Normalized defining polynomial
\( x^{9} - 4x^{8} - 25x^{7} - 145x^{6} - 343x^{5} - 352x^{4} - 166x^{3} - 43x^{2} - 13x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-2700223234725844672\)
\(\medspace = -\,2^{6}\cdot 13^{3}\cdot 79^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(111.67\) | 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: | $2^{2/3}13^{1/2}79^{5/6}\approx 218.28015290706028$ | ||
Ramified primes: |
\(2\), \(13\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1027}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13}a^{6}+\frac{6}{13}a^{5}-\frac{4}{13}a^{4}-\frac{2}{13}a^{3}-\frac{6}{13}a^{2}+\frac{1}{13}$, $\frac{1}{13}a^{7}-\frac{1}{13}a^{5}-\frac{4}{13}a^{4}+\frac{6}{13}a^{3}-\frac{3}{13}a^{2}+\frac{1}{13}a-\frac{6}{13}$, $\frac{1}{13}a^{8}+\frac{2}{13}a^{5}+\frac{2}{13}a^{4}-\frac{5}{13}a^{3}-\frac{5}{13}a^{2}-\frac{6}{13}a+\frac{1}{13}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{134}$, which has order $134$ (assuming GRH)
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: |
$a^{8}-4a^{7}-25a^{6}-145a^{5}-343a^{4}-352a^{3}-166a^{2}-43a-12$, $a$, $\frac{5}{13}a^{8}-\frac{23}{13}a^{7}-\frac{111}{13}a^{6}-\frac{659}{13}a^{5}-102a^{4}-\frac{994}{13}a^{3}-\frac{330}{13}a^{2}-\frac{92}{13}a-\frac{20}{13}$, $\frac{7}{13}a^{8}-\frac{33}{13}a^{7}-\frac{152}{13}a^{6}-\frac{904}{13}a^{5}-134a^{4}-\frac{1138}{13}a^{3}-\frac{168}{13}a^{2}+\frac{16}{13}a-\frac{12}{13}$, $\frac{35}{13}a^{8}-\frac{165}{13}a^{7}-\frac{760}{13}a^{6}-\frac{4520}{13}a^{5}-670a^{4}-\frac{5690}{13}a^{3}-\frac{840}{13}a^{2}+\frac{145}{13}a+\frac{18}{13}$
| 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: | \( 2971.7390850139386 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 2971.7390850139386 \cdot 134}{2\cdot\sqrt{2700223234725844672}}\cr\approx \mathstrut & 0.240444715204947 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3$ (as 9T4):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
3.3.6241.1, 3.1.4108.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | data not computed |
Minimal sibling: | 6.0.108164686537648.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(79\)
| 79.3.2.1 | $x^{3} + 79$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
79.6.5.5 | $x^{6} + 79$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |