Normalized defining polynomial
\( x^{9} - 2x^{7} + 2x^{5} - 5x^{4} + 2x^{3} + 3x^{2} - 3x + 1 \)
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: | \(-738391927\) \(\medspace = -\,7^{2}\cdot 13^{3}\cdot 19^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(9.67\) | 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: | $7^{2/3}13^{1/2}19^{1/2}\approx 57.51050351903955$ | ||
Ramified primes: | \(7\), \(13\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-247}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}-\frac{5}{11}a^{7}+\frac{1}{11}a^{6}-\frac{5}{11}a^{5}+\frac{5}{11}a^{4}+\frac{3}{11}a^{3}-\frac{2}{11}a^{2}+\frac{2}{11}a-\frac{2}{11}$
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: | $\frac{5}{11}a^{8}+\frac{8}{11}a^{7}-\frac{6}{11}a^{6}-\frac{14}{11}a^{5}+\frac{3}{11}a^{4}-\frac{7}{11}a^{3}-\frac{21}{11}a^{2}+\frac{10}{11}a+\frac{1}{11}$, $a^{8}-2a^{6}+2a^{4}-5a^{3}+2a^{2}+3a-3$, $\frac{5}{11}a^{8}+\frac{8}{11}a^{7}-\frac{6}{11}a^{6}-\frac{14}{11}a^{5}+\frac{3}{11}a^{4}-\frac{7}{11}a^{3}-\frac{21}{11}a^{2}-\frac{1}{11}a+\frac{1}{11}$, $\frac{16}{11}a^{8}+\frac{8}{11}a^{7}-\frac{28}{11}a^{6}-\frac{14}{11}a^{5}+\frac{25}{11}a^{4}-\frac{73}{11}a^{3}-\frac{10}{11}a^{2}+\frac{43}{11}a-\frac{21}{11}$, $\frac{12}{11}a^{8}-\frac{5}{11}a^{7}-\frac{32}{11}a^{6}+\frac{6}{11}a^{5}+\frac{38}{11}a^{4}-\frac{63}{11}a^{3}+\frac{31}{11}a^{2}+\frac{57}{11}a-\frac{46}{11}$ | 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: | \( 6.69242166606 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 6.69242166606 \cdot 1}{2\cdot\sqrt{738391927}}\cr\approx \mathstrut & 0.244365236486 \end{aligned}\]
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
3.1.247.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(13\) | 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(19\) | 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
19.6.3.1 | $x^{6} + 1444 x^{2} - 116603$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.247.2t1.a.a | $1$ | $ 13 \cdot 19 $ | \(\Q(\sqrt{-247}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1729.6t1.a.a | $1$ | $ 7 \cdot 13 \cdot 19 $ | 6.0.36181204423.4 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.1729.6t1.a.b | $1$ | $ 7 \cdot 13 \cdot 19 $ | 6.0.36181204423.4 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
* | 2.247.3t2.a.a | $2$ | $ 13 \cdot 19 $ | 3.1.247.1 | $S_3$ (as 3T2) | $1$ | $0$ |
2.12103.6t5.a.a | $2$ | $ 7^{2} \cdot 13 \cdot 19 $ | 6.0.36181204423.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
2.12103.6t5.a.b | $2$ | $ 7^{2} \cdot 13 \cdot 19 $ | 6.0.36181204423.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
3.20926087.18t86.c.a | $3$ | $ 7^{3} \cdot 13^{2} \cdot 19^{2}$ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.2989441.18t86.c.a | $3$ | $ 7^{2} \cdot 13^{2} \cdot 19^{2}$ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
* | 3.1729.9t20.c.a | $3$ | $ 7 \cdot 13 \cdot 19 $ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.12103.9t20.c.a | $3$ | $ 7^{2} \cdot 13 \cdot 19 $ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
* | 3.1729.9t20.c.b | $3$ | $ 7 \cdot 13 \cdot 19 $ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.427063.18t86.c.a | $3$ | $ 7 \cdot 13^{2} \cdot 19^{2}$ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.84721.9t20.c.a | $3$ | $ 7^{3} \cdot 13 \cdot 19 $ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.12103.9t20.c.b | $3$ | $ 7^{2} \cdot 13 \cdot 19 $ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.427063.18t86.c.b | $3$ | $ 7 \cdot 13^{2} \cdot 19^{2}$ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.20926087.18t86.c.b | $3$ | $ 7^{3} \cdot 13^{2} \cdot 19^{2}$ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.2989441.18t86.c.b | $3$ | $ 7^{2} \cdot 13^{2} \cdot 19^{2}$ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.84721.9t20.c.b | $3$ | $ 7^{3} \cdot 13 \cdot 19 $ | 9.3.738391927.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
6.36181204423.9t11.a.a | $6$ | $ 7^{4} \cdot 13^{3} \cdot 19^{3}$ | 9.1.8936757492481.1 | $C_3^2 : C_6$ (as 9T11) | $1$ | $0$ |