Normalized defining polynomial
\( x^{15} - 3 x^{14} - 114 x^{13} + 165 x^{12} + 5277 x^{11} + 309 x^{10} - 119015 x^{9} - 163851 x^{8} + \cdots + 831403 \)
Invariants
| Degree: | $15$ |
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| Signature: | $(15, 0)$ |
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| Discriminant: |
\(3091133177133909578645502426129\)
\(\medspace = 3^{20}\cdot 7^{10}\cdot 11^{12}\)
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| Root discriminant: | \(107.81\) |
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| Galois root discriminant: | $3^{4/3}7^{2/3}11^{4/5}\approx 107.81383996755932$ | ||
| Ramified primes: |
\(3\), \(7\), \(11\)
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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: | \(693=3^{2}\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{693}(64,·)$, $\chi_{693}(1,·)$, $\chi_{693}(529,·)$, $\chi_{693}(631,·)$, $\chi_{693}(466,·)$, $\chi_{693}(592,·)$, $\chi_{693}(625,·)$, $\chi_{693}(562,·)$, $\chi_{693}(499,·)$, $\chi_{693}(214,·)$, $\chi_{693}(247,·)$, $\chi_{693}(25,·)$, $\chi_{693}(58,·)$, $\chi_{693}(379,·)$, $\chi_{693}(190,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{21\cdots 74}a^{14}+\frac{44\cdots 57}{21\cdots 74}a^{13}-\frac{24\cdots 61}{21\cdots 74}a^{12}+\frac{20\cdots 85}{16\cdots 61}a^{11}+\frac{12\cdots 94}{10\cdots 87}a^{10}-\frac{73\cdots 71}{21\cdots 74}a^{9}-\frac{37\cdots 13}{10\cdots 87}a^{8}+\frac{62\cdots 53}{21\cdots 74}a^{7}+\frac{23\cdots 19}{10\cdots 87}a^{6}+\frac{26\cdots 76}{10\cdots 87}a^{5}+\frac{22\cdots 39}{10\cdots 87}a^{4}+\frac{34\cdots 52}{10\cdots 87}a^{3}-\frac{51\cdots 51}{10\cdots 87}a^{2}+\frac{12\cdots 55}{10\cdots 87}a+\frac{11\cdots 25}{32\cdots 22}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ (assuming GRH) |
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| Narrow class group: | $C_{3}$, which has order $3$ (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: |
$\frac{11\cdots 14}{10\cdots 87}a^{14}-\frac{57\cdots 74}{10\cdots 87}a^{13}-\frac{11\cdots 13}{10\cdots 87}a^{12}+\frac{64\cdots 96}{16\cdots 61}a^{11}+\frac{48\cdots 85}{10\cdots 87}a^{10}-\frac{10\cdots 74}{10\cdots 87}a^{9}-\frac{10\cdots 04}{10\cdots 87}a^{8}+\frac{59\cdots 04}{10\cdots 87}a^{7}+\frac{11\cdots 35}{10\cdots 87}a^{6}+\frac{88\cdots 88}{10\cdots 87}a^{5}-\frac{48\cdots 33}{10\cdots 87}a^{4}-\frac{88\cdots 40}{10\cdots 87}a^{3}-\frac{26\cdots 22}{10\cdots 87}a^{2}+\frac{21\cdots 34}{10\cdots 87}a+\frac{62\cdots 46}{16\cdots 61}$, $\frac{22\cdots 01}{10\cdots 87}a^{14}-\frac{12\cdots 35}{10\cdots 87}a^{13}-\frac{22\cdots 46}{10\cdots 87}a^{12}+\frac{13\cdots 34}{16\cdots 61}a^{11}+\frac{98\cdots 82}{10\cdots 87}a^{10}-\frac{22\cdots 62}{10\cdots 87}a^{9}-\frac{21\cdots 44}{10\cdots 