Normalized defining polynomial
\( x^{8} - 24x^{6} - 62x^{5} + 339x^{4} + 434x^{3} + 326x^{2} - 7130x + 8305 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(2364213760000\)
\(\medspace = 2^{12}\cdot 5^{4}\cdot 31^{4}\)
|
| |
| Root discriminant: | \(35.21\) |
| |
| Galois root discriminant: | $2^{3/2}5^{1/2}31^{1/2}\approx 35.21363372331802$ | ||
| Ramified primes: |
\(2\), \(5\), \(31\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $D_4$ |
| |
| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 4.0.12400.1$^{4}$, 4.0.307520.3$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{33}a^{5}+\frac{2}{11}a^{4}-\frac{13}{33}a^{3}+\frac{1}{3}a^{2}-\frac{5}{33}a+\frac{1}{3}$, $\frac{1}{165}a^{6}+\frac{1}{165}a^{5}+\frac{23}{165}a^{4}-\frac{23}{165}a^{3}+\frac{13}{55}a^{2}-\frac{2}{11}a-\frac{1}{3}$, $\frac{1}{23556885}a^{7}-\frac{20641}{7852295}a^{6}-\frac{58222}{7852295}a^{5}+\frac{1817194}{4711377}a^{4}+\frac{2277582}{7852295}a^{3}-\frac{8626931}{23556885}a^{2}+\frac{451100}{4711377}a+\frac{155927}{428307}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{4}\times C_{4}$, which has order $16$ |
| |
| Narrow class group: | $C_{4}\times C_{4}$, which has order $16$ |
| |
| Relative class number: | $16$ |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{916}{7852295}a^{7}+\frac{16244}{23556885}a^{6}+\frac{1442}{4711377}a^{5}-\frac{482081}{23556885}a^{4}-\frac{483162}{7852295}a^{3}+\frac{1128214}{23556885}a^{2}+\frac{2957722}{4711377}a-\frac{105142}{428307}$, $\frac{18430}{4711377}a^{7}+\frac{43237}{7852295}a^{6}-\frac{1979144}{23556885}a^{5}-\frac{8514827}{23556885}a^{4}+\frac{17402222}{23556885}a^{3}+\frac{62612194}{23556885}a^{2}+\frac{9884458}{1570459}a-\frac{2578981}{142769}$, $\frac{249886}{1570459}a^{7}-\frac{243789}{1570459}a^{6}-\frac{6530615}{1570459}a^{5}-\frac{8558641}{1570459}a^{4}+\frac{118614587}{1570459}a^{3}+\frac{22787583}{1570459}a^{2}-\frac{688734865}{1570459}a+\frac{66251637}{142769}$
|
| |
| Regulator: | \( 191.74010219 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 191.74010219 \cdot 16}{2\cdot\sqrt{2364213760000}}\cr\approx \mathstrut & 1.5548161556 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_4$ |
| Character table for $D_4$ |
Intermediate fields
| \(\Q(\sqrt{155}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{31}) \), \(\Q(\sqrt{5}, \sqrt{31})\), 4.0.12400.1 x2, 4.0.307520.3 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.0.307520.3, 4.0.12400.1 |
| Minimal sibling: | 4.0.12400.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.1.0.1}{1} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.12a1.1 | $x^{8} + 4 x^{7} + 12 x^{6} + 22 x^{5} + 31 x^{4} + 30 x^{3} + 22 x^{2} + 10 x + 5$ | $4$ | $2$ | $12$ | $D_4$ | $$[2, 2]^{2}$$ |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(31\)
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *8 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *8 | 1.124.2t1.a.a | $1$ | $ 2^{2} \cdot 31 $ | \(\Q(\sqrt{31}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.620.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 31 $ | \(\Q(\sqrt{155}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 2.2480.4t3.i.a | $2$ | $ 2^{4} \cdot 5 \cdot 31 $ | 8.0.2364213760000.6 | $D_4$ (as 8T4) | $1$ | $-2$ |
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 *.