Normalized defining polynomial
\( x^{8} - 3x^{7} + x^{6} + 11x^{5} - 31x^{4} + 28x^{3} + 24x^{2} - 56x + 16 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(1216265625\)
\(\medspace = 3^{4}\cdot 5^{6}\cdot 31^{2}\)
|
| |
| Root discriminant: | \(13.67\) |
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}31^{1/2}\approx 32.24548975496266$ | ||
| Ramified primes: |
\(3\), \(5\), \(31\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{6}-\frac{1}{12}a^{5}-\frac{5}{12}a^{4}+\frac{5}{12}a^{3}-\frac{1}{12}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{696}a^{7}-\frac{3}{232}a^{6}+\frac{57}{232}a^{5}+\frac{3}{8}a^{4}-\frac{89}{696}a^{3}-\frac{125}{348}a^{2}+\frac{2}{87}a-\frac{19}{87}$
| Monogenic: | Not computed | |
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{7}{116}a^{7}-\frac{11}{87}a^{6}-\frac{17}{174}a^{5}+\frac{2}{3}a^{4}-\frac{112}{87}a^{3}+\frac{173}{348}a^{2}+\frac{143}{58}a-\frac{218}{87}$, $\frac{19}{232}a^{7}-\frac{107}{696}a^{6}-\frac{55}{696}a^{5}+\frac{23}{24}a^{4}-\frac{1303}{696}a^{3}+\frac{77}{174}a^{2}+\frac{96}{29}a-\frac{358}{87}$, $\frac{17}{696}a^{7}+\frac{7}{232}a^{6}-\frac{17}{232}a^{5}+\frac{1}{8}a^{4}+\frac{53}{696}a^{3}-\frac{31}{87}a^{2}-\frac{19}{174}a+\frac{112}{87}$, $\frac{17}{696}a^{7}+\frac{7}{232}a^{6}-\frac{17}{232}a^{5}+\frac{1}{8}a^{4}+\frac{53}{696}a^{3}-\frac{31}{87}a^{2}+\frac{155}{174}a+\frac{25}{87}$, $\frac{5}{696}a^{7}-\frac{15}{232}a^{6}+\frac{53}{232}a^{5}-\frac{1}{8}a^{4}-\frac{445}{696}a^{3}+\frac{767}{348}a^{2}-\frac{251}{87}a+\frac{79}{87}$
|
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| Regulator: | \( 29.9069034439 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 29.9069034439 \cdot 1}{2\cdot\sqrt{1216265625}}\cr\approx \mathstrut & 0.270836352324 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T20):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_2^3: C_4$ |
| Character table for $C_2^3: C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Minimal sibling: | 8.0.46753250625.4 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.2.4.6a1.2 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |