Normalized defining polynomial
\( x^{15} - x^{14} - 294 x^{13} + 127 x^{12} + 29253 x^{11} - 353 x^{10} - 1219824 x^{9} - 638978 x^{8} + \cdots - 110965429 \)
Invariants
| Degree: | $15$ |
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| Signature: | $[15, 0]$ |
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| Discriminant: |
\(1586393893300992738237335763411468444721\)
\(\medspace = 631^{14}\)
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| Root discriminant: | \(410.54\) |
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| Galois root discriminant: | $631^{14/15}\approx 410.5449697876061$ | ||
| Ramified primes: |
\(631\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{15}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(631\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{631}(64,·)$, $\chi_{631}(1,·)$, $\chi_{631}(43,·)$, $\chi_{631}(228,·)$, $\chi_{631}(310,·)$, $\chi_{631}(8,·)$, $\chi_{631}(587,·)$, $\chi_{631}(242,·)$, $\chi_{631}(79,·)$, $\chi_{631}(562,·)$, $\chi_{631}(339,·)$, $\chi_{631}(512,·)$, $\chi_{631}(279,·)$, $\chi_{631}(344,·)$, $\chi_{631}(188,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{12599}a^{13}-\frac{2806}{12599}a^{12}-\frac{3292}{12599}a^{11}+\frac{1316}{12599}a^{10}+\frac{5115}{12599}a^{9}+\frac{94}{293}a^{8}+\frac{4496}{12599}a^{7}-\frac{2560}{12599}a^{6}-\frac{1857}{12599}a^{5}+\frac{5389}{12599}a^{4}-\frac{2118}{12599}a^{3}+\frac{2025}{12599}a^{2}-\frac{1648}{12599}a-\frac{3513}{12599}$, $\frac{1}{53\cdots 93}a^{14}+\frac{11\cdots 96}{53\cdots 93}a^{13}+\frac{15\cdots 35}{53\cdots 93}a^{12}+\frac{18\cdots 80}{53\cdots 93}a^{11}+\frac{11\cdots 23}{53\cdots 93}a^{10}+\frac{66\cdots 79}{53\cdots 93}a^{9}-\frac{12\cdots 18}{53\cdots 93}a^{8}+\frac{18\cdots 03}{53\cdots 93}a^{7}+\frac{25\cdots 62}{53\cdots 93}a^{6}+\frac{18\cdots 93}{53\cdots 93}a^{5}+\frac{12\cdots 10}{53\cdots 93}a^{4}-\frac{42\cdots 11}{53\cdots 93}a^{3}+\frac{16\cdots 19}{53\cdots 93}a^{2}-\frac{24\cdots 90}{53\cdots 93}a-\frac{21\cdots 50}{53\cdots 93}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{11}$, which has order $11$ (assuming GRH) |
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| Narrow class group: | $C_{11}$, which has order $11$ (assuming GRH) |
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Unit group
| Rank: | $14$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{10\cdots 95}{26\cdots 03}a^{14}-\frac{72\cdots 37}{26\cdots 03}a^{13}-\frac{42\cdots 58}{26\cdots 03}a^{12}+\frac{20\cdots 57}{26\cdots 03}a^{11}+\frac{71\cdots 59}{26\cdots 03}a^{10}-\frac{18\cdots 39}{26\cdots 03}a^{9}-\frac{54\cdots 39}{26\cdots 03}a^{8}+\frac{63\cdots 96}{26\cdots 03}a^{7}+\frac{19\cdots 38}{26\cdots 03}a^{6}-\frac{70\cdots 77}{26\cdots 03}a^{5}-\frac{23\cdots 32}{26\cdots 03}a^{4}+\frac{12\cdots 30}{26\cdots 03}a^{3}+\frac{51\cdots 52}{26\cdots 03}a^{2}-\frac{12\cdots 44}{26\cdots 03}a-\frac{11\cdots 21}{26\cdots 03}$, $\frac{10\cdots 65}{26\cdots 03}a^{14}+\frac{11\cdots 74}{26\cdots 03}a^{13}-\frac{27\cdots 06}{26\cdots 03}a^{12}-\frac{33\cdots 19}{26\cdots 03}a^{11}+\frac{22\cdots 40}{26\cdots 03}a^{10}+\frac{31\cdots 46}{26\cdots 03}a^{9}-\frac{52\cdots 68}{26\cdots 03}a^{8}-\frac{11\cdots 70}{26\cdots 03}a^{7}-\frac{61\cdots 04}{26\cdots 03}a^{6}+\frac{15\cdots 37}{26\cdots 03}a^{5}+\frac{20\cdots 32}{26\cdots 03}a^{4}-\frac{56\cdots 77}{26\cdots 03}a^{3}-\frac{67\cdots 89}{26\cdots 03}a^{2}+\frac{68\cdots 50}{26\cdots 03}a+\frac{58\cdots 99}{26\cdots 03}$, $\frac{20\cdots 28}{26\cdots 03}a^{14}-\frac{57\cdots 27}{26\cdots 03}a^{13}-\frac{13\cdots 17}{26\cdots 03}a^{12}+\frac{16\cdots 05}{26\cdots 03}a^{11}+\frac{36\cdots 72}{26\cdots 03}a^{10}-\frac{14\cdots 00}{26\cdots 03}a^{9}-\frac{34\cdots 42}{26\cdots 03}a^{8}+\frac{51\cdots 18}{26\cdots 03}a^{7}+\frac{13\cdots 29}{26\cdots 03}a^{6}-\frac{58\cdots 23}{26\cdots 03}a^{5}-\frac{16\cdots 07}{26\cdots 03}a^{4}+\frac{11\cdots 60}{26\cdots 03}a^{3}+\frac{36\cdots 68}{26\cdots 03}a^{2}-\frac{13\cdots 49}{26\cdots 03}a-\frac{89\cdots 16}{26\cdots 03}$, $\frac{21\cdots 60}{26\cdots 03}a^{14}-\frac{61\cdots 63}{26\cdots 03}a^{13}-\frac{70\cdots 64}{26\cdots 03}a^{12}+\frac{17\cdots 38}{26\cdots 03}a^{11}+\frac{93\cdots 99}{26\cdots 03}a^{10}-\frac{15\cdots 93}{26\cdots 03}a^{9}-\frac{60\cdots 07}{26\cdots 03}a^{8}+\frac{51\cdots 26}{26\cdots 03}a^{7}+\frac{18\cdots 34}{26\cdots 03}a^{6}-\frac{55\cdots 40}{26\cdots 03}a^{5}-\frac{21\cdots 00}{26\cdots 03}a^{4}+\frac{65\cdots 53}{26\cdots 03}a^{3}+\frac{44\cdots 63}{26\cdots 03}a^{2}+\frac{56\cdots 06}{26\cdots 03}a-\frac{63\cdots 34}{26\cdots 03}$, $\frac{53\cdots 55}{53\cdots 93}a^{14}+\frac{16\cdots 74}{53\cdots 93}a^{13}-\frac{14\cdots 29}{53\cdots 93}a^{12}-\frac{55\cdots 25}{53\cdots 93}a^{11}+\frac{13\cdots 31}{53\cdots 93}a^{10}+\frac{54\cdots 43}{53\cdots 93}a^{9}-\frac{42\cdots 03}{53\cdots 93}a^{8}-\frac{20\cdots 85}{53\cdots 93}a^{7}+\frac{32\cdots 10}{53\cdots 93}a^{6}+\frac{25\cdots 61}{53\cdots 93}a^{5}+\frac{16\cdots 17}{53\cdots 93}a^{4}-\frac{53\cdots 97}{53\cdots 93}a^{3}-\frac{58\cdots 09}{53\cdots 93}a^{2}+\frac{13\cdots 83}{53\cdots 93}a+\frac{13\cdots 53}{53\cdots 93}$, $\frac{80\cdots 77}{53\cdots 93}a^{14}-\frac{30\cdots 17}{53\cdots 93}a^{13}-\frac{24\cdots 51}{53\cdots 93}a^{12}+\frac{74\cdots 84}{53\cdots 93}a^{11}+\frac{25\cdots 63}{53\cdots 93}a^{10}-\frac{59\cdots 60}{53\cdots 93}a^{9}-\frac{11\cdots 48}{53\cdots 93}a^{8}+\frac{15\cdots 37}{53\cdots 93}a^{7}+\frac{24\cdots 01}{53\cdots 93}a^{6}-\frac{57\cdots 52}{53\cdots 93}a^{5}-\frac{22\cdots 43}{53\cdots 93}a^{4}-\frac{14\cdots 69}{53\cdots 93}a^{3}+\frac{50\cdots 53}{53\cdots 93}a^{2}+\frac{34\cdots 80}{53\cdots 93}a-\frac{27\cdots 93}{53\cdots 93}$, $\frac{52\cdots 10}{53\cdots 93}a^{14}+\frac{22\cdots 37}{53\cdots 93}a^{13}+\frac{31\cdots 31}{53\cdots 93}a^{12}-\frac{64\cdots 16}{53\cdots 93}a^{11}-\frac{12\cdots 87}{53\cdots 93}a^{10}+\frac{58\cdots 54}{53\cdots 93}a^{9}+\frac{13\cdots 44}{53\cdots 93}a^{8}-\frac{20\cdots 89}{53\cdots 93}a^{7}-\frac{55\cdots 06}{53\cdots 93}a^{6}+\frac{23\cdots 72}{53\cdots 93}a^{5}+\frac{69\cdots 33}{53\cdots 93}a^{4}-\frac{43\cdots 72}{53\cdots 93}a^{3}-\frac{15\cdots 59}{53\cdots 93}a^{2}+\frac{95\cdots 53}{53\cdots 93}a+\frac{36\cdots 36}{53\cdots 93}$, $\frac{20\cdots 68}{53\cdots 93}a^{14}+\frac{65\cdots 74}{53\cdots 93}a^{13}-\frac{57\cdots 20}{53\cdots 93}a^{12}-\frac{21\cdots 79}{53\cdots 93}a^{11}+\frac{50\cdots 51}{53\cdots 93}a^{10}+\frac{21\cdots 70}{53\cdots 93}a^{9}-\frac{16\cdots 52}{53\cdots 93}a^{8}-\frac{81\cdots 14}{53\cdots 93}a^{7}+\frac{12\cdots 58}{53\cdots 93}a^{6}+\frac{99\cdots 31}{53\cdots 93}a^{5}+\frac{68\cdots 03}{53\cdots 93}a^{4}-\frac{21\cdots 65}{53\cdots 93}a^{3}-\frac{23\cdots 10}{53\cdots 93}a^{2}+\frac{60\cdots 63}{53\cdots 93}a+\frac{63\cdots 07}{53\cdots 93}$, $\frac{37\cdots 35}{53\cdots 93}a^{14}-\frac{19\cdots 96}{12\cdots 51}a^{13}-\frac{10\cdots 11}{53\cdots 93}a^{12}+\frac{18\cdots 92}{53\cdots 93}a^{11}+\frac{10\cdots 23}{53\cdots 93}a^{10}-\frac{14\cdots 77}{53\cdots 93}a^{9}-\frac{42\cdots 75}{53\cdots 93}a^{8}+\frac{39\cdots 16}{53\cdots 93}a^{7}+\frac{73\cdots 87}{53\cdots 93}a^{6}-\frac{42\cdots 39}{53\cdots 93}a^{5}-\frac{50\cdots 79}{53\cdots 93}a^{4}+\frac{13\cdots 85}{53\cdots 93}a^{3}+\frac{10\cdots 51}{53\cdots 93}a^{2}-\frac{25\cdots 95}{53\cdots 93}a-\frac{34\cdots 89}{53\cdots 93}$, $\frac{54\cdots 74}{53\cdots 93}a^{14}-\frac{15\cdots 77}{53\cdots 93}a^{13}-\frac{15\cdots 19}{53\cdots 93}a^{12}+\frac{36\cdots 66}{53\cdots 93}a^{11}+\frac{15\cdots 87}{53\cdots 93}a^{10}-\frac{28\cdots 81}{53\cdots 93}a^{9}-\frac{60\cdots 90}{53\cdots 93}a^{8}+\frac{78\cdots 22}{53\cdots 93}a^{7}+\frac{10\cdots 43}{53\cdots 93}a^{6}-\frac{76\cdots 42}{53\cdots 93}a^{5}-\frac{75\cdots 78}{53\cdots 93}a^{4}+\frac{11\cdots 75}{53\cdots 93}a^{3}+\frac{33\cdots 01}{12\cdots 51}a^{2}-\frac{23\cdots 90}{53\cdots 93}a-\frac{32\cdots 38}{53\cdots 93}$, $\frac{57\cdots 35}{53\cdots 93}a^{14}-\frac{15\cdots 76}{53\cdots 93}a^{13}-\frac{16\cdots 25}{53\cdots 93}a^{12}+\frac{34\cdots 02}{53\cdots 93}a^{11}+\frac{16\cdots 80}{53\cdots 93}a^{10}-\frac{27\cdots 53}{53\cdots 93}a^{9}-\frac{64\cdots 04}{53\cdots 93}a^{8}+\frac{72\cdots 21}{53\cdots 93}a^{7}+\frac{11\cdots 60}{53\cdots 