Normalized defining polynomial
\( x^{9} - 3x^{7} - 3x^{6} - 12x^{4} - 20x^{3} + 36x^{2} + 3x - 13 \)
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: | \(-22347450531\) \(\medspace = -\,3^{7}\cdot 7^{3}\cdot 31^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.12\) | 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: | $3^{7/6}7^{1/2}31^{1/2}\approx 53.07271814040024$ | ||
Ramified primes: | \(3\), \(7\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-651}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{638763}a^{8}+\frac{122162}{638763}a^{7}+\frac{134272}{638763}a^{6}+\frac{140984}{638763}a^{5}-\frac{79361}{638763}a^{4}+\frac{246320}{638763}a^{3}+\frac{96416}{638763}a^{2}+\frac{220471}{638763}a-\frac{263590}{638763}$
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: | $\frac{3357}{212921}a^{8}+\frac{11988}{212921}a^{7}-\frac{2653}{212921}a^{6}-\frac{40095}{212921}a^{5}-\frac{50706}{212921}a^{4}-\frac{88924}{212921}a^{3}-\frac{184329}{212921}a^{2}-\frac{205170}{212921}a+\frac{240967}{212921}$, $\frac{24832}{638763}a^{8}+\frac{41297}{638763}a^{7}-\frac{100556}{638763}a^{6}-\frac{145315}{638763}a^{5}-\frac{108497}{638763}a^{4}-\frac{176248}{638763}a^{3}-\frac{1159138}{638763}a^{2}+\frac{536962}{638763}a+\frac{576344}{638763}$, $\frac{29114}{638763}a^{8}-\frac{7916}{638763}a^{7}-\frac{34552}{638763}a^{6}-\frac{82862}{638763}a^{5}-\frac{110383}{638763}a^{4}-\frac{670484}{638763}a^{3}-\frac{307961}{638763}a^{2}+\frac{1140833}{638763}a-\frac{60578}{638763}$, $\frac{41768}{638763}a^{8}+\frac{23572}{638763}a^{7}-\frac{66244}{638763}a^{6}-\frac{136385}{638763}a^{5}-\frac{209041}{638763}a^{4}-\frac{900644}{638763}a^{3}-\frac{1574753}{638763}a^{2}+\frac{225320}{638763}a-\frac{546815}{638763}$, $\frac{12178}{638763}a^{8}+\frac{9809}{638763}a^{7}-\frac{68864}{638763}a^{6}-\frac{91792}{638763}a^{5}-\frac{9839}{638763}a^{4}+\frac{53912}{638763}a^{3}+\frac{107654}{638763}a^{2}+\frac{174949}{638763}a-\frac{214945}{638763}$ | 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: | \( 80.8822807801 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 80.8822807801 \cdot 1}{2\cdot\sqrt{22347450531}}\cr\approx \mathstrut & 0.536832738763 \end{aligned}\]
Galois group
$C_3^2:D_6$ (as 9T18):
A solvable group of order 108 |
The 11 conjugacy class representatives for $C_3^2 : D_{6} $ |
Character table for $C_3^2 : D_{6} $ |
Intermediate fields
3.1.31.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 9 sibling: | data not computed |
Degree 18 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | R | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.6.3 | $x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(7\) | 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.31.2t1.a.a | $1$ | $ 31 $ | \(\Q(\sqrt{-31}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.651.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 31 $ | \(\Q(\sqrt{-651}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.13671.6t3.b.a | $2$ | $ 3^{2} \cdot 7^{2} \cdot 31 $ | 6.0.275894451.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.5859.6t3.d.a | $2$ | $ 3^{3} \cdot 7 \cdot 31 $ | 6.2.720885501.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.5859.3t2.a.a | $2$ | $ 3^{3} \cdot 7 \cdot 31 $ | 3.1.5859.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
* | 2.31.3t2.b.a | $2$ | $ 31 $ | 3.1.31.1 | $S_3$ (as 3T2) | $1$ | $0$ |
4.34327881.6t9.b.a | $4$ | $ 3^{6} \cdot 7^{2} \cdot 31^{2}$ | 6.2.720885501.2 | $S_3^2$ (as 6T9) | $1$ | $0$ | |
6.692770966461.18t51.b.a | $6$ | $ 3^{7} \cdot 7^{3} \cdot 31^{4}$ | 9.3.22347450531.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $-2$ | |
* | 6.720885501.9t18.b.a | $6$ | $ 3^{7} \cdot 7^{3} \cdot 31^{2}$ | 9.3.22347450531.1 | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $2$ |