Normalized defining polynomial
\( x^{10} + 20x^{8} + 120x^{6} + 225x^{4} + 90x^{2} + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-156250000000000\)
\(\medspace = -\,2^{10}\cdot 5^{16}\)
|
| |
| Root discriminant: | \(26.27\) |
| |
| Galois root discriminant: | $2\cdot 5^{8/5}\approx 26.26527804403767$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{10}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(100=2^{2}\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{100}(1,·)$, $\chi_{100}(71,·)$, $\chi_{100}(41,·)$, $\chi_{100}(11,·)$, $\chi_{100}(81,·)$, $\chi_{100}(51,·)$, $\chi_{100}(21,·)$, $\chi_{100}(91,·)$, $\chi_{100}(61,·)$, $\chi_{100}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-1}) \), 10.0.156250000000000.1$^{15}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}+\frac{1}{7}$, $\frac{1}{7}a^{7}+\frac{1}{7}a$, $\frac{1}{301}a^{8}+\frac{3}{301}a^{6}+\frac{16}{43}a^{4}+\frac{127}{301}a^{2}+\frac{38}{301}$, $\frac{1}{301}a^{9}+\frac{3}{301}a^{7}+\frac{16}{43}a^{5}+\frac{127}{301}a^{3}+\frac{38}{301}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{5}$, which has order $5$ |
| |
| Narrow class group: | $C_{5}$, which has order $5$ |
| |
| Relative class number: | $5$ |
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -\frac{25}{301} a^{9} - \frac{505}{301} a^{7} - \frac{443}{43} a^{5} - \frac{6185}{301} a^{3} - \frac{2885}{301} a \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{26}{301}a^{8}+\frac{508}{301}a^{6}+\frac{416}{43}a^{4}+\frac{4807}{301}a^{2}+\frac{816}{301}$, $\frac{5}{301}a^{8}+\frac{101}{301}a^{6}+\frac{80}{43}a^{4}+\frac{635}{301}a^{2}-\frac{25}{301}$, $\frac{8}{301}a^{8}+\frac{153}{301}a^{6}+\frac{128}{43}a^{4}+\frac{1919}{301}a^{2}+\frac{734}{301}$, $\frac{4}{301}a^{8}+\frac{14}{43}a^{6}+\frac{107}{43}a^{4}+\frac{1712}{301}a^{2}+\frac{34}{43}$
|
| |
| Regulator: | \( 257.113789169 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 257.113789169 \cdot 5}{4\cdot\sqrt{156250000000000}}\cr\approx \mathstrut & 0.251782018289 \end{aligned}\]
Galois group
| A cyclic group of order 10 |
| The 10 conjugacy class representatives for $C_{10}$ |
| Character table for $C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 5.5.390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{5}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.2.10a1.1 | $x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 5$ | $2$ | $5$ | $10$ | $C_{10}$ | $$[2]^{5}$$ |
|
\(5\)
| 5.1.5.8a4.1 | $x^{5} + 20 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $$[2]$$ |
| 5.1.5.8a4.1 | $x^{5} + 20 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $$[2]$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *10 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *10 | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *10 | 1.25.5t1.a.c | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| *10 | 1.100.10t1.a.d | $1$ | $ 2^{2} \cdot 5^{2}$ | 10.0.156250000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| *10 | 1.25.5t1.a.a | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| *10 | 1.100.10t1.a.b | $1$ | $ 2^{2} \cdot 5^{2}$ | 10.0.156250000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| *10 | 1.25.5t1.a.d | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| *10 | 1.100.10t1.a.c | $1$ | $ 2^{2} \cdot 5^{2}$ | 10.0.156250000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| *10 | 1.25.5t1.a.b | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
| *10 | 1.100.10t1.a.a | $1$ | $ 2^{2} \cdot 5^{2}$ | 10.0.156250000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
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 *.