Normalized defining polynomial
\( x^{28} - 259 x^{26} - 126 x^{25} + 27692 x^{24} + 25270 x^{23} - 1593004 x^{22} - 2067704 x^{21} + \cdots + 38174136301 \)
Invariants
| Degree: | $28$ |
| |
| Signature: | $[28, 0]$ |
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| Discriminant: |
\(14093520972294144372749526040992276772190449136895590400000000000000\)
\(\medspace = 2^{28}\cdot 3^{14}\cdot 5^{14}\cdot 7^{50}\)
|
| |
| Root discriminant: | \(250.14\) |
| |
| Galois root discriminant: | $2\cdot 3^{1/2}5^{1/2}7^{25/14}\approx 250.1377667287264$ | ||
| 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$$\chi_{2940}(1,·)$, $\chi_{2940}(2659,·)$, $\chi_{2940}(2561,·)$, $\chi_{2940}(841,·)$, $\chi_{2940}(139,·)$, $\chi_{2940}(461,·)$, $\chi_{2940}(239,·)$, $\chi_{2940}(1681,·)$, $\chi_{2940}(659,·)$, $\chi_{2940}(1301,·)$, $\chi_{2940}(2521,·)$, $\chi_{2940}(1819,·)$, $\chi_{2940}(2141,·)$, $\chi_{2940}(1499,·)$, $\chi_{2940}(2339,·)$, $\chi_{2940}(421,·)$, $\chi_{2940}(41,·)$, $\chi_{2940}(2759,·)$, $\chi_{2940}(1261,·)$, $\chi_{2940}(559,·)$, $\chi_{2940}(881,·)$, $\chi_{2940}(979,·)$, $\chi_{2940}(2101,·)$, $\chi_{2940}(1399,·)$, $\chi_{2940}(1079,·)$, $\chi_{2940}(1721,·)$, $\chi_{2940}(1919,·)$, $\chi_{2940}(2239,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{67}a^{22}-\frac{13}{67}a^{21}-\frac{25}{67}a^{19}-\frac{12}{67}a^{18}+\frac{20}{67}a^{17}-\frac{12}{67}a^{16}+\frac{31}{67}a^{15}+\frac{1}{67}a^{14}+\frac{10}{67}a^{13}+\frac{27}{67}a^{12}+\frac{8}{67}a^{11}-\frac{18}{67}a^{10}-\frac{20}{67}a^{9}-\frac{17}{67}a^{8}-\frac{3}{67}a^{7}+\frac{24}{67}a^{6}+\frac{26}{67}a^{5}+\frac{7}{67}a^{4}-\frac{22}{67}a^{3}-\frac{19}{67}a^{2}-\frac{12}{67}a+\frac{18}{67}$, $\frac{1}{67}a^{23}+\frac{32}{67}a^{21}-\frac{25}{67}a^{20}-\frac{2}{67}a^{19}-\frac{2}{67}a^{18}-\frac{20}{67}a^{17}+\frac{9}{67}a^{16}+\frac{2}{67}a^{15}+\frac{23}{67}a^{14}+\frac{23}{67}a^{13}+\frac{24}{67}a^{12}+\frac{19}{67}a^{11}+\frac{14}{67}a^{10}-\frac{9}{67}a^{9}-\frac{23}{67}a^{8}-\frac{15}{67}a^{7}+\frac{3}{67}a^{6}+\frac{10}{67}a^{5}+\frac{2}{67}a^{4}+\frac{30}{67}a^{3}+\frac{9}{67}a^{2}-\frac{4}{67}a+\frac{33}{67}$, $\frac{1}{67}a^{24}-\frac{11}{67}a^{21}-\frac{2}{67}a^{20}-\frac{6}{67}a^{19}+\frac{29}{67}a^{18}-\frac{28}{67}a^{17}-\frac{16}{67}a^{16}-\frac{31}{67}a^{15}-\frac{9}{67}a^{14}-\frac{28}{67}a^{13}+\frac{26}{67}a^{12}+\frac{26}{67}a^{11}+\frac{31}{67}a^{10}+\frac{14}{67}a^{9}-\frac{7}{67}a^{8}+\frac{32}{67}a^{7}-\frac{21}{67}a^{6}-\frac{26}{67}a^{5}+\frac{7}{67}a^{4}-\frac{24}{67}a^{3}+\frac{1}{67}a^{2}+\frac{15}{67}a+\frac{27}{67}$, $\frac{1}{67}a^{25}-\frac{11}{67}a^{21}-\frac{6}{67}a^{20}+\frac{22}{67}a^{19}-\frac{26}{67}a^{18}+\frac{3}{67}a^{17}-\frac{29}{67}a^{16}-\frac{3}{67}a^{15}-\frac{17}{67}a^{14}+\frac{2}{67}a^{13}-\frac{12}{67}a^{12}-\frac{15}{67}a^{11}+\frac{17}{67}a^{10}-\frac{26}{67}a^{9}-\frac{21}{67}a^{8}+\frac{13}{67}a^{7}-\frac{30}{67}a^{6}+\frac{25}{67}a^{5}-\frac{14}{67}a^{4}+\frac{27}{67}a^{3}+\frac{7}{67}a^{2}+\frac{29}{67}a-\frac{3}{67}$, $\frac{1}{48076319}a^{26}-\frac{332650}{48076319}a^{25}-\frac{317827}{48076319}a^{24}+\frac{161797}{48076319}a^{23}-\frac{94965}{48076319}a^{22}+\frac{5447216}{48076319}a^{21}-\frac{4574476}{48076319}a^{20}-\frac{69934}{48076319}a^{19}-\frac{6822267}{48076319}a^{18}+\frac{4368462}{48076319}a^{17}-\frac{2225245}{48076319}a^{16}-\frac{14868263}{48076319}a^{15}-\frac{15255835}{48076319}a^{14}+\frac{2377844}{48076319}a^{13}-\frac{9256310}{48076319}a^{12}+\frac{1828177}{48076319}a^{11}-\frac{18569321}{48076319}a^{10}-\frac{20407886}{48076319}a^{9}-\frac{11676962}{48076319}a^{8}-\frac{19813976}{48076319}a^{7}-\frac{6732373}{48076319}a^{6}+\frac{5006541}{48076319}a^{5}-\frac{15325999}{48076319}a^{4}-\frac{1140009}{48076319}a^{3}-\frac{12860553}{48076319}a^{2}-\frac{21397188}{48076319}a-\frac{16755945}{48076319}$, $\frac{1}{54\cdots 71}a^{27}+\frac{20\cdots 76}{54\cdots 71}a^{26}-\frac{12\cdots 90}{54\cdots 71}a^{25}-\frac{18\cdots 43}{54\cdots 71}a^{24}-\frac{37\cdots 76}{54\cdots 71}a^{23}+\frac{11\cdots 20}{54\cdots 71}a^{22}-\frac{15\cdots 11}{54\cdots 71}a^{21}+\frac{15\cdots 65}{54\cdots 71}a^{20}-\frac{85\cdots 21}{54\cdots 71}a^{19}+\frac{14\cdots 31}{54\cdots 71}a^{18}+\frac{57\cdots 79}{81\cdots 13}a^{17}+\frac{23\cdots 23}{54\cdots 71}a^{16}+\frac{13\cdots 70}{54\cdots 71}a^{15}-\frac{18\cdots 34}{54\cdots 71}a^{14}+\frac{12\cdots 75}{54\cdots 71}a^{13}-\frac{94\cdots 75}{54\cdots 71}a^{12}+\frac{22\cdots 82}{54\cdots 71}a^{11}-\frac{57\cdots 42}{54\cdots 71}a^{10}+\frac{69\cdots 16}{54\cdots 71}a^{9}-\frac{34\cdots 01}{17\cdots 41}a^{8}-\frac{17\cdots 42}{54\cdots 71}a^{7}-\frac{14\cdots 80}{54\cdots 71}a^{6}+\frac{30\cdots 31}{54\cdots 71}a^{5}+\frac{26\cdots 93}{54\cdots 71}a^{4}-\frac{22\cdots 28}{54\cdots 71}a^{3}-\frac{10\cdots 76}{54\cdots 71}a^{2}+\frac{23\cdots 63}{54\cdots 71}a+\frac{23\cdots 92}{83\cdots 93}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
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Unit group
| Rank: | $27$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: | not computed |
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| Regulator: | not computed |
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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 = \mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{14093520972294144372749526040992276772190449136895590400000000000000}}\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 | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{14}$ | ${\href{/padicField/23.14.0.1}{14} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{14}$ | ${\href{/padicField/37.14.0.1}{14} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{4}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $28$ | $2$ | $14$ | $28$ | |||
|
\(3\)
| 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 | $x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$ | $2$ | $7$ | $7$ | $C_{14}$ | $$[\ ]_{2}^{7}$$ | |
|
\(7\)
| 7.1.14.25a5.1 | $x^{14} + 42 x^{12} + 7$ | $14$ | $1$ | $25$ | $C_{14}$ | $$[2]_{2}$$ |
| 7.1.14.25a5.1 | $x^{14} + 42 x^{12} + 7$ | $14$ | $1$ | $25$ | $C_{14}$ | $$[2]_{2}$$ |