Normalized defining polynomial
\( x^{18} - 9 x^{17} + 54 x^{16} - 212 x^{15} + 642 x^{14} - 1504 x^{13} + 2820 x^{12} - 3902 x^{11} + 3025 x^{10} + 1721 x^{9} - 8946 x^{8} + 11988 x^{7} - 5076 x^{6} - 4594 x^{5} + 5540 x^{4} - 252 x^{3} - 1882 x^{2} + 610 x + 50 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(254118268458270476120520458240=2^{16}\cdot 5\cdot 13\cdot 19^{8}\cdot 37^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $43.01$ | 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: | $2, 5, 13, 19, 37$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{667535277668800710130963300445} a^{17} - \frac{244449915314818725869184430834}{667535277668800710130963300445} a^{16} - \frac{262805332658793308613996872481}{667535277668800710130963300445} a^{15} + \frac{114015633026809786851997388543}{667535277668800710130963300445} a^{14} - \frac{210901147936289791606354445008}{667535277668800710130963300445} a^{13} + \frac{221424390584951589437077216896}{667535277668800710130963300445} a^{12} + \frac{27001209980121896603900007802}{133507055533760142026192660089} a^{11} - \frac{190421770427895097829538709492}{667535277668800710130963300445} a^{10} + \frac{3664376593880573834786620605}{133507055533760142026192660089} a^{9} + \frac{51103387775663348908538634371}{667535277668800710130963300445} a^{8} - \frac{143206335187067957929546066486}{667535277668800710130963300445} a^{7} + \frac{261196305676442953616821451433}{667535277668800710130963300445} a^{6} - \frac{332236707231258476528733808736}{667535277668800710130963300445} a^{5} + \frac{94080449332601972023627994791}{667535277668800710130963300445} a^{4} - \frac{42681297378250567262632313819}{133507055533760142026192660089} a^{3} + \frac{42398520843850097625781558848}{667535277668800710130963300445} a^{2} + \frac{176603442844748564381558251688}{667535277668800710130963300445} a - \frac{50567424304924586769557646720}{133507055533760142026192660089}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 45307928.7387 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 27648 |
| The 88 conjugacy class representatives for t18n656 are not computed |
| Character table for t18n656 is not computed |
Intermediate fields
| 3.3.148.1, 9.9.62526089134336.1 |
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 | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | $18$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | R | $18$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.9.8.1 | $x^{9} - 2$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ |
| 2.9.8.1 | $x^{9} - 2$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ | |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.12.8.1 | $x^{12} - 114 x^{9} + 4332 x^{6} - 54872 x^{3} + 130321000$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ | |
| 37 | Data not computed | ||||||