Normalized defining polynomial
\( x^{9} - 4x^{8} + 11x^{7} - 34x^{6} + 71x^{5} - 104x^{4} + 117x^{3} - 82x^{2} + 60x - 24 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1311614290944\) \(\medspace = 2^{10}\cdot 3^{4}\cdot 251^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.20\) | 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: | $2^{4/3}3^{3/4}251^{1/2}\approx 91.00204026987298$ | ||
Ramified primes: | \(2\), \(3\), \(251\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{251}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{28}a^{7}-\frac{5}{28}a^{6}+\frac{1}{7}a^{4}-\frac{3}{28}a^{3}-\frac{11}{28}a^{2}-\frac{3}{14}a-\frac{3}{7}$, $\frac{1}{392}a^{8}+\frac{3}{196}a^{7}+\frac{71}{392}a^{6}+\frac{11}{49}a^{5}-\frac{29}{392}a^{4}+\frac{97}{196}a^{3}-\frac{99}{392}a^{2}+\frac{13}{49}a-\frac{19}{98}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{1}{196}a^{8}-\frac{1}{196}a^{7}+\frac{2}{49}a^{6}-\frac{5}{98}a^{5}+\frac{41}{196}a^{4}-\frac{79}{196}a^{3}-\frac{11}{98}a^{2}-\frac{25}{98}a+\frac{2}{49}$, $\frac{11}{392}a^{8}-\frac{23}{196}a^{7}+\frac{165}{392}a^{6}-\frac{101}{98}a^{5}+\frac{997}{392}a^{4}-\frac{823}{196}a^{3}+\frac{1515}{392}a^{2}-\frac{165}{49}a+\frac{127}{98}$, $\frac{1}{56}a^{8}-\frac{3}{28}a^{7}+\frac{19}{56}a^{6}-\frac{13}{14}a^{5}+\frac{17}{8}a^{4}-\frac{95}{28}a^{3}+\frac{229}{56}a^{2}-\frac{20}{7}a+\frac{17}{14}$, $\frac{10}{7}a^{8}-\frac{47}{7}a^{7}+\frac{271}{14}a^{6}-\frac{401}{7}a^{5}+\frac{1777}{14}a^{4}-\frac{389}{2}a^{3}+\frac{1482}{7}a^{2}-\frac{1725}{14}a+\frac{188}{7}$ | 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: | \( 1534.09394687 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 1534.09394687 \cdot 1}{2\cdot\sqrt{1311614290944}}\cr\approx \mathstrut & 2.08770129101 \end{aligned}\]
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
3.1.3012.1 |
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 | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.8.1 | $x^{6} + 2 x^{3} + 2$ | $6$ | $1$ | $8$ | $D_{6}$ | $[2]_{3}^{2}$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(251\) | $\Q_{251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.1004.2t1.a.a | $1$ | $ 2^{2} \cdot 251 $ | \(\Q(\sqrt{251}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.3012.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 251 $ | \(\Q(\sqrt{-753}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.3012.6t3.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 251 $ | 6.2.9108432576.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.3012.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 251 $ | 3.1.3012.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.9072144.6t8.a.a | $3$ | $ 2^{4} \cdot 3^{2} \cdot 251^{2}$ | 4.2.27108.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.27108.4t5.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 251 $ | 4.2.27108.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.9036.6t11.b.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 251 $ | 6.0.27216432.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.27216432.6t11.b.a | $3$ | $ 2^{4} \cdot 3^{3} \cdot 251^{2}$ | 6.0.27216432.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
6.987...832.18t319.a.a | $6$ | $ 2^{10} \cdot 3^{5} \cdot 251^{4}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.3919166208.18t311.a.a | $6$ | $ 2^{8} \cdot 3^{5} \cdot 251^{2}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
* | 6.435462912.9t31.a.a | $6$ | $ 2^{8} \cdot 3^{3} \cdot 251^{2}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.109...648.18t300.a.a | $6$ | $ 2^{10} \cdot 3^{3} \cdot 251^{4}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.298...936.24t2893.a.a | $8$ | $ 2^{14} \cdot 3^{6} \cdot 251^{6}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.11757498624.12t213.a.a | $8$ | $ 2^{8} \cdot 3^{6} \cdot 251^{2}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.809...088.36t2217.a.a | $12$ | $ 2^{16} \cdot 3^{9} \cdot 251^{7}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.387...056.36t2210.a.a | $12$ | $ 2^{18} \cdot 3^{10} \cdot 251^{6}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.321...272.36t2214.a.a | $12$ | $ 2^{14} \cdot 3^{9} \cdot 251^{5}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.809...088.36t2216.a.a | $12$ | $ 2^{16} \cdot 3^{9} \cdot 251^{7}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.321...272.18t315.a.a | $12$ | $ 2^{14} \cdot 3^{9} \cdot 251^{5}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.351...064.24t2912.a.a | $16$ | $ 2^{22} \cdot 3^{12} \cdot 251^{8}$ | 9.1.1311614290944.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |