Normalized defining polynomial
\( x^{4} - 2x^{3} - 12x^{2} + 10x + 53 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[0, 2]$ |
| |
| Discriminant: |
\(13120\)
\(\medspace = 2^{6}\cdot 5\cdot 41\)
|
| |
| Root discriminant: | \(10.70\) |
| |
| Galois root discriminant: | $2^{3/2}5^{1/2}41^{1/2}\approx 40.496913462633174$ | ||
| Ramified primes: |
\(2\), \(5\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{205}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-1}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{5}a^{3}-\frac{2}{5}a+\frac{1}{5}$
| Monogenic: | No | |
| 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: |
\( \frac{1}{5} a^{3} - \frac{7}{5} a - \frac{9}{5} \)
(order $4$)
|
| |
| Fundamental unit: |
$\frac{3}{5}a^{3}-3a^{2}+\frac{4}{5}a+\frac{43}{5}$
|
| |
| Regulator: | \( 5.9165496554 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 5.9165496554 \cdot 2}{4\cdot\sqrt{13120}}\cr\approx \mathstrut & 1.0196035462 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 8.0.7233948160000.2 |
| Degree 4 sibling: | 4.2.672400.2 |
| 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 | ${\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.6a1.1 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $$[2, 2]^{2}$$ |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(41\)
| 41.1.2.1a1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 41.2.1.0a1.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
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 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.205.2t1.a.a | $1$ | $ 5 \cdot 41 $ | \(\Q(\sqrt{205}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.820.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 41 $ | \(\Q(\sqrt{-205}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *8 | 2.3280.4t3.a.a | $2$ | $ 2^{4} \cdot 5 \cdot 41 $ | 4.0.13120.1 | $D_{4}$ (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 *.