Normalized defining polynomial
\( x^{20} - x^{19} + x^{18} - 93 x^{17} + 209 x^{16} - 800 x^{15} + 4981 x^{14} - 12556 x^{13} + 52672 x^{12} - 186410 x^{11} + 461402 x^{10} - 1486634 x^{9} + 3730667 x^{8} - 7453323 x^{7} + 16223240 x^{6} - 23433182 x^{5} + 16658019 x^{4} - 7158636 x^{3} + 5906450 x^{2} - 2576267 x + 1575113 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(396508891893913150393870750167904820789=11^{16}\cdot 29^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $85.10$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $11, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(319=11\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(133,·)$, $\chi_{319}(262,·)$, $\chi_{319}(202,·)$, $\chi_{319}(75,·)$, $\chi_{319}(12,·)$, $\chi_{319}(144,·)$, $\chi_{319}(273,·)$, $\chi_{319}(146,·)$, $\chi_{319}(278,·)$, $\chi_{319}(86,·)$, $\chi_{319}(157,·)$, $\chi_{319}(289,·)$, $\chi_{319}(291,·)$, $\chi_{319}(70,·)$, $\chi_{319}(104,·)$, $\chi_{319}(302,·)$, $\chi_{319}(115,·)$, $\chi_{319}(59,·)$, $\chi_{319}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} - \frac{7}{23} a^{13} + \frac{4}{23} a^{12} - \frac{5}{23} a^{11} - \frac{7}{23} a^{10} - \frac{7}{23} a^{9} + \frac{8}{23} a^{8} + \frac{1}{23} a^{7} - \frac{9}{23} a^{6} - \frac{7}{23} a^{5} - \frac{6}{23} a^{4} - \frac{4}{23} a^{2} + \frac{2}{23} a - \frac{10}{23}$, $\frac{1}{161} a^{15} + \frac{47}{161} a^{13} + \frac{1}{7} a^{12} + \frac{27}{161} a^{11} + \frac{36}{161} a^{10} - \frac{41}{161} a^{9} + \frac{11}{161} a^{8} + \frac{3}{23} a^{7} - \frac{1}{161} a^{6} - \frac{78}{161} a^{5} + \frac{4}{161} a^{4} - \frac{27}{161} a^{3} - \frac{26}{161} a^{2} - \frac{6}{23} a - \frac{1}{161}$, $\frac{1}{161} a^{16} - \frac{2}{161} a^{14} + \frac{44}{161} a^{13} - \frac{8}{161} a^{12} - \frac{41}{161} a^{11} - \frac{20}{161} a^{10} + \frac{32}{161} a^{9} - \frac{7}{23} a^{8} - \frac{50}{161} a^{7} + \frac{41}{161} a^{6} + \frac{25}{161} a^{5} - \frac{55}{161} a^{4} - \frac{26}{161} a^{3} - \frac{1}{23} a^{2} + \frac{62}{161} a + \frac{1}{23}$, $\frac{1}{161} a^{17} + \frac{2}{161} a^{14} + \frac{58}{161} a^{13} - \frac{2}{161} a^{12} - \frac{78}{161} a^{11} + \frac{76}{161} a^{10} + \frac{2}{161} a^{9} - \frac{6}{23} a^{8} + \frac{41}{161} a^{7} + \frac{79}{161} a^{6} - \frac{78}{161} a^{5} + \frac{73}{161} a^{4} - \frac{61}{161} a^{3} + \frac{17}{161} a^{2} - \frac{65}{161}$, $\frac{1}{3703} a^{18} + \frac{2}{3703} a^{17} + \frac{8}{3703} a^{15} - \frac{43}{3703} a^{14} - \frac{479}{3703} a^{13} + \frac{63}{529} a^{12} - \frac{1003}{3703} a^{11} - \frac{988}{3703} a^{10} - \frac{998}{3703} a^{9} + \frac{1276}{3703} a^{8} + \frac{233}{529} a^{7} + \frac{697}{3703} a^{6} + \frac{71}{161} a^{5} - \frac{1676}{3703} a^{4} + \frac{1343}{3703} a^{3} + \frac{1747}{3703} a^{2} + \frac{117}{3703} a - \frac{535}{3703}$, $\frac{1}{10902267357544506598826591424528294404112914508693958647875881} a^{19} - \frac{1306184448337962056009708088355406393696682444038309521689}{10902267357544506598826591424528294404112914508693958647875881} a^{18} - \frac{28785129107973221500788271469662715939842057955119100688654}{10902267357544506598826591424528294404112914508693958647875881} a^{17} - \frac{8922982604865467929515467533310602605685439783590606822582}{10902267357544506598826591424528294404112914508693958647875881} a^{16} + \frac{27814875789496862767472731453175305332280186097789382233753}{10902267357544506598826591424528294404112914508693958647875881} a^{15} + \frac{47182522936136956528421941530378638870190621793497772407004}{10902267357544506598826591424528294404112914508693958647875881} a^{14} - \frac{492311897655579366761832206698252953728162896554800921667220}{1557466765363500942689513060646899200587559215527708378267983} a^{13} + \frac{4657057598024356653585857367737702513576584491505495203150167}{10902267357544506598826591424528294404112914508693958647875881} a^{12} + \frac{5218742962061836744124261711267665651251567834302117739079645}{10902267357544506598826591424528294404112914508693958647875881} a^{11} + \frac{16817897754405850266963493564826976851462827539282345865094}{474011624241065504296808322805578017570126717769302549907647} a^{10} - \frac{3194226057408501459455525459426793974044353233550274578079185}{10902267357544506598826591424528294404112914508693958647875881} a^{9} + \frac{1067427303002632247179177026480368782780604661723204220506671}{10902267357544506598826591424528294404112914508693958647875881} a^{8} - \frac{2633042802473434046489214380075987873487858084026227337734430}{10902267357544506598826591424528294404112914508693958647875881} a^{7} - \frac{3602644426055422753788751324542232890244003660154421015401824}{10902267357544506598826591424528294404112914508693958647875881} a^{6} + \frac{1531665760409168525130623062292992836723842469816310407700932}{10902267357544506598826591424528294404112914508693958647875881} a^{5} + \frac{222436908487826967083882153561752990890981764396469431881366}{10902267357544506598826591424528294404112914508693958647875881} a^{4} - \frac{3972558657755358412380000437677566972692610929094885084475490}{10902267357544506598826591424528294404112914508693958647875881} a^{3} + \frac{4350295495665970914212032969698743917940668633340299587489710}{10902267357544506598826591424528294404112914508693958647875881} a^{2} + \frac{3988387430073185151550130335775884044674991876212690399639088}{10902267357544506598826591424528294404112914508693958647875881} a + \frac{5442246688387434765123047666994484438088491928704481272371712}{10902267357544506598826591424528294404112914508693958647875881}$
Class group and class number
$C_{8221}$, which has order $8221$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 15197121.6751 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.0.24389.1, \(\Q(\zeta_{11})^+\), 10.10.4396746947664269.1 |
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 | $20$ | $20$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ | R | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| 29 | Data not computed | ||||||