87}a^{8}+\frac{13\cdots 55}{10\cdots 87}a^{7}+\frac{24\cdots 11}{10\cdots 87}a^{6}+\frac{16\cdots 52}{10\cdots 87}a^{5}-\frac{98\cdots 05}{10\cdots 87}a^{4}-\frac{17\cdots 85}{10\cdots 87}a^{3}-\frac{49\cdots 61}{10\cdots 87}a^{2}+\frac{40\cdots 26}{10\cdots 87}a+\frac{12\cdots 70}{16\cdots 61}$, $\frac{24\cdots 07}{10\cdots 87}a^{14}-\frac{12\cdots 63}{10\cdots 87}a^{13}-\frac{24\cdots 50}{10\cdots 87}a^{12}+\frac{14\cdots 24}{16\cdots 61}a^{11}+\frac{10\cdots 04}{10\cdots 87}a^{10}-\frac{23\cdots 36}{10\cdots 87}a^{9}-\frac{23\cdots 61}{10\cdots 87}a^{8}+\frac{14\cdots 73}{10\cdots 87}a^{7}+\frac{25\cdots 01}{10\cdots 87}a^{6}+\frac{17\cdots 26}{10\cdots 87}a^{5}-\frac{10\cdots 99}{10\cdots 87}a^{4}-\frac{18\cdots 63}{10\cdots 87}a^{3}-\frac{53\cdots 70}{10\cdots 87}a^{2}+\frac{43\cdots 60}{10\cdots 87}a+\frac{13\cdots 27}{16\cdots 61}$, $\frac{15\cdots 60}{10\cdots 87}a^{14}-\frac{82\cdots 78}{10\cdots 87}a^{13}-\frac{15\cdots 31}{10\cdots 87}a^{12}+\frac{92\cdots 98}{16\cdots 61}a^{11}+\frac{67\cdots 93}{10\cdots 87}a^{10}-\frac{15\cdots 52}{10\cdots 87}a^{9}-\frac{15\cdots 50}{10\cdots 87}a^{8}+\frac{91\cdots 86}{10\cdots 87}a^{7}+\frac{16\cdots 49}{10\cdots 87}a^{6}+\frac{11\cdots 94}{10\cdots 87}a^{5}-\frac{67\cdots 54}{10\cdots 87}a^{4}-\frac{11\cdots 42}{10\cdots 87}a^{3}-\frac{35\cdots 95}{10\cdots 87}a^{2}+\frac{27\cdots 08}{10\cdots 87}a+\frac{86\cdots 48}{16\cdots 61}$, $\frac{78\cdots 12}{10\cdots 87}a^{14}-\frac{82\cdots 93}{21\cdots 74}a^{13}-\frac{16\cdots 87}{21\cdots 74}a^{12}+\frac{45\cdots 04}{16\cdots 61}a^{11}+\frac{69\cdots 79}{21\cdots 74}a^{10}-\frac{14\cdots 93}{21\cdots 74}a^{9}-\frac{76\cdots 05}{10\cdots 87}a^{8}+\frac{41\cdots 09}{10\cdots 87}a^{7}+\frac{16\cdots 89}{21\cdots 74}a^{6}+\frac{63\cdots 42}{10\cdots 87}a^{5}-\frac{69\cdots 39}{21\cdots 74}a^{4}-\frac{62\cdots 86}{10\cdots 87}a^{3}-\frac{18\cdots 64}{10\cdots 87}a^{2}+\frac{29\cdots 17}{21\cdots 74}a+\frac{89\cdots 11}{32\cdots 22}$, $\frac{27\cdots 82}{10\cdots 87}a^{14}-\frac{27\cdots 45}{21\cdots 74}a^{13}-\frac{58\cdots 13}{21\cdots 74}a^{12}+\frac{30\cdots 01}{32\cdots 22}a^{11}+\frac{25\cdots 89}{21\cdots 74}a^{10}-\frac{22\cdots 23}{10\cdots 87}a^{9}-\frac{27\cdots 95}{10\cdots 87}a^{8}+\frac{66\cdots 09}{10\cdots 87}a^{7}+\frac{59\cdots 51}{21\cdots 74}a^{6}+\frac{31\cdots 15}{10\cdots 87}a^{5}-\frac{10\cdots 98}{10\cdots 87}a^{4}-\frac{51\cdots 45}{21\cdots 74}a^{3}-\frac{36\cdots 29}{21\cdots 74}a^{2}+\frac{56\cdots 05}{21\cdots 74}a+\frac{81\cdots 83}{16\cdots 61}$, $\frac{49\cdots 91}{21\cdots 74}a^{14}-\frac{33\cdots 45}{21\cdots 74}a^{13}-\frac{21\cdots 17}{10\cdots 87}a^{12}+\frac{37\cdots 59}{32\cdots 22}a^{11}+\frac{16\cdots 35}{21\cdots 74}a^{10}-\frac{32\cdots 97}{10\cdots 87}a^{9}-\frac{17\cdots 39}{10\cdots 87}a^{8}+\frac{62\cdots 