}{169} a^{10} + \frac{75}{169} a^{9} - \frac{19}{169} a^{8} - \frac{35}{169} a^{7} - \frac{63}{169} a^{6} + \frac{84}{169} a^{5} + \frac{33}{169} a^{4} + \frac{49}{169} a^{3} + \frac{32}{169} a^{2} + \frac{75}{169} a - \frac{3}{169}$
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: | \( 34139.7681346 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 165888 |
| The 192 conjugacy class representatives for t18n839 are not computed |
| Character table for t18n839 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.5.16721334721.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | 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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | $18$ | $18$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | R | $18$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.3.4 | $x^{4} + 104$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.6.4.1 | $x^{6} + 232 x^{3} + 22707$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |