Normalized defining polynomial
\( x^{4} + 3x^{2} - 62x + 258 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[0, 2]$ |
| |
| Discriminant: |
\(1491472\)
\(\medspace = 2^{4}\cdot 31^{2}\cdot 97\)
|
| |
| Root discriminant: | \(34.95\) |
| |
| Galois root discriminant: | $2\cdot 31^{1/2}97^{1/2}\approx 109.67223896684156$ | ||
| Ramified primes: |
\(2\), \(31\), \(97\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{97}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 4.0.1166716.1$^{2}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{18}a^{3}-\frac{1}{18}a^{2}+\frac{2}{9}a+\frac{1}{3}$
| Monogenic: | No | |
| Index: | $3$ | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{22}$, which has order $22$ |
| |
| Narrow class group: | $C_{22}$, which has order $22$ |
| |
| Relative class number: | $22$ |
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$\frac{91}{6}a^{3}-\frac{91}{6}a^{2}-\frac{637}{3}a+792$
|
| |
| Regulator: | \( 16.0392253724 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 16.0392253724 \cdot 22}{2\cdot\sqrt{1491472}}\cr\approx \mathstrut & 5.70332682757 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{31}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 8.0.20930214430310656.1 |
| Degree 4 sibling: | 4.0.1166716.1 |
| Minimal sibling: | 4.0.1166716.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.1.0.1}{1} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }$ | R | ${\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
|
\(31\)
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(97\)
| 97.1.2.1a1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 97.2.1.0a1.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
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) | $1$ | $1$ |
| 1.97.2t1.a.a | $1$ | $ 97 $ | \(\Q(\sqrt{97}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.12028.2t1.a.a | $1$ | $ 2^{2} \cdot 31 \cdot 97 $ | \(\Q(\sqrt{3007}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *8 | 2.12028.4t3.d.a | $2$ | $ 2^{2} \cdot 31 \cdot 97 $ | 4.0.1491472.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ |
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 *.