Normalized defining polynomial
\( x^{9} - 5x^{7} - 2x^{6} - x^{5} + 2x^{4} + 12x^{3} - 4x^{2} - 3x + 1 \)
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: |
\(-6240926503\)
\(\medspace = -\,7^{6}\cdot 53047\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.26\) | 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: | $7^{2/3}53047^{1/2}\approx 842.8088888755792$ | ||
Ramified primes: |
\(7\), \(53047\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-53047}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{83}a^{8}+\frac{36}{83}a^{7}-\frac{37}{83}a^{6}-\frac{6}{83}a^{5}+\frac{32}{83}a^{4}-\frac{8}{83}a^{3}-\frac{27}{83}a^{2}+\frac{20}{83}a-\frac{30}{83}$
Monogenic: | Not computed | |
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: |
$\frac{52}{83}a^{8}+\frac{46}{83}a^{7}-\frac{264}{83}a^{6}-\frac{312}{83}a^{5}-\frac{162}{83}a^{4}-\frac{1}{83}a^{3}+\frac{754}{83}a^{2}+\frac{293}{83}a-\frac{149}{83}$, $\frac{52}{83}a^{8}+\frac{46}{83}a^{7}-\frac{264}{83}a^{6}-\frac{312}{83}a^{5}-\frac{162}{83}a^{4}-\frac{1}{83}a^{3}+\frac{754}{83}a^{2}+\frac{293}{83}a-\frac{232}{83}$, $a^{8}-5a^{6}-2a^{5}-a^{4}+2a^{3}+12a^{2}-4a-3$, $\frac{135}{83}a^{8}+\frac{46}{83}a^{7}-\frac{679}{83}a^{6}-\frac{478}{83}a^{5}-\frac{245}{83}a^{4}+\frac{165}{83}a^{3}+\frac{1750}{83}a^{2}-\frac{39}{83}a-\frac{398}{83}$, $\frac{70}{83}a^{8}+\frac{30}{83}a^{7}-\frac{349}{83}a^{6}-\frac{254}{83}a^{5}-\frac{167}{83}a^{4}+\frac{21}{83}a^{3}+\frac{849}{83}a^{2}-\frac{94}{83}a-\frac{191}{83}$, $\frac{152}{83}a^{8}+\frac{77}{83}a^{7}-\frac{727}{83}a^{6}-\frac{663}{83}a^{5}-\frac{448}{83}a^{4}+\frac{29}{83}a^{3}+\frac{1789}{83}a^{2}+\frac{301}{83}a-\frac{410}{83}$, $\frac{294}{83}a^{8}+\frac{126}{83}a^{7}-\frac{1416}{83}a^{6}-\frac{1183}{83}a^{5}-\frac{801}{83}a^{4}+\frac{221}{83}a^{3}+\frac{3599}{83}a^{2}+\frac{319}{83}a-\frac{769}{83}$
| 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: | \( 39.0906023051 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{1}\cdot 39.0906023051 \cdot 1}{2\cdot\sqrt{6240926503}}\cr\approx \mathstrut & 0.198979195379 \end{aligned}\]
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(53047\)
| $\Q_{53047}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{53047}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{53047}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{53047}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{53047}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{53047}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{53047}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |