Normalized defining polynomial
\( x^{8} - 2x^{7} + 5x^{6} - 4x^{5} + 20x^{4} + 4x^{3} + 22x^{2} + 40x + 14 \)
Invariants
Degree: | $8$ |
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Signature: | $[0, 4]$ |
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Discriminant: |
\(186624000000\)
\(\medspace = 2^{14}\cdot 3^{6}\cdot 5^{6}\)
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Root discriminant: | \(25.64\) |
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Galois root discriminant: | $2^{31/12}3^{43/36}5^{23/20}\approx 141.7004270627813$ | ||
Ramified primes: |
\(2\), \(3\), \(5\)
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Discriminant root field: | \(\Q\) | ||
$\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1210}a^{7}-\frac{1}{110}a^{6}+\frac{52}{605}a^{5}+\frac{27}{121}a^{4}+\frac{1}{121}a^{3}-\frac{43}{605}a^{2}-\frac{207}{605}a+\frac{68}{605}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Ideal class group: | Trivial group, which has order $1$ |
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Narrow class group: | Trivial group, which has order $1$ |
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Unit group
Rank: | $3$ |
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Torsion generator: |
\( -1 \)
(order $2$)
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Fundamental units: |
$\frac{2093}{1210}a^{7}-\frac{443}{110}a^{6}+\frac{5986}{605}a^{5}-\frac{1206}{121}a^{4}+\frac{4513}{121}a^{3}-\frac{2879}{605}a^{2}+\frac{22314}{605}a+\frac{35844}{605}$, $\frac{27}{242}a^{7}-\frac{5}{22}a^{6}-\frac{48}{121}a^{5}+\frac{136}{121}a^{4}-\frac{107}{121}a^{3}+\frac{49}{121}a^{2}+\frac{340}{121}a+\frac{142}{121}$, $\frac{714}{605}a^{7}-\frac{54}{55}a^{6}+\frac{446}{605}a^{5}-\frac{1132}{121}a^{4}-\frac{750}{121}a^{3}-\frac{10584}{605}a^{2}-\frac{16086}{605}a-\frac{6351}{605}$
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Regulator: | \( 1590.41162767 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1590.41162767 \cdot 1}{2\cdot\sqrt{186624000000}}\cr\approx \mathstrut & 2.86889909299 \end{aligned}\]
Galois group
A non-solvable group of order 20160 |
The 14 conjugacy class representatives for $A_8$ |
Character table for $A_8$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 15 siblings: | deg 15, deg 15 |
Degree 28 sibling: | deg 28 |
Degree 35 sibling: | deg 35 |
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 | R | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.1.2.3a1.1 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
2.1.6.11a1.15 | $x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $$[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$$ | |
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
3.1.6.6a1.1 | $x^{6} + 3 x + 3$ | $6$ | $1$ | $6$ | $C_3^2:D_4$ | $$[\frac{5}{4}, \frac{5}{4}]_{4}^{2}$$ | |
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
5.1.5.5a1.4 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |