Normalized defining polynomial
\( x^{9} - 3x^{8} + 3x^{7} - x^{6} - 3x^{5} + 12x^{4} - 20x^{3} + 15x^{2} - 6x + 1 \)
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: | \(-21705152688\) \(\medspace = -\,2^{4}\cdot 3^{9}\cdot 41^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.08\) | 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: | $2^{2/3}3^{7/6}41^{1/2}\approx 36.620144695373874$ | ||
Ramified primes: | \(2\), \(3\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-123}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $a^{8}-2a^{7}+a^{6}-3a^{4}+9a^{3}-11a^{2}+4a-2$, $a$, $\frac{3}{2}a^{8}-3a^{7}+a^{6}+\frac{1}{2}a^{5}-4a^{4}+\frac{27}{2}a^{3}-\frac{31}{2}a^{2}+\frac{7}{2}a-1$, $\frac{5}{2}a^{8}-\frac{11}{2}a^{7}+2a^{6}+\frac{3}{2}a^{5}-\frac{15}{2}a^{4}+\frac{47}{2}a^{3}-28a^{2}+5a+\frac{3}{2}$, $a^{8}-\frac{5}{2}a^{7}+2a^{6}-\frac{7}{2}a^{4}+10a^{3}-\frac{31}{2}a^{2}+\frac{15}{2}a-\frac{3}{2}$ | 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: | \( 31.3245429739 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 31.3245429739 \cdot 2}{2\cdot\sqrt{21705152688}}\cr\approx \mathstrut & 0.421922720531 \end{aligned}\]
Galois group
$C_3^2:S_3$ (as 9T12):
A solvable group of order 54 |
The 10 conjugacy class representatives for $(C_3^2:C_3):C_2$ |
Character table for $(C_3^2:C_3):C_2$ |
Intermediate fields
3.1.1107.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 27 sibling: | data not computed |
Minimal sibling: | 9.3.9646734528.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(3\) | 3.9.9.1 | $x^{9} + 90 x^{7} - 207 x^{6} + 540 x^{5} + 324 x^{4} + 243 x^{3} + 324 x^{2} + 162 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ |
\(41\) | 41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
41.6.3.2 | $x^{6} + 1681 x^{2} - 2412235$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.123.2t1.a.a | $1$ | $ 3 \cdot 41 $ | \(\Q(\sqrt{-123}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.1107.3t2.a.a | $2$ | $ 3^{3} \cdot 41 $ | 3.1.1107.1 | $S_3$ (as 3T2) | $1$ | $0$ |
2.4428.3t2.a.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 41 $ | 3.1.4428.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.4428.3t2.b.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 41 $ | 3.1.4428.2 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.492.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 41 $ | 3.1.492.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 3.4428.9t12.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 41 $ | 9.3.21705152688.1 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $1$ |
* | 3.4428.9t12.a.b | $3$ | $ 2^{2} \cdot 3^{3} \cdot 41 $ | 9.3.21705152688.1 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $1$ |
3.544644.18t24.c.a | $3$ | $ 2^{2} \cdot 3^{4} \cdot 41^{2}$ | 9.3.21705152688.1 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $-1$ | |
3.544644.18t24.c.b | $3$ | $ 2^{2} \cdot 3^{4} \cdot 41^{2}$ | 9.3.21705152688.1 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $-1$ |