Normalized defining polynomial
\( x^{10} - 3x^{9} + 14x^{8} - 32x^{7} + 70x^{6} - 118x^{5} + 166x^{4} - 166x^{3} + 108x^{2} - 36x + 4 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(2, 4)$ |
| |
| Discriminant: |
\(47762759097344\)
\(\medspace = 2^{10}\cdot 19^{4}\cdot 71^{3}\)
|
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| Root discriminant: | \(23.33\) |
| |
| Galois root discriminant: | $2^{4/3}19^{1/2}71^{1/2}\approx 92.55061359288983$ | ||
| Ramified primes: |
\(2\), \(19\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{71}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{1754}a^{9}+\frac{213}{1754}a^{8}+\frac{209}{877}a^{7}+\frac{401}{877}a^{6}-\frac{172}{877}a^{5}-\frac{377}{877}a^{4}+\frac{212}{877}a^{3}+\frac{105}{877}a^{2}-\frac{68}{877}a+\frac{203}{877}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{155}{877}a^{9}-\frac{311}{877}a^{8}+\frac{1646}{877}a^{7}-\frac{2855}{877}a^{6}+\frac{5439}{877}a^{5}-\frac{8122}{877}a^{4}+\frac{6961}{877}a^{3}-\frac{2530}{877}a^{2}-\frac{6171}{877}a+\frac{4171}{877}$, $\frac{1981}{1754}a^{9}-\frac{2573}{877}a^{8}+\frac{25603}{1754}a^{7}-\frac{26491}{877}a^{6}+\frac{58303}{877}a^{5}-\frac{92595}{877}a^{4}+\frac{125300}{877}a^{3}-\frac{110346}{877}a^{2}+\frac{58232}{877}a-\frac{8293}{877}$, $\frac{154}{877}a^{9}-\frac{524}{877}a^{8}+\frac{2105}{877}a^{7}-\frac{5411}{877}a^{6}+\frac{10168}{877}a^{5}-\frac{18769}{877}a^{4}+\frac{23200}{877}a^{3}-\frac{22911}{877}a^{2}+\frac{10628}{877}a-\frac{1497}{877}$, $\frac{815}{877}a^{9}-\frac{1805}{877}a^{8}+\frac{10041}{877}a^{7}-\frac{18152}{877}a^{6}+\frac{43253}{877}a^{5}-\frac{62000}{877}a^{4}+\frac{87722}{877}a^{3}-\frac{67394}{877}a^{2}+\frac{34742}{877}a-\frac{5001}{877}$, $\frac{156}{877}a^{9}+\frac{681}{1754}a^{8}+\frac{1497}{1754}a^{7}+\frac{3209}{877}a^{6}-\frac{2798}{877}a^{5}+\frac{8664}{877}a^{4}-\frac{18048}{877}a^{3}+\frac{19605}{877}a^{2}-\frac{8061}{877}a+\frac{1069}{877}$
|
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| Regulator: | \( 4832.21327999 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4832.21327999 \cdot 1}{2\cdot\sqrt{47762759097344}}\cr\approx \mathstrut & 2.17946982398 \end{aligned}\]
Galois group
| A non-solvable group of order 120 |
| The 7 conjugacy class representatives for $S_5$ |
| Character table for $S_5$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 5 sibling: | 5.1.410096.1 |
| Degree 6 sibling: | 6.2.33076702976.2 |
| Degree 10 sibling: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 15 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | 5.1.410096.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 2.1.3.2a1.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 2.1.6.8a1.2 | $x^{6} + 2 x^{3} + 6$ | $6$ | $1$ | $8$ | $D_{6}$ | $$[2]_{3}^{2}$$ | |
|
\(19\)
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 19.2.2.2a1.1 | $x^{4} + 36 x^{3} + 328 x^{2} + 91 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
| 19.2.2.2a1.1 | $x^{4} + 36 x^{3} + 328 x^{2} + 91 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 71.3.1.0a1.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 71.3.2.3a1.1 | $x^{6} + 8 x^{4} + 128 x^{3} + 16 x^{2} + 583 x + 4096$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |