Normalized defining polynomial
\( x^{4} + 19x^{2} + 91 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[0, 2]$ |
| |
| Discriminant: |
\(13104\)
\(\medspace = 2^{4}\cdot 3^{2}\cdot 7\cdot 13\)
|
| |
| Root discriminant: | \(10.70\) |
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}7^{1/2}13^{1/2}\approx 46.73328578219169$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{91}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -a^{2} - 9 \)
(order $6$)
|
| |
| Fundamental unit: |
$12a^{3}-9a^{2}+111a-118$
|
| |
| Regulator: | \( 8.40429713417 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 8.40429713417 \cdot 2}{6\cdot\sqrt{13104}}\cr\approx \mathstrut & 0.966135273148 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 8.0.22751526260736.2 |
| Degree 4 sibling: | 4.2.1589952.4 |
| 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 | R | R | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a2.1 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 3$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(13\)
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
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 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.364.2t1.a.a | $1$ | $ 2^{2} \cdot 7 \cdot 13 $ | \(\Q(\sqrt{91}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.1092.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 7 \cdot 13 $ | \(\Q(\sqrt{-273}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *8 | 2.4368.4t3.d.a | $2$ | $ 2^{4} \cdot 3 \cdot 7 \cdot 13 $ | 4.0.13104.2 | $D_{4}$ (as 4T3) | $1$ | $0$ |
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 *.