Normalized defining polynomial
\( x^{18} - 8 x^{17} + 15 x^{16} + 7 x^{15} - 50 x^{14} + 96 x^{13} - 105 x^{12} + 61 x^{11} + 11 x^{10} - 39 x^{9} + 11 x^{8} + 61 x^{7} - 105 x^{6} + 96 x^{5} - 50 x^{4} + 7 x^{3} + 15 x^{2} - 8 x + 1 \)
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: | \(1852297436979562450373120000=2^{12}\cdot 5^{4}\cdot 17\cdot 37^{8}\cdot 59^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.72$ | 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: | $2, 5, 17, 37, 59$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{109903} a^{16} - \frac{46072}{109903} a^{15} + \frac{33692}{109903} a^{14} - \frac{1946}{109903} a^{13} + \frac{35857}{109903} a^{12} + \frac{17381}{109903} a^{11} - \frac{30991}{109903} a^{10} + \frac{22037}{109903} a^{9} - \frac{17258}{109903} a^{8} + \frac{22037}{109903} a^{7} - \frac{30991}{109903} a^{6} + \frac{17381}{109903} a^{5} + \frac{35857}{109903} a^{4} - \frac{1946}{109903} a^{3} + \frac{33692}{109903} a^{2} - \frac{46072}{109903} a + \frac{1}{109903}$, $\frac{1}{2527769} a^{17} - \frac{9}{2527769} a^{16} + \frac{1255019}{2527769} a^{15} - \frac{1196546}{2527769} a^{14} - \frac{31796}{2527769} a^{13} - \frac{363524}{2527769} a^{12} - \frac{1152373}{2527769} a^{11} - \frac{425941}{2527769} a^{10} - \frac{870259}{2527769} a^{9} + \frac{874406}{2527769} a^{8} + \frac{874456}{2527769} a^{7} - \frac{870209}{2527769} a^{6} - \frac{426107}{2527769} a^{5} - \frac{1152172}{2527769} a^{4} - \frac{363670}{2527769} a^{3} - \frac{31739}{2527769} a^{2} - \frac{1196538}{2527769} a + \frac{1254996}{2527769}$
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: | \( 6373008.2472 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 82944 |
| The 144 conjugacy class representatives for t18n772 are not computed |
| Character table for t18n772 is not computed |
Intermediate fields
| 3.3.148.1, 9.9.10438327105600.1 |
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 | $18$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ | $18$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.4.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.12.8.1 | $x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
| $5$ | 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.4.1 | $x^{6} + 25 x^{3} + 200$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $17$ | 17.2.1.1 | $x^{2} - 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.12.0.1 | $x^{12} + 3 x^{2} - 2 x + 5$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $37$ | 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.6.3.1 | $x^{6} - 74 x^{4} + 1369 x^{2} - 202612$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 37.6.3.1 | $x^{6} - 74 x^{4} + 1369 x^{2} - 202612$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 59 | Data not computed | ||||||