Normalized defining polynomial
\( x^{9} - 3x^{8} - 3x^{5} + 6x^{3} + 33x^{2} + 27x - 7 \)
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: | \(-475099770627\) \(\medspace = -\,3^{9}\cdot 17^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.83\) | 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: | $3^{79/72}17^{5/6}\approx 35.39012890556844$ | ||
Ramified primes: | \(3\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{63}a^{8}+\frac{1}{9}a^{5}-\frac{1}{21}a^{4}-\frac{1}{7}a^{3}+\frac{4}{9}a^{2}+\frac{4}{21}a$
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: | $\frac{1}{21}a^{8}-\frac{2}{9}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{2}{63}a^{4}+\frac{5}{21}a^{3}+\frac{43}{63}a$, $\frac{4}{63}a^{8}-\frac{2}{9}a^{7}+\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{47}{63}a^{4}+\frac{34}{63}a^{3}+\frac{1}{9}a^{2}+\frac{34}{63}a+\frac{7}{9}$, $\frac{8}{63}a^{8}-\frac{1}{3}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{8}{21}a^{4}+\frac{5}{63}a^{3}+\frac{5}{9}a^{2}+\frac{34}{7}a+\frac{44}{9}$, $\frac{10}{63}a^{8}-\frac{4}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{37}{63}a^{4}-\frac{13}{63}a^{3}+\frac{13}{9}a^{2}+\frac{344}{63}a+\frac{44}{9}$, $\frac{1}{63}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{4}{9}a^{5}-\frac{31}{63}a^{4}-\frac{23}{63}a^{3}+\frac{1}{9}a^{2}-\frac{16}{63}a+\frac{1}{9}$ | 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: | \( 749.899873552 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 749.899873552 \cdot 1}{2\cdot\sqrt{475099770627}}\cr\approx \mathstrut & 1.07946972465 \end{aligned}\]
Galois group
$C_3^2:\GL(2,3)$ (as 9T26):
A solvable group of order 432 |
The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$ |
Character table for $((C_3^2:Q_8):C_3):C_2$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.9.12 | $x^{9} + 6 x + 3$ | $9$ | $1$ | $9$ | $(C_3^2:C_8):C_2$ | $[9/8, 9/8]_{8}^{2}$ |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.6.5.1 | $x^{6} + 17$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
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) | $1$ | $-1$ | |
2.867.3t2.b.a | $2$ | $ 3 \cdot 17^{2}$ | 3.1.867.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.2601.24t22.c.a | $2$ | $ 3^{2} \cdot 17^{2}$ | 8.2.52788863403.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.2601.24t22.c.b | $2$ | $ 3^{2} \cdot 17^{2}$ | 8.2.52788863403.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.7803.4t5.a.a | $3$ | $ 3^{3} \cdot 17^{2}$ | 4.2.7803.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.2601.6t8.a.a | $3$ | $ 3^{2} \cdot 17^{2}$ | 4.2.7803.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
4.6765201.8t23.a.a | $4$ | $ 3^{4} \cdot 17^{4}$ | 8.2.52788863403.1 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ | |
* | 8.475099770627.9t26.a.a | $8$ | $ 3^{9} \cdot 17^{6}$ | 9.3.475099770627.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $2$ |
8.475099770627.18t157.a.a | $8$ | $ 3^{9} \cdot 17^{6}$ | 9.3.475099770627.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $-2$ | |
16.652...281.24t1334.a.a | $16$ | $ 3^{18} \cdot 17^{14}$ | 9.3.475099770627.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $0$ |