Normalized defining polynomial
\( x^{9} - 4x^{7} - 3x^{6} + 2x^{5} + 8x^{4} - 4x^{3} - 2x^{2} - 4x - 1 \)
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: | \(-8934171875\) \(\medspace = -\,5^{6}\cdot 83^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.75\) | 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: | $5^{2/3}83^{1/2}\approx 26.63906938822806$ | ||
Ramified primes: | \(5\), \(83\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-83}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{431}a^{8}-\frac{6}{431}a^{7}+\frac{32}{431}a^{6}-\frac{195}{431}a^{5}-\frac{121}{431}a^{4}-\frac{128}{431}a^{3}-\frac{98}{431}a^{2}+\frac{155}{431}a-\frac{72}{431}$
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: | $\frac{2}{431}a^{8}-\frac{12}{431}a^{7}+\frac{64}{431}a^{6}+\frac{41}{431}a^{5}-\frac{242}{431}a^{4}-\frac{256}{431}a^{3}+\frac{235}{431}a^{2}+\frac{310}{431}a-\frac{144}{431}$, $a$, $\frac{12}{431}a^{8}-\frac{72}{431}a^{7}-\frac{47}{431}a^{6}+\frac{246}{431}a^{5}+\frac{272}{431}a^{4}+\frac{188}{431}a^{3}-\frac{745}{431}a^{2}-\frac{295}{431}a-\frac{433}{431}$, $\frac{158}{431}a^{8}-\frac{86}{431}a^{7}-\frac{547}{431}a^{6}-\frac{209}{431}a^{5}+\frac{277}{431}a^{4}+\frac{1326}{431}a^{3}-\frac{1261}{431}a^{2}-\frac{77}{431}a-\frac{601}{431}$, $\frac{142}{431}a^{8}+\frac{10}{431}a^{7}-\frac{628}{431}a^{6}-\frac{537}{431}a^{5}+\frac{489}{431}a^{4}+\frac{1650}{431}a^{3}-\frac{555}{431}a^{2}-\frac{833}{431}a-\frac{311}{431}$ | 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: | \( 33.8634528155 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 33.8634528155 \cdot 1}{2\cdot\sqrt{8934171875}}\cr\approx \mathstrut & 0.355470596347 \end{aligned}\]
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
3.1.2075.3 |
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: | 9.3.357366875.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.9.6.1 | $x^{9} + 9 x^{7} + 24 x^{6} + 27 x^{5} + 9 x^{4} - 186 x^{3} + 216 x^{2} - 504 x + 647$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(83\) | 83.3.0.1 | $x^{3} + 3 x + 81$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
83.6.3.2 | $x^{6} + 255 x^{4} + 162 x^{3} + 20676 x^{2} - 39852 x + 537761$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.83.2t1.a.a | $1$ | $ 83 $ | \(\Q(\sqrt{-83}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.2075.3t2.a.a | $2$ | $ 5^{2} \cdot 83 $ | 3.1.2075.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.83.3t2.a.a | $2$ | $ 83 $ | 3.1.83.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 2.2075.3t2.b.a | $2$ | $ 5^{2} \cdot 83 $ | 3.1.2075.3 | $S_3$ (as 3T2) | $1$ | $0$ |
2.2075.3t2.c.a | $2$ | $ 5^{2} \cdot 83 $ | 3.1.2075.2 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 3.2075.9t12.b.a | $3$ | $ 5^{2} \cdot 83 $ | 9.3.8934171875.3 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $1$ |
* | 3.2075.9t12.b.b | $3$ | $ 5^{2} \cdot 83 $ | 9.3.8934171875.3 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $1$ |
3.172225.18t24.b.a | $3$ | $ 5^{2} \cdot 83^{2}$ | 9.3.8934171875.3 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $-1$ | |
3.172225.18t24.b.b | $3$ | $ 5^{2} \cdot 83^{2}$ | 9.3.8934171875.3 | $(C_3^2:C_3):C_2$ (as 9T12) | $0$ | $-1$ |