Normalized defining polynomial
\( x^{18} - 6 x^{17} - 39 x^{16} + 288 x^{15} + 228 x^{14} - 3564 x^{13} + 245 x^{12} + 20271 x^{11} - 4368 x^{10} - 61071 x^{9} + 8136 x^{8} + 98508 x^{7} + 1667 x^{6} - 78813 x^{5} - 11949 x^{4} + 25836 x^{3} + 5586 x^{2} - 1620 x + 83 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(70019845241007744545157057521088=2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $58.77$ | 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: | $2, 3, 7, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7762681495045790022475199981} a^{17} + \frac{2800621122973384969988339369}{7762681495045790022475199981} a^{16} - \frac{928100433583461397252730795}{7762681495045790022475199981} a^{15} + \frac{1224216816914850934456508343}{7762681495045790022475199981} a^{14} - \frac{1553070724393907881440052927}{7762681495045790022475199981} a^{13} - \frac{3042871719317637556818718942}{7762681495045790022475199981} a^{12} - \frac{676624057794984426510755985}{7762681495045790022475199981} a^{11} - \frac{1698364705622673368285873605}{7762681495045790022475199981} a^{10} - \frac{917380154751931419976344132}{7762681495045790022475199981} a^{9} - \frac{2990867732416223447547905731}{7762681495045790022475199981} a^{8} + \frac{3649024873647103325784686064}{7762681495045790022475199981} a^{7} + \frac{1400350732910003803066081821}{7762681495045790022475199981} a^{6} + \frac{1808516270746318652452340306}{7762681495045790022475199981} a^{5} + \frac{988275664194328033652146198}{7762681495045790022475199981} a^{4} - \frac{2635449905522225813343280700}{7762681495045790022475199981} a^{3} - \frac{2592969145005301835573204250}{7762681495045790022475199981} a^{2} + \frac{3483314170824477833161010972}{7762681495045790022475199981} a - \frac{1259105387812269669476745377}{7762681495045790022475199981}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 25313634976.0 \) (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 t18n766 are not computed |
| Character table for t18n766 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.9.13632439166829.1 |
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 | R | $18$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | $18$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.6.6 | $x^{6} - 13 x^{4} + 7 x^{2} - 3$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
| 2.12.0.1 | $x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $3$ | 3.9.9.2 | $x^{9} + 18 x^{3} + 27 x + 27$ | $3$ | $3$ | $9$ | $C_3^2 : S_3 $ | $[3/2, 3/2]_{2}^{3}$ |
| 3.9.9.2 | $x^{9} + 18 x^{3} + 27 x + 27$ | $3$ | $3$ | $9$ | $C_3^2 : S_3 $ | $[3/2, 3/2]_{2}^{3}$ | |
| $7$ | 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.12.11.2 | $x^{12} + 56$ | $12$ | $1$ | $11$ | $D_4 \times C_3$ | $[\ ]_{12}^{2}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.6.5.2 | $x^{6} + 58$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |