Normalized defining polynomial
\( x^{9} - 2x^{8} + 8x^{7} - 21x^{6} + 21x^{5} - 7x^{4} - 7x^{3} + 8x^{2} + 5x + 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(941192000000\) \(\medspace = 2^{9}\cdot 5^{6}\cdot 7^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.40\) | 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: | $2^{3/2}5^{2/3}7^{5/6}\approx 41.857507436411844$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{371}a^{8}-\frac{122}{371}a^{7}+\frac{179}{371}a^{6}+\frac{17}{371}a^{5}-\frac{164}{371}a^{4}+\frac{10}{371}a^{3}-\frac{94}{371}a^{2}+\frac{158}{371}a-\frac{34}{371}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $a$, $\frac{167}{371}a^{8}-\frac{340}{371}a^{7}+\frac{1326}{371}a^{6}-\frac{3468}{371}a^{5}+\frac{3405}{371}a^{4}-\frac{556}{371}a^{3}-\frac{2342}{371}a^{2}+\frac{1900}{371}a+\frac{1000}{371}$, $\frac{213}{371}a^{8}-\frac{387}{371}a^{7}+\frac{1769}{371}a^{6}-\frac{4541}{371}a^{5}+\frac{4765}{371}a^{4}-\frac{4177}{371}a^{3}+\frac{1867}{371}a^{2}+\frac{1748}{371}a+\frac{549}{371}$, $\frac{271}{371}a^{8}-\frac{414}{371}a^{7}+\frac{2134}{371}a^{6}-\frac{5039}{371}a^{5}+\frac{4528}{371}a^{4}-\frac{3226}{371}a^{3}-\frac{246}{371}a^{2}+\frac{2379}{371}a+\frac{432}{371}$ | 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: | \( 230.851078365 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 230.851078365 \cdot 1}{2\cdot\sqrt{941192000000}}\cr\approx \mathstrut & 0.370861941440 \end{aligned}\]
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
3.1.175.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.1 | $x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(5\) | 5.9.6.1 | $x^{9} + 9 x^{7} + 24 x^{6} + 27 x^{5} + 9 x^{4} - 186 x^{3} + 216 x^{2} - 504 x + 647$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.56.2t1.b.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{-14}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.11200.6t3.b.a | $2$ | $ 2^{6} \cdot 5^{2} \cdot 7 $ | 6.2.15680000.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.9800.6t3.h.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 7^{2}$ | 6.0.5378240000.1 | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
2.9800.3t2.a.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 7^{2}$ | 3.3.9800.1 | $S_3$ (as 3T2) | $1$ | $2$ | |
* | 2.175.3t2.b.a | $2$ | $ 5^{2} \cdot 7 $ | 3.1.175.1 | $S_3$ (as 3T2) | $1$ | $0$ |
4.3841600.6t9.b.a | $4$ | $ 2^{6} \cdot 5^{2} \cdot 7^{4}$ | 6.0.215129600.1 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
6.5378240000.18t51.b.a | $6$ | $ 2^{9} \cdot 5^{4} \cdot 7^{5}$ | 9.1.941192000000.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ | |
* | 6.5378240000.9t18.b.a | $6$ | $ 2^{9} \cdot 5^{4} \cdot 7^{5}$ | 9.1.941192000000.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ |