Normalized defining polynomial
\( x^{9} - 3x^{8} + 4x^{7} - 2x^{6} + 2x^{5} - 2x^{4} - 2x^{3} + 8x^{2} - 3x - 1 \)
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: | \(-14873341696\) \(\medspace = -\,2^{8}\cdot 31\cdot 37^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.50\) | 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: | $2^{8/9}31^{1/2}37^{1/2}\approx 62.71391706256195$ | ||
Ramified primes: | \(2\), \(31\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13}a^{8}-\frac{1}{13}a^{7}+\frac{2}{13}a^{6}+\frac{2}{13}a^{5}+\frac{6}{13}a^{4}-\frac{3}{13}a^{3}+\frac{5}{13}a^{2}+\frac{5}{13}a-\frac{6}{13}$
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: | $a$, $\frac{7}{13}a^{8}-\frac{20}{13}a^{7}+\frac{27}{13}a^{6}-\frac{12}{13}a^{5}+\frac{16}{13}a^{4}-\frac{8}{13}a^{3}-\frac{17}{13}a^{2}+\frac{61}{13}a-\frac{3}{13}$, $\frac{1}{13}a^{8}-\frac{1}{13}a^{7}+\frac{2}{13}a^{6}+\frac{2}{13}a^{5}+\frac{6}{13}a^{4}-\frac{3}{13}a^{3}+\frac{5}{13}a^{2}+\frac{5}{13}a+\frac{20}{13}$, $\frac{2}{13}a^{8}-\frac{15}{13}a^{7}+\frac{30}{13}a^{6}-\frac{22}{13}a^{5}+\frac{12}{13}a^{4}-\frac{19}{13}a^{3}+\frac{10}{13}a^{2}+\frac{49}{13}a-\frac{38}{13}$, $\frac{5}{13}a^{8}-\frac{18}{13}a^{7}+\frac{23}{13}a^{6}-\frac{16}{13}a^{5}+\frac{17}{13}a^{4}-\frac{28}{13}a^{3}-\frac{1}{13}a^{2}+\frac{38}{13}a-\frac{17}{13}$ | 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: | \( 55.0188542047 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 55.0188542047 \cdot 1}{2\cdot\sqrt{14873341696}}\cr\approx \mathstrut & 0.447617297090 \end{aligned}\]
Galois group
$S_3\wr S_3$ (as 9T31):
A solvable group of order 1296 |
The 22 conjugacy class representatives for $S_3\wr S_3$ |
Character table for $S_3\wr S_3$ is not computed |
Intermediate fields
3.3.148.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | R | R | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.8.1 | $x^{9} + 2$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.37.2t1.a.a | $1$ | $ 37 $ | \(\Q(\sqrt{37}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.1147.2t1.a.a | $1$ | $ 31 \cdot 37 $ | \(\Q(\sqrt{-1147}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.142228.6t3.a.a | $2$ | $ 2^{2} \cdot 31^{2} \cdot 37 $ | 6.0.652542064.2 | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
* | 2.148.3t2.a.a | $2$ | $ 2^{2} \cdot 37 $ | 3.3.148.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.142228.4t5.a.a | $3$ | $ 2^{2} \cdot 31^{2} \cdot 37 $ | 4.4.142228.1 | $S_4$ (as 4T5) | $1$ | $3$ | |
3.4588.6t11.a.a | $3$ | $ 2^{2} \cdot 31 \cdot 37 $ | 6.0.25123888.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-3$ | |
3.5262436.6t8.a.a | $3$ | $ 2^{2} \cdot 31^{2} \cdot 37^{2}$ | 4.4.142228.1 | $S_4$ (as 4T5) | $1$ | $3$ | |
3.169756.6t11.a.a | $3$ | $ 2^{2} \cdot 31 \cdot 37^{2}$ | 6.0.25123888.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-3$ | |
* | 6.100495552.9t31.a.a | $6$ | $ 2^{6} \cdot 31 \cdot 37^{3}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.928...592.18t300.a.a | $6$ | $ 2^{6} \cdot 31^{5} \cdot 37^{3}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.100495552.18t319.a.a | $6$ | $ 2^{6} \cdot 31 \cdot 37^{3}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.928...592.18t311.a.a | $6$ | $ 2^{6} \cdot 31^{5} \cdot 37^{3}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.443...536.24t2893.a.a | $8$ | $ 2^{8} \cdot 31^{4} \cdot 37^{4}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.443...536.12t213.a.a | $8$ | $ 2^{8} \cdot 31^{4} \cdot 37^{4}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.605...888.36t2217.a.a | $12$ | $ 2^{10} \cdot 31^{8} \cdot 37^{5}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.829...672.36t2214.a.a | $12$ | $ 2^{10} \cdot 31^{8} \cdot 37^{7}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.932...784.36t2210.a.a | $12$ | $ 2^{12} \cdot 31^{6} \cdot 37^{6}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.897...032.36t2216.a.a | $12$ | $ 2^{10} \cdot 31^{4} \cdot 37^{7}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.655...328.18t315.a.a | $12$ | $ 2^{10} \cdot 31^{4} \cdot 37^{5}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
16.490...824.24t2912.a.a | $16$ | $ 2^{14} \cdot 31^{8} \cdot 37^{8}$ | 9.3.14873341696.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |