Normalized defining polynomial
\( x^{8} - x^{7} + 2x^{6} + 2x^{5} - 5x^{4} + 13x^{3} - 13x^{2} + x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(56953125\)
\(\medspace = 3^{6}\cdot 5^{7}\)
|
| |
| Root discriminant: | \(9.32\) |
| |
| Galois root discriminant: | $3^{3/4}5^{7/8}\approx 9.320510388204081$ | ||
| Ramified primes: |
\(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{571}a^{7}+\frac{12}{571}a^{6}+\frac{158}{571}a^{5}-\frac{228}{571}a^{4}-\frac{114}{571}a^{3}+\frac{244}{571}a^{2}-\frac{267}{571}a-\frac{44}{571}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{47}{571}a^{7}-\frac{7}{571}a^{6}+\frac{3}{571}a^{5}+\frac{133}{571}a^{4}-\frac{219}{571}a^{3}+\frac{48}{571}a^{2}+\frac{13}{571}a-\frac{355}{571}$, $\frac{355}{571}a^{7}-\frac{308}{571}a^{6}+\frac{703}{571}a^{5}+\frac{713}{571}a^{4}-\frac{1642}{571}a^{3}+\frac{4396}{571}a^{2}-\frac{4567}{571}a+\frac{368}{571}$, $\frac{366}{571}a^{7}-\frac{176}{571}a^{6}+\frac{728}{571}a^{5}+\frac{1060}{571}a^{4}-\frac{1183}{571}a^{3}+\frac{4225}{571}a^{2}-\frac{2936}{571}a-\frac{116}{571}$, $\frac{51}{571}a^{7}+\frac{41}{571}a^{6}+\frac{64}{571}a^{5}+\frac{363}{571}a^{4}-\frac{104}{571}a^{3}+\frac{453}{571}a^{2}+\frac{658}{571}a-\frac{531}{571}$, $\frac{132}{571}a^{7}-\frac{129}{571}a^{6}+\frac{300}{571}a^{5}+\frac{167}{571}a^{4}-\frac{773}{571}a^{3}+\frac{1945}{571}a^{2}-\frac{2126}{571}a-\frac{98}{571}$
|
| |
| Regulator: | \( 5.73476282291 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 5.73476282291 \cdot 1}{2\cdot\sqrt{56953125}}\cr\approx \mathstrut & 0.239997327439 \end{aligned}\]
Galois group
$\OD_{16}$ (as 8T7):
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $C_8:C_2$ |
| Character table for $C_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 16.0.3243658447265625.1 |
| 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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.4.6a1.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 16$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $$[\ ]_{4}^{4}$$ |
|
\(5\)
| 5.1.8.7a1.4 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *16 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *16 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ | |
| *16 | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| *16 | 2.225.8t7.a.a | $2$ | $ 3^{2} \cdot 5^{2}$ | 8.4.56953125.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
| *16 | 2.225.8t7.a.b | $2$ | $ 3^{2} \cdot 5^{2}$ | 8.4.56953125.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
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 *.