Normalized defining polynomial
\( x^{9} - 2x^{8} - 13x^{7} + 30x^{6} + 39x^{5} - 118x^{4} + 3x^{3} + 96x^{2} - 36x - 18 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2877061966848\) \(\medspace = 2^{10}\cdot 3^{3}\cdot 101^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.23\) | 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: | $2^{19/12}3^{1/2}101^{1/2}\approx 52.161748485858105$ | ||
Ramified primes: | \(2\), \(3\), \(101\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{159}a^{7}-\frac{13}{159}a^{6}+\frac{79}{159}a^{5}-\frac{17}{159}a^{4}+\frac{13}{159}a^{3}-\frac{10}{53}a^{2}-\frac{4}{53}a+\frac{3}{53}$, $\frac{1}{477}a^{8}-\frac{1}{477}a^{7}-\frac{77}{477}a^{6}-\frac{23}{477}a^{5}+\frac{127}{477}a^{4}-\frac{64}{159}a^{3}+\frac{35}{159}a^{2}-\frac{15}{53}a+\frac{12}{53}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{4}{159}a^{7}+\frac{1}{159}a^{6}-\frac{55}{159}a^{5}-\frac{5}{53}a^{4}+\frac{88}{53}a^{3}+\frac{92}{159}a^{2}-\frac{175}{53}a-\frac{41}{53}$, $\frac{11}{159}a^{8}-\frac{1}{53}a^{7}-\frac{52}{53}a^{6}+\frac{61}{159}a^{5}+\frac{625}{159}a^{4}-\frac{259}{159}a^{3}-\frac{225}{53}a^{2}+\frac{56}{53}a+\frac{49}{53}$, $\frac{62}{477}a^{8}-\frac{113}{477}a^{7}-\frac{772}{477}a^{6}+\frac{1700}{477}a^{5}+\frac{2063}{477}a^{4}-\frac{2122}{159}a^{3}+\frac{454}{159}a^{2}+\frac{410}{53}a-\frac{367}{53}$, $\frac{43}{159}a^{8}-\frac{31}{53}a^{7}-\frac{541}{159}a^{6}+\frac{456}{53}a^{5}+\frac{1435}{159}a^{4}-\frac{1732}{53}a^{3}+\frac{980}{159}a^{2}+\frac{1180}{53}a-\frac{669}{53}$, $\frac{85}{477}a^{8}-\frac{67}{477}a^{7}-\frac{1214}{477}a^{6}+\frac{1057}{477}a^{5}+\frac{5083}{477}a^{4}-\frac{427}{53}a^{3}-\frac{782}{53}a^{2}+\frac{79}{53}a+\frac{137}{53}$, $\frac{4}{477}a^{8}+\frac{32}{477}a^{7}-\frac{140}{477}a^{6}-\frac{269}{477}a^{5}+\frac{1009}{477}a^{4}-\frac{47}{159}a^{3}-\frac{326}{159}a^{2}+\frac{51}{53}a+\frac{31}{53}$ | 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: | \( 4948.19109892 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 4948.19109892 \cdot 1}{2\cdot\sqrt{2877061966848}}\cr\approx \mathstrut & 1.84268701349 \end{aligned}\]
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 $S_3\wr S_3$ |
Intermediate fields
3.3.404.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.8.2 | $x^{6} + 2 x^{4} + 2 x^{3} + 2$ | $6$ | $1$ | $8$ | $S_4\times C_2$ | $[4/3, 4/3, 2]_{3}^{2}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(101\) | $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
101.6.3.1 | $x^{6} + 24543 x^{5} + 200786592 x^{4} + 547549249797 x^{3} + 20884233801 x^{2} + 1662760800612 x + 54226912599047$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.101.2t1.a.a | $1$ | $ 101 $ | \(\Q(\sqrt{101}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.1212.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 101 $ | \(\Q(\sqrt{303}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.14544.6t3.a.a | $2$ | $ 2^{4} \cdot 3^{2} \cdot 101 $ | 6.6.70509312.1 | $D_{6}$ (as 6T3) | $1$ | $2$ | |
* | 2.404.3t2.a.a | $2$ | $ 2^{2} \cdot 101 $ | 3.3.404.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.1616.4t5.a.a | $3$ | $ 2^{4} \cdot 101 $ | 4.0.1616.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.174528.6t11.b.a | $3$ | $ 2^{6} \cdot 3^{3} \cdot 101 $ | 6.2.7121440512.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.163216.6t8.a.a | $3$ | $ 2^{4} \cdot 101^{2}$ | 4.0.1616.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.17627328.6t11.b.a | $3$ | $ 2^{6} \cdot 3^{3} \cdot 101^{2}$ | 6.2.7121440512.2 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.7121440512.9t31.a.a | $6$ | $ 2^{8} \cdot 3^{3} \cdot 101^{3}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ |
6.28485762048.18t300.a.a | $6$ | $ 2^{10} \cdot 3^{3} \cdot 101^{3}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
6.7121440512.18t319.a.a | $6$ | $ 2^{8} \cdot 3^{3} \cdot 101^{3}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
6.28485762048.18t311.a.a | $6$ | $ 2^{10} \cdot 3^{3} \cdot 101^{3}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
8.345...176.24t2893.a.a | $8$ | $ 2^{12} \cdot 3^{4} \cdot 101^{4}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.345...176.12t213.a.a | $8$ | $ 2^{12} \cdot 3^{4} \cdot 101^{4}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.803...904.36t2217.a.a | $12$ | $ 2^{20} \cdot 3^{6} \cdot 101^{5}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.819...704.36t2214.a.a | $12$ | $ 2^{20} \cdot 3^{6} \cdot 101^{7}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.811...304.36t2210.a.a | $12$ | $ 2^{20} \cdot 3^{6} \cdot 101^{6}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-4$ | |
12.204...176.36t2216.a.a | $12$ | $ 2^{18} \cdot 3^{6} \cdot 101^{7}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.200...976.18t315.a.a | $12$ | $ 2^{18} \cdot 3^{6} \cdot 101^{5}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
16.476...904.24t2912.a.a | $16$ | $ 2^{26} \cdot 3^{8} \cdot 101^{8}$ | 9.5.2877061966848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |