Normalized defining polynomial
\( x^{4} - x^{3} - 33x^{2} + 31x + 211 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[4, 0]$ |
| |
| Discriminant: |
\(90725\)
\(\medspace = 5^{2}\cdot 19\cdot 191\)
|
| |
| Root discriminant: | \(17.36\) |
| |
| Galois root discriminant: | $5^{1/2}19^{1/2}191^{1/2}\approx 134.70337783441067$ | ||
| Ramified primes: |
\(5\), \(19\), \(191\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{3629}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{17}a^{3}+\frac{5}{17}a^{2}-\frac{3}{17}a-\frac{4}{17}$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{17}a^{3}+\frac{5}{17}a^{2}-\frac{20}{17}a-\frac{89}{17}$, $\frac{9}{17}a^{3}+\frac{28}{17}a^{2}-\frac{180}{17}a-\frac{461}{17}$, $\frac{21}{17}a^{3}+\frac{71}{17}a^{2}-\frac{369}{17}a-\frac{951}{17}$
|
| |
| Regulator: | \( 10.1360083557 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{0}\cdot 10.1360083557 \cdot 1}{2\cdot\sqrt{90725}}\cr\approx \mathstrut & 0.269211407384 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 8 |
| Degree 4 sibling: | 4.4.65848205.1 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.1.2.1a1.2 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(191\)
| 191.1.2.1a1.2 | $x^{2} + 3629$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 191.2.1.0a1.1 | $x^{2} + 190 x + 19$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.3629.2t1.a.a | $1$ | $ 19 \cdot 191 $ | \(\Q(\sqrt{3629}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.18145.2t1.a.a | $1$ | $ 5 \cdot 19 \cdot 191 $ | \(\Q(\sqrt{18145}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 2.18145.4t3.c.a | $2$ | $ 5 \cdot 19 \cdot 191 $ | 4.4.90725.1 | $D_{4}$ (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 *.