Normalized defining polynomial
\( x^{8} - 2x^{7} - 4x^{6} + 14x^{5} - 4x^{4} - 22x^{3} + 20x^{2} - 6x + 7 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(454201344\)
\(\medspace = 2^{12}\cdot 3^{4}\cdot 37^{2}\)
|
| |
| Root discriminant: | \(12.08\) |
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}37^{2/3}\approx 54.39681060743917$ | ||
| Ramified primes: |
\(2\), \(3\), \(37\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}-\frac{1}{6}a^{3}-\frac{1}{6}a-\frac{1}{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $3$ |
| |
| Torsion generator: |
\( -\frac{1}{6} a^{7} + \frac{7}{6} a^{5} - \frac{3}{2} a^{4} - \frac{11}{6} a^{3} + 4 a^{2} - \frac{11}{6} a + \frac{11}{6} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{6}a^{7}-\frac{2}{3}a^{5}+a^{4}+\frac{5}{6}a^{3}-2a^{2}+\frac{1}{3}a-\frac{4}{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-2a^{5}+\frac{9}{2}a^{4}+\frac{1}{2}a^{3}-\frac{13}{2}a^{2}+5a-\frac{9}{2}$, $\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}+\frac{5}{2}a^{4}-\frac{13}{6}a^{3}-\frac{5}{2}a^{2}+\frac{17}{6}a+\frac{1}{6}$
|
| |
| Regulator: | \( 41.9593457515 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 41.9593457515 \cdot 1}{6\cdot\sqrt{454201344}}\cr\approx \mathstrut & 0.511414130385 \end{aligned}\]
Galois group
$A_4\wr C_2$ (as 8T42):
| A solvable group of order 288 |
| The 14 conjugacy class representatives for $A_4\wr C_2$ |
| Character table for $A_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
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 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 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 | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.12a3.1 | $x^{8} + 4 x^{7} + 12 x^{6} + 22 x^{5} + 33 x^{4} + 34 x^{3} + 28 x^{2} + 14 x + 7$ | $4$ | $2$ | $12$ | $A_4\times C_2$ | $$[2, 2]^{6}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 37.1.3.2a1.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *288 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *288 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.111.6t1.b.a | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.37.3t1.a.a | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.111.6t1.b.b | $1$ | $ 3 \cdot 37 $ | 6.0.50602347.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.37.3t1.a.b | $1$ | $ 37 $ | 3.3.1369.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 2.4107.3t2.a.a | $2$ | $ 3 \cdot 37^{2}$ | 3.1.4107.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.111.6t5.a.a | $2$ | $ 3 \cdot 37 $ | 6.0.36963.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| 2.111.6t5.a.b | $2$ | $ 3 \cdot 37 $ | 6.0.36963.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| *288 | 6.151400448.8t42.a.a | $6$ | $ 2^{12} \cdot 3^{3} \cdot 37^{2}$ | 8.0.454201344.2 | $A_4\wr C_2$ (as 8T42) | $1$ | $0$ |
| 6.766...544.24t702.a.a | $6$ | $ 2^{12} \cdot 3^{3} \cdot 37^{5}$ | 8.0.454201344.2 | $A_4\wr C_2$ (as 8T42) | $0$ | $0$ | |
| 6.766...544.24t702.a.b | $6$ | $ 2^{12} \cdot 3^{3} \cdot 37^{5}$ | 8.0.454201344.2 | $A_4\wr C_2$ (as 8T42) | $0$ | $0$ | |
| 9.283...128.12t128.a.a | $9$ | $ 2^{12} \cdot 3^{3} \cdot 37^{6}$ | 8.0.454201344.2 | $A_4\wr C_2$ (as 8T42) | $1$ | $3$ | |
| 9.766...456.18t112.a.a | $9$ | $ 2^{12} \cdot 3^{6} \cdot 37^{6}$ | 8.0.454201344.2 | $A_4\wr C_2$ (as 8T42) | $1$ | $-3$ |
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 *.