Normalized defining polynomial
\( x^{9} - x^{8} + 4x^{7} - x^{6} + 2x^{5} + 5x^{4} - 12x^{3} - x^{2} + 5x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1861324829\) \(\medspace = 7^{6}\cdot 13\cdot 1217\) | 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: | \(10.71\) | 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: | $7^{2/3}13^{1/2}1217^{1/2}\approx 460.2731685624242$ | ||
Ramified primes: | \(7\), \(13\), \(1217\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{15821}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{293}a^{8}+\frac{84}{293}a^{7}+\frac{112}{293}a^{6}+\frac{143}{293}a^{5}+\frac{144}{293}a^{4}-\frac{61}{293}a^{3}+\frac{77}{293}a^{2}+\frac{98}{293}a+\frac{131}{293}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{91}{293}a^{8}+\frac{26}{293}a^{7}+\frac{230}{293}a^{6}+\frac{414}{293}a^{5}-\frac{81}{293}a^{4}+\frac{895}{293}a^{3}-\frac{904}{293}a^{2}-\frac{1337}{293}a+\frac{494}{293}$, $\frac{284}{293}a^{8}-\frac{170}{293}a^{7}+\frac{1043}{293}a^{6}+\frac{178}{293}a^{5}+\frac{462}{293}a^{4}+\frac{1721}{293}a^{3}-\frac{3037}{293}a^{2}-\frac{1468}{293}a+\frac{872}{293}$, $\frac{257}{293}a^{8}-\frac{94}{293}a^{7}+\frac{949}{293}a^{6}+\frac{419}{293}a^{5}+\frac{676}{293}a^{4}+\frac{1903}{293}a^{3}-\frac{1893}{293}a^{2}-\frac{1477}{293}a+\frac{851}{293}$, $a$, $\frac{210}{293}a^{8}-\frac{233}{293}a^{7}+\frac{959}{293}a^{6}-\frac{442}{293}a^{5}+\frac{940}{293}a^{4}+\frac{668}{293}a^{3}-\frac{1996}{293}a^{2}+\frac{70}{293}a+\frac{261}{293}$, $\frac{348}{293}a^{8}-\frac{68}{293}a^{7}+\frac{1179}{293}a^{6}+\frac{833}{293}a^{5}+\frac{595}{293}a^{4}+\frac{2798}{293}a^{3}-\frac{2797}{293}a^{2}-\frac{2814}{293}a+\frac{1052}{293}$ | 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: | \( 14.5686204324 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 14.5686204324 \cdot 1}{2\cdot\sqrt{1861324829}}\cr\approx \mathstrut & 0.213298137133 \end{aligned}\]
Galois group
$S_3\wr C_3$ (as 9T28):
A solvable group of order 648 |
The 17 conjugacy class representatives for $S_3 \wr C_3 $ |
Character table for $S_3 \wr C_3 $ |
Intermediate fields
\(\Q(\zeta_{7})^+\) |
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(1217\) | $\Q_{1217}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1217}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |