Normalized defining polynomial
\( x^{8} - 3x^{7} + 64x^{6} - 192x^{5} + 1836x^{4} - 4635x^{3} + 39520x^{2} - 110850x + 565795 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(1686221298140625\)
\(\medspace = 3^{6}\cdot 5^{6}\cdot 23^{6}\)
|
| |
| Root discriminant: | \(80.05\) |
| |
| Galois root discriminant: | $3^{3/4}5^{3/4}23^{3/4}\approx 80.05055021202112$ | ||
| Ramified primes: |
\(3\), \(5\), \(23\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $Q_8$ |
| |
| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 8.0.1686221298140625.1$^{8}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4552}a^{6}+\frac{229}{4552}a^{5}+\frac{1045}{4552}a^{4}-\frac{775}{4552}a^{3}-\frac{2059}{4552}a^{2}+\frac{729}{2276}a-\frac{1943}{4552}$, $\frac{1}{285557347664}a^{7}+\frac{2961779}{71389336916}a^{6}-\frac{1113940305}{35694668458}a^{5}-\frac{33644557125}{71389336916}a^{4}-\frac{3078513299}{71389336916}a^{3}+\frac{103105931573}{285557347664}a^{2}+\frac{30215055775}{285557347664}a-\frac{72836877657}{285557347664}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{6}\times C_{6}$, which has order $72$ |
| |
| Narrow class group: | $C_{2}\times C_{6}\times C_{6}$, which has order $72$ |
| |
| Relative class number: | $72$ |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{601}{69920996}a^{7}+\frac{2709}{69920996}a^{6}+\frac{23787}{69920996}a^{5}+\frac{130509}{69920996}a^{4}+\frac{162427}{69920996}a^{3}+\frac{365709}{34960498}a^{2}+\frac{5822035}{69920996}a-\frac{13451789}{34960498}$, $\frac{9015}{125464564}a^{7}+\frac{40635}{125464564}a^{6}+\frac{725457}{125464564}a^{5}+\frac{114375}{125464564}a^{4}+\frac{13495965}{125464564}a^{3}-\frac{7417185}{62732282}a^{2}+\frac{140785065}{125464564}a-\frac{742343551}{62732282}$, $\frac{3985231}{35694668458}a^{7}+\frac{17963379}{35694668458}a^{6}-\frac{331715375}{35694668458}a^{5}+\frac{3312640039}{35694668458}a^{4}-\frac{13606355723}{35694668458}a^{3}+\frac{19555660399}{17847334229}a^{2}-\frac{32363896855}{35694668458}a+\frac{51429790346}{17847334229}$
|
| |
| Regulator: | \( 235.623843305 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 235.623843305 \cdot 72}{2\cdot\sqrt{1686221298140625}}\cr\approx \mathstrut & 0.321946660474 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{69}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{345}) \), \(\Q(\sqrt{5}, \sqrt{69})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.4.6a1.3 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(5\)
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(23\)
| 23.2.4.6a1.3 | $x^{8} + 84 x^{7} + 2666 x^{6} + 38304 x^{5} + 221091 x^{4} + 191520 x^{3} + 66650 x^{2} + 10546 x + 1039$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
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 | 1.69.2t1.a.a | $1$ | $ 3 \cdot 23 $ | \(\Q(\sqrt{69}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.345.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 23 $ | \(\Q(\sqrt{345}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *16 | 2.119025.8t5.b.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 23^{2}$ | 8.0.1686221298140625.1 | $Q_8$ (as 8T5) | $-1$ | $-2$ |
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 *.