Normalized defining polynomial
\( x^{28} - x^{27} + 88 x^{26} - 88 x^{25} + 3394 x^{24} - 3394 x^{23} + 75865 x^{22} - 67948 x^{21} + \cdots + 24797115841 \)
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6963275855162665256600064094162323796066183958530426025390625\)
\(\medspace = 3^{14}\cdot 5^{21}\cdot 29^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(148.92\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{1/2}5^{3/4}29^{27/28}\approx 148.92157770926886$ | ||
| Ramified primes: |
\(3\), \(5\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{145}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $28$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(435=3\cdot 5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(4,·)$, $\chi_{435}(98,·)$, $\chi_{435}(263,·)$, $\chi_{435}(136,·)$, $\chi_{435}(226,·)$, $\chi_{435}(398,·)$, $\chi_{435}(109,·)$, $\chi_{435}(16,·)$, $\chi_{435}(274,·)$, $\chi_{435}(278,·)$, $\chi_{435}(154,·)$, $\chi_{435}(286,·)$, $\chi_{435}(287,·)$, $\chi_{435}(289,·)$, $\chi_{435}(34,·)$, $\chi_{435}(293,·)$, $\chi_{435}(64,·)$, $\chi_{435}(338,·)$, $\chi_{435}(302,·)$, $\chi_{435}(47,·)$, $\chi_{435}(392,·)$, $\chi_{435}(242,·)$, $\chi_{435}(181,·)$, $\chi_{435}(182,·)$, $\chi_{435}(188,·)$, $\chi_{435}(317,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{8192}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17}a^{14}-\frac{6}{17}a^{13}+\frac{2}{17}a^{12}-\frac{4}{17}a^{11}-\frac{7}{17}a^{10}+\frac{7}{17}a^{9}-\frac{6}{17}a^{8}-\frac{3}{17}a^{7}+\frac{6}{17}a^{6}+\frac{1}{17}a^{5}+\frac{4}{17}a^{4}+\frac{7}{17}a^{3}-\frac{2}{17}a$, $\frac{1}{17}a^{15}+\frac{8}{17}a^{12}+\frac{3}{17}a^{11}-\frac{1}{17}a^{10}+\frac{2}{17}a^{9}-\frac{5}{17}a^{8}+\frac{5}{17}a^{7}+\frac{3}{17}a^{6}-\frac{7}{17}a^{5}-\frac{3}{17}a^{4}+\frac{8}{17}a^{3}-\frac{2}{17}a^{2}+\frac{5}{17}a$, $\frac{1}{17}a^{16}+\frac{8}{17}a^{13}+\frac{3}{17}a^{12}-\frac{1}{17}a^{11}+\frac{2}{17}a^{10}-\frac{5}{17}a^{9}+\frac{5}{17}a^{8}+\frac{3}{17}a^{7}-\frac{7}{17}a^{6}-\frac{3}{17}a^{5}+\frac{8}{17}a^{4}-\frac{2}{17}a^{3}+\frac{5}{17}a^{2}$, $\frac{1}{17}a^{17}-\frac{1}{17}a$, $\frac{1}{17}a^{18}-\frac{1}{17}a^{2}$, $\frac{1}{17}a^{19}-\frac{1}{17}a^{3}$, $\frac{1}{17}a^{20}-\frac{1}{17}a^{4}$, $\frac{1}{17}a^{21}-\frac{1}{17}a^{5}$, $\frac{1}{17051}a^{22}-\frac{405}{17051}a^{21}+\frac{1}{1003}a^{20}-\frac{284}{17051}a^{19}-\frac{161}{17051}a^{18}-\frac{208}{17051}a^{17}+\frac{264}{17051}a^{16}-\frac{200}{17051}a^{15}+\frac{420}{17051}a^{14}+\frac{406}{1003}a^{13}-\frac{1702}{17051}a^{12}+\frac{4749}{17051}a^{11}-\frac{2365}{17051}a^{10}-\frac{6158}{17051}a^{9}-\frac{64}{17051}a^{8}-\frac{6126}{17051}a^{7}-\frac{1884}{17051}a^{6}-\frac{4432}{17051}a^{5}+\frac{8149}{17051}a^{4}-\frac{740}{17051}a^{3}-\frac{6347}{17051}a^{2}+\frac{367}{1003}a$, $\frac{1}{17051}a^{23}+\frac{484}{17051}a^{21}-\frac{420}{17051}a^{20}+\frac{164}{17051}a^{19}-\frac{218}{17051}a^{18}+\frac{276}{17051}a^{17}+\frac{402}{17051}a^{16}-\frac{20}{1003}a^{15}+\frac{474}{17051}a^{14}+\frac{1256}{17051}a^{13}+\frac{1491}{17051}a^{12}+\frac{7256}{17051}a^{11}-\frac{4112}{17051}a^{10}-\frac{2602}{17051}a^{9}-\frac{7974}{17051}a^{8}+\frac{4526}{17051}a^{7}-\frac{6175}{17051}a^{6}-\frac{468}{17051}a^{5}-\frac{8289}{17051}a^{4}-\frac{1135}{17051}a^{3}+\frac{3384}{17051}a^{2}+\frac{486}{1003}a$, $\frac{1}{17051}a^{24}+\frac{15}{17051}a^{21}-\frac{40}{17051}a^{20}-\frac{173}{17051}a^{19}-\frac{2}{1003}a^{18}-\frac{229}{17051}a^{17}+\frac{268}{17051}a^{16}-\frac{1}{1003}a^{15}-\frac{421}{17051}a^{14}+\frac{7934}{17051}a^{13}+\frac{5555}{17051}a^{12}+\frac{260}{17051}a^{11}-\frac{359}{17051}a^{10}+\frac{609}{17051}a^{9}+\frac{5412}{17051}a^{8}+\frac{1965}{17051}a^{7}-\frac{7360}{17051}a^{6}-\frac{594}{17051}a^{5}-\frac{3461}{17051}a^{4}+\frac{5479}{17051}a^{3}+\frac{5012}{17051}a^{2}-\frac{333}{1003}a$, $\frac{1}{45509119}a^{25}+\frac{1038}{45509119}a^{24}-\frac{1077}{45509119}a^{23}+\frac{1211}{45509119}a^{22}-\frac{253916}{45509119}a^{21}-\frac{812741}{45509119}a^{20}+\frac{547820}{45509119}a^{19}+\frac{412924}{45509119}a^{18}-\frac{843634}{45509119}a^{17}-\frac{1183063}{45509119}a^{16}-\frac{599205}{45509119}a^{15}-\frac{1105240}{45509119}a^{14}-\frac{10220737}{45509119}a^{13}+\frac{6744769}{45509119}a^{12}-\frac{14688742}{45509119}a^{11}+\frac{19065403}{45509119}a^{10}-\frac{17432461}{45509119}a^{9}+\frac{2324762}{45509119}a^{8}-\frac{12361446}{45509119}a^{7}+\frac{19539079}{45509119}a^{6}+\frac{30317}{289867}a^{5}-\frac{205185}{771341}a^{4}+\frac{19861942}{45509119}a^{3}+\frac{21688147}{45509119}a^{2}-\frac{1279449}{2677007}a+\frac{8}{17}$, $\frac{1}{2685038021}a^{26}+\frac{28}{2685038021}a^{25}+\frac{39495}{2685038021}a^{24}+\frac{34726}{2685038021}a^{23}-\frac{57118}{2685038021}a^{22}+\frac{464667}{2685038021}a^{21}+\frac{10683459}{2685038021}a^{20}+\frac{69263962}{2685038021}a^{19}+\frac{62963511}{2685038021}a^{18}+\frac{52680123}{2685038021}a^{17}-\frac{48560122}{2685038021}a^{16}-\frac{33300087}{2685038021}a^{15}+\frac{2837973}{157943413}a^{14}+\frac{75775475}{2685038021}a^{13}-\frac{1326400698}{2685038021}a^{12}-\frac{917891458}{2685038021}a^{11}+\frac{16838790}{2685038021}a^{10}-\frac{302241693}{2685038021}a^{9}+\frac{1049746178}{2685038021}a^{8}-\frac{520069750}{2685038021}a^{7}+\frac{167522170}{2685038021}a^{6}+\frac{175576909}{2685038021}a^{5}-\frac{1272588396}{2685038021}a^{4}-\frac{1280476692}{2685038021}a^{3}+\frac{1072976089}{2685038021}a^{2}+\frac{797880}{2677007}a-\frac{7}{17}$, $\frac{1}{17\!\cdots\!99}a^{27}-\frac{28\!\cdots\!25}{17\!\cdots\!99}a^{26}+\frac{10\!\cdots\!52}{17\!\cdots\!99}a^{25}+\frac{26\!\cdots\!17}{17\!\cdots\!99}a^{24}-\frac{23\!\cdots\!19}{17\!\cdots\!99}a^{23}+\frac{23\!\cdots\!98}{17\!\cdots\!99}a^{22}-\frac{15\!\cdots\!42}{17\!\cdots\!99}a^{21}+\frac{46\!\cdots\!76}{17\!\cdots\!99}a^{20}-\frac{30\!\cdots\!70}{17\!\cdots\!99}a^{19}-\frac{89\!\cdots\!54}{17\!\cdots\!99}a^{18}-\frac{42\!\cdots\!43}{17\!\cdots\!99}a^{17}+\frac{49\!\cdots\!97}{17\!\cdots\!99}a^{16}-\frac{18\!\cdots\!59}{17\!\cdots\!99}a^{15}+\frac{10\!\cdots\!41}{17\!\cdots\!99}a^{14}-\frac{12\!\cdots\!96}{29\!\cdots\!61}a^{13}-\frac{56\!\cdots\!73}{17\!\cdots\!99}a^{12}-\frac{49\!\cdots\!06}{17\!\cdots\!99}a^{11}-\frac{32\!\cdots\!96}{17\!\cdots\!99}a^{10}+\frac{28\!\cdots\!61}{17\!\cdots\!99}a^{9}-\frac{57\!\cdots\!32}{17\!\cdots\!99}a^{8}+\frac{93\!\cdots\!41}{17\!\cdots\!99}a^{7}+\frac{10\!\cdots\!87}{17\!\cdots\!99}a^{6}+\frac{32\!\cdots\!94}{10\!\cdots\!47}a^{5}-\frac{36\!\cdots\!93}{17\!\cdots\!99}a^{4}-\frac{28\!\cdots\!26}{17\!\cdots\!99}a^{3}+\frac{38\!\cdots\!64}{17\!\cdots\!99}a^{2}-\frac{52\!\cdots\!05}{11\!\cdots\!69}a-\frac{20\!\cdots\!86}{70\!\cdots\!39}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $17$ |
Class group and class number
not computed
Unit group
| Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
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 $
Galois group
| A cyclic group of order 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{145}) \), 4.0.27437625.2, 7.7.594823321.1, 14.14.801611618199890796015625.1 |
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 | ${\href{/padicField/2.7.0.1}{7} }^{4}$ | R | R | $28$ | $28$ | $28$ | ${\href{/padicField/17.1.0.1}{1} }^{28}$ | $28$ | $28$ | R | $28$ | ${\href{/padicField/37.7.0.1}{7} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{4}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | $28$ | ${\href{/padicField/59.1.0.1}{1} }^{28}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.14.7.2 | $x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
| 3.14.7.2 | $x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(5\)
| Deg $28$ | $4$ | $7$ | $21$ | |||
|
\(29\)
| Deg $28$ | $28$ | $1$ | $27$ |