93}a^{6}-\frac{65\cdots 74}{53\cdots 93}a^{5}-\frac{82\cdots 01}{53\cdots 93}a^{4}+\frac{57\cdots 53}{53\cdots 93}a^{3}+\frac{16\cdots 37}{53\cdots 93}a^{2}-\frac{29\cdots 80}{53\cdots 93}a-\frac{36\cdots 09}{53\cdots 93}$, $\frac{64\cdots 63}{53\cdots 93}a^{14}-\frac{14\cdots 84}{53\cdots 93}a^{13}-\frac{18\cdots 71}{53\cdots 93}a^{12}+\frac{32\cdots 33}{53\cdots 93}a^{11}+\frac{18\cdots 77}{53\cdots 93}a^{10}-\frac{25\cdots 90}{53\cdots 93}a^{9}-\frac{79\cdots 49}{53\cdots 93}a^{8}+\frac{68\cdots 96}{53\cdots 93}a^{7}+\frac{15\cdots 94}{53\cdots 93}a^{6}-\frac{59\cdots 01}{53\cdots 93}a^{5}-\frac{11\cdots 71}{53\cdots 93}a^{4}-\frac{25\cdots 96}{53\cdots 93}a^{3}+\frac{24\cdots 54}{53\cdots 93}a^{2}+\frac{13\cdots 01}{53\cdots 93}a-\frac{54\cdots 96}{53\cdots 93}$, $\frac{76\cdots 04}{53\cdots 93}a^{14}+\frac{22\cdots 21}{53\cdots 93}a^{13}-\frac{21\cdots 42}{53\cdots 93}a^{12}-\frac{75\cdots 00}{53\cdots 93}a^{11}+\frac{18\cdots 98}{53\cdots 93}a^{10}+\frac{74\cdots 33}{53\cdots 93}a^{9}-\frac{57\cdots 65}{53\cdots 93}a^{8}-\frac{27\cdots 66}{53\cdots 93}a^{7}+\frac{39\cdots 27}{53\cdots 93}a^{6}+\frac{31\cdots 60}{53\cdots 93}a^{5}+\frac{27\cdots 56}{53\cdots 93}a^{4}-\frac{39\cdots 23}{53\cdots 93}a^{3}-\frac{33\cdots 60}{53\cdots 93}a^{2}+\frac{27\cdots 89}{53\cdots 93}a+\frac{11\cdots 55}{53\cdots 93}$, $\frac{99\cdots 22}{12\cdots 51}a^{14}-\frac{57\cdots 39}{53\cdots 93}a^{13}-\frac{12\cdots 94}{53\cdots 93}a^{12}+\frac{96\cdots 59}{53\cdots 93}a^{11}+\frac{12\cdots 69}{53\cdots 93}a^{10}-\frac{41\cdots 76}{53\cdots 93}a^{9}-\frac{11\cdots 91}{12\cdots 51}a^{8}-\frac{11\cdots 58}{53\cdots 93}a^{7}+\frac{82\cdots 28}{53\cdots 93}a^{6}+\frac{65\cdots 21}{53\cdots 93}a^{5}-\frac{49\cdots 03}{53\cdots 93}a^{4}-\frac{67\cdots 89}{53\cdots 93}a^{3}+\frac{10\cdots 33}{53\cdots 93}a^{2}+\frac{15\cdots 44}{53\cdots 93}a+\frac{10\cdots 98}{53\cdots 93}$
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| Regulator: | \( 40655123886363.37 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 40655123886363.37 \cdot 11}{2\cdot\sqrt{1586393893300992738237335763411468444721}}\cr\approx \mathstrut & 0.183959575421441 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 15 |
| The 15 conjugacy class representatives for $C_{15}$ |
| Character table for $C_{15}$ |
Intermediate fields
| 3.3.398161.1, 5.5.158532181921.1 |
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 | $15$ | $15$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | $15$ | $15$ | $15$ | $15$ | $15$ | $15$ | $15$ | $15$ | ${\href{/padicField/37.3.0.1}{3} }^{5}$ | ${\href{/padicField/41.3.0.1}{3} }^{5}$ | ${\href{/padicField/43.1.0.1}{1} }^{15}$ | $15$ | $15$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(631\)
| Deg $15$ | $15$ | $1$ | $14$ |