Normalized defining polynomial
\( x^{18} - 4 x^{17} + 7 x^{16} - 9 x^{15} + 15 x^{14} - 24 x^{13} + 28 x^{12} - 30 x^{11} + 36 x^{10} - 39 x^{9} + 36 x^{8} - 30 x^{7} + 28 x^{6} - 24 x^{5} + 15 x^{4} - 9 x^{3} + 7 x^{2} - 4 x + 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-11436479389438477403\) \(\medspace = -\,23^{6}\cdot 43^{2}\cdot 347^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.45\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $23^{1/2}43^{1/2}347^{1/2}\approx 585.8182311946258$ | ||
Ramified primes: | \(23\), \(43\), \(347\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-347}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a^{17}-2a^{16}+2a^{14}+a^{13}-7a^{11}+8a^{10}-5a^{9}+7a^{8}-8a^{7}+9a^{6}+2a^{4}-3a^{3}-a^{2}+2a-1$, $a$, $a^{15}-2a^{14}+2a^{12}+a^{11}-7a^{9}+8a^{8}-6a^{7}+10a^{6}-12a^{5}+13a^{4}-5a^{3}+7a^{2}-7a+3$, $a^{17}-2a^{16}+3a^{14}-3a^{13}+7a^{12}-15a^{11}+19a^{10}-22a^{9}+27a^{8}-27a^{7}+29a^{6}-21a^{5}+20a^{4}-14a^{3}+9a^{2}-5a+2$, $2a^{17}-9a^{16}+18a^{15}-24a^{14}+33a^{13}-48a^{12}+57a^{11}-57a^{10}+57a^{9}-56a^{8}+49a^{7}-33a^{6}+23a^{5}-19a^{4}+9a^{3}-2a^{2}-a$, $2a^{17}-9a^{16}+18a^{15}-23a^{14}+29a^{13}-41a^{12}+49a^{11}-46a^{10}+40a^{9}-36a^{8}+30a^{7}-14a^{6}+4a^{5}-2a^{4}-2a^{3}+6a^{2}-7a+3$, $a^{17}-2a^{16}+3a^{14}-3a^{13}+7a^{12}-15a^{11}+19a^{10}-22a^{9}+27a^{8}-27a^{7}+29a^{6}-21a^{5}+20a^{4}-14a^{3}+9a^{2}-5a+3$, $3a^{17}-11a^{16}+15a^{15}-12a^{14}+21a^{13}-37a^{12}+32a^{11}-24a^{10}+35a^{9}-37a^{8}+25a^{7}-14a^{6}+20a^{5}-22a^{4}+4a^{3}-2a^{2}+7a-4$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 88.267766218 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 88.267766218 \cdot 1}{2\cdot\sqrt{11436479389438477403}}\cr\approx \mathstrut & 0.19917956512 \end{aligned}\]
Galois group
$D_6\wr S_3$ (as 18T556):
A solvable group of order 10368 |
The 98 conjugacy class representatives for $D_6\wr S_3$ are not computed |
Character table for $D_6\wr S_3$ is not computed |
Intermediate fields
3.1.23.1, 6.0.183563.3, 9.3.181543807.1 |
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 |
Degree 36 siblings: | data not computed |
Minimal sibling: | 18.0.1417200615982289707.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $18$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
23.6.3.2 | $x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(43\) | 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(347\) | $\Q_{347}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{347}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |