Normalized defining polynomial
\( x^{9} - 3x^{8} + 6x^{7} - 4x^{6} - x^{5} + 7x^{4} - 8x^{3} - x^{2} + 3x + 1 \)
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: | \(3323228821\) \(\medspace = 11^{4}\cdot 61^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.43\) | 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: | $11^{3/4}61^{1/2}\approx 47.1747308877165$ | ||
Ramified primes: | \(11\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{43}a^{8}+\frac{9}{43}a^{7}-\frac{15}{43}a^{6}-\frac{12}{43}a^{5}-\frac{16}{43}a^{4}-\frac{13}{43}a^{3}+\frac{8}{43}a^{2}+\frac{9}{43}a-\frac{18}{43}$
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: | $a$, $\frac{17}{43}a^{8}-\frac{62}{43}a^{7}+\frac{132}{43}a^{6}-\frac{118}{43}a^{5}-\frac{14}{43}a^{4}+\frac{166}{43}a^{3}-\frac{208}{43}a^{2}+\frac{24}{43}a+\frac{81}{43}$, $\frac{20}{43}a^{8}-\frac{78}{43}a^{7}+\frac{173}{43}a^{6}-\frac{197}{43}a^{5}+\frac{67}{43}a^{4}+\frac{127}{43}a^{3}-\frac{270}{43}a^{2}+\frac{137}{43}a+\frac{70}{43}$, $\frac{18}{43}a^{8}-\frac{53}{43}a^{7}+\frac{117}{43}a^{6}-\frac{87}{43}a^{5}+\frac{13}{43}a^{4}+\frac{153}{43}a^{3}-\frac{157}{43}a^{2}+\frac{33}{43}a+\frac{63}{43}$ | 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: | \( 14.0819594408 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 14.0819594408 \cdot 1}{2\cdot\sqrt{3323228821}}\cr\approx \mathstrut & 0.380717149113 \end{aligned}\]
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
3.1.671.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | R | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.4.3.2 | $x^{4} + 22$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(61\) | $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.61.2t1.a.a | $1$ | $ 61 $ | \(\Q(\sqrt{61}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.671.2t1.a.a | $1$ | $ 11 \cdot 61 $ | \(\Q(\sqrt{-671}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.671.6t3.b.a | $2$ | $ 11 \cdot 61 $ | 6.2.27464701.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.671.3t2.a.a | $2$ | $ 11 \cdot 61 $ | 3.1.671.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.450241.6t8.a.a | $3$ | $ 11^{2} \cdot 61^{2}$ | 4.2.81191.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.81191.4t5.a.a | $3$ | $ 11^{3} \cdot 61 $ | 4.2.81191.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.7381.6t11.a.a | $3$ | $ 11^{2} \cdot 61 $ | 6.0.4952651.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.4952651.6t11.a.a | $3$ | $ 11^{3} \cdot 61^{2}$ | 6.0.4952651.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
6.222...891.18t319.a.a | $6$ | $ 11^{5} \cdot 61^{4}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.599270771.18t311.a.a | $6$ | $ 11^{5} \cdot 61^{2}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
* | 6.4952651.9t31.a.a | $6$ | $ 11^{3} \cdot 61^{2}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.18428814371.18t300.a.a | $6$ | $ 11^{3} \cdot 61^{4}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.912...521.24t2893.a.a | $8$ | $ 11^{6} \cdot 61^{6}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.6591978481.12t213.a.a | $8$ | $ 11^{6} \cdot 61^{2}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.741...511.36t2217.a.a | $12$ | $ 11^{9} \cdot 61^{7}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.133...961.36t2210.a.a | $12$ | $ 11^{10} \cdot 61^{6}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.199...991.36t2214.a.a | $12$ | $ 11^{9} \cdot 61^{5}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.741...511.36t2216.a.a | $12$ | $ 11^{9} \cdot 61^{7}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.199...991.18t315.a.a | $12$ | $ 11^{9} \cdot 61^{5}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.601...601.24t2912.a.a | $16$ | $ 11^{12} \cdot 61^{8}$ | 9.1.3323228821.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |