Normalized defining polynomial
\( x^{18} - 3 x^{17} - 10 x^{16} + 30 x^{15} + 25 x^{14} - 142 x^{13} + 77 x^{12} + 476 x^{11} - 581 x^{10} - 468 x^{9} + 950 x^{8} - 1021 x^{7} + 1054 x^{6} - 223 x^{5} - 251 x^{4} + 156 x^{3} - 53 x^{2} + 13 x - 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(103814649619199728852328413=7^{12}\cdot 13^{9}\cdot 29^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.88$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{377} a^{16} + \frac{148}{377} a^{15} + \frac{79}{377} a^{14} + \frac{166}{377} a^{13} + \frac{76}{377} a^{12} + \frac{7}{377} a^{11} - \frac{82}{377} a^{10} + \frac{46}{377} a^{9} + \frac{137}{377} a^{8} - \frac{121}{377} a^{7} + \frac{7}{29} a^{6} - \frac{47}{377} a^{5} + \frac{41}{377} a^{4} - \frac{66}{377} a^{3} + \frac{60}{377} a^{2} + \frac{93}{377} a - \frac{165}{377}$, $\frac{1}{26933471296454013551} a^{17} + \frac{2118419580950533}{26933471296454013551} a^{16} + \frac{12293804163017152572}{26933471296454013551} a^{15} + \frac{8544254906847022696}{26933471296454013551} a^{14} + \frac{8936218708628811552}{26933471296454013551} a^{13} + \frac{7153414412829017123}{26933471296454013551} a^{12} + \frac{3317615775590489862}{26933471296454013551} a^{11} + \frac{10168117742775372473}{26933471296454013551} a^{10} + \frac{1803747781553560719}{26933471296454013551} a^{9} - \frac{10518836337833805233}{26933471296454013551} a^{8} - \frac{2291015231159848302}{26933471296454013551} a^{7} + \frac{253981760303058677}{928740389532897019} a^{6} - \frac{7730589724869409250}{26933471296454013551} a^{5} - \frac{12957095264360854365}{26933471296454013551} a^{4} + \frac{3240357919855243015}{26933471296454013551} a^{3} - \frac{7356524240403155468}{26933471296454013551} a^{2} + \frac{424539615376487017}{2071805484342616427} a - \frac{7083598753816903950}{26933471296454013551}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 3376176.15476 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 648 |
| The 26 conjugacy class representatives for t18n201 |
| Character table for t18n201 is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 6.6.5274997.1, 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 | $18$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | $18$ | R | $18$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | $18$ |
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 | ||||||
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | 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 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.3.2.1 | $x^{3} - 29$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 29.3.2.1 | $x^{3} - 29$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |