Normalized defining polynomial
\( x^{9} - 3x^{8} - x^{7} + 20x^{6} - 30x^{5} - 19x^{4} + 99x^{3} - 91x^{2} + 16x + 3 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-4633831094976\) \(\medspace = -\,2^{6}\cdot 3^{4}\cdot 19^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(25.55\) | 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: | $2^{2/3}3^{1/2}19^{7/8}\approx 36.15402816127715$ | ||
Ramified primes: | \(2\), \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{1214}a^{8}+\frac{97}{607}a^{7}-\frac{12}{607}a^{6}+\frac{74}{607}a^{5}+\frac{597}{1214}a^{4}-\frac{84}{607}a^{3}-\frac{219}{1214}a^{2}+\frac{235}{607}a-\frac{265}{1214}$
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)
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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: | $\frac{137}{607}a^{8}-\frac{867}{1214}a^{7}-\frac{253}{607}a^{6}+\frac{5953}{1214}a^{5}-\frac{3798}{607}a^{4}-\frac{4199}{607}a^{3}+\frac{14308}{607}a^{2}-\frac{17507}{1214}a-\frac{1099}{607}$, $\frac{47}{607}a^{8}+\frac{13}{607}a^{7}-\frac{435}{1214}a^{6}+\frac{279}{607}a^{5}+\frac{881}{1214}a^{4}-\frac{1219}{607}a^{3}-\frac{581}{607}a^{2}+\frac{238}{607}a-\frac{23}{1214}$, $\frac{499}{607}a^{8}-\frac{2449}{1214}a^{7}-\frac{1050}{607}a^{6}+\frac{18413}{1214}a^{5}-\frac{10453}{607}a^{4}-\frac{13420}{607}a^{3}+\frac{41255}{607}a^{2}-\frac{54781}{1214}a-\frac{1123}{607}$, $\frac{49}{1214}a^{8}-\frac{103}{607}a^{7}+\frac{19}{607}a^{6}+\frac{591}{607}a^{5}-\frac{2311}{1214}a^{4}-\frac{474}{607}a^{3}+\frac{6265}{1214}a^{2}-\frac{3053}{607}a+\frac{1583}{1214}$, $\frac{483}{1214}a^{8}-\frac{495}{607}a^{7}-\frac{1273}{1214}a^{6}+\frac{4178}{607}a^{5}-\frac{3629}{607}a^{4}-\frac{7187}{607}a^{3}+\frac{35047}{1214}a^{2}-\frac{7895}{607}a-\frac{2387}{607}$ | 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: | \( 1684.63900421 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1684.63900421 \cdot 1}{2\cdot\sqrt{4633831094976}}\cr\approx \mathstrut & 0.776490860378 \end{aligned}\]
Galois group
$C_3^2:\GL(2,3)$ (as 9T26):
A solvable group of order 432 |
The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$ |
Character table for $((C_3^2:Q_8):C_3):C_2$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 sibling: | data not computed |
Degree 24 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 | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
19.8.7.2 | $x^{8} + 38$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.76.3t2.a.a | $2$ | $ 2^{2} \cdot 19 $ | 3.1.76.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.12996.24t22.a.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 19^{2}$ | 8.2.1158457773744.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.12996.24t22.a.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 19^{2}$ | 8.2.1158457773744.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.27436.4t5.a.a | $3$ | $ 2^{2} \cdot 19^{3}$ | 4.2.27436.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.1444.6t8.a.a | $3$ | $ 2^{2} \cdot 19^{2}$ | 4.2.27436.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
4.42224004.8t23.a.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 19^{4}$ | 8.2.1158457773744.1 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ | |
* | 8.463...976.9t26.a.a | $8$ | $ 2^{6} \cdot 3^{4} \cdot 19^{7}$ | 9.3.4633831094976.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $2$ |
8.463...976.18t157.a.a | $8$ | $ 2^{6} \cdot 3^{4} \cdot 19^{7}$ | 9.3.4633831094976.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $-2$ | |
16.536...144.24t1334.a.a | $16$ | $ 2^{10} \cdot 3^{8} \cdot 19^{14}$ | 9.3.4633831094976.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $0$ |