Normalized defining polynomial
\( x^{9} - 2x^{7} - 6x^{6} - 4x^{5} + 4x^{4} + 11x^{3} + 12x^{2} + 6x + 1 \)
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: | \(-969508459\) \(\medspace = -\,19^{2}\cdot 139^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.97\) | 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: | $19^{2/3}139^{1/2}\approx 83.94789309014644$ | ||
Ramified primes: | \(19\), \(139\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-139}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $a^{8}-a^{7}-a^{6}-5a^{5}+a^{4}+4a^{3}+7a^{2}+4a-1$, $a^{8}-2a^{6}-6a^{5}-4a^{4}+4a^{3}+11a^{2}+12a+5$, $2a^{8}-2a^{7}-3a^{6}-8a^{5}+2a^{4}+9a^{3}+11a^{2}+8a+1$, $4a^{8}-2a^{7}-7a^{6}-21a^{5}-5a^{4}+19a^{3}+36a^{2}+30a+8$, $a^{7}-a^{6}-a^{5}-5a^{4}+a^{3}+4a^{2}+7a+4$ | 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: | \( 7.22483093173 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 7.22483093173 \cdot 1}{2\cdot\sqrt{969508459}}\cr\approx \mathstrut & 0.230224470974 \end{aligned}\]
Galois group
$C_3\wr S_3$ (as 9T20):
A solvable group of order 162 |
The 22 conjugacy class representatives for $C_3 \wr S_3 $ |
Character table for $C_3 \wr S_3 $ |
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 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(139\) | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.139.2t1.a.a | $1$ | $ 139 $ | \(\Q(\sqrt{-139}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2641.6t1.a.a | $1$ | $ 19 \cdot 139 $ | 6.0.349992553699.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.2641.6t1.a.b | $1$ | $ 19 \cdot 139 $ | 6.0.349992553699.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
* | 2.139.3t2.a.a | $2$ | $ 139 $ | 3.1.139.1 | $S_3$ (as 3T2) | $1$ | $0$ |
2.50179.6t5.a.a | $2$ | $ 19^{2} \cdot 139 $ | 6.0.349992553699.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
2.50179.6t5.a.b | $2$ | $ 19^{2} \cdot 139 $ | 6.0.349992553699.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
3.132522739.18t86.a.a | $3$ | $ 19^{3} \cdot 139^{2}$ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.6974881.18t86.a.a | $3$ | $ 19^{2} \cdot 139^{2}$ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.50179.9t20.a.a | $3$ | $ 19^{2} \cdot 139 $ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.953401.9t20.a.a | $3$ | $ 19^{3} \cdot 139 $ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.50179.9t20.a.b | $3$ | $ 19^{2} \cdot 139 $ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
* | 3.2641.9t20.a.a | $3$ | $ 19 \cdot 139 $ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.367099.18t86.a.a | $3$ | $ 19 \cdot 139^{2}$ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.367099.18t86.a.b | $3$ | $ 19 \cdot 139^{2}$ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.132522739.18t86.a.b | $3$ | $ 19^{3} \cdot 139^{2}$ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.953401.9t20.a.b | $3$ | $ 19^{3} \cdot 139 $ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
* | 3.2641.9t20.a.b | $3$ | $ 19 \cdot 139 $ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.6974881.18t86.a.b | $3$ | $ 19^{2} \cdot 139^{2}$ | 9.3.969508459.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
6.349992553699.9t11.a.a | $6$ | $ 19^{4} \cdot 139^{3}$ | 9.1.48648964964161.1 | $C_3^2 : C_6$ (as 9T11) | $1$ | $0$ |