11}{21\cdots 74}a^{7}+\frac{19\cdots 86}{10\cdots 87}a^{6}+\frac{67\cdots 45}{21\cdots 74}a^{5}-\frac{88\cdots 89}{10\cdots 87}a^{4}-\frac{10\cdots 57}{10\cdots 87}a^{3}+\frac{46\cdots 65}{21\cdots 74}a^{2}+\frac{77\cdots 65}{21\cdots 74}a-\frac{48\cdots 05}{16\cdots 61}$, $\frac{45\cdots 25}{10\cdots 87}a^{14}-\frac{49\cdots 95}{21\cdots 74}a^{13}-\frac{45\cdots 42}{10\cdots 87}a^{12}+\frac{55\cdots 27}{32\cdots 22}a^{11}+\frac{38\cdots 47}{21\cdots 74}a^{10}-\frac{92\cdots 33}{21\cdots 74}a^{9}-\frac{42\cdots 61}{10\cdots 87}a^{8}+\frac{31\cdots 69}{10\cdots 87}a^{7}+\frac{93\cdots 33}{21\cdots 74}a^{6}+\frac{55\cdots 99}{21\cdots 74}a^{5}-\frac{38\cdots 25}{21\cdots 74}a^{4}-\frac{64\cdots 77}{21\cdots 74}a^{3}-\frac{16\cdots 05}{21\cdots 74}a^{2}+\frac{14\cdots 51}{21\cdots 74}a+\frac{43\cdots 21}{32\cdots 22}$, $\frac{24\cdots 75}{21\cdots 74}a^{14}-\frac{10\cdots 23}{21\cdots 74}a^{13}-\frac{12\cdots 25}{10\cdots 87}a^{12}+\frac{11\cdots 23}{32\cdots 22}a^{11}+\frac{57\cdots 20}{10\cdots 87}a^{10}-\frac{16\cdots 65}{21\cdots 74}a^{9}-\frac{13\cdots 48}{10\cdots 87}a^{8}-\frac{10\cdots 85}{21\cdots 74}a^{7}+\frac{14\cdots 68}{10\cdots 87}a^{6}+\frac{34\cdots 13}{21\cdots 74}a^{5}-\frac{56\cdots 71}{10\cdots 87}a^{4}-\frac{26\cdots 73}{21\cdots 74}a^{3}-\frac{10\cdots 05}{21\cdots 74}a^{2}+\frac{31\cdots 40}{10\cdots 87}a+\frac{20\cdots 19}{32\cdots 22}$, $\frac{22\cdots 49}{10\cdots 87}a^{14}-\frac{27\cdots 05}{21\cdots 74}a^{13}-\frac{22\cdots 44}{10\cdots 87}a^{12}+\frac{15\cdots 18}{16\cdots 61}a^{11}+\frac{90\cdots 27}{10\cdots 87}a^{10}-\frac{57\cdots 09}{21\cdots 74}a^{9}-\frac{19\cdots 53}{10\cdots 87}a^{8}+\frac{26\cdots 57}{10\cdots 87}a^{7}+\frac{42\cdots 39}{21\cdots 74}a^{6}+\frac{53\cdots 67}{21\cdots 74}a^{5}-\frac{88\cdots 01}{10\cdots 87}a^{4}-\frac{22\cdots 15}{21\cdots 74}a^{3}-\frac{19\cdots 91}{10\cdots 87}a^{2}+\frac{28\cdots 08}{10\cdots 87}a+\frac{92\cdots 79}{32\cdots 22}$, $\frac{13\cdots 07}{10\cdots 87}a^{14}-\frac{73\cdots 24}{10\cdots 87}a^{13}-\frac{27\cdots 09}{21\cdots 74}a^{12}+\frac{16\cdots 07}{32\cdots 22}a^{11}+\frac{58\cdots 20}{10\cdots 87}a^{10}-\frac{13\cdots 02}{10\cdots 87}a^{9}-\frac{12\cdots 89}{10\cdots 87}a^{8}+\frac{84\cdots 61}{10\cdots 87}a^{7}+\frac{14\cdots 12}{10\cdots 87}a^{6}+\frac{19\cdots 45}{21\cdots 74}a^{5}-\frac{58\cdots 90}{10\cdots 87}a^{4}-\frac{20\cdots 11}{21\cdots 74}a^{3}-\frac{58\cdots 65}{21\cdots 74}a^{2}+\frac{24\cdots 35}{10\cdots 87}a+\frac{71\cdots 73}{16\cdots 61}$, $\frac{11\cdots 83}{10\cdots 87}a^{14}-\frac{58\cdots 81}{10\cdots 87}a^{13}-\frac{22\cdots 31}{21\cdots 74}a^{12}+\frac{12\cdots 31}{32\cdots 22}a^{11}+\frac{96\cdots 39}{21\cdots 74}a^{10}-\frac{20\cdots 59}{21\cdots 74}a^{9}-\frac{10\cdots 85}{10\cdots 87}a^{8}+\frac{56\cdots 26}{10\cdots 87}a^{7}+\frac{11\cdots 69}{10\cdots 87}a^{6}+\frac{18\cdots 33}{21\cdots 74}a^{5}-\frac{48\cdots 68}{10\cdots 87}a^{4}-\frac{88\cdots 17}{10\cdots 87}a^{3}-\frac{56\cdots 95}{21\cdots 74}a^{2}+\frac{41\cdots 93}{21\cdots 74}a+\frac{14\cdots 21}{32\cdots 22}$, $\frac{98\cdots 11}{21\cdots 74}a^{14}-\frac{52\cdots 49}{21\cdots 74}a^{13}-\frac{10\cdots 37}{21\cdots 74}a^{12}+\frac{59\cdots 83}{32\cdots 22}a^{11}+\frac{21\cdots 25}{10\cdots 87}a^{10}-\frac{48\cdots 36}{10\cdots 87}a^{9}-\frac{47\cdots 58}{10\cdots 87}a^{8}+\frac{59\cdots 11}{21\cdots 74}a^{7}+\frac{52\cdots 36}{10\cdots 87}a^{6}+\frac{35\cdots 95}{10\cdots 87}a^{5}-\frac{43\cdots 89}{21\cdots 74}a^{4}-\frac{75\cdots 19}{21\cdots 74}a^{3}-\frac{20\cdots 63}{21\cdots 74}a^{2}+\frac{94\cdots 48}{10\cdots 87}a+\frac{27\cdots 53}{16\cdots 61}$, $\frac{16\cdots 54}{10\cdots 87}a^{14}-\frac{11\cdots 46}{10\cdots 87}a^{13}-\frac{30\cdots 73}{21\cdots 74}a^{12}+\frac{13\cdots 22}{16\cdots 61}a^{11}+\frac{11\cdots 89}{21\cdots 74}a^{10}-\frac{25\cdots 80}{10\cdots 87}a^{9}-\frac{12\cdots 04}{10\cdots 87}a^{8}+\frac{28\cdots 42}{10\cdots 87}a^{7}+\frac{14\cdots 47}{10\cdots 87}a^{6}-\frac{15\cdots 75}{21\cdots 74}a^{5}-\frac{14\cdots 35}{21\cdots 74}a^{4}-\frac{80\cdots 21}{21\cdots 74}a^{3}+\frac{70\cdots 11}{10\cdots 87}a^{2}+\frac{73\cdots 59}{21\cdots 74}a-\frac{37\cdots 53}{16\cdots 61}$
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| Regulator: | \( 5252645612.166227 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 5252645612.166227 \cdot 3}{2\cdot\sqrt{3091133177133909578645502426129}}\cr\approx \mathstrut & 0.146845430370137 \end{aligned}\] (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
| 3.3.3969.1, \(\Q(\zeta_{11})^+\) |
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 | ${\href{/padicField/2.5.0.1}{5} }^{3}$ | R | $15$ | R | R | $15$ | $15$ | $15$ | ${\href{/padicField/23.3.0.1}{3} }^{5}$ | $15$ | ${\href{/padicField/31.5.0.1}{5} }^{3}$ | $15$ | $15$ | ${\href{/padicField/43.3.0.1}{3} }^{5}$ | ${\href{/padicField/47.5.0.1}{5} }^{3}$ | $15$ | ${\href{/padicField/59.5.0.1}{5} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
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\(3\)
| 3.5.3.20a2.3 | $x^{15} + 6 x^{11} + 9 x^{10} + 12 x^{7} + 36 x^{6} + 15 x^{5} + 8 x^{3} + 36 x^{2} + 30 x + 28$ | $3$ | $5$ | $20$ | $C_{15}$ | $$[2]^{5}$$ |
|
\(7\)
| 7.5.3.10a1.1 | $x^{15} + 3 x^{11} + 12 x^{10} + 3 x^{7} + 24 x^{6} + 48 x^{5} + x^{3} + 19 x^{2} + 48 x + 64$ | $3$ | $5$ | $10$ | $C_{15}$ | $$[\ ]_{3}^{5}$$ |
|
\(11\)
| 11.3.5.12a1.4 | $x^{15} + 10 x^{13} + 45 x^{12} + 40 x^{11} + 360 x^{10} + 890 x^{9} + 1080 x^{8} + 4940 x^{7} + 8730 x^{6} + 9752 x^{5} + 29880 x^{4} + 39285 x^{3} + 29160 x^{2} + 65610 x + 59060$ | $5$ | $3$ | $12$ | $C_{15}$ | $$[\ ]_{5}^{3}$$ |