Normalized defining polynomial
\( x^{20} - 7 x^{19} + 400 x^{18} - 879 x^{17} + 62865 x^{16} - 1013 x^{15} + 7876524 x^{14} + \cdots + 45\!\cdots\!60 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(0, 10)$ |
| |
| Discriminant: |
\(1956418642373887901494433553988783092785816063629150390625\)
\(\medspace = 5^{15}\cdot 13^{15}\cdot 47^{18}\)
|
| |
| Root discriminant: | \(732.10\) |
| |
| Galois root discriminant: | $5^{3/4}13^{3/4}47^{9/10}\approx 732.1045224212894$ | ||
| Ramified primes: |
\(5\), \(13\), \(47\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{65}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 4.0.606646625.2 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{15}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{10\cdots 44}a^{19}-\frac{22\cdots 71}{11\cdots 28}a^{18}-\frac{10\cdots 01}{54\cdots 72}a^{17}-\frac{67\cdots 45}{10\cdots 44}a^{16}+\frac{32\cdots 92}{13\cdots 43}a^{15}-\frac{20\cdots 53}{10\cdots 44}a^{14}+\frac{35\cdots 79}{10\cdots 44}a^{13}-\frac{10\cdots 57}{27\cdots 86}a^{12}-\frac{36\cdots 67}{10\cdots 44}a^{11}-\frac{34\cdots 35}{10\cdots 44}a^{10}-\frac{13\cdots 21}{78\cdots 96}a^{9}-\frac{59\cdots 01}{10\cdots 44}a^{8}-\frac{31\cdots 79}{10\cdots 44}a^{7}-\frac{21\cdots 95}{27\cdots 86}a^{6}+\frac{13\cdots 37}{15\cdots 92}a^{5}-\frac{34\cdots 57}{54\cdots 72}a^{4}-\frac{41\cdots 03}{13\cdots 43}a^{3}-\frac{44\cdots 47}{10\cdots 44}a^{2}+\frac{40\cdots 05}{27\cdots 86}a-\frac{13\cdots 75}{65\cdots 02}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
|
Unit group
| Rank: | $9$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: | not computed |
| |
| Regulator: | not computed |
|
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 = \mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot R \cdot h}{2\cdot\sqrt{1956418642373887901494433553988783092785816063629150390625}}\cr\mathstrut & \text{
Galois group
$C_4\times F_5$ (as 20T20):
| A solvable group of order 80 |
| The 20 conjugacy class representatives for $C_4\times F_5$ |
| Character table for $C_4\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{65}) \), 4.0.606646625.2, 5.1.103083261125.1, 10.2.690700317070720857265625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.4.15651349138991103211955468431910264742286528509033203125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.2 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.4.4.12a1.4 | $x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$ | $4$ | $4$ | $12$ | $C_4^2$ | $$[\ ]_{4}^{4}$$ | |
|
\(13\)
| 13.1.4.3a1.4 | $x^{4} + 104$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 13.4.4.12a1.4 | $x^{16} + 12 x^{14} + 48 x^{13} + 62 x^{12} + 432 x^{11} + 1044 x^{10} + 1584 x^{9} + 5505 x^{8} + 9936 x^{7} + 11592 x^{6} + 23904 x^{5} + 31352 x^{4} + 15552 x^{3} + 3552 x^{2} + 384 x + 29$ | $4$ | $4$ | $12$ | $C_4^2$ | $$[\ ]_{4}^{4}$$ | |
|
\(47\)
| 47.1.10.9a1.2 | $x^{10} + 235$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ |
| 47.1.10.9a1.2 | $x^{10} + 235$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ |