Normalized defining polynomial
\( x^{8} - 4x^{7} + 11x^{6} - 14x^{5} - 24x^{4} + 36x^{3} + 10x^{2} + 8 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(174478792704\)
\(\medspace = 2^{11}\cdot 3\cdot 73^{4}\)
|
| |
| Root discriminant: | \(25.42\) |
| |
| Galois root discriminant: | $2^{19/8}3^{1/2}73^{1/2}\approx 76.7658913717416$ | ||
| Ramified primes: |
\(2\), \(3\), \(73\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{6}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2492}a^{7}-\frac{5}{89}a^{6}+\frac{361}{2492}a^{5}-\frac{129}{623}a^{4}+\frac{94}{623}a^{3}+\frac{44}{89}a^{2}-\frac{289}{1246}a-\frac{284}{623}$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{823}{2492}a^{7}-\frac{110}{89}a^{6}+\frac{8031}{2492}a^{5}-\frac{2126}{623}a^{4}-\frac{6120}{623}a^{3}+\frac{968}{89}a^{2}+\frac{10107}{1246}a-\frac{730}{623}$, $\frac{53}{623}a^{7}-\frac{73}{178}a^{6}+\frac{1509}{1246}a^{5}-\frac{1182}{623}a^{4}-\frac{631}{623}a^{3}+\frac{339}{89}a^{2}-\frac{730}{623}a+\frac{223}{623}$, $\frac{32}{623}a^{7}-\frac{17}{89}a^{6}+\frac{338}{623}a^{5}-\frac{314}{623}a^{4}-\frac{1051}{623}a^{3}+\frac{292}{89}a^{2}-\frac{1052}{623}a+\frac{405}{623}$, $\frac{30}{623}a^{7}-\frac{43}{178}a^{6}+\frac{1101}{1246}a^{5}-\frac{1151}{623}a^{4}+\frac{689}{623}a^{3}+\frac{29}{89}a^{2}-\frac{519}{623}a+\frac{185}{623}$, $\frac{1613}{2492}a^{7}-\frac{144}{89}a^{6}+\frac{11625}{2492}a^{5}-\frac{1241}{623}a^{4}-\frac{11604}{623}a^{3}-\frac{495}{89}a^{2}-\frac{2645}{1246}a-\frac{2056}{623}$
|
| |
| Regulator: | \( 4370.22085972 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 4370.22085972 \cdot 1}{2\cdot\sqrt{174478792704}}\cr\approx \mathstrut & 3.30431761532 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
| A solvable group of order 1152 |
| The 20 conjugacy class representatives for $S_4\wr C_2$ |
| Character table for $S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{73}) \) |
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 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 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 | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.1.4.6a2.1 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $$[2, 2]^{3}$$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.3.1.0a1.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(73\)
| 73.4.2.4a1.2 | $x^{8} + 32 x^{6} + 112 x^{5} + 266 x^{4} + 1792 x^{3} + 3296 x^{2} + 560 x + 98$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *1152 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.1752.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 73 $ | \(\Q(\sqrt{438}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *1152 | 1.73.2t1.a.a | $1$ | $ 73 $ | \(\Q(\sqrt{73}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 2.1752.4t3.d.a | $2$ | $ 2^{3} \cdot 3 \cdot 73 $ | 4.0.42048.4 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| 4.42048.6t13.b.a | $4$ | $ 2^{6} \cdot 3^{2} \cdot 73 $ | 6.2.1009152.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.73668096.12t34.b.a | $4$ | $ 2^{9} \cdot 3^{3} \cdot 73^{2}$ | 6.2.1009152.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.224073792.12t34.b.a | $4$ | $ 2^{6} \cdot 3^{2} \cdot 73^{3}$ | 6.2.1009152.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 4.127896.6t13.b.a | $4$ | $ 2^{3} \cdot 3 \cdot 73^{2}$ | 6.2.1009152.2 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
| 6.516266016768.12t201.a.a | $6$ | $ 2^{14} \cdot 3^{4} \cdot 73^{3}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 6.123...432.12t202.a.a | $6$ | $ 2^{17} \cdot 3^{5} \cdot 73^{3}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
| *1152 | 6.2390120448.8t47.a.a | $6$ | $ 2^{11} \cdot 3 \cdot 73^{3}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ |
| 6.57362890752.12t200.a.a | $6$ | $ 2^{14} \cdot 3^{2} \cdot 73^{3}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
| 9.220...768.16t1294.a.a | $9$ | $ 2^{21} \cdot 3^{3} \cdot 73^{3}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.475...888.18t272.a.a | $9$ | $ 2^{24} \cdot 3^{6} \cdot 73^{3}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.185...096.18t273.a.a | $9$ | $ 2^{24} \cdot 3^{6} \cdot 73^{6}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 9.856...056.18t274.a.a | $9$ | $ 2^{21} \cdot 3^{3} \cdot 73^{6}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
| 12.710...864.36t1763.a.a | $12$ | $ 2^{31} \cdot 3^{7} \cdot 73^{6}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
| 12.123...064.24t2821.a.a | $12$ | $ 2^{25} \cdot 3^{5} \cdot 73^{6}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
| 18.407...728.36t1758.a.a | $18$ | $ 2^{45} \cdot 3^{9} \cdot 73^{9}$ | 8.4.174478792704.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ |
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 *.