Normalized defining polynomial
\( x^{8} - 3x^{6} + 9x^{4} - 4x^{2} + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(20025646144\)
\(\medspace = 2^{6}\cdot 7^{4}\cdot 19^{4}\)
|
| |
| Root discriminant: | \(19.40\) |
| |
| Galois root discriminant: | $2^{3/2}7^{1/2}19^{2/3}\approx 53.28395024995941$ | ||
| Ramified primes: |
\(2\), \(7\), \(19\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{7}-3a^{5}+9a^{3}-4a$, $\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-a^{5}-a^{4}+\frac{7}{2}a^{3}+\frac{7}{2}a^{2}+\frac{3}{2}a+\frac{1}{2}$, $\frac{3}{2}a^{7}+\frac{1}{2}a^{6}-4a^{5}-2a^{4}+\frac{25}{2}a^{3}+\frac{11}{2}a^{2}-\frac{7}{2}a-\frac{5}{2}$
|
| |
| Regulator: | \( 25.7973960594 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 25.7973960594 \cdot 4}{2\cdot\sqrt{20025646144}}\cr\approx \mathstrut & 0.568240353031 \end{aligned}\]
Galois group
| A solvable group of order 96 |
| The 11 conjugacy class representatives for $((C_2 \times D_4): C_2):C_3$ |
| Character table for $((C_2 \times D_4): C_2):C_3$ |
Intermediate fields
| 4.0.17689.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.3.2.6a3.2 | $x^{6} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 6 x + 5$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $$[2, 2]^{6}$$ | |
|
\(7\)
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *96 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 3.1132096.4t4.a.a | $3$ | $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ | 4.4.1132096.2 | $A_4$ (as 4T4) | $1$ | $3$ | |
| 3.1132096.4t4.c.a | $3$ | $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ | 4.0.1132096.1 | $A_4$ (as 4T4) | $1$ | $-1$ | |
| 3.23104.4t4.b.a | $3$ | $ 2^{6} \cdot 19^{2}$ | 4.0.23104.1 | $A_4$ (as 4T4) | $1$ | $-1$ | |
| 3.1132096.4t4.f.a | $3$ | $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ | 4.0.1132096.4 | $A_4$ (as 4T4) | $1$ | $-1$ | |
| *96 | 3.17689.4t4.b.a | $3$ | $ 7^{2} \cdot 19^{2}$ | 4.0.17689.1 | $A_4$ (as 4T4) | $1$ | $-1$ |
| *96 | 4.1132096.8t32.a.a | $4$ | $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ | 8.0.20025646144.1 | $((C_2 \times D_4): C_2):C_3$ (as 8T32) | $1$ | $0$ |
| 4.21509824.24t97.a.a | $4$ | $ 2^{6} \cdot 7^{2} \cdot 19^{3}$ | 8.0.20025646144.1 | $((C_2 \times D_4): C_2):C_3$ (as 8T32) | $0$ | $0$ | |
| 4.21509824.24t97.a.b | $4$ | $ 2^{6} \cdot 7^{2} \cdot 19^{3}$ | 8.0.20025646144.1 | $((C_2 \times D_4): C_2):C_3$ (as 8T32) | $0$ | $0$ |
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 *.