Normalized defining polynomial
\( x^{24} - x^{23} + 26 x^{22} + 4324 x^{21} - 36164 x^{20} + 228200 x^{19} + 2423260 x^{18} + \cdots - 526693993977600 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
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| Discriminant: |
\(77516311234764156989663655746389565161872035940177738666534423828125\)
\(\medspace = 5^{33}\cdot 109^{22}\)
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| |
| Root discriminant: | \(674.10\) |
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| Galois root discriminant: | $5^{31/20}109^{19/20}\approx 1044.6208257588585$ | ||
| Ramified primes: |
\(5\), \(109\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}$, $\frac{1}{545}a^{14}+\frac{31}{545}a^{13}+\frac{34}{545}a^{12}+\frac{69}{545}a^{11}+\frac{25}{109}a^{10}-\frac{33}{109}a^{9}+\frac{23}{109}a^{8}-\frac{33}{109}a^{7}+\frac{42}{109}a^{6}-\frac{4}{109}a^{5}+\frac{38}{109}a^{4}+\frac{3}{109}a^{3}+\frac{12}{109}a^{2}-\frac{43}{109}a+\frac{3}{109}$, $\frac{1}{545}a^{15}+\frac{54}{545}a^{13}-\frac{4}{545}a^{12}-\frac{54}{109}a^{11}-\frac{116}{545}a^{10}-\frac{44}{109}a^{9}+\frac{17}{109}a^{8}-\frac{25}{109}a^{7}+\frac{2}{109}a^{6}+\frac{53}{109}a^{5}+\frac{24}{109}a^{4}+\frac{28}{109}a^{3}+\frac{21}{109}a^{2}+\frac{28}{109}a+\frac{16}{109}$, $\frac{1}{545}a^{16}-\frac{43}{545}a^{13}-\frac{7}{109}a^{12}-\frac{49}{109}a^{11}+\frac{6}{545}a^{10}-\frac{54}{109}a^{9}+\frac{41}{109}a^{8}+\frac{40}{109}a^{7}-\frac{35}{109}a^{6}+\frac{22}{109}a^{5}+\frac{47}{109}a^{4}-\frac{32}{109}a^{3}+\frac{34}{109}a^{2}+\frac{49}{109}a-\frac{53}{109}$, $\frac{1}{3270}a^{17}-\frac{1}{3270}a^{16}+\frac{1}{1635}a^{14}+\frac{34}{545}a^{13}+\frac{23}{327}a^{12}-\frac{611}{1635}a^{11}-\frac{253}{545}a^{10}-\frac{41}{327}a^{9}-\frac{28}{327}a^{8}-\frac{17}{327}a^{7}-\frac{62}{327}a^{6}-\frac{44}{109}a^{5}-\frac{2}{327}a^{4}-\frac{21}{109}a^{3}-\frac{104}{327}a^{2}+\frac{143}{654}a-\frac{5}{109}$, $\frac{1}{6540}a^{18}-\frac{1}{6540}a^{17}-\frac{1}{1090}a^{16}-\frac{1}{1635}a^{15}+\frac{19}{327}a^{13}-\frac{19}{1635}a^{12}+\frac{266}{545}a^{11}+\frac{1109}{3270}a^{10}+\frac{35}{654}a^{9}+\frac{79}{654}a^{8}+\frac{158}{327}a^{7}+\frac{87}{218}a^{6}+\frac{181}{654}a^{5}+\frac{33}{218}a^{4}-\frac{71}{654}a^{3}-\frac{19}{1308}a^{2}-\frac{37}{218}a+\frac{22}{109}$, $\frac{1}{13080}a^{19}-\frac{1}{13080}a^{18}-\frac{1}{6540}a^{17}-\frac{1}{1635}a^{16}-\frac{1}{1090}a^{15}-\frac{1}{1635}a^{14}-\frac{103}{3270}a^{13}+\frac{29}{327}a^{12}-\frac{317}{2180}a^{11}-\frac{151}{1308}a^{10}-\frac{481}{1308}a^{9}+\frac{21}{218}a^{8}+\frac{337}{1308}a^{7}-\frac{547}{1308}a^{6}-\frac{203}{436}a^{5}-\frac{553}{1308}a^{4}-\frac{739}{2616}a^{3}-\frac{413}{1308}a^{2}+\frac{131}{654}a-\frac{8}{109}$, $\frac{1}{26160}a^{20}-\frac{1}{26160}a^{19}-\frac{1}{13080}a^{18}+\frac{1}{6540}a^{16}-\frac{1}{3270}a^{15}+\frac{1}{2180}a^{14}-\frac{161}{3270}a^{13}+\frac{1253}{13080}a^{12}-\frac{349}{4360}a^{11}+\frac{1399}{13080}a^{10}+\frac{133}{1308}a^{9}-\frac{1243}{2616}a^{8}+\frac{913}{2616}a^{7}-\frac{805}{2616}a^{6}-\frac{853}{2616}a^{5}+\frac{55}{1744}a^{4}+\frac{619}{2616}a^{3}-\frac{55}{436}a^{2}+\frac{35}{654}a-\frac{6}{109}$, $\frac{1}{261600}a^{21}+\frac{1}{87200}a^{20}-\frac{1}{130800}a^{19}-\frac{1}{65400}a^{18}+\frac{1}{13080}a^{17}-\frac{1}{1308}a^{16}-\frac{1}{2616}a^{15}+\frac{1}{6540}a^{14}-\frac{141}{8720}a^{13}+\frac{233}{26160}a^{12}+\frac{413}{1744}a^{11}-\frac{1751}{13080}a^{10}-\frac{3557}{8720}a^{9}+\frac{1111}{8720}a^{8}-\frac{1697}{5232}a^{7}+\frac{1207}{5232}a^{6}+\frac{1441}{10464}a^{5}+\frac{1045}{5232}a^{4}+\frac{467}{1308}a^{3}-\frac{19}{1308}a^{2}-\frac{2}{327}a-\frac{5}{109}$, $\frac{1}{523200}a^{22}-\frac{1}{523200}a^{21}+\frac{1}{87200}a^{20}-\frac{1}{32700}a^{19}-\frac{1}{130800}a^{18}+\frac{1}{13080}a^{17}-\frac{1}{26160}a^{16}-\frac{1}{2616}a^{15}+\frac{11}{17440}a^{14}+\frac{519}{17440}a^{13}-\frac{4061}{52320}a^{12}+\frac{967}{8720}a^{11}+\frac{1681}{10464}a^{10}-\frac{541}{17440}a^{9}-\frac{233}{480}a^{8}-\frac{2833}{10464}a^{7}+\frac{2369}{20928}a^{6}-\frac{3137}{10464}a^{5}-\frac{695}{5232}a^{4}+\frac{169}{654}a^{3}+\frac{197}{436}a^{2}+\frac{16}{327}a-\frac{30}{109}$, $\frac{1}{26\cdots 00}a^{23}-\frac{14\cdots 03}{26\cdots 00}a^{22}-\frac{16\cdots 39}{33\cdots 00}a^{21}+\frac{34\cdots 91}{22\cdots 00}a^{20}+\frac{23\cdots 17}{67\cdots 00}a^{19}+\frac{19\cdots 77}{42\cdots 50}a^{18}+\frac{10\cdots 47}{13\cdots 60}a^{17}-\frac{54\cdots 79}{56\cdots 40}a^{16}+\frac{14\cdots 27}{89\cdots 40}a^{15}+\frac{10\cdots 57}{40\cdots 60}a^{14}+\frac{86\cdots 61}{53\cdots 44}a^{13}-\frac{21\cdots 97}{67\cdots 80}a^{12}+\frac{84\cdots 41}{53\cdots 44}a^{11}+\frac{12\cdots 59}{53\cdots 44}a^{10}+\frac{67\cdots 09}{26\cdots 20}a^{9}-\frac{78\cdots 51}{26\cdots 20}a^{8}-\frac{35\cdots 79}{10\cdots 88}a^{7}-\frac{25\cdots 21}{89\cdots 24}a^{6}+\frac{73\cdots 71}{16\cdots 42}a^{5}+\frac{32\cdots 31}{67\cdots 68}a^{4}+\frac{47\cdots 73}{16\cdots 42}a^{3}+\frac{71\cdots 74}{41\cdots 71}a^{2}-\frac{13\cdots 01}{56\cdots 14}a-\frac{73\cdots 15}{28\cdots 57}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
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| Narrow class group: | not computed |
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Unit group
| Rank: | $13$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: | not computed |
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| Regulator: | not computed |
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| Unit signature rank: | 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^{4}\cdot(2\pi)^{10}\cdot R \cdot h}{2\cdot\sqrt{77516311234764156989663655746389565161872035940177738666534423828125}}\cr\mathstrut & \text{
Galois group
$\GL(2,5)$ (as 24T1353):
| A non-solvable group of order 480 |
| The 24 conjugacy class representatives for $\GL(2,5)$ |
| Character table for $\GL(2,5)$ |
Intermediate fields
| 6.2.11027981328125.2, 12.4.7224620588965201519805908203125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{3}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{3}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | $24$ | $24$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/41.12.0.1}{12} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{3}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.20.31a1.1 | $x^{20} + 10 x^{12} + 5$ | $20$ | $1$ | $31$ | 20T5 | $$[\frac{7}{4}]_{4}$$ | |
|
\(109\)
| 109.1.4.3a1.1 | $x^{4} + 109$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 109.1.20.19a1.3 | $x^{20} + 3924$ | $20$ | $1$ | $19$ | 20T6 | $$[\ ]_{20}^{2}$$ |