Normalized defining polynomial
\( x^{9} - 3x^{8} + 4x^{7} - 6x^{6} - 5x^{5} + 7x^{4} + 16x^{3} + 9x^{2} + x - 1 \)
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: | \(-69274613043\) \(\medspace = -\,3^{3}\cdot 37^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.01\) | 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: | $3^{1/2}37^{2/3}\approx 19.232176827720277$ | ||
Ramified primes: | \(3\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{101}a^{8}-\frac{35}{101}a^{7}+\frac{13}{101}a^{6}-\frac{18}{101}a^{5}-\frac{35}{101}a^{4}+\frac{16}{101}a^{3}+\frac{9}{101}a^{2}+\frac{24}{101}a+\frac{41}{101}$
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: | $\frac{16}{101}a^{8}-\frac{55}{101}a^{7}+\frac{107}{101}a^{6}-\frac{187}{101}a^{5}+\frac{46}{101}a^{4}+\frac{54}{101}a^{3}+\frac{43}{101}a^{2}+\frac{283}{101}a+\frac{151}{101}$, $a^{8}-3a^{7}+4a^{6}-6a^{5}-5a^{4}+7a^{3}+16a^{2}+9a+1$, $\frac{62}{101}a^{8}-\frac{251}{101}a^{7}+\frac{503}{101}a^{6}-\frac{813}{101}a^{5}+\frac{355}{101}a^{4}+\frac{386}{101}a^{3}+\frac{255}{101}a^{2}-\frac{27}{101}a+\frac{17}{101}$, $\frac{17}{101}a^{8}-\frac{90}{101}a^{7}+\frac{221}{101}a^{6}-\frac{407}{101}a^{5}+\frac{415}{101}a^{4}-\frac{132}{101}a^{3}+\frac{153}{101}a^{2}-\frac{198}{101}a+\frac{91}{101}$, $\frac{1}{101}a^{8}+\frac{66}{101}a^{7}-\frac{290}{101}a^{6}+\frac{588}{101}a^{5}-\frac{944}{101}a^{4}+\frac{420}{101}a^{3}+\frac{615}{101}a^{2}+\frac{125}{101}a-\frac{60}{101}$ | 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: | \( 81.8898092201 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 81.8898092201 \cdot 1}{2\cdot\sqrt{69274613043}}\cr\approx \mathstrut & 0.308704102791 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 9T4):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
3.3.1369.1, 3.1.4107.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 6 sibling: | 6.0.36963.1 |
Minimal sibling: | 6.0.36963.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(37\) | 37.3.2.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
37.3.2.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
37.3.2.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.37.3t1.a.a | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ |
1.111.6t1.b.a | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.111.6t1.b.b | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.37.3t1.a.b | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 2.4107.3t2.a.a | $2$ | $ 3 \cdot 37^{2}$ | 3.1.4107.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.111.6t5.b.a | $2$ | $ 3 \cdot 37 $ | 9.3.69274613043.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
* | 2.111.6t5.b.b | $2$ | $ 3 \cdot 37 $ | 9.3.69274613043.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |