Normalized defining polynomial
\( x^{10} + 10x^{8} - 10x^{7} + 20x^{6} - 46x^{5} + 50x^{4} + 130x^{3} - 85x^{2} - 90x + 4 \)
Invariants
| Degree: | $10$ |
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| Signature: | $[4, 3]$ |
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| Discriminant: |
\(-45562500000000\)
\(\medspace = -\,2^{8}\cdot 3^{6}\cdot 5^{12}\)
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| Root discriminant: | \(23.22\) |
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| Galois root discriminant: | $2^{3/2}3^{25/18}5^{271/200}\approx 115.16426990932663$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{6}-\frac{1}{4}a^{3}+\frac{3}{8}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{7}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{8}+\frac{1}{32}a^{6}-\frac{1}{16}a^{5}+\frac{3}{32}a^{4}+\frac{1}{8}a^{3}+\frac{7}{32}a^{2}+\frac{3}{16}a+\frac{1}{8}$, $\frac{1}{7904}a^{9}-\frac{69}{7904}a^{8}+\frac{25}{608}a^{7}+\frac{289}{7904}a^{6}+\frac{333}{7904}a^{5}+\frac{129}{608}a^{4}-\frac{561}{7904}a^{3}+\frac{307}{7904}a^{2}-\frac{1}{304}a+\frac{179}{1976}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $6$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{25}{416}a^{9}-\frac{11}{208}a^{8}+\frac{21}{32}a^{7}-\frac{121}{104}a^{6}+\frac{967}{416}a^{5}-\frac{75}{16}a^{4}+\frac{3031}{416}a^{3}+\frac{77}{52}a^{2}-\frac{27}{4}a+\frac{47}{52}$, $\frac{5}{3952}a^{9}+\frac{51}{7904}a^{8}+\frac{11}{304}a^{7}+\frac{667}{7904}a^{6}+\frac{231}{988}a^{5}+\frac{93}{608}a^{4}+\frac{1147}{3952}a^{3}-\frac{2611}{7904}a^{2}+\frac{237}{304}a+\frac{1543}{1976}$, $\frac{363}{1976}a^{9}-\frac{1141}{7904}a^{8}+\frac{297}{152}a^{7}-\frac{26701}{7904}a^{6}+\frac{25139}{3952}a^{5}-\frac{8259}{608}a^{4}+\frac{9913}{494}a^{3}+\frac{62173}{7904}a^{2}-\frac{6259}{304}a-\frac{677}{1976}$, $\frac{47}{7904}a^{9}-\frac{1}{247}a^{8}+\frac{35}{608}a^{7}-\frac{495}{3952}a^{6}+\frac{1325}{7904}a^{5}-\frac{85}{152}a^{4}+\frac{5249}{7904}a^{3}+\frac{669}{3952}a^{2}-\frac{185}{152}a-\frac{29}{247}$, $\frac{675}{1976}a^{9}-\frac{1791}{7904}a^{8}+\frac{277}{76}a^{7}-\frac{45915}{7904}a^{6}+\frac{45705}{3952}a^{5}-\frac{14505}{608}a^{4}+\frac{69877}{1976}a^{3}+\frac{141499}{7904}a^{2}-\frac{10661}{304}a+\frac{2885}{1976}$, $\frac{1741}{1976}a^{9}-\frac{2891}{3952}a^{8}+\frac{180}{19}a^{7}-\frac{65929}{3952}a^{6}+\frac{15819}{494}a^{5}-\frac{20597}{304}a^{4}+\frac{100621}{988}a^{3}+\frac{107897}{3952}a^{2}-\frac{14179}{152}a+\frac{1579}{988}$
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| Regulator: | \( 2282.15038065 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 2282.15038065 \cdot 1}{2\cdot\sqrt{45562500000000}}\cr\approx \mathstrut & 0.670918979589 \end{aligned}\]
Galois group
$S_5\wr C_2$ (as 10T43):
| A non-solvable group of order 28800 |
| The 35 conjugacy class representatives for $S_5^2 \wr C_2$ |
| Character table for $S_5^2 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 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 | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.2.4a1.2 | $x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$ | $2$ | $2$ | $4$ | $C_4$ | $$[2]^{2}$$ | |
| 2.2.2.4a2.2 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ | |
|
\(3\)
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 3.2.3.6a5.1 | $x^{6} + 6 x^{5} + 18 x^{4} + 35 x^{3} + 42 x^{2} + 30 x + 11$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$ | |
|
\(5\)
| 5.1.10.12a1.1 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $$[\frac{11}{8}, \frac{11}{8}]_{8}^{2}$$ |