Normalized defining polynomial
\( x^{18} - 2 x^{17} - 27 x^{16} + 57 x^{15} + 211 x^{14} - 400 x^{13} - 784 x^{12} + 1078 x^{11} + 1630 x^{10} - 615 x^{9} - 1592 x^{8} - 1711 x^{7} + 46 x^{6} + 2054 x^{5} + 420 x^{4} - 526 x^{3} - 50 x^{2} + 12 x + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(409618234651336442976143958821=7^{13}\cdot 83^{5}\cdot 181^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.17$ | 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: | $7, 83, 181$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{76087707703712875843} a^{17} - \frac{30248598915793303911}{76087707703712875843} a^{16} - \frac{13751213617473897743}{76087707703712875843} a^{15} + \frac{32250277471107990882}{76087707703712875843} a^{14} + \frac{25392550843453769522}{76087707703712875843} a^{13} - \frac{6865541052621662645}{76087707703712875843} a^{12} + \frac{188270677498051031}{76087707703712875843} a^{11} - \frac{14414916147006334712}{76087707703712875843} a^{10} + \frac{31490043086467986384}{76087707703712875843} a^{9} + \frac{27406351954598014283}{76087707703712875843} a^{8} + \frac{6887995149879104550}{76087707703712875843} a^{7} - \frac{14192404391416613645}{76087707703712875843} a^{6} + \frac{14634198798516579570}{76087707703712875843} a^{5} + \frac{28395261289354207230}{76087707703712875843} a^{4} - \frac{6905236994847179045}{76087707703712875843} a^{3} - \frac{9424405927919803286}{76087707703712875843} a^{2} + \frac{5477999011775977325}{76087707703712875843} a - \frac{32891954177646544936}{76087707703712875843}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 585630503.845 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 165888 |
| The 192 conjugacy class representatives for t18n839 are not computed |
| Character table for t18n839 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.9.26552265046321.1 |
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 | $18$ | $18$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | $18$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} - 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.5.2 | $x^{6} - 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 83 | Data not computed | ||||||
| $181$ | $\Q_{181}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{181}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 181.2.0.1 | $x^{2} - x + 18$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} - x + 18$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} - x + 18$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} - x + 18$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.4.2.1 | $x^{4} + 6335 x^{2} + 10614564$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |