Normalized defining polynomial
\( x^{9} - 3x^{7} - 7x^{6} + 3x^{5} + 14x^{4} - 8x^{3} - 7x^{2} + 7x - 1 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(83032187200\) \(\medspace = 2^{6}\cdot 5^{2}\cdot 373^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.34\) | 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: | $2^{2/3}5^{1/2}373^{1/2}\approx 68.55293951471283$ | ||
Ramified primes: | \(2\), \(5\), \(373\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{373}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{47}a^{8}-\frac{4}{47}a^{7}+\frac{13}{47}a^{6}-\frac{12}{47}a^{5}+\frac{4}{47}a^{4}-\frac{2}{47}a^{3}-\frac{7}{47}a-\frac{12}{47}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $a$, $\frac{7}{47}a^{8}+\frac{19}{47}a^{7}-\frac{3}{47}a^{6}-\frac{84}{47}a^{5}-\frac{160}{47}a^{4}-\frac{14}{47}a^{3}+2a^{2}-\frac{2}{47}a-\frac{37}{47}$, $\frac{10}{47}a^{8}+\frac{7}{47}a^{7}-\frac{11}{47}a^{6}-\frac{73}{47}a^{5}-\frac{54}{47}a^{4}-\frac{20}{47}a^{3}-2a^{2}+\frac{24}{47}a+\frac{68}{47}$, $\frac{19}{47}a^{8}-\frac{29}{47}a^{7}-\frac{35}{47}a^{6}-\frac{87}{47}a^{5}+\frac{217}{47}a^{4}+\frac{103}{47}a^{3}-5a^{2}+\frac{196}{47}a-\frac{87}{47}$, $a-1$, $\frac{66}{47}a^{8}+\frac{18}{47}a^{7}-\frac{176}{47}a^{6}-\frac{510}{47}a^{5}+\frac{29}{47}a^{4}+\frac{808}{47}a^{3}-6a^{2}-\frac{415}{47}a+\frac{242}{47}$ | 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: | \( 136.131778389 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 136.131778389 \cdot 1}{2\cdot\sqrt{83032187200}}\cr\approx \mathstrut & 0.298411804183 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 9T29):
A solvable group of order 648 |
The 17 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Intermediate fields
3.3.1492.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 27 siblings: | 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 | ${\href{/padicField/3.9.0.1}{9} }$ | R | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(373\) | Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.373.2t1.a.a | $1$ | $ 373 $ | \(\Q(\sqrt{373}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 2.1492.3t2.a.a | $2$ | $ 2^{2} \cdot 373 $ | 3.3.1492.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.37300.4t5.a.a | $3$ | $ 2^{2} \cdot 5^{2} \cdot 373 $ | 4.0.37300.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.13912900.6t8.a.a | $3$ | $ 2^{2} \cdot 5^{2} \cdot 373^{2}$ | 4.0.37300.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
4.5189511700.24t1527.a.a | $4$ | $ 2^{2} \cdot 5^{2} \cdot 373^{3}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
4.37300.12t175.a.a | $4$ | $ 2^{2} \cdot 5^{2} \cdot 373 $ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
4.5189511700.24t1527.a.b | $4$ | $ 2^{2} \cdot 5^{2} \cdot 373^{3}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
4.37300.12t175.a.b | $4$ | $ 2^{2} \cdot 5^{2} \cdot 373 $ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
6.774...400.18t220.a.a | $6$ | $ 2^{4} \cdot 5^{2} \cdot 373^{4}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $2$ | |
* | 6.55651600.9t29.a.a | $6$ | $ 2^{4} \cdot 5^{2} \cdot 373^{2}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $2$ |
6.518951170000.36t1131.a.a | $6$ | $ 2^{4} \cdot 5^{4} \cdot 373^{3}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $-2$ | |
6.518951170000.36t1131.a.b | $6$ | $ 2^{4} \cdot 5^{4} \cdot 373^{3}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $-2$ | |
8.774...000.24t1540.a.a | $8$ | $ 2^{6} \cdot 5^{4} \cdot 373^{4}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
8.774...000.24t1540.a.b | $8$ | $ 2^{6} \cdot 5^{4} \cdot 373^{4}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $0$ | $0$ | |
12.288...000.18t219.a.a | $12$ | $ 2^{8} \cdot 5^{6} \cdot 373^{5}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $0$ | |
12.401...000.36t1126.a.a | $12$ | $ 2^{8} \cdot 5^{6} \cdot 373^{7}$ | 9.5.83032187200.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T29) | $1$ | $0$ |