Normalized defining polynomial
\( x^{18} - 9 x^{17} + 42 x^{16} - 132 x^{15} + 310 x^{14} - 574 x^{13} + 865 x^{12} - 1082 x^{11} + \cdots + 1 \)
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: | \(-6646436567333419399\) \(\medspace = -\,17^{2}\cdot 43^{2}\cdot 2311\cdot 73363^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.11\) | 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: | $17^{1/2}43^{1/2}2311^{1/2}73363^{1/2}\approx 352044.20714308025$ | ||
Ramified primes: | \(17\), \(43\), \(2311\), \(73363\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2311}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7}a^{17}+\frac{2}{7}a^{16}+\frac{1}{7}a^{15}-\frac{2}{7}a^{14}+\frac{1}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{7}a^{11}-\frac{1}{7}a^{10}+\frac{1}{7}a^{9}-\frac{1}{7}a^{8}-\frac{1}{7}a^{7}+\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{1}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$
Monogenic: | Not computed | |
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: | $a^{2}-a+1$, $2a^{16}-16a^{15}+66a^{14}-182a^{13}+371a^{12}-588a^{11}+745a^{10}-766a^{9}+647a^{8}-456a^{7}+279a^{6}-158a^{5}+83a^{4}-36a^{3}+13a^{2}-4a$, $a^{14}-7a^{13}+26a^{12}-65a^{11}+121a^{10}-176a^{9}+206a^{8}-197a^{7}+157a^{6}-106a^{5}+64a^{4}-35a^{3}+17a^{2}-6a+3$, $a$, $a-1$, $\frac{2}{7}a^{17}-\frac{24}{7}a^{16}+\frac{128}{7}a^{15}-\frac{431}{7}a^{14}+\frac{1038}{7}a^{13}-\frac{1903}{7}a^{12}+\frac{2742}{7}a^{11}-\frac{3152}{7}a^{10}+\frac{2893}{7}a^{9}-\frac{2102}{7}a^{8}+\frac{1188}{7}a^{7}-\frac{526}{7}a^{6}+\frac{204}{7}a^{5}-\frac{85}{7}a^{4}+\frac{33}{7}a^{3}-\frac{19}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{19}{7}a^{17}-\frac{165}{7}a^{16}+\frac{740}{7}a^{15}-\frac{2229}{7}a^{14}+\frac{4996}{7}a^{13}-\frac{8779}{7}a^{12}+\frac{12462}{7}a^{11}-\frac{14544}{7}a^{10}+\frac{14124}{7}a^{9}-\frac{11527}{7}a^{8}+\frac{8031}{7}a^{7}-\frac{4906}{7}a^{6}+\frac{2701}{7}a^{5}-\frac{1336}{7}a^{4}+\frac{583}{7}a^{3}-\frac{212}{7}a^{2}+\frac{62}{7}a-\frac{10}{7}$, $\frac{20}{7}a^{17}-\frac{163}{7}a^{16}+\frac{692}{7}a^{15}-\frac{1986}{7}a^{14}+\frac{4269}{7}a^{13}-\frac{7249}{7}a^{12}+\frac{10032}{7}a^{11}-\frac{11535}{7}a^{10}+\frac{11178}{7}a^{9}-\frac{9232}{7}a^{8}+\frac{6595}{7}a^{7}-\frac{4147}{7}a^{6}+\frac{2313}{7}a^{5}-\frac{1144}{7}a^{4}+\frac{526}{7}a^{3}-\frac{218}{7}a^{2}+\frac{66}{7}a-\frac{19}{7}$ | 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: | \( 61.0464139366 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 61.0464139366 \cdot 1}{2\cdot\sqrt{6646436567333419399}}\cr\approx \mathstrut & 0.180698498816 \end{aligned}\]
Galois group
$C_2^9.S_9$ (as 18T968):
A non-solvable group of order 185794560 |
The 300 conjugacy class representatives for $C_2^9.S_9$ |
Character table for $C_2^9.S_9$ |
Intermediate fields
9.1.53628353.1 |
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 |
Degree 36 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.7.0.1 | $x^{7} + 12 x + 14$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
17.7.0.1 | $x^{7} + 12 x + 14$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(43\) | 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(2311\) | $\Q_{2311}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2311}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(73363\) | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |