Normalized defining polynomial
\( x^{9} + 6x^{7} - 11x^{6} - 3x^{5} - 69x^{4} - 15x^{3} - 99x^{2} + 99x - 33 \)
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: | \(9042800765625\) \(\medspace = 3^{14}\cdot 5^{6}\cdot 11^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.52\) | 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: | $3^{121/54}5^{23/20}11^{2/3}\approx 369.13253807620333$ | ||
Ramified primes: | \(3\), \(5\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{18890}a^{8}+\frac{881}{18890}a^{7}-\frac{2101}{18890}a^{6}-\frac{355}{1889}a^{5}-\frac{1385}{3778}a^{4}+\frac{238}{9445}a^{3}+\frac{3761}{18890}a^{2}+\frac{1907}{9445}a-\frac{1193}{3778}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{291}{18890}a^{8}-\frac{533}{18890}a^{7}+\frac{129}{3778}a^{6}-\frac{828}{9445}a^{5}+\frac{1211}{3778}a^{4}+\frac{3143}{9445}a^{3}+\frac{25277}{18890}a^{2}+\frac{292}{1889}a-\frac{5491}{18890}$, $\frac{159}{3778}a^{8}-\frac{2313}{18890}a^{7}+\frac{7137}{18890}a^{6}-\frac{11376}{9445}a^{5}+\frac{5879}{3778}a^{4}-\frac{5605}{1889}a^{3}+\frac{77157}{18890}a^{2}-\frac{6469}{9445}a+\frac{78553}{18890}$, $\frac{878}{9445}a^{8}+\frac{917}{9445}a^{7}-\frac{1014}{9445}a^{6}-\frac{1939}{9445}a^{5}-\frac{5181}{1889}a^{4}-\frac{7097}{9445}a^{3}-\frac{11148}{9445}a^{2}+\frac{16496}{9445}a-\frac{6629}{9445}$, $\frac{274}{1889}a^{8}-\frac{7657}{9445}a^{7}-\frac{12757}{9445}a^{6}-\frac{21993}{9445}a^{5}+\frac{6662}{1889}a^{4}+\frac{34085}{1889}a^{3}+\frac{348843}{9445}a^{2}+\frac{289223}{9445}a-\frac{175913}{9445}$ | 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: | \( 3766.63780033 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 3766.63780033 \cdot 1}{2\cdot\sqrt{9042800765625}}\cr\approx \mathstrut & 1.95218896127 \end{aligned}\]
Galois group
A non-solvable group of order 181440 |
The 18 conjugacy class representatives for $A_9$ |
Character table for $A_9$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 36 sibling: | 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 | ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | R | R | ${\href{/padicField/7.9.0.1}{9} }$ | R | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ |
3.6.9.16 | $x^{6} + 3 x^{5} + 3 x^{4} + 12$ | $6$ | $1$ | $9$ | $S_3^2$ | $[3/2, 2]_{2}^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ | |
\(11\) | 11.3.2.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |