Normalized defining polynomial
\( x^{9} - x^{8} - x^{7} + 6x^{6} - 6x^{5} - 3x^{4} + 9x^{3} - x^{2} - 6x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-6350622912\) \(\medspace = -\,2^{6}\cdot 3^{4}\cdot 107^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.28\) | 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: | $2^{2/3}3^{1/2}107^{1/2}\approx 28.44062787690364$ | ||
Ramified primes: | \(2\), \(3\), \(107\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-107}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{109}a^{8}-\frac{10}{109}a^{7}-\frac{20}{109}a^{6}-\frac{32}{109}a^{5}-\frac{45}{109}a^{4}-\frac{34}{109}a^{3}-\frac{12}{109}a^{2}-\frac{2}{109}a+\frac{12}{109}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{15}{109}a^{8}-\frac{41}{109}a^{7}+\frac{27}{109}a^{6}+\frac{65}{109}a^{5}-\frac{239}{109}a^{4}+\frac{253}{109}a^{3}-\frac{71}{109}a^{2}-\frac{139}{109}a+\frac{71}{109}$, $a$, $\frac{51}{109}a^{8}-\frac{74}{109}a^{7}-\frac{39}{109}a^{6}+\frac{330}{109}a^{5}-\frac{442}{109}a^{4}-\frac{99}{109}a^{3}+\frac{587}{109}a^{2}-\frac{211}{109}a-\frac{369}{109}$, $\frac{35}{109}a^{8}-\frac{23}{109}a^{7}-\frac{46}{109}a^{6}+\frac{188}{109}a^{5}-\frac{158}{109}a^{4}-\frac{100}{109}a^{3}+\frac{234}{109}a^{2}-\frac{70}{109}a-\frac{125}{109}$, $\frac{23}{109}a^{8}-\frac{12}{109}a^{7}-\frac{24}{109}a^{6}+\frac{136}{109}a^{5}-\frac{54}{109}a^{4}-\frac{128}{109}a^{3}+\frac{269}{109}a^{2}+\frac{63}{109}a-\frac{160}{109}$ | 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: | \( 37.0352534519 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 37.0352534519 \cdot 1}{2\cdot\sqrt{6350622912}}\cr\approx \mathstrut & 0.461112350465 \end{aligned}\]
Galois group
$C_3^2:D_6$ (as 9T18):
A solvable group of order 108 |
The 11 conjugacy class representatives for $C_3^2 : D_{6} $ |
Character table for $C_3^2 : D_{6} $ |
Intermediate fields
3.3.321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 9.1.2116874304.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(107\) | $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
107.4.2.1 | $x^{4} + 206 x^{3} + 10827 x^{2} + 22454 x + 1146188$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.321.2t1.a.a | $1$ | $ 3 \cdot 107 $ | \(\Q(\sqrt{321}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.107.2t1.a.a | $1$ | $ 107 $ | \(\Q(\sqrt{-107}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.321.6t3.b.a | $2$ | $ 3 \cdot 107 $ | 6.0.11025387.1 | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
2.963.6t3.b.a | $2$ | $ 3^{2} \cdot 107 $ | 6.2.33076161.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.107.3t2.a.a | $2$ | $ 107 $ | 3.1.107.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 2.321.3t2.a.a | $2$ | $ 3 \cdot 107 $ | 3.3.321.1 | $S_3$ (as 3T2) | $1$ | $2$ |
4.103041.6t9.a.a | $4$ | $ 3^{2} \cdot 107^{2}$ | 6.0.309123.2 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
6.226505550528.18t51.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 107^{4}$ | 9.3.6350622912.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ | |
* | 6.19783872.9t18.a.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 107^{2}$ | 9.3.6350622912.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ |