Normalized defining polynomial
\( x^{26} - x^{25} - 184 x^{24} + 183 x^{23} + 14268 x^{22} - 14086 x^{21} - 612490 x^{20} + 598585 x^{19} + 16089370 x^{18} - 15388122 x^{17} - 270313194 x^{16} + 246233523 x^{15} + 2954365608 x^{14} - 2442913785 x^{13} - 20986142706 x^{12} + 14563237893 x^{11} + 95061660594 x^{10} - 48083263759 x^{9} - 262326941485 x^{8} + 68701741726 x^{7} + 399116329504 x^{6} + 10753142193 x^{5} - 256395661245 x^{4} - 85248420923 x^{3} + 7165503790 x^{2} + 643114124 x - 4674283 \)
Invariants
| Degree: | $26$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[26, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3874006276495686122600289871040620757839501838095241811329=13^{13}\cdot 53^{25}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $164.03$ | 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: | $13, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(689=13\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{689}(64,·)$, $\chi_{689}(1,·)$, $\chi_{689}(66,·)$, $\chi_{689}(324,·)$, $\chi_{689}(261,·)$, $\chi_{689}(521,·)$, $\chi_{689}(651,·)$, $\chi_{689}(272,·)$, $\chi_{689}(467,·)$, $\chi_{689}(599,·)$, $\chi_{689}(664,·)$, $\chi_{689}(25,·)$, $\chi_{689}(90,·)$, $\chi_{689}(222,·)$, $\chi_{689}(417,·)$, $\chi_{689}(38,·)$, $\chi_{689}(168,·)$, $\chi_{689}(428,·)$, $\chi_{689}(365,·)$, $\chi_{689}(623,·)$, $\chi_{689}(688,·)$, $\chi_{689}(625,·)$, $\chi_{689}(183,·)$, $\chi_{689}(248,·)$, $\chi_{689}(441,·)$, $\chi_{689}(506,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{83} a^{23} + \frac{8}{83} a^{22} + \frac{15}{83} a^{21} + \frac{17}{83} a^{20} - \frac{20}{83} a^{19} + \frac{10}{83} a^{18} + \frac{28}{83} a^{17} - \frac{37}{83} a^{16} - \frac{1}{83} a^{15} - \frac{6}{83} a^{14} + \frac{37}{83} a^{13} - \frac{29}{83} a^{12} + \frac{3}{83} a^{11} - \frac{29}{83} a^{10} - \frac{20}{83} a^{9} + \frac{30}{83} a^{8} - \frac{18}{83} a^{7} + \frac{39}{83} a^{6} - \frac{33}{83} a^{5} - \frac{22}{83} a^{4} + \frac{20}{83} a^{3} + \frac{14}{83} a^{2} - \frac{12}{83} a + \frac{5}{83}$, $\frac{1}{8881} a^{24} + \frac{2}{8881} a^{23} + \frac{1046}{8881} a^{22} + \frac{2666}{8881} a^{21} - \frac{1118}{8881} a^{20} + \frac{2039}{8881} a^{19} - \frac{2688}{8881} a^{18} - \frac{4438}{8881} a^{17} + \frac{1798}{8881} a^{16} + \frac{50}{107} a^{15} - \frac{1006}{8881} a^{14} - \frac{1330}{8881} a^{13} + \frac{3248}{8881} a^{12} - \frac{545}{8881} a^{11} + \frac{3059}{8881} a^{10} + \frac{4217}{8881} a^{9} - \frac{2522}{8881} a^{8} + \frac{728}{8881} a^{7} + \frac{2555}{8881} a^{6} - \frac{322}{8881} a^{5} + \frac{1231}{8881} a^{4} - \frac{3094}{8881} a^{3} + \frac{1481}{8881} a^{2} - \frac{3077}{8881} a + \frac{3788}{8881}$, $\frac{1}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{25} - \frac{262229766182189033459803762371596067811086621655366099705684538252595028931390434036314433846673362687718035946901655520677306}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{24} - \frac{4516343873933743257797341492629611374717699526199185136886719272664013921856870670777143823861956947186435304804944476639703774}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{23} + \frac{1347680273374078846793406141885124617685911329638319175270143816027364429343369336522594219243500317783238514928649249534125215233}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{22} + \frac{1959496744653875652349679161340000036583818459075355948449913938777592686246070336386346093866041685851758190758569871018289697953}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{21} + \frac{1501714589747840550965531477501187149702200037652194569171322066989794343330794040168234783413881656503953611420331216482025129143}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{20} - \frac{2325744477725484911842517997116442533665765099581325743901885601007769914257140955499892964954352488398173356944089549446204156187}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{19} - \frac{2214836688591980236712937688433222156181007974526021694334674945241206907558954839038229009728269539534868872392685941098109621861}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{18} - \frac{1126213944359866884427288943980693230427719535568621787880152491363558445838198833109290874657809978627602766333095060745641538659}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{17} + \frac{1764006264688503183911107556820853206357248568874103620389043134954961880142687337384869972920236017216983349098252103222227556042}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{16} - \frac{2201306257585932817025446685676105777954801327019018768710967644986833618023380353464182499147540447875057372410023085813571321746}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{15} - \frac{1985773236696781286862128048113470399372485247649072851436115847943904474731377976069902732676415796938845463713666132740553321851}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{14} - \frac{2574878087724596423748818165781705528915895026502117652316156585092209754663626795818595303916488706594394932835793139493138106078}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{13} + \frac{647178750140386983916191813327144697178371176338114967383944300346833224418237822130955844573256808324224491641104273169739149475}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{12} - \frac{2237041774157855036824302910511791397677919143088196116541169143141226428366429479986905637971216762961648342728844142896487134711}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{11} - \frac{1475284666277259328405730182228635995632928899065973842923896211314243343366846061567201458916923352826527988651888769519577560559}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{10} - \frac{160201204688896196759879488153765242219177142590822630184996682371177175625702388880965140477661250390643823008790379664392446047}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{9} + \frac{580959796299775223248911098977382245974127160572206833365689921357599152146334724713920789464431044326237009274467962283032007640}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{8} - \frac{1948398863764371881353196572542722677518149321664073976844223493282272971443664976028942238519681218074613242120203101585141356139}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{7} + \frac{355954833051993597274346194163320976889256329490758284010367785595041926849584797800395547151157021522956323490463645009895909899}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{6} + \frac{155444304639652019910306099328976248101734112768791155062417404989420481023556780139542577094869173571958714777625143485009133255}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{5} + \frac{812627618128078519678725025141857672008195832448419967938812341406334578259863707458409078074312665642920768006895964662621566836}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{4} - \frac{2442416408998338018095348577074754572364396431221815740723897469357908845915805844727510386060219614829339250541129723552714617679}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{3} - \frac{1055073772691166358752456719170832228032999413254617292153226678857718546533993537142688155471982866154837917752526061692143791682}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a^{2} - \frac{1307040137091849162089739997566299851014923669742281916475358137015213555135430008143484871925695795322278213868942727397856418684}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173} a + \frac{889040991639096509675273415893943500901210309823045088344156152136600345631027490536974673707426371834004594559738109061421188116}{5259745386174104707723473151149924289252408284119806784796722389693094243044776960617096879313908155297064936682556923819381339173}$
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $25$ | 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: | \( 88301907040780570000 \) (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
| \(\Q(\sqrt{689}) \), 13.13.491258904256726154641.1 |
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 | ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ | $26$ | $26$ | R | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | $26$ | R | $26$ |
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 |
|---|---|---|---|---|---|---|---|
| 13 | Data not computed | ||||||
| 53 | Data not computed | ||||||