Normalized defining polynomial
\( x^{9} - 7x^{7} - x^{6} + 29x^{5} + 30x^{4} - 18x^{3} - 35x^{2} - 5x + 7 \)
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)
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Discriminant: | \(-24087491072\) \(\medspace = -\,2^{9}\cdot 19^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.24\) | 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: | $2^{3/2}19^{2/3}\approx 20.13944017607579$ | ||
Ramified primes: | \(2\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{74}a^{8}-\frac{5}{37}a^{7}-\frac{9}{37}a^{6}-\frac{3}{37}a^{5}+\frac{15}{74}a^{4}+\frac{14}{37}a^{3}+\frac{35}{74}a^{2}+\frac{11}{37}a+\frac{17}{37}$
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: | $\frac{19}{37}a^{8}-\frac{47}{74}a^{7}-\frac{203}{74}a^{6}+\frac{108}{37}a^{5}+\frac{396}{37}a^{4}+\frac{139}{74}a^{3}-\frac{705}{74}a^{2}-\frac{311}{74}a+\frac{145}{74}$, $\frac{65}{74}a^{8}-\frac{29}{37}a^{7}-\frac{215}{37}a^{6}+\frac{175}{37}a^{5}+\frac{1715}{74}a^{4}+\frac{133}{37}a^{3}-\frac{1943}{74}a^{2}-\frac{321}{37}a+\frac{328}{37}$, $\frac{11}{37}a^{8}+\frac{1}{37}a^{7}-\frac{87}{37}a^{6}+\frac{8}{37}a^{5}+\frac{350}{37}a^{4}+\frac{234}{37}a^{3}-\frac{244}{37}a^{2}-\frac{239}{37}a+\frac{4}{37}$, $\frac{30}{37}a^{8}-\frac{41}{37}a^{7}-\frac{170}{37}a^{6}+\frac{227}{37}a^{5}+\frac{635}{37}a^{4}-\frac{85}{37}a^{3}-\frac{689}{37}a^{2}-\frac{117}{37}a+\frac{206}{37}$, $\frac{31}{74}a^{8}-\frac{51}{74}a^{7}-\frac{151}{74}a^{6}+\frac{129}{37}a^{5}+\frac{539}{74}a^{4}-\frac{131}{74}a^{3}-\frac{253}{37}a^{2}-\frac{21}{74}a+\frac{55}{74}$ | 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: | \( 56.715041346 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 56.715041346 \cdot 1}{2\cdot\sqrt{24087491072}}\cr\approx \mathstrut & 0.36257851514 \end{aligned}\]
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
3.3.361.1, 3.1.2888.1 |
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: | 6.0.184832.1 |
Minimal sibling: | 6.0.184832.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
2.6.9.5 | $x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(19\) | 19.9.6.2 | $x^{9} + 981 x^{7} + 108 x^{6} + 316911 x^{5} + 20529 x^{4} + 34115982 x^{3} + 10990188 x^{2} + 130942880 x + 566550143$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
1.152.6t1.c.a | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.152.6t1.c.b | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 2.2888.3t2.b.a | $2$ | $ 2^{3} \cdot 19^{2}$ | 3.1.2888.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.152.6t5.b.a | $2$ | $ 2^{3} \cdot 19 $ | 9.3.24087491072.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
* | 2.152.6t5.b.b | $2$ | $ 2^{3} \cdot 19 $ | 9.3.24087491072.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |