Normalized defining polynomial
\( x^{9} - 2x^{8} - x^{7} - 2x^{6} + 11x^{5} + 2x^{4} - 21x^{3} + 12x^{2} + 6x - 2 \)
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: | \(-21484952000\) \(\medspace = -\,2^{6}\cdot 5^{3}\cdot 139^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.06\) | 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: | $2^{2/3}5^{1/2}139^{1/2}\approx 41.84842003414076$ | ||
Ramified primes: | \(2\), \(5\), \(139\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-695}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{206}a^{8}-\frac{5}{103}a^{7}-\frac{12}{103}a^{6}-\frac{8}{103}a^{5}-\frac{67}{206}a^{4}-\frac{40}{103}a^{3}-\frac{51}{103}a^{2}+\frac{2}{103}a-\frac{13}{103}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{31}{103}a^{8}-\frac{105}{206}a^{7}-\frac{23}{103}a^{6}-\frac{84}{103}a^{5}+\frac{292}{103}a^{4}+\frac{87}{206}a^{3}-\frac{484}{103}a^{2}+\frac{330}{103}a+\frac{18}{103}$, $\frac{20}{103}a^{8}-\frac{91}{206}a^{7}-\frac{33}{206}a^{6}-\frac{125}{206}a^{5}+\frac{513}{206}a^{4}+\frac{48}{103}a^{3}-\frac{289}{103}a^{2}+\frac{183}{103}a-\frac{5}{103}$, $\frac{51}{206}a^{8}-\frac{49}{103}a^{7}-\frac{91}{206}a^{6}-\frac{95}{206}a^{5}+\frac{300}{103}a^{4}+\frac{349}{206}a^{3}-\frac{541}{103}a^{2}+\frac{102}{103}a+\frac{58}{103}$, $\frac{101}{206}a^{8}-\frac{93}{103}a^{7}-\frac{55}{206}a^{6}-\frac{277}{206}a^{5}+\frac{479}{103}a^{4}+\frac{57}{206}a^{3}-\frac{825}{103}a^{2}+\frac{717}{103}a+\frac{26}{103}$, $\frac{23}{206}a^{8}-\frac{12}{103}a^{7}-\frac{37}{206}a^{6}-\frac{59}{206}a^{5}+\frac{105}{103}a^{4}+\frac{117}{206}a^{3}-\frac{246}{103}a^{2}-\frac{57}{103}a+\frac{10}{103}$ | 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: | \( 111.645348361 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 111.645348361 \cdot 1}{2\cdot\sqrt{21484952000}}\cr\approx \mathstrut & 0.755741090043 \end{aligned}\]
Galois group
$C_3^2:D_6$ (as 9T18):
A solvable group of order 108 |
The 11 conjugacy class representatives for $C_3^2 : D_{6} $ |
Character table for $C_3^2 : D_{6} $ |
Intermediate fields
3.1.139.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | 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 | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(139\) | 139.3.0.1 | $x^{3} + 6 x + 137$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
139.6.3.2 | $x^{6} + 429 x^{4} + 274 x^{3} + 57999 x^{2} - 112614 x + 2477540$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.139.2t1.a.a | $1$ | $ 139 $ | \(\Q(\sqrt{-139}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.695.2t1.a.a | $1$ | $ 5 \cdot 139 $ | \(\Q(\sqrt{-695}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.3475.6t3.b.a | $2$ | $ 5^{2} \cdot 139 $ | 6.0.335702375.2 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.695.6t3.a.a | $2$ | $ 5 \cdot 139 $ | 6.2.2415125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.695.3t2.a.a | $2$ | $ 5 \cdot 139 $ | 3.1.695.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 2.139.3t2.a.a | $2$ | $ 139 $ | 3.1.139.1 | $S_3$ (as 3T2) | $1$ | $0$ |
4.483025.6t9.a.a | $4$ | $ 5^{2} \cdot 139^{2}$ | 6.2.2415125.3 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
6.298...000.18t51.a.a | $6$ | $ 2^{6} \cdot 5^{3} \cdot 139^{4}$ | 9.3.21484952000.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $-2$ | |
* | 6.154568000.9t18.a.a | $6$ | $ 2^{6} \cdot 5^{3} \cdot 139^{2}$ | 9.3.21484952000.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $2$ |