Normalized defining polynomial
\( x^{9} - 3x^{7} - 4x^{6} + 3x^{5} + 8x^{4} - 10x^{3} - 4x^{2} + 9x + 4 \)
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: |
\(-69534993539\)
\(\medspace = -\,11\cdot 43^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.02\) | 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: | $11^{1/2}43^{2/3}\approx 40.70758261752865$ | ||
Ramified primes: |
\(11\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$
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: |
$12a^{8}-7a^{7}-32a^{6}-\frac{59}{2}a^{5}+\frac{107}{2}a^{4}+\frac{131}{2}a^{3}-158a^{2}+\frac{87}{2}a+83$, $\frac{21}{2}a^{8}-6a^{7}-28a^{6}-26a^{5}+46a^{4}+57a^{3}-137a^{2}+\frac{77}{2}a+73$, $\frac{41}{2}a^{8}-12a^{7}-\frac{109}{2}a^{6}-50a^{5}+91a^{4}+\frac{221}{2}a^{3}-\frac{541}{2}a^{2}+76a+141$, $a^{2}-a-1$, $22a^{8}-\frac{25}{2}a^{7}-\frac{117}{2}a^{6}-\frac{109}{2}a^{5}+96a^{4}+\frac{239}{2}a^{3}-288a^{2}+77a+151$
| 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: | \( 186.606756006 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 186.606756006 \cdot 1}{2\cdot\sqrt{69534993539}}\cr\approx \mathstrut & 0.702142482130 \end{aligned}\]
Galois group
$S_3\wr C_3$ (as 9T28):
A solvable group of order 648 |
The 17 conjugacy class representatives for $S_3 \wr C_3 $ |
Character table for $S_3 \wr C_3 $ |
Intermediate fields
3.3.1849.1 |
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 | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(43\)
| 43.3.2.1 | $x^{3} + 43$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
43.6.4.1 | $x^{6} + 126 x^{5} + 5301 x^{4} + 74930 x^{3} + 21321 x^{2} + 227916 x + 3171406$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.473.6t1.b.a | $1$ | $ 11 \cdot 43 $ | 6.0.4550424131.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
* | 1.43.3t1.a.a | $1$ | $ 43 $ | 3.3.1849.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.43.3t1.a.b | $1$ | $ 43 $ | 3.3.1849.1 | $C_3$ (as 3T1) | $0$ | $1$ |
1.473.6t1.b.b | $1$ | $ 11 \cdot 43 $ | 6.0.4550424131.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
3.20339.6t6.a.a | $3$ | $ 11 \cdot 43^{2}$ | 6.0.37606811.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-3$ | |
3.223729.4t4.a.a | $3$ | $ 11^{2} \cdot 43^{2}$ | 4.4.223729.1 | $A_4$ (as 4T4) | $1$ | $3$ | |
6.4550424131.18t197.a.a | $6$ | $ 11^{3} \cdot 43^{4}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
* | 6.37606811.9t28.a.a | $6$ | $ 11 \cdot 43^{4}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ |
6.550601319851.18t202.a.a | $6$ | $ 11^{5} \cdot 43^{4}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
6.4550424131.18t197.b.a | $6$ | $ 11^{3} \cdot 43^{4}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
8.50054665441.12t176.a.a | $8$ | $ 11^{4} \cdot 43^{4}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
8.925...409.24t1539.a.a | $8$ | $ 11^{4} \cdot 43^{6}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
8.925...409.24t1539.a.b | $8$ | $ 11^{4} \cdot 43^{6}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
12.171...241.18t206.a.a | $12$ | $ 11^{4} \cdot 43^{8}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
12.250...481.36t1101.a.a | $12$ | $ 11^{8} \cdot 43^{8}$ | 9.3.69534993539.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ |