Normalized defining polynomial
\( x^{9} - x^{8} - 4x^{7} + 4x^{6} + x^{5} - 7x^{4} + 3x^{3} - 8x^{2} - 13x + 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-816174527\) \(\medspace = -\,7^{2}\cdot 23^{3}\cdot 37^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(9.78\) | 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: | $7^{2/3}23^{1/2}37^{2/3}\approx 194.86346797384473$ | ||
Ramified primes: | \(7\), \(23\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{607}a^{8}-\frac{249}{607}a^{7}-\frac{166}{607}a^{6}-\frac{104}{607}a^{5}+\frac{299}{607}a^{4}-\frac{105}{607}a^{3}-\frac{58}{607}a^{2}-\frac{192}{607}a+\frac{257}{607}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{271}{607}a^{8}-\frac{709}{607}a^{7}-\frac{68}{607}a^{6}+\frac{1559}{607}a^{5}-\frac{2130}{607}a^{4}+\frac{681}{607}a^{3}+\frac{671}{607}a^{2}-\frac{3472}{607}a+\frac{1056}{607}$, $\frac{214}{607}a^{8}-\frac{477}{607}a^{7}-\frac{318}{607}a^{6}+\frac{1417}{607}a^{5}-\frac{1570}{607}a^{4}-\frac{11}{607}a^{3}+\frac{942}{607}a^{2}-\frac{2847}{607}a+\frac{975}{607}$, $\frac{49}{607}a^{8}-\frac{61}{607}a^{7}-\frac{243}{607}a^{6}+\frac{367}{607}a^{5}+\frac{83}{607}a^{4}-\frac{289}{607}a^{3}+\frac{193}{607}a^{2}-\frac{910}{607}a-\frac{154}{607}$, $\frac{352}{607}a^{8}-\frac{847}{607}a^{7}-\frac{160}{607}a^{6}+\frac{1633}{607}a^{5}-\frac{2191}{607}a^{4}+\frac{674}{607}a^{3}+\frac{222}{607}a^{2}-\frac{3242}{607}a+\frac{21}{607}$, $\frac{352}{607}a^{8}-\frac{847}{607}a^{7}-\frac{160}{607}a^{6}+\frac{1633}{607}a^{5}-\frac{2191}{607}a^{4}+\frac{674}{607}a^{3}+\frac{222}{607}a^{2}-\frac{3242}{607}a+\frac{628}{607}$ | 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: | \( 6.74074904797 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 6.74074904797 \cdot 1}{2\cdot\sqrt{816174527}}\cr\approx \mathstrut & 0.234107990602 \end{aligned}\]
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
3.1.23.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }$ | R | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(37\) | 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |