Normalized defining polynomial
\( x^{8} - 2x^{7} + 3x^{6} - 4x^{5} + 3x^{4} - 4x^{3} + 3x^{2} - 2x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-4616192\)
\(\medspace = -\,2^{12}\cdot 7^{2}\cdot 23\)
|
| |
| Root discriminant: | \(6.81\) |
| |
| Galois root discriminant: | $2^{3/2}7^{1/2}23^{1/2}\approx 35.888716889852724$ | ||
| Ramified primes: |
\(2\), \(7\), \(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
| Monogenic: | Yes | |
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{7}-2a^{6}+3a^{5}-4a^{4}+3a^{3}-4a^{2}+3a-2$, $a^{7}-2a^{6}+3a^{5}-3a^{4}+2a^{3}-3a^{2}+2a-2$, $a^{7}-a^{6}+a^{5}-a^{4}-a^{3}-a^{2}$, $a^{7}-2a^{6}+2a^{5}-3a^{4}+2a^{3}-3a^{2}+3a$
|
| |
| Regulator: | \( 0.873136825319 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 0.873136825319 \cdot 1}{2\cdot\sqrt{4616192}}\cr\approx \mathstrut & 0.2016090272056 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.2.448.1 |
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 16 siblings: | data not computed |
| Degree 32 siblings: | 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }$ | R | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.2.12a1.9 | $x^{8} + 2 x^{5} + 6 x^{4} + x^{2} + 6 x + 7$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $$[3]^{4}$$ |
|
\(7\)
| 7.4.1.0a1.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 23.1.2.1a1.1 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *128 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *128 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.56.2t1.b.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{-14}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.184.2t1.b.a | $1$ | $ 2^{3} \cdot 23 $ | \(\Q(\sqrt{-46}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.1288.2t1.a.a | $1$ | $ 2^{3} \cdot 7 \cdot 23 $ | \(\Q(\sqrt{322}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.161.2t1.a.a | $1$ | $ 7 \cdot 23 $ | \(\Q(\sqrt{161}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.1288.4t3.d.a | $2$ | $ 2^{3} \cdot 7 \cdot 23 $ | 4.0.10304.1 | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| 2.9016.4t3.m.a | $2$ | $ 2^{3} \cdot 7^{2} \cdot 23 $ | 4.2.72128.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.184.4t3.c.a | $2$ | $ 2^{3} \cdot 23 $ | 4.2.1472.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| *128 | 2.56.4t3.c.a | $2$ | $ 2^{3} \cdot 7 $ | 4.2.448.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.29624.4t3.a.a | $2$ | $ 2^{3} \cdot 7 \cdot 23^{2}$ | 4.2.236992.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.1288.4t3.c.a | $2$ | $ 2^{3} \cdot 7 \cdot 23 $ | 4.4.10304.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
| *128 | 4.10304.8t35.a.a | $4$ | $ 2^{6} \cdot 7 \cdot 23 $ | 8.2.4616192.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $0$ |
| 4.13271552.8t29.e.a | $4$ | $ 2^{9} \cdot 7^{2} \cdot 23^{2}$ | 8.0.81288256.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
| 4.5450816.8t35.a.a | $4$ | $ 2^{6} \cdot 7 \cdot 23^{3}$ | 8.2.4616192.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $0$ | |
| 4.267089984.8t35.a.a | $4$ | $ 2^{6} \cdot 7^{3} \cdot 23^{3}$ | 8.2.4616192.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $0$ | |
| 4.504896.8t35.a.a | $4$ | $ 2^{6} \cdot 7^{3} \cdot 23 $ | 8.2.4616192.1 | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $0$ | |
| 4.207368.8t29.e.a | $4$ | $ 2^{3} \cdot 7^{2} \cdot 23^{2}$ | 8.0.81288256.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ |
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 *.