Normalized defining polynomial
\( x^{18} - 8 x^{16} - 5 x^{15} + 27 x^{14} + 36 x^{13} - 49 x^{12} - 95 x^{11} + 31 x^{10} + 147 x^{9} + 7 x^{8} - 126 x^{7} - 35 x^{6} + 76 x^{5} + 17 x^{4} - 24 x^{3} - 3 x^{2} + 5 x - 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1546727421952108776391=-\,11^{4}\cdot 37^{4}\cdot 139^{4}\cdot 151\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.04$ | 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: | $11, 37, 139, 151$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{28419} a^{17} - \frac{7423}{28419} a^{16} - \frac{3520}{28419} a^{15} + \frac{11894}{28419} a^{14} + \frac{8698}{28419} a^{13} + \frac{2750}{28419} a^{12} - \frac{2819}{9473} a^{11} - \frac{1355}{28419} a^{10} - \frac{710}{9473} a^{9} + \frac{3391}{9473} a^{8} - \frac{4889}{28419} a^{7} - \frac{142}{28419} a^{6} + \frac{2528}{28419} a^{5} - \frac{8728}{28419} a^{4} - \frac{2453}{9473} a^{3} + \frac{1505}{9473} a^{2} - \frac{2949}{9473} a - \frac{5023}{28419}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $10$ | 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: | \( 4246.06379679 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 92897280 |
| The 168 conjugacy class representatives for t18n966 are not computed |
| Character table for t18n966 is not computed |
Intermediate fields
| 9.5.3200504329.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | $18$ | R | $18$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | R | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | 11.6.4.1 | $x^{6} + 220 x^{3} + 41503$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 11.12.0.1 | $x^{12} - x + 7$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $37$ | $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $139$ | 139.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 139.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 139.3.0.1 | $x^{3} - x + 5$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 139.3.0.1 | $x^{3} - x + 5$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 139.8.4.1 | $x^{8} + 77284 x^{4} - 2685619 x^{2} + 1493204164$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $151$ | $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{151}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 151.2.1.2 | $x^{2} + 755$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.3.0.1 | $x^{3} - x + 5$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 151.3.0.1 | $x^{3} - x + 5$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 151.6.0.1 | $x^{6} - x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |