Normalized defining polynomial
\( x^{9} - x^{8} - 1045 x^{7} + 29560 x^{6} - 417628 x^{5} + 3585287 x^{4} - 19479671 x^{3} + \cdots + 102451091 \)
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: |
\(-3873496509817600597571142141632\)
\(\medspace = -\,2^{6}\cdot 7^{7}\cdot 53^{3}\cdot 337^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(2504.25\) | 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: | $2^{2/3}7^{5/6}53^{1/2}337^{5/6}\approx 7471.902840325843$ | ||
Ramified primes: |
\(2\), \(7\), \(53\), \(337\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-125027}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{53}a^{6}+\frac{17}{53}a^{5}+\frac{4}{53}a^{4}+\frac{9}{53}a^{3}+\frac{10}{53}a^{2}-\frac{26}{53}a+\frac{10}{53}$, $\frac{1}{53}a^{7}-\frac{20}{53}a^{5}-\frac{6}{53}a^{4}+\frac{16}{53}a^{3}+\frac{16}{53}a^{2}-\frac{25}{53}a-\frac{11}{53}$, $\frac{1}{53}a^{8}+\frac{16}{53}a^{5}-\frac{10}{53}a^{4}-\frac{16}{53}a^{3}+\frac{16}{53}a^{2}-\frac{1}{53}a-\frac{12}{53}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}\times C_{6}\times C_{283896}$, which has order $10220256$ (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: |
$a^{8}+5a^{7}-1015a^{6}+23470a^{5}-276808a^{4}+1924439a^{3}-7933037a^{2}+17917121a-17075182$, $a^{8}+4a^{7}-1025a^{6}+24435a^{5}-295453a^{4}+2108022a^{3}-8939561a^{2}+20817538a-20490217$, $\frac{48}{53}a^{8}+\frac{242}{53}a^{7}-\frac{48699}{53}a^{6}+\frac{1124646}{53}a^{5}-\frac{13250363}{53}a^{4}+\frac{92019338}{53}a^{3}-\frac{378875691}{53}a^{2}+\frac{854609101}{53}a-\frac{813346723}{53}$, $\frac{48}{53}a^{8}+\frac{190}{53}a^{7}-\frac{49220}{53}a^{6}+\frac{1174811}{53}a^{5}-\frac{14219014}{53}a^{4}+\frac{101551432}{53}a^{3}-\frac{431101732}{53}a^{2}+\frac{1004981290}{53}a-\frac{990224074}{53}$, $\frac{175}{53}a^{8}+\frac{825}{53}a^{7}-\frac{178175}{53}a^{6}+\frac{4154725}{53}a^{5}-\frac{49329675}{53}a^{4}+\frac{345265325}{53}a^{3}-\frac{1433217575}{53}a^{2}+\frac{3259715400}{53}a-\frac{3127599047}{53}$
| 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: | \( 50251.72122232476 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 50251.72122232476 \cdot 10220256}{2\cdot\sqrt{3873496509817600597571142141632}}\cr\approx \mathstrut & 0.258917096379420 \end{aligned}\] (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
3.3.5564881.2, 3.1.500108.1 |
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: | 6.0.174015244432792030843568.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | R | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(53\)
| $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
53.2.1.2 | $x^{2} + 106$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.2.1.2 | $x^{2} + 106$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.2.1.2 | $x^{2} + 106$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(337\)
| Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $6$ | $6$ | $1$ | $5$ |