Normalized defining polynomial
\( x^{28} - 14 x^{27} + 231 x^{26} - 2184 x^{25} + 20125 x^{24} - 140490 x^{23} + 910749 x^{22} + \cdots + 45398629 \)
Invariants
| Degree: | $28$ |
| |
| Signature: | $[0, 14]$ |
| |
| Discriminant: |
\(287622876985594783117337266142699525963070390548889600000000000000\)
\(\medspace = 2^{28}\cdot 3^{14}\cdot 5^{14}\cdot 7^{48}\)
|
| |
| Root discriminant: | \(217.68\) |
| |
| Galois root discriminant: | $2\cdot 3^{1/2}5^{1/2}7^{12/7}\approx 217.67829262168013$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_{14}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2940=2^{2}\cdot 3\cdot 5\cdot 7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2940}(1,·)$, $\chi_{2940}(1891,·)$, $\chi_{2940}(2129,·)$, $\chi_{2940}(2311,·)$, $\chi_{2940}(1289,·)$, $\chi_{2940}(2549,·)$, $\chi_{2940}(1261,·)$, $\chi_{2940}(1681,·)$, $\chi_{2940}(659,·)$, $\chi_{2940}(2521,·)$, $\chi_{2940}(1051,·)$, $\chi_{2940}(29,·)$, $\chi_{2940}(1499,·)$, $\chi_{2940}(869,·)$, $\chi_{2940}(2339,·)$, $\chi_{2940}(421,·)$, $\chi_{2940}(841,·)$, $\chi_{2940}(2731,·)$, $\chi_{2940}(1709,·)$, $\chi_{2940}(2759,·)$, $\chi_{2940}(239,·)$, $\chi_{2940}(449,·)$, $\chi_{2940}(211,·)$, $\chi_{2940}(2101,·)$, $\chi_{2940}(1079,·)$, $\chi_{2940}(631,·)$, $\chi_{2940}(1919,·)$, $\chi_{2940}(1471,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19}a^{20}+\frac{9}{19}a^{19}+\frac{8}{19}a^{18}+\frac{4}{19}a^{17}+\frac{5}{19}a^{16}-\frac{4}{19}a^{15}+\frac{1}{19}a^{13}+\frac{6}{19}a^{12}+\frac{9}{19}a^{11}+\frac{9}{19}a^{10}-\frac{2}{19}a^{9}-\frac{5}{19}a^{8}+\frac{6}{19}a^{7}-\frac{6}{19}a^{6}+\frac{6}{19}a^{5}-\frac{5}{19}a^{4}-\frac{1}{19}a^{3}+\frac{8}{19}a^{2}+\frac{8}{19}a-\frac{1}{19}$, $\frac{1}{19}a^{21}+\frac{3}{19}a^{19}+\frac{8}{19}a^{18}+\frac{7}{19}a^{17}+\frac{8}{19}a^{16}-\frac{2}{19}a^{15}+\frac{1}{19}a^{14}-\frac{3}{19}a^{13}-\frac{7}{19}a^{12}+\frac{4}{19}a^{11}-\frac{7}{19}a^{10}-\frac{6}{19}a^{9}-\frac{6}{19}a^{8}-\frac{3}{19}a^{7}+\frac{3}{19}a^{6}-\frac{2}{19}a^{5}+\frac{6}{19}a^{4}-\frac{2}{19}a^{3}-\frac{7}{19}a^{2}+\frac{3}{19}a+\frac{9}{19}$, $\frac{1}{19}a^{22}+\frac{2}{19}a^{18}-\frac{4}{19}a^{17}+\frac{2}{19}a^{16}-\frac{6}{19}a^{15}-\frac{3}{19}a^{14}+\frac{9}{19}a^{13}+\frac{5}{19}a^{12}+\frac{4}{19}a^{11}+\frac{5}{19}a^{10}-\frac{7}{19}a^{8}+\frac{4}{19}a^{7}-\frac{3}{19}a^{6}+\frac{7}{19}a^{5}-\frac{6}{19}a^{4}-\frac{4}{19}a^{3}-\frac{2}{19}a^{2}+\frac{4}{19}a+\frac{3}{19}$, $\frac{1}{19}a^{23}+\frac{2}{19}a^{19}-\frac{4}{19}a^{18}+\frac{2}{19}a^{17}-\frac{6}{19}a^{16}-\frac{3}{19}a^{15}+\frac{9}{19}a^{14}+\frac{5}{19}a^{13}+\frac{4}{19}a^{12}+\frac{5}{19}a^{11}-\frac{7}{19}a^{9}+\frac{4}{19}a^{8}-\frac{3}{19}a^{7}+\frac{7}{19}a^{6}-\frac{6}{19}a^{5}-\frac{4}{19}a^{4}-\frac{2}{19}a^{3}+\frac{4}{19}a^{2}+\frac{3}{19}a$, $\frac{1}{884089}a^{24}-\frac{12}{884089}a^{23}+\frac{7290}{884089}a^{22}+\frac{13378}{884089}a^{21}-\frac{11097}{884089}a^{20}-\frac{360802}{884089}a^{19}+\frac{429468}{884089}a^{18}-\frac{330395}{884089}a^{17}+\frac{167467}{884089}a^{16}+\frac{15213}{884089}a^{15}+\frac{35276}{884089}a^{14}+\frac{299454}{884089}a^{13}+\frac{325517}{884089}a^{12}+\frac{248474}{884089}a^{11}+\frac{34502}{884089}a^{10}-\frac{354819}{884089}a^{9}+\frac{314026}{884089}a^{8}+\frac{98453}{884089}a^{7}-\frac{410030}{884089}a^{6}-\frac{404468}{884089}a^{5}+\frac{35848}{884089}a^{4}+\frac{324267}{884089}a^{3}+\frac{380347}{884089}a^{2}-\frac{66331}{884089}a-\frac{124162}{884089}$, $\frac{1}{884089}a^{25}+\frac{7146}{884089}a^{23}+\frac{7796}{884089}a^{22}+\frac{9846}{884089}a^{21}+\frac{17875}{884089}a^{20}+\frac{287634}{884089}a^{19}-\frac{341720}{884089}a^{18}+\frac{297455}{884089}a^{17}-\frac{255202}{884089}a^{16}-\frac{107885}{884089}a^{15}-\frac{21730}{884089}a^{14}-\frac{408418}{884089}a^{13}-\frac{219236}{884089}a^{12}-\frac{380573}{884089}a^{11}-\frac{126919}{884089}a^{10}+\frac{290519}{884089}a^{9}+\frac{144285}{884089}a^{8}+\frac{352627}{884089}a^{7}+\frac{305423}{884089}a^{6}-\frac{350792}{884089}a^{5}-\frac{315770}{884089}a^{4}-\frac{9301}{884089}a^{3}+\frac{30857}{884089}a^{2}-\frac{268700}{884089}a-\frac{952}{884089}$, $\frac{1}{16\cdots 13}a^{26}-\frac{13}{16\cdots 13}a^{25}+\frac{19\cdots 84}{16\cdots 13}a^{24}-\frac{22\cdots 58}{16\cdots 13}a^{23}-\frac{23\cdots 96}{16\cdots 13}a^{22}-\frac{24\cdots 11}{16\cdots 13}a^{21}-\frac{10\cdots 35}{52\cdots 23}a^{20}+\frac{41\cdots 82}{16\cdots 13}a^{19}+\frac{20\cdots 39}{16\cdots 13}a^{18}-\frac{17\cdots 88}{16\cdots 13}a^{17}+\frac{98\cdots 25}{16\cdots 13}a^{16}-\frac{50\cdots 86}{16\cdots 13}a^{15}+\frac{46\cdots 05}{16\cdots 13}a^{14}+\frac{26\cdots 58}{16\cdots 13}a^{13}-\frac{43\cdots 13}{16\cdots 13}a^{12}+\frac{74\cdots 17}{16\cdots 13}a^{11}-\frac{22\cdots 48}{16\cdots 13}a^{10}+\frac{41\cdots 10}{16\cdots 13}a^{9}-\frac{33\cdots 29}{16\cdots 13}a^{8}+\frac{42\cdots 51}{16\cdots 13}a^{7}+\frac{50\cdots 64}{16\cdots 13}a^{6}-\frac{20\cdots 18}{16\cdots 13}a^{5}+\frac{59\cdots 26}{16\cdots 13}a^{4}-\frac{59\cdots 29}{16\cdots 13}a^{3}-\frac{25\cdots 45}{16\cdots 13}a^{2}-\frac{22\cdots 02}{52\cdots 23}a+\frac{77\cdots 60}{16\cdots 13}$, $\frac{1}{10\cdots 27}a^{27}+\frac{332926}{10\cdots 27}a^{26}+\frac{33\cdots 61}{10\cdots 27}a^{25}-\frac{43\cdots 53}{10\cdots 27}a^{24}-\frac{22\cdots 87}{10\cdots 27}a^{23}+\frac{10\cdots 51}{57\cdots 33}a^{22}-\frac{18\cdots 50}{10\cdots 27}a^{21}+\frac{12\cdots 79}{10\cdots 27}a^{20}+\frac{25\cdots 30}{10\cdots 27}a^{19}+\frac{35\cdots 89}{10\cdots 27}a^{18}-\frac{11\cdots 07}{10\cdots 27}a^{17}-\frac{34\cdots 78}{10\cdots 27}a^{16}+\frac{33\cdots 96}{10\cdots 27}a^{15}+\frac{28\cdots 05}{10\cdots 27}a^{14}+\frac{39\cdots 65}{10\cdots 27}a^{13}-\frac{39\cdots 13}{10\cdots 27}a^{12}-\frac{36\cdots 26}{10\cdots 27}a^{11}-\frac{16\cdots 44}{10\cdots 27}a^{10}-\frac{48\cdots 62}{10\cdots 27}a^{9}-\frac{18\cdots 07}{10\cdots 27}a^{8}-\frac{10\cdots 04}{57\cdots 33}a^{7}+\frac{69\cdots 90}{10\cdots 27}a^{6}+\frac{19\cdots 32}{10\cdots 27}a^{5}-\frac{74\cdots 58}{10\cdots 27}a^{4}-\frac{36\cdots 35}{10\cdots 27}a^{3}-\frac{43\cdots 83}{10\cdots 27}a^{2}+\frac{40\cdots 86}{10\cdots 27}a+\frac{28\cdots 31}{10\cdots 27}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
| |
| Relative class number: | data not computed |
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( \frac{15795504631906004542769610}{2753041279013794033518958125503} a^{27} - \frac{213239312530731061327389735}{2753041279013794033518958125503} a^{26} + \frac{67249723064341867027269576705}{52307784301262086636860204384557} a^{25} - \frac{621162745824729287224764439875}{52307784301262086636860204384557} a^{24} + \frac{300954060242413253239407840640}{2753041279013794033518958125503} a^{23} - \frac{39203996745841553886139617419815}{52307784301262086636860204384557} a^{22} + \frac{252804444134442193749543806216725}{52307784301262086636860204384557} a^{21} - \frac{1342899403468932465366496405999985}{52307784301262086636860204384557} a^{20} + \frac{6563972626823531673234701676462762}{52307784301262086636860204384557} a^{19} - \frac{1453273764305538925061246576201686}{2753041279013794033518958125503} a^{18} + \frac{105653946338703437077520595186124934}{52307784301262086636860204384557} a^{17} - \frac{354777638672675461679502343670748152}{52307784301262086636860204384557} a^{16} + \frac{1073403201314384185328989169967566304}{52307784301262086636860204384557} a^{15} - \frac{2867598510805650252410950083629048343}{52307784301262086636860204384557} a^{14} + \frac{6826329386925709932622952560366110319}{52307784301262086636860204384557} a^{13} - \frac{14295620155511572697431141310175782688}{52307784301262086636860204384557} a^{12} + \frac{26273052474072957748797660159212220886}{52307784301262086636860204384557} a^{11} - \frac{41914977663744480790014427481353633903}{52307784301262086636860204384557} a^{10} + \frac{57431295914556149060197347597426278734}{52307784301262086636860204384557} a^{9} - \frac{66672506534727593272151262097815306496}{52307784301262086636860204384557} a^{8} + \frac{64578063084576967415597838873770210010}{52307784301262086636860204384557} a^{7} - \frac{51262306519950275592508482217986506887}{52307784301262086636860204384557} a^{6} + \frac{32556667405334932659688253130236608466}{52307784301262086636860204384557} a^{5} - \frac{16004013522282706670618202951958886454}{52307784301262086636860204384557} a^{4} + \frac{5778438597361123166296802330604985285}{52307784301262086636860204384557} a^{3} - \frac{14459789158269996713192820117962238}{539255508260433882854228911181} a^{2} + \frac{194816515603723673545778491414099067}{52307784301262086636860204384557} a - \frac{10574032324567158408877033504854132}{52307784301262086636860204384557} \)
(order $4$)
|
| |
| 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)^{14}\cdot R \cdot h}{4\cdot\sqrt{287622876985594783117337266142699525963070390548889600000000000000}}\cr\mathstrut & \text{
Galois group
$C_2\times C_{14}$ (as 28T2):
| An abelian group of order 28 |
| The 28 conjugacy class representatives for $C_2\times C_{14}$ |
| Character table for $C_2\times C_{14}$ |
Intermediate fields
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 | R | R | R | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{14}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{14}$ | ${\href{/padicField/37.14.0.1}{14} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }^{4}$ | ${\href{/padicField/59.14.0.1}{14} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.2.14a1.1 | $x^{14} + 2 x^{8} + 4 x^{7} + x^{2} + 4 x + 5$ | $2$ | $7$ | $14$ | $C_{14}$ | $$[2]^{7}$$ |
| 2.7.2.14a1.1 | $x^{14} + 2 x^{8} + 4 x^{7} + x^{2} + 4 x + 5$ | $2$ | $7$ | $14$ | $C_{14}$ | $$[2]^{7}$$ | |
|
\(3\)
| Deg $28$ | $2$ | $14$ | $14$ | |||
|
\(5\)
| 5.7.2.7a1.1 | $x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$ | $2$ | $7$ | $7$ | $C_{14}$ | $$[\ ]_{2}^{7}$$ |
| 5.7.2.7a1.1 | $x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$ | $2$ | $7$ | $7$ | $C_{14}$ | $$[\ ]_{2}^{7}$$ | |
|
\(7\)
| 7.2.7.24a6.1 | $x^{14} + 42 x^{13} + 819 x^{12} + 9828 x^{11} + 80325 x^{10} + 463806 x^{9} + 1898127 x^{8} + 5419224 x^{7} + 10517283 x^{6} + 13522950 x^{5} + 11384793 x^{4} + 6184836 x^{3} + 2087127 x^{2} + 398034 x + 32812$ | $7$ | $2$ | $24$ | $C_{14}$ | $$[2]^{2}$$ |
| 7.2.7.24a6.1 | $x^{14} + 42 x^{13} + 819 x^{12} + 9828 x^{11} + 80325 x^{10} + 463806 x^{9} + 1898127 x^{8} + 5419224 x^{7} + 10517283 x^{6} + 13522950 x^{5} + 11384793 x^{4} + 6184836 x^{3} + 2087127 x^{2} + 398034 x + 32812$ | $7$ | $2$ | $24$ | $C_{14}$ | $$[2]^{2}$$ |