Normalized defining polynomial
\( x^{9} - 3x^{8} + 18x^{7} + 147x^{6} + 378x^{5} - 5670x^{4} + 5040x^{3} - 1188x^{2} + 44388x - 36252 \)
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: |
\(-827185934225070000\)
\(\medspace = -\,2^{4}\cdot 3^{15}\cdot 5^{4}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(97.91\) | 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: | $2^{2/3}3^{97/54}5^{2/3}7^{8/9}\approx 188.32850860699145$ | ||
Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{249027177851052}a^{8}-\frac{7584440804755}{249027177851052}a^{7}-\frac{1371555755043}{20752264820921}a^{6}-\frac{17182636881857}{249027177851052}a^{5}+\frac{3793865475297}{20752264820921}a^{4}+\frac{1263584240156}{20752264820921}a^{3}-\frac{12995734288799}{41504529641842}a^{2}-\frac{4540388596315}{20752264820921}a-\frac{298101231660}{1092224464259}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{63}$, which has order $189$ (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: |
$\frac{6740327117}{249027177851052}a^{8}-\frac{5594817169}{83009059283684}a^{7}+\frac{20427362710}{62256794462763}a^{6}+\frac{378634175745}{83009059283684}a^{5}+\frac{230290746036}{20752264820921}a^{4}-\frac{3340226093872}{20752264820921}a^{3}-\frac{229071291387}{41504529641842}a^{2}+\frac{19500578551863}{20752264820921}a-\frac{847345699961}{1092224464259}$, $\frac{765962511509}{249027177851052}a^{8}-\frac{82497398221}{249027177851052}a^{7}+\frac{2258089213925}{41504529641842}a^{6}+\frac{50684563562585}{83009059283684}a^{5}+\frac{121475060928283}{41504529641842}a^{4}-\frac{185882034495062}{20752264820921}a^{3}-\frac{424125141510521}{41504529641842}a^{2}-\frac{674287047814289}{20752264820921}a+\frac{41726157600649}{1092224464259}$, $\frac{36462092175}{83009059283684}a^{8}-\frac{305734801657}{249027177851052}a^{7}+\frac{143066745881}{20752264820921}a^{6}+\frac{5446315133211}{83009059283684}a^{5}+\frac{6881135040081}{41504529641842}a^{4}-\frac{108347601169719}{41504529641842}a^{3}+\frac{35513304626673}{41504529641842}a^{2}+\frac{24924931761784}{20752264820921}a+\frac{24858487240841}{1092224464259}$, $\frac{6740327117}{249027177851052}a^{8}-\frac{5594817169}{83009059283684}a^{7}+\frac{20427362710}{62256794462763}a^{6}+\frac{378634175745}{83009059283684}a^{5}+\frac{230290746036}{20752264820921}a^{4}-\frac{3340226093872}{20752264820921}a^{3}-\frac{229071291387}{41504529641842}a^{2}-\frac{1251686269058}{20752264820921}a+\frac{1337103228557}{1092224464259}$, $\frac{79707761219}{249027177851052}a^{8}-\frac{11388763347}{83009059283684}a^{7}+\frac{656924523547}{124513588925526}a^{6}+\frac{5198087296217}{83009059283684}a^{5}+\frac{5745344311580}{20752264820921}a^{4}-\frac{46132515500931}{41504529641842}a^{3}-\frac{35331531687473}{41504529641842}a^{2}-\frac{42205113659737}{20752264820921}a+\frac{4080891917806}{1092224464259}$
| 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: | \( 2696.43507296 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 2696.43507296 \cdot 189}{2\cdot\sqrt{827185934225070000}}\cr\approx \mathstrut & 0.555968002644 \end{aligned}\] (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
3.1.1323.1 |
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 | R | R | R | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\)
| 3.9.15.39 | $x^{9} + 3 x^{8} + 6 x^{7} + 3 x^{3} + 3$ | $9$ | $1$ | $15$ | $C_3 \wr S_3 $ | $[3/2, 3/2, 2]_{2}^{3}$ |
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(7\)
| 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |