Normalized defining polynomial
\( x^{18} + 66 x^{16} + 1747 x^{14} + 23660 x^{12} + 173859 x^{10} + 685517 x^{8} + 1396052 x^{6} + 1433185 x^{4} + 689767 x^{2} + 122317 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-869245000449362794439473301291008=-\,2^{18}\cdot 7^{16}\cdot 13^{9}\cdot 97^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $67.60$ | 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: | $2, 7, 13, 97$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{8468023282276528361} a^{16} - \frac{2417029537151522425}{8468023282276528361} a^{14} + \frac{2340049044940468715}{8468023282276528361} a^{12} - \frac{1694815517852212598}{8468023282276528361} a^{10} + \frac{4215072589687785715}{8468023282276528361} a^{8} + \frac{1654617785185197218}{8468023282276528361} a^{6} - \frac{268655090913950801}{8468023282276528361} a^{4} + \frac{23302060152200207}{66677348679342743} a^{2} - \frac{32501020456052097}{87299209095634313}$, $\frac{1}{8468023282276528361} a^{17} - \frac{2417029537151522425}{8468023282276528361} a^{15} + \frac{2340049044940468715}{8468023282276528361} a^{13} - \frac{1694815517852212598}{8468023282276528361} a^{11} + \frac{4215072589687785715}{8468023282276528361} a^{9} + \frac{1654617785185197218}{8468023282276528361} a^{7} - \frac{268655090913950801}{8468023282276528361} a^{5} + \frac{23302060152200207}{66677348679342743} a^{3} - \frac{32501020456052097}{87299209095634313} a$
Class group and class number
$C_{2}\times C_{4}\times C_{1584}$, which has order $12672$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 89545.2892541 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 13824 |
| The 80 conjugacy class representatives for t18n586 are not computed |
| Character table for t18n586 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.9.164648481361.1 |
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 | $18$ | $18$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 | ||||||
| $7$ | 7.9.8.2 | $x^{9} - 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
| 7.9.8.2 | $x^{9} - 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ | |
| $13$ | 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.5.4 | $x^{6} + 26$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $97$ | $\Q_{97}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{97}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97.3.0.1 | $x^{3} - x + 5$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 97.3.0.1 | $x^{3} - x + 5$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 97.6.0.1 | $x^{6} - x + 10$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |