Normalized defining polynomial
\( x^{4} - 34x^{2} + 316 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[0, 2]$ |
| |
| Discriminant: |
\(45504\)
\(\medspace = 2^{6}\cdot 3^{2}\cdot 79\)
|
| |
| Root discriminant: | \(14.61\) |
| |
| Galois root discriminant: | $2^{2}3^{1/2}79^{1/2}\approx 61.57921727336261$ | ||
| Ramified primes: |
\(2\), \(3\), \(79\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{79}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{6}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{3}-\frac{1}{3}a$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( \frac{1}{6} a^{2} - \frac{7}{3} \)
(order $6$)
|
| |
| Fundamental unit: |
$377a^{3}-\frac{8890}{3}a^{2}+5194a+\frac{9641}{3}$
|
| |
| Regulator: | \( 23.0103087205 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 23.0103087205 \cdot 1}{6\cdot\sqrt{45504}}\cr\approx \mathstrut & 0.709751214662 \end{aligned}\]
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: | 8.0.206763233181696.2 |
| Degree 4 sibling: | 4.2.4793088.2 |
| 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 | R | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.6a1.3 | $x^{4} + 6 x^{3} + 7 x^{2} + 6 x + 3$ | $2$ | $2$ | $6$ | $D_{4}$ | $$[2, 3]^{2}$$ |
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
|
\(79\)
| 79.1.2.1a1.1 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 79.2.1.0a1.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
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) | $1$ | $-1$ |
| 1.316.2t1.a.a | $1$ | $ 2^{2} \cdot 79 $ | \(\Q(\sqrt{79}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.948.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 79 $ | \(\Q(\sqrt{-237}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *8 | 2.15168.4t3.c.a | $2$ | $ 2^{6} \cdot 3 \cdot 79 $ | 4.0.45504.1 | $D_{4}$ (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 *.