Normalized defining polynomial
\( x^{9} - 4x^{8} + 7x^{7} - 9x^{6} + 8x^{5} - 21x^{4} + 14x^{3} - 15x^{2} + 14x - 4 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(97123852609\) \(\medspace = 7^{2}\cdot 211^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.63\) | 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: | $7^{1/2}211^{1/2}\approx 38.43175770115127$ | ||
Ramified primes: | \(7\), \(211\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}+\frac{1}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{238}a^{8}-\frac{1}{17}a^{7}+\frac{45}{238}a^{6}+\frac{1}{14}a^{5}+\frac{4}{119}a^{4}+\frac{69}{238}a^{3}-\frac{32}{119}a^{2}+\frac{13}{238}a-\frac{1}{17}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{165}{238}a^{8}-\frac{271}{119}a^{7}+\frac{761}{238}a^{6}-\frac{55}{14}a^{5}+\frac{337}{119}a^{4}-\frac{433}{34}a^{3}+\frac{143}{119}a^{2}-\frac{2241}{238}a+\frac{460}{119}$, $\frac{201}{119}a^{8}-\frac{706}{119}a^{7}+\frac{1072}{119}a^{6}-\frac{78}{7}a^{5}+\frac{1030}{119}a^{4}-\frac{3777}{119}a^{3}+\frac{974}{119}a^{2}-\frac{2708}{119}a+\frac{1623}{119}$, $\frac{165}{238}a^{8}-\frac{271}{119}a^{7}+\frac{761}{238}a^{6}-\frac{55}{14}a^{5}+\frac{337}{119}a^{4}-\frac{433}{34}a^{3}+\frac{143}{119}a^{2}-\frac{2479}{238}a+\frac{579}{119}$, $\frac{22}{17}a^{8}-\frac{558}{119}a^{7}+\frac{123}{17}a^{6}-\frac{61}{7}a^{5}+\frac{790}{119}a^{4}-\frac{2889}{119}a^{3}+\frac{1041}{119}a^{2}-\frac{1857}{119}a+\frac{1363}{119}$ | 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: | \( 101.473000238 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 101.473000238 \cdot 1}{2\cdot\sqrt{97123852609}}\cr\approx \mathstrut & 0.507466086575 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 9T30):
A solvable group of order 648 |
The 14 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
3.1.211.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | 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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(211\) | $\Q_{211}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.211.2t1.a.a | $1$ | $ 211 $ | \(\Q(\sqrt{-211}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.211.3t2.a.a | $2$ | $ 211 $ | 3.1.211.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.2181529.6t8.a.a | $3$ | $ 7^{2} \cdot 211^{2}$ | 4.2.10339.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.10339.4t5.a.a | $3$ | $ 7^{2} \cdot 211 $ | 4.2.10339.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
6.460302619.18t217.a.a | $6$ | $ 7^{2} \cdot 211^{3}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
* | 6.460302619.9t30.a.a | $6$ | $ 7^{2} \cdot 211^{3}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ |
6.22554828331.36t1121.a.a | $6$ | $ 7^{4} \cdot 211^{3}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
6.22554828331.36t1121.a.b | $6$ | $ 7^{4} \cdot 211^{3}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.475...841.12t177.a.a | $8$ | $ 7^{4} \cdot 211^{4}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.475...841.12t177.b.a | $8$ | $ 7^{4} \cdot 211^{4}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.475...841.12t178.a.a | $8$ | $ 7^{4} \cdot 211^{4}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
12.219...579.36t1123.a.a | $12$ | $ 7^{6} \cdot 211^{7}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $-2$ | |
12.492...099.18t218.a.a | $12$ | $ 7^{6} \cdot 211^{5}$ | 9.1.97123852609.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $2$ |