Normalized defining polynomial
\( x^{9} - 3x^{8} + 5x^{7} - 8x^{6} + x^{5} - 9x^{3} + 4x^{2} + 2x - 1 \)
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: | \(-1352000000\) \(\medspace = -\,2^{9}\cdot 5^{6}\cdot 13^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.34\) | 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: | $2^{3/2}5^{2/3}13^{2/3}\approx 45.72501934899875$ | ||
Ramified primes: | \(2\), \(5\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$
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: | $\frac{3}{4}a^{8}-\frac{3}{2}a^{7}+\frac{7}{4}a^{6}-\frac{11}{4}a^{5}-5a^{4}+\frac{1}{2}a^{3}-\frac{35}{4}a^{2}-\frac{9}{4}a+\frac{9}{4}$, $a$, $a^{8}-\frac{11}{4}a^{7}+\frac{17}{4}a^{6}-7a^{5}-\frac{1}{4}a^{4}-\frac{3}{4}a^{3}-\frac{31}{4}a^{2}+\frac{3}{2}a+\frac{11}{4}$, $\frac{7}{4}a^{8}-\frac{17}{4}a^{7}+6a^{6}-\frac{39}{4}a^{5}-\frac{21}{4}a^{4}-\frac{1}{4}a^{3}-\frac{33}{2}a^{2}-\frac{3}{4}a+5$, $\frac{9}{4}a^{8}-6a^{7}+\frac{37}{4}a^{6}-\frac{59}{4}a^{5}-3a^{4}-a^{3}-\frac{83}{4}a^{2}+\frac{7}{4}a+\frac{21}{4}$ | 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: | \( 9.29588797346 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 9.29588797346 \cdot 1}{2\cdot\sqrt{1352000000}}\cr\approx \mathstrut & 0.250843084187 \end{aligned}\]
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
3.1.200.1 |
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 | R | ${\href{/padicField/3.9.0.1}{9} }$ | R | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.9.0.1}{9} }$ | R | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.5 | $x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(5\) | 5.9.6.1 | $x^{9} + 9 x^{7} + 24 x^{6} + 27 x^{5} + 9 x^{4} - 186 x^{3} + 216 x^{2} - 504 x + 647$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(13\) | 13.3.2.1 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.104.6t1.d.a | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.104.6t1.d.b | $1$ | $ 2^{3} \cdot 13 $ | 6.0.14623232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
* | 2.200.3t2.b.a | $2$ | $ 2^{3} \cdot 5^{2}$ | 3.1.200.1 | $S_3$ (as 3T2) | $1$ | $0$ |
2.33800.6t5.a.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 13^{2}$ | 6.0.9139520000.4 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
2.33800.6t5.a.b | $2$ | $ 2^{3} \cdot 5^{2} \cdot 13^{2}$ | 6.0.9139520000.4 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
3.3515200.18t86.c.a | $3$ | $ 2^{6} \cdot 5^{2} \cdot 13^{3}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
* | 3.2600.9t20.c.a | $3$ | $ 2^{3} \cdot 5^{2} \cdot 13 $ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.20800.18t86.c.a | $3$ | $ 2^{6} \cdot 5^{2} \cdot 13 $ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.33800.9t20.c.a | $3$ | $ 2^{3} \cdot 5^{2} \cdot 13^{2}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.439400.9t20.c.a | $3$ | $ 2^{3} \cdot 5^{2} \cdot 13^{3}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.33800.9t20.c.b | $3$ | $ 2^{3} \cdot 5^{2} \cdot 13^{2}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
* | 3.2600.9t20.c.b | $3$ | $ 2^{3} \cdot 5^{2} \cdot 13 $ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.270400.18t86.c.a | $3$ | $ 2^{6} \cdot 5^{2} \cdot 13^{2}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.270400.18t86.c.b | $3$ | $ 2^{6} \cdot 5^{2} \cdot 13^{2}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.3515200.18t86.c.b | $3$ | $ 2^{6} \cdot 5^{2} \cdot 13^{3}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.20800.18t86.c.b | $3$ | $ 2^{6} \cdot 5^{2} \cdot 13 $ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.439400.9t20.c.b | $3$ | $ 2^{3} \cdot 5^{2} \cdot 13^{3}$ | 9.3.1352000000.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
6.9139520000.9t11.a.a | $6$ | $ 2^{9} \cdot 5^{4} \cdot 13^{4}$ | 9.1.1827904000000.1 | $C_3^2 : C_6$ (as 9T11) | $1$ | $0$ |