Normalized defining polynomial
\( x^{9} - x^{8} - 7x^{7} + 4x^{6} + 15x^{5} - 3x^{4} - 6x^{3} + 2x^{2} - 3x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[7, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-9305918251\) \(\medspace = -\,7^{6}\cdot 83\cdot 953\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.81\) | 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: | $7^{2/3}83^{1/2}953^{1/2}\approx 1029.163060638494$ | ||
Ramified primes: | \(7\), \(83\), \(953\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-79099}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $a^{4}-a^{3}-3a^{2}+a+1$, $a^{8}-2a^{7}-5a^{6}+8a^{5}+8a^{4}-7a^{3}-2a^{2}+a-1$, $a$, $a^{7}-2a^{6}-4a^{5}+7a^{4}+4a^{3}-4a^{2}-1$, $a^{8}-2a^{7}-4a^{6}+7a^{5}+3a^{4}-4a^{3}+4a^{2}-1$, $a^{4}-a^{3}-3a^{2}+2a$, $a^{8}-2a^{7}-5a^{6}+8a^{5}+8a^{4}-7a^{3}-a^{2}-2$ | 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: | \( 48.5682110199 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{1}\cdot 48.5682110199 \cdot 1}{2\cdot\sqrt{9305918251}}\cr\approx \mathstrut & 0.202456783410 \end{aligned}\]
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
\(83\) | $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(953\) | $\Q_{953}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{953}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |