Normalized defining polynomial
\( x^{9} - 2x^{8} + x^{7} - 7x^{6} + 7x^{5} + 14x^{3} - x^{2} - 5x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(10699470656\) \(\medspace = 2^{6}\cdot 7^{8}\cdot 29\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.01\) | 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: | $2^{2/3}7^{8/9}29^{1/2}\approx 48.20409353864055$ | ||
Ramified primes: | \(2\), \(7\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13}a^{8}+\frac{6}{13}a^{7}-\frac{3}{13}a^{6}-\frac{5}{13}a^{5}+\frac{6}{13}a^{4}-\frac{4}{13}a^{3}-\frac{5}{13}a^{2}-\frac{2}{13}a+\frac{5}{13}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{22}{13}a^{8}-\frac{50}{13}a^{7}+\frac{38}{13}a^{6}-\frac{162}{13}a^{5}+\frac{197}{13}a^{4}-\frac{62}{13}a^{3}+\frac{306}{13}a^{2}-\frac{122}{13}a-\frac{72}{13}$, $\frac{22}{13}a^{8}-\frac{50}{13}a^{7}+\frac{38}{13}a^{6}-\frac{162}{13}a^{5}+\frac{197}{13}a^{4}-\frac{62}{13}a^{3}+\frac{306}{13}a^{2}-\frac{122}{13}a-\frac{85}{13}$, $\frac{72}{13}a^{8}-\frac{166}{13}a^{7}+\frac{122}{13}a^{6}-\frac{542}{13}a^{5}+\frac{666}{13}a^{4}-\frac{197}{13}a^{3}+\frac{1070}{13}a^{2}-\frac{378}{13}a-\frac{238}{13}$, $\frac{4}{13}a^{8}-\frac{15}{13}a^{7}+\frac{14}{13}a^{6}-\frac{33}{13}a^{5}+\frac{76}{13}a^{4}-\frac{29}{13}a^{3}+\frac{71}{13}a^{2}-\frac{99}{13}a-\frac{32}{13}$, $\frac{28}{13}a^{8}-\frac{66}{13}a^{7}+\frac{46}{13}a^{6}-\frac{205}{13}a^{5}+\frac{259}{13}a^{4}-\frac{60}{13}a^{3}+\frac{406}{13}a^{2}-\frac{147}{13}a-\frac{94}{13}$, $\frac{36}{13}a^{8}-\frac{83}{13}a^{7}+\frac{61}{13}a^{6}-\frac{271}{13}a^{5}+\frac{333}{13}a^{4}-\frac{92}{13}a^{3}+\frac{535}{13}a^{2}-\frac{202}{13}a-\frac{132}{13}$ | 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: | \( 42.1102547294 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 42.1102547294 \cdot 1}{2\cdot\sqrt{10699470656}}\cr\approx \mathstrut & 0.257149948628 \end{aligned}\]
Galois group
$S_3\wr C_3$ (as 9T28):
A solvable group of order 648 |
The 17 conjugacy class representatives for $S_3 \wr C_3 $ |
Character table for $S_3 \wr C_3 $ |
Intermediate fields
\(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }$ | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(7\) | 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.3.0.1 | $x^{3} + 2 x + 27$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.203.6t1.b.a | $1$ | $ 7 \cdot 29 $ | 6.6.58557989.1 | $C_6$ (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) | $0$ | $1$ |
1.203.6t1.b.b | $1$ | $ 7 \cdot 29 $ | 6.6.58557989.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
3.1421.6t6.a.a | $3$ | $ 7^{2} \cdot 29 $ | 6.2.69629.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ | |
3.41209.4t4.a.a | $3$ | $ 7^{2} \cdot 29^{2}$ | 4.0.41209.1 | $A_4$ (as 4T4) | $1$ | $-1$ | |
6.183637853504.18t197.a.a | $6$ | $ 2^{6} \cdot 7^{6} \cdot 29^{3}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $-2$ | |
* | 6.218356544.9t28.a.a | $6$ | $ 2^{6} \cdot 7^{6} \cdot 29 $ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $2$ |
6.154...864.18t202.a.a | $6$ | $ 2^{6} \cdot 7^{6} \cdot 29^{5}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $2$ | |
6.183637853504.18t197.b.a | $6$ | $ 2^{6} \cdot 7^{6} \cdot 29^{3}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $-2$ | |
8.260...184.12t176.a.a | $8$ | $ 2^{6} \cdot 7^{8} \cdot 29^{4}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
8.372...312.24t1539.a.a | $8$ | $ 2^{6} \cdot 7^{7} \cdot 29^{4}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
8.372...312.24t1539.a.b | $8$ | $ 2^{6} \cdot 7^{7} \cdot 29^{4}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
12.127...016.18t206.a.a | $12$ | $ 2^{6} \cdot 7^{10} \cdot 29^{4}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
12.904...496.36t1101.a.a | $12$ | $ 2^{6} \cdot 7^{10} \cdot 29^{8}$ | 9.5.10699470656.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ |