Normalized defining polynomial
\( x^{8} - x^{7} + 8x^{6} - 9x^{5} + 29x^{4} - 23x^{3} + 50x^{2} - 25x + 51 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(23973872753\)
\(\medspace = 17^{3}\cdot 47^{4}\)
|
| |
| Root discriminant: | \(19.84\) |
| |
| Galois root discriminant: | $17^{1/2}47^{1/2}\approx 28.26658805020514$ | ||
| Ramified primes: |
\(17\), \(47\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-47}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{63}a^{7}-\frac{2}{21}a^{6}-\frac{4}{63}a^{5}-\frac{10}{63}a^{4}-\frac{26}{63}a^{3}-\frac{19}{63}a^{2}-\frac{2}{63}a-\frac{5}{21}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{5}$, which has order $5$ |
| |
| Narrow class group: | $C_{5}$, which has order $5$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{63}a^{7}+\frac{1}{7}a^{6}+\frac{13}{63}a^{5}+\frac{43}{63}a^{4}-\frac{10}{63}a^{3}+\frac{88}{63}a^{2}-\frac{25}{63}a+\frac{53}{21}$, $\frac{5}{63}a^{7}-\frac{1}{7}a^{6}+\frac{43}{63}a^{5}-\frac{71}{63}a^{4}+\frac{143}{63}a^{3}-\frac{137}{63}a^{2}+\frac{137}{63}a-\frac{4}{21}$, $\frac{5}{63}a^{7}-\frac{1}{7}a^{6}+\frac{43}{63}a^{5}-\frac{50}{63}a^{4}+\frac{143}{63}a^{3}-\frac{53}{63}a^{2}+\frac{95}{63}a+\frac{38}{21}$
|
| |
| Regulator: | \( 39.0059184349 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 39.0059184349 \cdot 5}{2\cdot\sqrt{23973872753}}\cr\approx \mathstrut & 0.981569132061 \end{aligned}\]
Galois group
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $D_{8}$ |
| Character table for $D_{8}$ |
Intermediate fields
| \(\Q(\sqrt{-47}) \), 4.0.37553.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 16.0.166101760110563345913601.2 |
| Degree 8 sibling: | 8.2.8671400783.1 |
| Minimal sibling: | 8.2.8671400783.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.1.2.1a1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(47\)
| 47.1.2.1a1.1 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 47.1.2.1a1.1 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 47.1.2.1a1.1 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 47.1.2.1a1.1 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *16 | 1.47.2t1.a.a | $1$ | $ 47 $ | \(\Q(\sqrt{-47}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.799.2t1.a.a | $1$ | $ 17 \cdot 47 $ | \(\Q(\sqrt{-799}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 2.799.4t3.a.a | $2$ | $ 17 \cdot 47 $ | 4.2.13583.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| *16 | 2.799.8t6.b.a | $2$ | $ 17 \cdot 47 $ | 8.0.23973872753.1 | $D_{8}$ (as 8T6) | $1$ | $0$ |
| *16 | 2.799.8t6.b.b | $2$ | $ 17 \cdot 47 $ | 8.0.23973872753.1 | $D_{8}$ (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 *.