Normalized defining polynomial
\( x^{9} - 2x^{7} - 5x^{6} + 4x^{5} + 2x^{4} - x^{3} + 4x^{2} + 2x - 1 \)
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: | \(-20751532992\) \(\medspace = -\,2^{6}\cdot 3^{3}\cdot 229^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.01\) | 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: | $2^{2/3}3^{1/2}229^{1/2}\approx 41.6068686944302$ | ||
Ramified primes: | \(2\), \(3\), \(229\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-687}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{46}a^{8}-\frac{11}{46}a^{7}-\frac{19}{46}a^{6}+\frac{10}{23}a^{5}+\frac{7}{23}a^{4}-\frac{7}{23}a^{3}+\frac{15}{46}a^{2}-\frac{1}{2}a-\frac{21}{46}$
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: | $a$, $\frac{29}{46}a^{8}+\frac{3}{46}a^{7}-\frac{45}{46}a^{6}-\frac{78}{23}a^{5}+\frac{42}{23}a^{4}+\frac{4}{23}a^{3}+\frac{21}{46}a^{2}+\frac{3}{2}a+\frac{81}{46}$, $\frac{16}{23}a^{8}+\frac{8}{23}a^{7}-\frac{28}{23}a^{6}-\frac{94}{23}a^{5}+\frac{17}{23}a^{4}+\frac{52}{23}a^{3}+\frac{10}{23}a^{2}+2a+\frac{32}{23}$, $\frac{16}{23}a^{8}+\frac{8}{23}a^{7}-\frac{28}{23}a^{6}-\frac{94}{23}a^{5}+\frac{17}{23}a^{4}+\frac{52}{23}a^{3}+\frac{10}{23}a^{2}+3a+\frac{55}{23}$, $\frac{14}{23}a^{8}+\frac{7}{23}a^{7}-\frac{36}{23}a^{6}-\frac{88}{23}a^{5}+\frac{35}{23}a^{4}+\frac{103}{23}a^{3}-\frac{20}{23}a^{2}+\frac{28}{23}$ | 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: | \( 83.7740992114 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 83.7740992114 \cdot 1}{2\cdot\sqrt{20751532992}}\cr\approx \mathstrut & 0.577011236549 \end{aligned}\]
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
3.3.229.1 |
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 | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(229\) | Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.229.2t1.a.a | $1$ | $ 229 $ | \(\Q(\sqrt{229}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.687.2t1.a.a | $1$ | $ 3 \cdot 229 $ | \(\Q(\sqrt{-687}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.2061.6t3.b.a | $2$ | $ 3^{2} \cdot 229 $ | 6.0.324242703.2 | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
2.687.6t3.a.a | $2$ | $ 3 \cdot 229 $ | 6.0.1415907.3 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.687.3t2.a.a | $2$ | $ 3 \cdot 229 $ | 3.1.687.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 2.229.3t2.a.a | $2$ | $ 229 $ | 3.3.229.1 | $S_3$ (as 3T2) | $1$ | $2$ |
4.471969.6t9.a.a | $4$ | $ 3^{2} \cdot 229^{2}$ | 6.0.1415907.2 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
6.475...168.18t51.b.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 229^{4}$ | 9.3.20751532992.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ | |
* | 6.90618048.9t18.b.a | $6$ | $ 2^{6} \cdot 3^{3} \cdot 229^{2}$ | 9.3.20751532992.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ |