Normalized defining polynomial
\( x^{9} - 3x^{8} + 7x^{6} - 3x^{5} - 12x^{4} + 8x^{3} + 3x - 2 \)
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: | \(-17964142659\) \(\medspace = -\,3^{9}\cdot 97^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.78\) | 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: | $3^{7/6}97^{1/2}\approx 35.4835719013513$ | ||
Ramified primes: | \(3\), \(97\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-291}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{25}a^{8}+\frac{7}{25}a^{5}-\frac{7}{25}a^{4}-\frac{8}{25}a^{3}+\frac{9}{25}a^{2}+\frac{2}{25}a+\frac{9}{25}$
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: | $\frac{32}{25}a^{8}-\frac{16}{5}a^{7}-\frac{8}{5}a^{6}+\frac{204}{25}a^{5}+\frac{6}{25}a^{4}-\frac{381}{25}a^{3}+\frac{78}{25}a^{2}+\frac{14}{25}a+\frac{103}{25}$, $a-1$, $\frac{32}{25}a^{8}-3a^{7}-2a^{6}+\frac{199}{25}a^{5}+\frac{26}{25}a^{4}-\frac{381}{25}a^{3}+\frac{38}{25}a^{2}+\frac{39}{25}a+\frac{63}{25}$, $\frac{3}{5}a^{8}-\frac{8}{5}a^{7}-\frac{4}{5}a^{6}+\frac{24}{5}a^{5}-\frac{3}{5}a^{4}-\frac{44}{5}a^{3}+\frac{16}{5}a^{2}+\frac{16}{5}a+\frac{1}{5}$, $\frac{27}{25}a^{8}-\frac{13}{5}a^{7}-\frac{9}{5}a^{6}+\frac{179}{25}a^{5}+\frac{26}{25}a^{4}-\frac{341}{25}a^{3}+\frac{38}{25}a^{2}+\frac{79}{25}a+\frac{63}{25}$ | 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: | \( 76.803675927 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 76.803675927 \cdot 1}{2\cdot\sqrt{17964142659}}\cr\approx \mathstrut & 0.56856268741 \end{aligned}\]
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.9.6 | $x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ |
\(97\) | $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
97.4.2.1 | $x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.291.2t1.a.a | $1$ | $ 3 \cdot 97 $ | \(\Q(\sqrt{-291}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.2619.3t2.a.a | $2$ | $ 3^{3} \cdot 97 $ | 3.1.2619.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.2619.24t22.c.a | $2$ | $ 3^{3} \cdot 97 $ | 8.2.1996015851.3 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.2619.24t22.c.b | $2$ | $ 3^{3} \cdot 97 $ | 8.2.1996015851.3 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.2619.4t5.a.a | $3$ | $ 3^{3} \cdot 97 $ | 4.2.2619.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.762129.6t8.a.a | $3$ | $ 3^{4} \cdot 97^{2}$ | 4.2.2619.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
4.762129.8t23.c.a | $4$ | $ 3^{4} \cdot 97^{2}$ | 8.2.1996015851.3 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ | |
* | 8.17964142659.9t26.a.a | $8$ | $ 3^{9} \cdot 97^{3}$ | 9.3.17964142659.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $2$ |
8.152...779.18t157.a.a | $8$ | $ 3^{11} \cdot 97^{5}$ | 9.3.17964142659.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $-2$ | |
16.303...929.24t1334.a.a | $16$ | $ 3^{18} \cdot 97^{8}$ | 9.3.17964142659.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $0$ |