Normalized defining polynomial
\( x^{9} - 6x^{7} - 6x^{6} + 12x^{5} + 24x^{4} - 18x^{3} - 24x^{2} + 42x + 36 \)
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: | \(-15869558403072\) \(\medspace = -\,2^{12}\cdot 3^{7}\cdot 11^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.29\) | 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: | $2^{19/12}3^{7/8}11^{2/3}\approx 38.759102715644154$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{834}a^{8}-\frac{6}{139}a^{7}-\frac{63}{139}a^{6}+\frac{43}{139}a^{5}-\frac{17}{139}a^{4}+\frac{60}{139}a^{3}+\frac{61}{139}a^{2}+\frac{24}{139}a-\frac{23}{139}$
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: | $a+1$, $\frac{31}{139}a^{8}-\frac{4}{139}a^{7}-\frac{181}{139}a^{6}-\frac{203}{139}a^{5}+\frac{313}{139}a^{4}+\frac{874}{139}a^{3}+\frac{87}{139}a^{2}-\frac{679}{139}a-\frac{247}{139}$, $\frac{184}{139}a^{8}-\frac{230}{139}a^{7}-\frac{886}{139}a^{6}-\frac{66}{139}a^{5}+\frac{2499}{139}a^{4}+\frac{1744}{139}a^{3}-\frac{5492}{139}a^{2}+\frac{1893}{139}a+\frac{6161}{139}$, $\frac{69}{139}a^{8}-\frac{121}{139}a^{7}-\frac{228}{139}a^{6}+\frac{10}{139}a^{5}+\frac{885}{139}a^{4}+\frac{98}{139}a^{3}-\frac{1434}{139}a^{2}+\frac{901}{139}a+\frac{1459}{139}$, $\frac{932}{417}a^{8}-\frac{342}{139}a^{7}-\frac{1506}{139}a^{6}-\frac{51}{139}a^{5}+\frac{3479}{139}a^{4}+\frac{3281}{139}a^{3}-\frac{7782}{139}a^{2}+\frac{395}{139}a+\frac{11477}{139}$ | 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: | \( 8000.74076133 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 8000.74076133 \cdot 1}{2\cdot\sqrt{15869558403072}}\cr\approx \mathstrut & 1.99272502197 \end{aligned}\]
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 | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.8.12.30 | $x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ | $8$ | $1$ | $12$ | $\textrm{GL(2,3)}$ | $[4/3, 4/3, 2]_{3}^{2}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.8.7.2 | $x^{8} + 6$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
\(11\) | 11.9.6.1 | $x^{9} + 6 x^{7} + 60 x^{6} + 12 x^{5} + 42 x^{4} - 1465 x^{3} + 240 x^{2} - 1560 x + 8088$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
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) | $1$ | $-1$ | |
2.1452.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 11^{2}$ | 3.1.1452.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.17424.24t22.a.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 11^{2}$ | 8.2.131153375232.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.17424.24t22.a.b | $2$ | $ 2^{4} \cdot 3^{2} \cdot 11^{2}$ | 8.2.131153375232.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.52272.4t5.b.a | $3$ | $ 2^{4} \cdot 3^{3} \cdot 11^{2}$ | 4.2.52272.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.17424.6t8.a.a | $3$ | $ 2^{4} \cdot 3^{2} \cdot 11^{2}$ | 4.2.52272.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
4.2509056.8t23.a.a | $4$ | $ 2^{8} \cdot 3^{4} \cdot 11^{2}$ | 8.2.131153375232.1 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ | |
* | 8.158...072.9t26.a.a | $8$ | $ 2^{12} \cdot 3^{7} \cdot 11^{6}$ | 9.3.15869558403072.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $2$ |
8.158...072.18t157.a.a | $8$ | $ 2^{12} \cdot 3^{7} \cdot 11^{6}$ | 9.3.15869558403072.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $-2$ | |
16.832...816.24t1334.a.a | $16$ | $ 2^{26} \cdot 3^{14} \cdot 11^{10}$ | 9.3.15869558403072.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $0$ |