Normalized defining polynomial
\( x^{9} - 3x^{8} + 3x^{7} - 12x^{6} + 39x^{5} - 45x^{4} + 63x^{3} - 216x^{2} + 153x + 42 \)
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: | \(-38483081420787\) \(\medspace = -\,3^{13}\cdot 17^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.32\) | 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))
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Galois root discriminant: | $3^{37/24}17^{2/3}\approx 35.963439071886725$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6744893}a^{8}+\frac{3214700}{6744893}a^{7}+\frac{3260972}{6744893}a^{6}+\frac{2127951}{6744893}a^{5}-\frac{721045}{6744893}a^{4}-\frac{2341193}{6744893}a^{3}+\frac{1479183}{6744893}a^{2}-\frac{2047781}{6744893}a+\frac{1099217}{6744893}$
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: | $\frac{359365}{6744893}a^{8}-\frac{1117754}{6744893}a^{7}+\frac{1258281}{6744893}a^{6}-\frac{4622546}{6744893}a^{5}+\frac{13707742}{6744893}a^{4}-\frac{18594090}{6744893}a^{3}+\frac{28561037}{6744893}a^{2}-\frac{73462123}{6744893}a+\frac{66162697}{6744893}$, $\frac{295481}{6744893}a^{8}-\frac{510490}{6744893}a^{7}+\frac{88231}{6744893}a^{6}-\frac{3325815}{6744893}a^{5}+\frac{9327332}{6744893}a^{4}-\frac{8332967}{6744893}a^{3}+\frac{21640302}{6744893}a^{2}-\frac{63475561}{6744893}a+\frac{4160855}{6744893}$, $\frac{64864}{6744893}a^{8}-\frac{66295}{6744893}a^{7}-\frac{156672}{6744893}a^{6}-\frac{76688}{6744893}a^{5}-\frac{774818}{6744893}a^{4}+\frac{2123143}{6744893}a^{3}-\frac{376813}{6744893}a^{2}-\frac{88935}{6744893}a-\frac{652415}{6744893}$, $\frac{69247}{6744893}a^{8}-\frac{117672}{6744893}a^{7}+\frac{255337}{6744893}a^{6}-\frac{1454474}{6744893}a^{5}+\frac{2239764}{6744893}a^{4}-\frac{343523}{6744893}a^{3}+\frac{7784996}{6744893}a^{2}-\frac{25040047}{6744893}a-\frac{5382799}{6744893}$, $\frac{136790}{6744893}a^{8}-\frac{1231028}{6744893}a^{7}+\frac{1606218}{6744893}a^{6}-\frac{185018}{6744893}a^{5}+\frac{5569682}{6744893}a^{4}-\frac{24505509}{6744893}a^{3}+\frac{17632142}{6744893}a^{2}-\frac{7301593}{6744893}a+\frac{11483567}{6744893}$ | 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: | \( 6309.07358735 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 6309.07358735 \cdot 1}{2\cdot\sqrt{38483081420787}}\cr\approx \mathstrut & 1.00909085307 \end{aligned}\]
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 | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.13.6 | $x^{9} + 3 x^{5} + 3$ | $9$ | $1$ | $13$ | $(C_3^2:C_8):C_2$ | $[13/8, 13/8]_{8}^{2}$ |
\(17\) | 17.3.2.1 | $x^{3} + 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
17.3.2.1 | $x^{3} + 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
17.3.2.1 | $x^{3} + 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
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.b.a | $2$ | $ 3^{2} \cdot 17^{2}$ | 8.2.182660427.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.2601.24t22.b.b | $2$ | $ 3^{2} \cdot 17^{2}$ | 8.2.182660427.2 | $\textrm{GL(2,3)}$ (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) | $1$ | $-1$ | |
4.23409.8t23.b.a | $4$ | $ 3^{4} \cdot 17^{2}$ | 8.2.182660427.2 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ | |
* | 8.384...787.9t26.a.a | $8$ | $ 3^{13} \cdot 17^{6}$ | 9.3.38483081420787.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $2$ |
8.384...787.18t157.a.a | $8$ | $ 3^{13} \cdot 17^{6}$ | 9.3.38483081420787.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $-2$ | |
16.512...721.24t1334.a.a | $16$ | $ 3^{26} \cdot 17^{10}$ | 9.3.38483081420787.2 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $0$ |