Normalized defining polynomial
\( x^{9} - 2x^{8} + 2x^{7} - 2x^{5} + 2x^{4} - x + 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(33860761\) \(\medspace = 11^{2}\cdot 23^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(6.86\) | 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: | $11^{1/2}23^{3/4}\approx 34.833107461122644$ | ||
Ramified primes: | \(11\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $a$, $a^{8}-a^{7}+a^{6}-a^{4}+a-1$, $a^{8}-a^{7}+a^{6}-a^{4}+a^{3}+a$, $a-1$ | 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: | \( 0.744153660637 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 0.744153660637 \cdot 1}{2\cdot\sqrt{33860761}}\cr\approx \mathstrut & 0.1993121338299 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 9T30):
A solvable group of order 648 |
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Intermediate fields
3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | 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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.3.1 | $x^{4} + 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.23.3t2.b.a | $2$ | $ 23 $ | 3.1.23.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.64009.6t8.a.a | $3$ | $ 11^{2} \cdot 23^{2}$ | 4.2.1472207.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.1472207.4t5.a.a | $3$ | $ 11^{2} \cdot 23^{3}$ | 4.2.1472207.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
6.778797503.18t217.a.a | $6$ | $ 11^{2} \cdot 23^{5}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
* | 6.1472207.9t30.a.a | $6$ | $ 11^{2} \cdot 23^{3}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ |
6.94234497863.36t1121.a.a | $6$ | $ 11^{4} \cdot 23^{5}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
6.94234497863.36t1121.a.b | $6$ | $ 11^{4} \cdot 23^{5}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.216...849.12t177.a.a | $8$ | $ 11^{4} \cdot 23^{6}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.216...849.12t177.b.a | $8$ | $ 11^{4} \cdot 23^{6}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.216...849.12t178.a.a | $8$ | $ 11^{4} \cdot 23^{6}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
12.319...743.36t1123.a.a | $12$ | $ 11^{6} \cdot 23^{9}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $-2$ | |
12.319...743.18t218.a.a | $12$ | $ 11^{6} \cdot 23^{9}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $2$ |