Normalized defining polynomial
\( x^{22} - 9 x^{21} - 12 x^{20} + 337 x^{19} - 383 x^{18} - 5876 x^{17} + 18523 x^{16} + 16196 x^{15} - 84841 x^{14} - 300133 x^{13} + 1731641 x^{12} - 3103687 x^{11} + 4035441 x^{10} - 12380083 x^{9} + 63615149 x^{8} - 174132886 x^{7} + 358371438 x^{6} - 556676137 x^{5} + 1436734353 x^{4} - 2616581008 x^{3} + 5644816006 x^{2} - 5254555364 x + 5354971187 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-277408439698811100133951455282636371147535786589811=-\,11^{11}\cdot 89^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $196.28$ | 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: | $11, 89$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(979=11\cdot 89\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{979}(1,·)$, $\chi_{979}(67,·)$, $\chi_{979}(714,·)$, $\chi_{979}(395,·)$, $\chi_{979}(716,·)$, $\chi_{979}(461,·)$, $\chi_{979}(78,·)$, $\chi_{979}(331,·)$, $\chi_{979}(210,·)$, $\chi_{979}(846,·)$, $\chi_{979}(153,·)$, $\chi_{979}(538,·)$, $\chi_{979}(32,·)$, $\chi_{979}(802,·)$, $\chi_{979}(868,·)$, $\chi_{979}(364,·)$, $\chi_{979}(45,·)$, $\chi_{979}(879,·)$, $\chi_{979}(186,·)$, $\chi_{979}(892,·)$, $\chi_{979}(573,·)$, $\chi_{979}(639,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37} a^{16} - \frac{14}{37} a^{15} - \frac{14}{37} a^{14} + \frac{15}{37} a^{13} + \frac{7}{37} a^{12} + \frac{17}{37} a^{11} + \frac{12}{37} a^{10} - \frac{4}{37} a^{9} - \frac{3}{37} a^{8} + \frac{9}{37} a^{7} - \frac{7}{37} a^{6} - \frac{5}{37} a^{5} + \frac{4}{37} a^{4} - \frac{18}{37} a^{3} + \frac{18}{37} a^{2} - \frac{12}{37} a$, $\frac{1}{37} a^{17} + \frac{12}{37} a^{15} + \frac{4}{37} a^{14} - \frac{5}{37} a^{13} + \frac{4}{37} a^{12} - \frac{9}{37} a^{11} + \frac{16}{37} a^{10} + \frac{15}{37} a^{9} + \frac{4}{37} a^{8} + \frac{8}{37} a^{7} + \frac{8}{37} a^{6} + \frac{8}{37} a^{5} + \frac{1}{37} a^{4} - \frac{12}{37} a^{3} + \frac{18}{37} a^{2} + \frac{17}{37} a$, $\frac{1}{37} a^{18} - \frac{13}{37} a^{15} + \frac{15}{37} a^{14} + \frac{9}{37} a^{13} + \frac{18}{37} a^{12} - \frac{3}{37} a^{11} - \frac{18}{37} a^{10} + \frac{15}{37} a^{9} + \frac{7}{37} a^{8} + \frac{11}{37} a^{7} + \frac{18}{37} a^{6} - \frac{13}{37} a^{5} + \frac{14}{37} a^{4} + \frac{12}{37} a^{3} - \frac{14}{37} a^{2} - \frac{4}{37} a$, $\frac{1}{6623} a^{19} - \frac{67}{6623} a^{18} - \frac{22}{6623} a^{17} + \frac{69}{6623} a^{16} + \frac{2323}{6623} a^{15} - \frac{3120}{6623} a^{14} + \frac{903}{6623} a^{13} - \frac{279}{6623} a^{12} - \frac{1}{6623} a^{11} + \frac{2667}{6623} a^{10} + \frac{2340}{6623} a^{9} - \frac{459}{6623} a^{8} + \frac{435}{6623} a^{7} + \frac{3063}{6623} a^{6} - \frac{1810}{6623} a^{5} - \frac{3210}{6623} a^{4} + \frac{3113}{6623} a^{3} - \frac{1131}{6623} a^{2} + \frac{1241}{6623} a + \frac{25}{179}$, $\frac{1}{1347040840969097} a^{20} - \frac{47019940027}{1347040840969097} a^{19} + \frac{6175507262437}{1347040840969097} a^{18} - \frac{4939204332881}{1347040840969097} a^{17} - \frac{5524616660646}{1347040840969097} a^{16} - \frac{304509124608480}{1347040840969097} a^{15} + \frac{385765121712685}{1347040840969097} a^{14} - \frac{13874633745510}{36406509215381} a^{13} - \frac{416181452762108}{1347040840969097} a^{12} - \frac{623203886486345}{1347040840969097} a^{11} + \frac{26275425283012}{1347040840969097} a^{10} - \frac{392757813602394}{1347040840969097} a^{9} - \frac{414495262542772}{1347040840969097} a^{8} - \frac{331677529813}{3110948824409} a^{7} - \frac{569390113394577}{1347040840969097} a^{6} - \frac{572811759753226}{1347040840969097} a^{5} - \frac{16556202528082}{1347040840969097} a^{4} + \frac{5232897180586}{13337038029397} a^{3} - \frac{176688735160743}{1347040840969097} a^{2} - \frac{352198913104622}{1347040840969097} a + \frac{37332421934}{84079697957}$, $\frac{1}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{21} + \frac{39976470317438379166369529325464192553329007469177760962}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{20} - \frac{18339945106025534775126370187958785245977650311462923237739864295353}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{19} + \frac{3713028964230726283280898251510710678564570087134600532058064241999344}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{18} - \frac{4235384374555256649773047588824679266122994421395971808950152195555778}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{17} - \frac{2883901466628334546853679593569177944032901850229526481530530953876007}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{16} - \frac{172110079641869508420976645065617818515411476805323674707764945272755735}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{15} + \frac{28258429228794845903826206184866642249437854139102135729568574950810290}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{14} + \frac{13530008858511971107829859866576679878850923183848908627473011544707442}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{13} - \frac{107736702170676564626426639037844867599355920379620273704292393789518491}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{12} - \frac{23583937945922881385349227337124080234523481468745849669908113440319615}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{11} + \frac{152885035673082420744676724162510589020013165205931987950374770551669433}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{10} + \frac{41791659610443168981789548474856800434103989151312505241956003445467187}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{9} + \frac{77262385895881588719139705365313284811004752706972194242255027011525648}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{8} + \frac{109958880328987800067928763096295091122169851916106989847759819183047991}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{7} - \frac{66288710277796257653948157260494725807516263022395325094509770032852782}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{6} + \frac{62649445645562878426279846206678171029808501860563262954249066258079314}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{5} + \frac{47768993913867278731106860756963116048834048410073592798980970484985383}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{4} + \frac{106963489841270110736461155576033420664620193459844799318639769788278890}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{3} - \frac{36356380176272497323023083439646758526230153006008298660129403887524408}{427444632439259046265593692751404965520448772591636228076471526035520841} a^{2} - \frac{62544473956957089054607805550061731754818668769813882091045584728525403}{427444632439259046265593692751404965520448772591636228076471526035520841} a + \frac{6641953833679543462201797284950536520660481620919248066746230359147}{26680271670885652972073758988290678829064900604933289312556739656421}$
Class group and class number
$C_{23}\times C_{517891}$, which has order $11911493$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 866679281.3791491 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 11.11.31181719929966183601.1 |
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 | $22$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ | $22$ | $22$ | ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| $89$ | 89.11.10.1 | $x^{11} - 89$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 89.11.10.1 | $x^{11} - 89$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |