Normalized defining polynomial
\( x^{8} + 16x^{6} + 94x^{4} + 240x^{2} + 252 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(116530348032\)
\(\medspace = 2^{22}\cdot 3^{4}\cdot 7^{3}\)
|
| |
| Root discriminant: | \(24.17\) |
| |
| Galois root discriminant: | $2^{3}3^{1/2}7^{1/2}\approx 36.66060555964672$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{7}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{6}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{18}a^{7}+\frac{1}{18}a^{5}+\frac{2}{9}a^{3}-\frac{1}{3}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
\( -\frac{1}{6} a^{4} - \frac{4}{3} a^{2} - 2 \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{11}{18}a^{7}-\frac{5}{2}a^{6}+\frac{94}{9}a^{5}-\frac{104}{3}a^{4}+\frac{550}{9}a^{3}-\frac{460}{3}a^{2}+\frac{370}{3}a-193$, $\frac{11}{18}a^{7}+\frac{5}{2}a^{6}+\frac{94}{9}a^{5}+\frac{104}{3}a^{4}+\frac{550}{9}a^{3}+\frac{460}{3}a^{2}+\frac{370}{3}a+193$, $\frac{19}{3}a^{7}+17a^{6}+\frac{301}{3}a^{5}+226a^{4}+\frac{1618}{3}a^{3}+946a^{2}+968a+1051$
|
| |
| Regulator: | \( 2153.21165776 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2153.21165776 \cdot 1}{6\cdot\sqrt{116530348032}}\cr\approx \mathstrut & 1.63845841704 \end{aligned}\]
Galois group
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.4032.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 16 |
| Degree 8 sibling: | 8.2.1087616581632.6 |
| 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 | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.76 | $x^{8} + 12 x^{7} + 38 x^{6} + 80 x^{5} + 111 x^{4} + 120 x^{3} + 86 x^{2} + 44 x + 11$ | $4$ | $2$ | $22$ | $D_{8}$ | $$[2, 3, 4]^{2}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(7\)
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *16 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.28.2t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\sqrt{7}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.84.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 7 $ | \(\Q(\sqrt{-21}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 2.1344.4t3.f.a | $2$ | $ 2^{6} \cdot 3 \cdot 7 $ | 4.0.4032.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| *16 | 2.5376.8t6.d.a | $2$ | $ 2^{8} \cdot 3 \cdot 7 $ | 8.0.116530348032.6 | $D_{8}$ (as 8T6) | $1$ | $0$ |
| *16 | 2.5376.8t6.d.b | $2$ | $ 2^{8} \cdot 3 \cdot 7 $ | 8.0.116530348032.6 | $D_{8}$ (as 8T6) | $1$ | $0$ |
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 *.