Normalized defining polynomial
\( x^{9} + 9x^{7} - 63x^{6} - 81x^{5} - 1296x^{4} - 1359x^{3} - 6885x^{2} - 5265x - 1125 \)
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: | \(47091202575755625\) \(\medspace = 3^{22}\cdot 5^{4}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(71.21\) | 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: | $3^{22/9}5^{1/2}7^{1/2}\approx 86.76217340159114$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{28733890485}a^{8}-\frac{472207895}{5746778097}a^{7}+\frac{1067277793}{9577963495}a^{6}+\frac{832518307}{28733890485}a^{5}+\frac{2670865738}{9577963495}a^{4}-\frac{1679177157}{9577963495}a^{3}-\frac{2901029378}{9577963495}a^{2}-\frac{34988759}{1915592699}a-\frac{100071575}{1915592699}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{9}$, which has order $9$
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: | $\frac{157716848}{28733890485}a^{8}+\frac{76832660}{5746778097}a^{7}-\frac{119397156}{9577963495}a^{6}-\frac{3987574258}{9577963495}a^{5}-\frac{13759048286}{9577963495}a^{4}-\frac{35347383621}{9577963495}a^{3}-\frac{54328419504}{9577963495}a^{2}+\frac{19220361822}{1915592699}a+\frac{8631688384}{1915592699}$, $\frac{22806}{31403159}a^{8}-\frac{43274}{31403159}a^{7}+\frac{94590}{31403159}a^{6}-\frac{228276}{31403159}a^{5}-\frac{4190106}{31403159}a^{4}+\frac{2221154}{31403159}a^{3}-\frac{29231910}{31403159}a^{2}+\frac{9486540}{31403159}a+\frac{9225761}{31403159}$, $\frac{37427474}{9577963495}a^{8}+\frac{179478355}{5746778097}a^{7}-\frac{231413672}{28733890485}a^{6}-\frac{2286057967}{9577963495}a^{5}-\frac{21430951319}{9577963495}a^{4}-\frac{48737265659}{9577963495}a^{3}-\frac{249559609156}{9577963495}a^{2}-\frac{36978920944}{1915592699}a-\frac{7791350039}{1915592699}$, $\frac{99851844}{9577963495}a^{8}+\frac{634434895}{1915592699}a^{7}+\frac{3255622571}{9577963495}a^{6}-\frac{7548249687}{9577963495}a^{5}-\frac{237696792044}{9577963495}a^{4}-\frac{572069708694}{9577963495}a^{3}-\frac{2275119449031}{9577963495}a^{2}-\frac{326547301971}{1915592699}a-\frac{67637451004}{1915592699}$ | 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: | \( 24173.5233815 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 24173.5233815 \cdot 9}{2\cdot\sqrt{47091202575755625}}\cr\approx \mathstrut & 1.56254325072 \end{aligned}\]
Galois group
A solvable group of order 54 |
The 10 conjugacy class representatives for $(C_9:C_3):C_2$ |
Character table for $(C_9:C_3):C_2$ |
Intermediate fields
3.1.2835.1 |
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 27 sibling: | 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | R | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.315.6t1.h.a | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.315.6t1.h.b | $1$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.281302875.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
* | 2.2835.3t2.a.a | $2$ | $ 3^{4} \cdot 5 \cdot 7 $ | 3.1.2835.1 | $S_3$ (as 3T2) | $1$ | $0$ |
2.315.6t5.a.a | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.3472875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
2.315.6t5.a.b | $2$ | $ 3^{2} \cdot 5 \cdot 7 $ | 6.0.3472875.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
* | 6.166...875.9t10.a.a | $6$ | $ 3^{18} \cdot 5^{3} \cdot 7^{3}$ | 9.1.47091202575755625.1 | $(C_9:C_3):C_2$ (as 9T10) | $1$ | $0$ |