Normalized defining polynomial
\( x^{9} - 63x^{7} - 63x^{6} + 1323x^{5} + 2646x^{4} - 5061x^{3} - 27783x^{2} - 88200x + 98000 \)
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: |
\(-3230456496696835875\)
\(\medspace = -\,3^{22}\cdot 5^{3}\cdot 7^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(113.92\) | 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: | $3^{22/9}5^{1/2}7^{5/6}\approx 165.97006698500886$ | ||
Ramified primes: |
\(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-35}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{17360}a^{6}-\frac{3}{1240}a^{4}+\frac{1131}{2480}a^{3}+\frac{63}{2480}a^{2}+\frac{1049}{2480}a-\frac{61}{124}$, $\frac{1}{17360}a^{7}-\frac{3}{1240}a^{5}+\frac{1131}{2480}a^{4}+\frac{63}{2480}a^{3}+\frac{1049}{2480}a^{2}-\frac{61}{124}a$, $\frac{1}{21075040}a^{8}+\frac{65}{4215008}a^{7}-\frac{1}{376340}a^{6}+\frac{319101}{3010720}a^{5}-\frac{16669}{48560}a^{4}+\frac{82657}{301072}a^{3}-\frac{71}{4960}a^{2}-\frac{496993}{1505360}a+\frac{9851}{75268}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{3}$, which has order $9$ (assuming GRH)
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: |
$\frac{11}{8680}a^{6}-\frac{33}{620}a^{4}+\frac{41}{1240}a^{3}+\frac{693}{1240}a^{2}-\frac{861}{1240}a+\frac{11}{62}$, $\frac{104471}{21075040}a^{8}+\frac{691903}{21075040}a^{7}-\frac{72367}{752680}a^{6}-\frac{3053197}{3010720}a^{5}-\frac{235891}{301072}a^{4}+\frac{9429107}{1505360}a^{3}+\frac{661}{32}a^{2}+\frac{51872067}{1505360}a-\frac{7642529}{75268}$, $\frac{24307}{2107504}a^{8}-\frac{711}{12140}a^{7}-\frac{69913}{150536}a^{6}+\frac{2952293}{1505360}a^{5}+\frac{8152037}{1505360}a^{4}-\frac{5756469}{1505360}a^{3}-\frac{8987}{310}a^{2}-\frac{7484579}{37634}a+\frac{3615873}{18817}$, $\frac{10207}{21075040}a^{8}-\frac{113489}{21075040}a^{7}-\frac{2019}{150536}a^{6}+\frac{479771}{3010720}a^{5}+\frac{624349}{1505360}a^{4}-\frac{149549}{301072}a^{3}-\frac{7017}{4960}a^{2}-\frac{51860157}{1505360}a+\frac{2318347}{75268}$, $\frac{227491}{21075040}a^{8}+\frac{1637233}{21075040}a^{7}-\frac{109599}{752680}a^{6}-\frac{6150757}{3010720}a^{5}-\frac{1326241}{752680}a^{4}+\frac{5436611}{376340}a^{3}+\frac{267741}{4960}a^{2}+\frac{32609751}{301072}a-\frac{10610169}{75268}$
| 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: | \( 386610.364683 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 386610.364683 \cdot 9}{2\cdot\sqrt{3230456496696835875}}\cr\approx \mathstrut & 1.92080664848 \end{aligned}\] (assuming GRH)
Galois group
$C_3\wr S_3$ (as 9T20):
A solvable group of order 162 |
The 22 conjugacy class representatives for $C_3 \wr S_3 $ |
Character table for $C_3 \wr S_3 $ |
Intermediate fields
3.1.2835.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 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 | ${\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ | R | R | R | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\)
| 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.6.5.3 | $x^{6} + 35$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |