Normalized defining polynomial
\( x^{18} - 9 x^{17} + 43 x^{16} - 140 x^{15} + 335 x^{14} - 609 x^{13} + 841 x^{12} - 847 x^{11} + 528 x^{10} - 461 x^{8} + 639 x^{7} - 547 x^{6} + 349 x^{5} - 175 x^{4} + 67 x^{3} - 13 x^{2} - 2 x + 1 \)
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: | \(229758542086710080000000=2^{12}\cdot 5^{7}\cdot 23^{4}\cdot 37^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.85$ | 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: | $2, 5, 23, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{575} a^{16} - \frac{8}{575} a^{15} + \frac{16}{575} a^{14} + \frac{28}{575} a^{13} + \frac{59}{575} a^{12} + \frac{68}{575} a^{11} - \frac{212}{575} a^{10} - \frac{51}{575} a^{9} - \frac{19}{115} a^{8} - \frac{12}{25} a^{7} - \frac{82}{575} a^{6} + \frac{51}{575} a^{5} - \frac{88}{575} a^{4} - \frac{133}{575} a^{3} + \frac{214}{575} a^{2} - \frac{67}{575} a - \frac{121}{575}$, $\frac{1}{2875} a^{17} - \frac{1}{2875} a^{16} - \frac{8}{575} a^{15} + \frac{28}{575} a^{14} + \frac{51}{575} a^{13} - \frac{669}{2875} a^{12} - \frac{311}{2875} a^{11} + \frac{38}{575} a^{10} + \frac{698}{2875} a^{9} + \frac{784}{2875} a^{8} + \frac{1436}{2875} a^{7} + \frac{1202}{2875} a^{6} + \frac{844}{2875} a^{5} + \frac{401}{2875} a^{4} + \frac{1008}{2875} a^{3} + \frac{856}{2875} a^{2} - \frac{118}{575} a + \frac{303}{2875}$
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: | \( 86008.1199938 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 82944 |
| The 80 conjugacy class representatives for t18n782 are not computed |
| Character table for t18n782 is not computed |
Intermediate fields
| 3.3.148.1, 9.5.42872699200.1 |
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 | R | $18$ | R | $18$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 | Data not computed | ||||||
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.6.0.1 | $x^{6} - x + 15$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 23.8.4.1 | $x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $37$ | 37.6.0.1 | $x^{6} - x + 20$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 37.12.6.1 | $x^{12} + 2026120 x^{6} - 69343957 x^{2} + 1026290563600$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |