Normalized defining polynomial
\( x^{9} - 3x^{8} + 4x^{7} - x^{6} - 7x^{5} + 11x^{4} - 9x^{3} + 3x^{2} - x + 1 \)
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: | \(-114479303\) \(\medspace = -\,23^{3}\cdot 97^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.86\) | 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: | $23^{1/2}97^{2/3}\approx 101.2461243234224$ | ||
Ramified primes: | \(23\), \(97\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{37}a^{8}-\frac{5}{37}a^{7}+\frac{14}{37}a^{6}+\frac{8}{37}a^{5}+\frac{14}{37}a^{4}-\frac{17}{37}a^{3}-\frac{12}{37}a^{2}-\frac{10}{37}a-\frac{18}{37}$
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: | $\frac{29}{37}a^{8}-\frac{71}{37}a^{7}+\frac{73}{37}a^{6}+\frac{10}{37}a^{5}-\frac{186}{37}a^{4}+\frac{210}{37}a^{3}-\frac{163}{37}a^{2}+\frac{43}{37}a-\frac{4}{37}$, $\frac{28}{37}a^{8}-\frac{66}{37}a^{7}+\frac{59}{37}a^{6}+\frac{39}{37}a^{5}-\frac{200}{37}a^{4}+\frac{190}{37}a^{3}-\frac{77}{37}a^{2}-\frac{58}{37}a+\frac{14}{37}$, $\frac{15}{37}a^{8}-\frac{38}{37}a^{7}+\frac{25}{37}a^{6}+\frac{46}{37}a^{5}-\frac{123}{37}a^{4}+\frac{78}{37}a^{3}+\frac{42}{37}a^{2}-\frac{76}{37}a-\frac{11}{37}$, $\frac{20}{37}a^{8}-\frac{26}{37}a^{7}+\frac{21}{37}a^{6}+\frac{12}{37}a^{5}-\frac{90}{37}a^{4}+\frac{30}{37}a^{3}-\frac{92}{37}a^{2}+\frac{22}{37}a-\frac{27}{37}$, $\frac{8}{37}a^{8}-\frac{40}{37}a^{7}+\frac{75}{37}a^{6}-\frac{47}{37}a^{5}-\frac{73}{37}a^{4}+\frac{197}{37}a^{3}-\frac{207}{37}a^{2}+\frac{68}{37}a-\frac{33}{37}$ | 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: | \( 1.99189617114 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1.99189617114 \cdot 1}{2\cdot\sqrt{114479303}}\cr\approx \mathstrut & 0.184715272479 \end{aligned}\]
Galois group
$C_3\wr S_3$ (as 9T20):
A solvable group of order 162 |
The 22 conjugacy class representatives for $C_3 \wr S_3 $ |
Character table for $C_3 \wr S_3 $ |
Intermediate fields
3.1.23.1 |
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 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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | R | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(97\) | 97.3.2.2 | $x^{3} + 194$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
97.6.0.1 | $x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2231.6t1.a.a | $1$ | $ 23 \cdot 97 $ | 6.0.1077135761927.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.2231.6t1.a.b | $1$ | $ 23 \cdot 97 $ | 6.0.1077135761927.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.97.3t1.a.a | $1$ | $ 97 $ | 3.3.9409.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.97.3t1.a.b | $1$ | $ 97 $ | 3.3.9409.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
* | 2.23.3t2.b.a | $2$ | $ 23 $ | 3.1.23.1 | $S_3$ (as 3T2) | $1$ | $0$ |
2.216407.6t5.a.a | $2$ | $ 23 \cdot 97^{2}$ | 6.0.1077135761927.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
2.216407.6t5.a.b | $2$ | $ 23 \cdot 97^{2}$ | 6.0.1077135761927.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
3.216407.9t20.b.a | $3$ | $ 23 \cdot 97^{2}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
* | 3.2231.9t20.b.a | $3$ | $ 23 \cdot 97 $ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.51313.18t86.b.a | $3$ | $ 23^{2} \cdot 97 $ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.482804017.18t86.b.a | $3$ | $ 23^{2} \cdot 97^{3}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.482804017.18t86.b.b | $3$ | $ 23^{2} \cdot 97^{3}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
* | 3.2231.9t20.b.b | $3$ | $ 23 \cdot 97 $ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ |
3.216407.9t20.b.b | $3$ | $ 23 \cdot 97^{2}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.4977361.18t86.b.a | $3$ | $ 23^{2} \cdot 97^{2}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.20991479.9t20.b.a | $3$ | $ 23 \cdot 97^{3}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.4977361.18t86.b.b | $3$ | $ 23^{2} \cdot 97^{2}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
3.20991479.9t20.b.b | $3$ | $ 23 \cdot 97^{3}$ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $1$ | |
3.51313.18t86.b.b | $3$ | $ 23^{2} \cdot 97 $ | 9.3.114479303.1 | $C_3 \wr S_3 $ (as 9T20) | $0$ | $-1$ | |
6.107...927.9t11.a.a | $6$ | $ 23^{3} \cdot 97^{4}$ | 9.1.24774122524321.1 | $C_3^2 : C_6$ (as 9T11) | $1$ | $0$ |