Normalized defining polynomial
\( x^{8} + 3x^{6} - 112x^{4} + 299x^{2} + 25 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(32793672683776\)
\(\medspace = 2^{8}\cdot 71^{6}\)
|
| |
| Root discriminant: | \(48.92\) |
| |
| Galois root discriminant: | $2^{2}71^{3/4}\approx 97.83714091452357$ | ||
| Ramified primes: |
\(2\), \(71\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{218}a^{6}-\frac{23}{218}a^{4}-\frac{1}{2}a^{3}+\frac{25}{109}a^{2}-\frac{1}{2}a-\frac{10}{109}$, $\frac{1}{2180}a^{7}-\frac{1}{436}a^{6}-\frac{33}{545}a^{5}-\frac{43}{218}a^{4}-\frac{411}{1090}a^{3}+\frac{42}{109}a^{2}+\frac{89}{2180}a-\frac{89}{436}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{23}{545}a^{7}+\frac{234}{545}a^{5}-\frac{921}{545}a^{3}-\frac{133}{545}a$, $\frac{8}{545}a^{7}-\frac{1}{218}a^{6}+\frac{34}{545}a^{5}+\frac{23}{218}a^{4}-\frac{1707}{1090}a^{3}-\frac{25}{109}a^{2}+\frac{4149}{1090}a+\frac{10}{109}$, $\frac{8}{545}a^{7}+\frac{1}{218}a^{6}+\frac{34}{545}a^{5}-\frac{23}{218}a^{4}-\frac{1707}{1090}a^{3}+\frac{25}{109}a^{2}+\frac{4149}{1090}a-\frac{10}{109}$, $\frac{441}{1090}a^{7}+\frac{1414}{545}a^{5}-\frac{19931}{545}a^{3}-\frac{3261}{1090}a$, $\frac{161273}{2180}a^{7}-\frac{59399}{436}a^{6}+\frac{257701}{545}a^{5}-\frac{94919}{109}a^{4}-\frac{3641569}{545}a^{3}+\frac{1341211}{109}a^{2}-\frac{1182473}{2180}a+\frac{437515}{436}$
|
| |
| Regulator: | \( 47240.8489934 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 47240.8489934 \cdot 1}{2\cdot\sqrt{32793672683776}}\cr\approx \mathstrut & 2.60538788211 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 8T39):
| A solvable group of order 192 |
| The 13 conjugacy class representatives for $C_2^3:S_4$ |
| Character table for $C_2^3:S_4$ |
Intermediate fields
| 4.4.2863288.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.0.26021561344.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 2.1.4.8b1.2 | $x^{4} + 2 x^{2} + 4 x + 10$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ | |
|
\(71\)
| 71.2.4.6a1.2 | $x^{8} + 276 x^{7} + 28594 x^{6} + 1319832 x^{5} + 23067339 x^{4} + 9238824 x^{3} + 1401106 x^{2} + 94668 x + 2472$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *192 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.568.2t1.a.a | $1$ | $ 2^{3} \cdot 71 $ | \(\Q(\sqrt{142}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.568.3t2.a.a | $2$ | $ 2^{3} \cdot 71 $ | 3.3.568.1 | $S_3$ (as 3T2) | $1$ | $2$ | |
| 3.2272.4t5.a.a | $3$ | $ 2^{5} \cdot 71 $ | 4.0.2272.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.1290496.6t8.b.a | $3$ | $ 2^{8} \cdot 71^{2}$ | 4.0.11453152.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *192 | 3.2863288.4t5.a.a | $3$ | $ 2^{3} \cdot 71^{3}$ | 4.4.2863288.1 | $S_4$ (as 4T5) | $1$ | $3$ |
| 3.322624.6t8.a.a | $3$ | $ 2^{6} \cdot 71^{2}$ | 4.4.2863288.1 | $S_4$ (as 4T5) | $1$ | $3$ | |
| 3.11453152.4t5.a.a | $3$ | $ 2^{5} \cdot 71^{3}$ | 4.0.11453152.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.1290496.6t8.a.a | $3$ | $ 2^{8} \cdot 71^{2}$ | 4.0.2272.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *192 | 4.11453152.8t39.c.a | $4$ | $ 2^{5} \cdot 71^{3}$ | 8.4.32793672683776.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |
| 4.733001728.8t39.c.a | $4$ | $ 2^{11} \cdot 71^{3}$ | 8.4.32793672683776.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ | |
| 6.147...392.8t34.a.a | $6$ | $ 2^{13} \cdot 71^{5}$ | 8.0.8395180207046656.8 | $V_4^2:S_3$ (as 8T34) | $1$ | $-2$ | |
| 8.839...656.24t333.f.a | $8$ | $ 2^{16} \cdot 71^{6}$ | 8.4.32793672683776.1 | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.