Normalized defining polynomial
\( x^{9} - 6x^{7} + 20x^{5} - 16x^{4} - 4x^{3} - 4x^{2} + 20x - 12 \)
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: | \(39217774848\) \(\medspace = 2^{8}\cdot 3\cdot 7^{3}\cdot 53^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.03\) | 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: | $2^{8/9}3^{1/2}7^{1/2}53^{1/2}\approx 61.777424773632966$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{1113}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{78}a^{8}+\frac{11}{78}a^{7}-\frac{1}{39}a^{6}+\frac{17}{78}a^{5}+\frac{2}{13}a^{4}+\frac{19}{39}a^{3}+\frac{4}{13}a^{2}+\frac{1}{3}a-\frac{1}{13}$
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: | $a^{8}+a^{7}-5a^{6}-5a^{5}+15a^{4}-a^{3}-5a^{2}-9a+11$, $\frac{1}{13}a^{8}-\frac{2}{13}a^{7}-\frac{17}{26}a^{6}+\frac{21}{26}a^{5}+\frac{38}{13}a^{4}-\frac{40}{13}a^{3}-\frac{28}{13}a^{2}-a+\frac{59}{13}$, $\frac{35}{78}a^{8}+\frac{17}{39}a^{7}-\frac{187}{78}a^{6}-\frac{185}{78}a^{5}+\frac{96}{13}a^{4}+\frac{2}{39}a^{3}-\frac{55}{13}a^{2}-\frac{13}{3}a+\frac{69}{13}$, $\frac{10}{13}a^{8}+\frac{51}{26}a^{7}-\frac{27}{26}a^{6}-\frac{167}{26}a^{5}+\frac{16}{13}a^{4}+\frac{42}{13}a^{3}+\frac{32}{13}a^{2}-4a+\frac{5}{13}$ | 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: | \( 181.78033132 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 181.78033132 \cdot 1}{2\cdot\sqrt{39217774848}}\cr\approx \mathstrut & 1.4306219682 \end{aligned}\]
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
3.1.1484.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | R | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{9} + 2$ | $9$ | $1$ | $8$ | $(C_9:C_3):C_2$ | $[\ ]_{9}^{6}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(53\) | $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
53.6.3.1 | $x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.371.2t1.a.a | $1$ | $ 7 \cdot 53 $ | \(\Q(\sqrt{-371}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1113.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 53 $ | \(\Q(\sqrt{1113}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.13356.6t3.a.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $ | 6.2.22059998352.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.1484.3t2.a.a | $2$ | $ 2^{2} \cdot 7 \cdot 53 $ | 3.1.1484.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.13356.4t5.a.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $ | 4.2.13356.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.1651692.6t11.a.a | $3$ | $ 2^{2} \cdot 3 \cdot 7^{2} \cdot 53^{2}$ | 6.0.6606768.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
3.4955076.6t8.a.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 53^{2}$ | 4.2.13356.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.4452.6t11.a.a | $3$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 53 $ | 6.0.6606768.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
* | 6.26427072.9t31.a.a | $6$ | $ 2^{6} \cdot 3 \cdot 7^{2} \cdot 53^{2}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
6.294...312.18t300.a.a | $6$ | $ 2^{6} \cdot 3^{5} \cdot 7^{4} \cdot 53^{4}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.363...152.18t319.a.a | $6$ | $ 2^{6} \cdot 3 \cdot 7^{4} \cdot 53^{4}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
6.2140592832.18t311.a.a | $6$ | $ 2^{6} \cdot 3^{5} \cdot 7^{2} \cdot 53^{2}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.540...656.24t2893.a.a | $8$ | $ 2^{8} \cdot 3^{4} \cdot 7^{6} \cdot 53^{6}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
8.2854123776.12t213.a.a | $8$ | $ 2^{8} \cdot 3^{4} \cdot 7^{2} \cdot 53^{2}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.649...824.36t2217.a.a | $12$ | $ 2^{10} \cdot 3^{8} \cdot 7^{7} \cdot 53^{7}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
12.472...864.36t2214.a.a | $12$ | $ 2^{10} \cdot 3^{8} \cdot 7^{5} \cdot 53^{5}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.778...464.36t2210.a.a | $12$ | $ 2^{12} \cdot 3^{6} \cdot 7^{6} \cdot 53^{6}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
12.802...504.36t2216.a.a | $12$ | $ 2^{10} \cdot 3^{4} \cdot 7^{7} \cdot 53^{7}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
12.582...344.18t315.a.a | $12$ | $ 2^{10} \cdot 3^{4} \cdot 7^{5} \cdot 53^{5}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
16.385...264.24t2912.a.a | $16$ | $ 2^{14} \cdot 3^{8} \cdot 7^{8} \cdot 53^{8}$ | 9.1.39217774848.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |