Normalized defining polynomial
\( x^{9} - 3x^{8} + 6x^{7} - x^{6} - 15x^{5} + 39x^{4} - 45x^{3} + 24x^{2} - 6x + 1 \)
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: | \(22140698112\) \(\medspace = 2^{9}\cdot 3^{9}\cdot 13^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.11\) | 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: | $2^{3/2}3^{7/6}13^{1/2}\approx 36.74160581422285$ | ||
Ramified primes: | \(2\), \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{78}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{91}a^{8}-\frac{33}{91}a^{7}-\frac{5}{91}a^{6}-\frac{33}{91}a^{5}-\frac{2}{7}a^{4}-\frac{45}{91}a^{2}+\frac{9}{91}a-\frac{3}{91}$
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: | $a$, $a^{8}-2a^{7}+4a^{6}+3a^{5}-12a^{4}+27a^{3}-18a^{2}+6a-1$, $\frac{76}{91}a^{8}-\frac{233}{91}a^{7}+\frac{439}{91}a^{6}-\frac{51}{91}a^{5}-\frac{96}{7}a^{4}+32a^{3}-\frac{3238}{91}a^{2}+\frac{1230}{91}a-\frac{137}{91}$, $\frac{7}{13}a^{8}-\frac{23}{13}a^{7}+\frac{43}{13}a^{6}-\frac{10}{13}a^{5}-9a^{4}+22a^{3}-\frac{341}{13}a^{2}+\frac{154}{13}a-\frac{8}{13}$ | 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: | \( 40.9777379892 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 40.9777379892 \cdot 1}{2\cdot\sqrt{22140698112}}\cr\approx \mathstrut & 0.429211944592 \end{aligned}\]
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
3.1.351.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 | $x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(3\) | 3.9.9.6 | $x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
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) | $1$ | $-1$ | |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.312.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 13 $ | \(\Q(\sqrt{78}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.22464.6t3.e.a | $2$ | $ 2^{6} \cdot 3^{3} \cdot 13 $ | 6.2.2460077568.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.2808.6t3.h.a | $2$ | $ 2^{3} \cdot 3^{3} \cdot 13 $ | 6.0.307509696.1 | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
2.2808.3t2.a.a | $2$ | $ 2^{3} \cdot 3^{3} \cdot 13 $ | 3.3.2808.1 | $S_3$ (as 3T2) | $1$ | $2$ | |
* | 2.351.3t2.b.a | $2$ | $ 3^{3} \cdot 13 $ | 3.1.351.1 | $S_3$ (as 3T2) | $1$ | $0$ |
4.876096.6t9.b.a | $4$ | $ 2^{6} \cdot 3^{4} \cdot 13^{2}$ | 6.0.7008768.1 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
6.95943025152.18t51.b.a | $6$ | $ 2^{9} \cdot 3^{8} \cdot 13^{4}$ | 9.1.22140698112.2 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ | |
* | 6.63078912.9t18.b.a | $6$ | $ 2^{9} \cdot 3^{6} \cdot 13^{2}$ | 9.1.22140698112.2 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ |