Normalized defining polynomial
\( x^{26} - x^{25} + 81 x^{24} + 127 x^{23} + 2487 x^{22} + 10345 x^{21} + 50834 x^{20} + 263974 x^{19} + 916837 x^{18} + 3605259 x^{17} + 11849642 x^{16} + 33958807 x^{15} + 95600897 x^{14} + 229517195 x^{13} + 508321268 x^{12} + 1046082342 x^{11} + 1876093298 x^{10} + 3098569897 x^{9} + 4639987670 x^{8} + 5825110231 x^{7} + 6533079805 x^{6} + 6257078964 x^{5} + 4117698851 x^{4} + 3036419703 x^{3} + 4545159242 x^{2} + 2836865762 x + 672625141 \)
Invariants
| Degree: | $26$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 13]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-336734286382459698713579075785535859888935158690185546875=-\,5^{13}\cdot 79^{25}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $149.32$ | 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: | $5, 79$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(395=5\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{395}(1,·)$, $\chi_{395}(131,·)$, $\chi_{395}(196,·)$, $\chi_{395}(69,·)$, $\chi_{395}(326,·)$, $\chi_{395}(199,·)$, $\chi_{395}(264,·)$, $\chi_{395}(394,·)$, $\chi_{395}(141,·)$, $\chi_{395}(14,·)$, $\chi_{395}(294,·)$, $\chi_{395}(146,·)$, $\chi_{395}(21,·)$, $\chi_{395}(219,·)$, $\chi_{395}(349,·)$, $\chi_{395}(94,·)$, $\chi_{395}(229,·)$, $\chi_{395}(101,·)$, $\chi_{395}(166,·)$, $\chi_{395}(301,·)$, $\chi_{395}(46,·)$, $\chi_{395}(176,·)$, $\chi_{395}(374,·)$, $\chi_{395}(249,·)$, $\chi_{395}(381,·)$, $\chi_{395}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{103} a^{22} + \frac{17}{103} a^{21} - \frac{13}{103} a^{20} - \frac{17}{103} a^{19} - \frac{34}{103} a^{18} - \frac{7}{103} a^{17} + \frac{45}{103} a^{16} - \frac{51}{103} a^{15} + \frac{36}{103} a^{14} + \frac{29}{103} a^{13} - \frac{9}{103} a^{12} + \frac{3}{103} a^{11} + \frac{14}{103} a^{10} - \frac{3}{103} a^{9} + \frac{31}{103} a^{8} + \frac{30}{103} a^{7} - \frac{24}{103} a^{6} + \frac{41}{103} a^{5} - \frac{28}{103} a^{4} + \frac{48}{103} a^{3} + \frac{22}{103} a^{2} - \frac{29}{103} a + \frac{1}{103}$, $\frac{1}{103} a^{23} + \frac{7}{103} a^{21} - \frac{2}{103} a^{20} + \frac{49}{103} a^{19} - \frac{47}{103} a^{18} - \frac{42}{103} a^{17} + \frac{8}{103} a^{16} - \frac{24}{103} a^{15} + \frac{35}{103} a^{14} + \frac{13}{103} a^{13} - \frac{50}{103} a^{12} - \frac{37}{103} a^{11} - \frac{35}{103} a^{10} - \frac{21}{103} a^{9} + \frac{18}{103} a^{8} - \frac{19}{103} a^{7} + \frac{37}{103} a^{6} - \frac{4}{103} a^{5} + \frac{9}{103} a^{4} + \frac{30}{103} a^{3} + \frac{9}{103} a^{2} - \frac{21}{103} a - \frac{17}{103}$, $\frac{1}{4738103} a^{24} + \frac{1843}{4738103} a^{23} - \frac{13470}{4738103} a^{22} + \frac{1599474}{4738103} a^{21} - \frac{1877518}{4738103} a^{20} - \frac{1912847}{4738103} a^{19} + \frac{2257382}{4738103} a^{18} - \frac{1789782}{4738103} a^{17} - \frac{1509578}{4738103} a^{16} - \frac{973455}{4738103} a^{15} + \frac{592969}{4738103} a^{14} - \frac{98609}{4738103} a^{13} + \frac{1940580}{4738103} a^{12} + \frac{1758012}{4738103} a^{11} - \frac{691984}{4738103} a^{10} + \frac{627780}{4738103} a^{9} + \frac{1062312}{4738103} a^{8} - \frac{552796}{4738103} a^{7} + \frac{1824880}{4738103} a^{6} + \frac{2124775}{4738103} a^{5} - \frac{572888}{4738103} a^{4} - \frac{1449484}{4738103} a^{3} + \frac{2142220}{4738103} a^{2} - \frac{1527637}{4738103} a - \frac{1729064}{4738103}$, $\frac{1}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{25} - \frac{2131161523329516444924962438409745042807266688915761367032990740981119655523916124681573054710917136349185421113}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{24} + \frac{733407331098205491393505688547327613545566628081458784357178274909702916866627583686271167815989712629166265474699815}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{23} + \frac{430329478019464649488126077975270712846630498137654507290280328615253029562394486041448260508112763051451783545511884}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{22} + \frac{14023152577344451641921153503366525022067365658918142181139541270330955510314830288725475361495914982019410393047323266}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{21} - \frac{24696509816369338772119456856951404908349202812250295051408685473957459344061369405784458377594529578475296618520495530}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{20} - \frac{52294754252206334565171962952635460575720266902356400358141453916528968199519169593089382971251143229551975173678735944}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{19} - \frac{51063956094442441284682258553405660359393767410101842044213852408552210470804037373116687211152425321831139508658632}{123601948146718910895864888141077291935475972448062651891411847113122040164057689600088096734463616489089546738466057} a^{18} - \frac{75687806906761516478995283017995192218504613565755401881454494065665698647550998540095606739672883524477107018802575042}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{17} + \frac{16946537126266432177314871953566707567206240911660508101858515576896661312216003694278204828141954643800277584058317567}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{16} + \frac{21509352087586947555686050228045185651720898637249911003978450145916532280578368879316664226874621592547138889323128161}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{15} - \frac{19392556663181006234974321383032885487061930399869109027507249198612313963677500276178503051605560562825416759442795430}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{14} + \frac{23127741594326563675968861768847967732028146609667979985314931940173170031818176632008909508395152814986884336507709461}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{13} - \frac{29066244187648252877810391739692950719793747928553707974522330749266527323436712449465616356625574336722532155102779410}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{12} + \frac{40225968552821348859949984853910628335992218454360003569189916068120356604184459327808841112477586139209673588142889167}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{11} - \frac{8514170474027165959087420703290158866873389006087739666498778423280139705755279457263835484683356339368355138352179655}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{10} - \frac{65528473371988292144686100282985825270055740522989642624497321463561957813761385722317198888769800529340489460824857656}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{9} - \frac{13510329265162754427097226399153177197828092428493061283288795084240519311905147565162514076416710125950249426799863423}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{8} + \frac{10096199408434197447742798297902482388691673599970936006328743825779210454246940616154604086904378504254389635920884173}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{7} - \frac{71620234091115369349478714119415787021756872300724969316503341390550033915912324214987884573698524898992354719284173836}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{6} - \frac{60787254162524059760398615286904455452996166663804448736489638243106336051317759876306907636224713195571391655995823451}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{5} - \frac{23384493499673357088592153949374270001866432202226460084684438068901929750020504135095264301517459000062877492391966827}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{4} + \frac{48333365015238228629214610562414723542104010476209243990489885252738686096038465072116660981395708141806076143058193917}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{3} - \frac{80377147210293204994252077042825862577049353019216968047653662315397760232806659397196570623838181971494433521435982559}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a^{2} - \frac{69914127326446929339012808886049581938017273106304642195105657497166587807497870372183694592049911669546742220682929744}{163030969605522243471645787458080948062892807658994637844772226342207970976392092582516199592757510149109112148036729183} a + \frac{65773333943027925593834956076152076917949289021484069015003044871493762580081300352657225589771791644615019740865003}{900723588980785875533954626840226232391672970491683082015316167636508126941392776698984528136781824028227138939429443}$
Class group and class number
$C_{4885304}$, which has order $4885304$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 57529828940.82975 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-395}) \), 13.13.59091511031674153381441.1 |
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 | $26$ | ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ | R | ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | $26$ | ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ | $26$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 79 | Data not computed | ||||||