Normalized defining polynomial
\( x^{26} - x^{25} + 187 x^{24} - 188 x^{23} + 14268 x^{22} - 14457 x^{21} + 578049 x^{20} - 592696 x^{19} + 13558779 x^{18} - 13959401 x^{17} + 188306247 x^{16} - 192097412 x^{15} + 1512632896 x^{14} - 1643661436 x^{13} + 6589281884 x^{12} - 10923776128 x^{11} + 18513107422 x^{10} - 64903945574 x^{9} + 87181434018 x^{8} - 204155575160 x^{7} + 296287639624 x^{6} - 177883731071 x^{5} + 788591474591 x^{4} - 1442009661016 x^{3} + 1728793845279 x^{2} + 1756646750735 x + 1140470028259 \)
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: | \(-24893336381914519275474376221073664531912183161424560546875=-\,3^{13}\cdot 5^{13}\cdot 53^{25}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $176.20$ | 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: | $3, 5, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(795=3\cdot 5\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{795}(256,·)$, $\chi_{795}(1,·)$, $\chi_{795}(331,·)$, $\chi_{795}(646,·)$, $\chi_{795}(449,·)$, $\chi_{795}(329,·)$, $\chi_{795}(779,·)$, $\chi_{795}(269,·)$, $\chi_{795}(526,·)$, $\chi_{795}(749,·)$, $\chi_{795}(16,·)$, $\chi_{795}(466,·)$, $\chi_{795}(149,·)$, $\chi_{795}(121,·)$, $\chi_{795}(794,·)$, $\chi_{795}(539,·)$, $\chi_{795}(346,·)$, $\chi_{795}(29,·)$, $\chi_{795}(736,·)$, $\chi_{795}(464,·)$, $\chi_{795}(674,·)$, $\chi_{795}(301,·)$, $\chi_{795}(46,·)$, $\chi_{795}(494,·)$, $\chi_{795}(59,·)$, $\chi_{795}(766,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{107} a^{22} + \frac{15}{107} a^{21} + \frac{5}{107} a^{20} - \frac{5}{107} a^{19} - \frac{8}{107} a^{18} + \frac{41}{107} a^{17} - \frac{51}{107} a^{16} + \frac{39}{107} a^{15} - \frac{30}{107} a^{14} + \frac{1}{107} a^{13} + \frac{25}{107} a^{12} - \frac{2}{107} a^{11} - \frac{50}{107} a^{10} - \frac{17}{107} a^{9} - \frac{32}{107} a^{8} + \frac{6}{107} a^{7} + \frac{38}{107} a^{6} - \frac{30}{107} a^{5} + \frac{40}{107} a^{4} - \frac{29}{107} a^{3} - \frac{26}{107} a^{2} + \frac{27}{107} a + \frac{27}{107}$, $\frac{1}{107} a^{23} - \frac{6}{107} a^{21} + \frac{27}{107} a^{20} - \frac{40}{107} a^{19} - \frac{53}{107} a^{18} - \frac{24}{107} a^{17} - \frac{52}{107} a^{16} + \frac{27}{107} a^{15} + \frac{23}{107} a^{14} + \frac{10}{107} a^{13} + \frac{51}{107} a^{12} - \frac{20}{107} a^{11} - \frac{16}{107} a^{10} + \frac{9}{107} a^{9} - \frac{49}{107} a^{8} - \frac{52}{107} a^{7} + \frac{42}{107} a^{6} - \frac{45}{107} a^{5} + \frac{13}{107} a^{4} - \frac{19}{107} a^{3} - \frac{11}{107} a^{2} + \frac{50}{107} a + \frac{23}{107}$, $\frac{1}{107} a^{24} + \frac{10}{107} a^{21} - \frac{10}{107} a^{20} + \frac{24}{107} a^{19} + \frac{35}{107} a^{18} - \frac{20}{107} a^{17} + \frac{42}{107} a^{16} + \frac{43}{107} a^{15} + \frac{44}{107} a^{14} - \frac{50}{107} a^{13} + \frac{23}{107} a^{12} - \frac{28}{107} a^{11} + \frac{30}{107} a^{10} - \frac{44}{107} a^{9} - \frac{30}{107} a^{8} - \frac{29}{107} a^{7} - \frac{31}{107} a^{6} + \frac{47}{107} a^{5} + \frac{7}{107} a^{4} + \frac{29}{107} a^{3} + \frac{1}{107} a^{2} - \frac{29}{107} a - \frac{52}{107}$, $\frac{1}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{25} + \frac{37046697581470926352380020534358821469671882109137262472374385268190839639736926941670469986697058795046511766169572472813717538765159741647865187574074995183332}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{24} - \frac{37764463076106303336388647400085299476660253860718379387573510637903623916747921521395883016640375528944953482917215014553622345609721718547897370275810101132139}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{23} - \frac{17571687201134098458726419068573712941365297484256163597730178743147460462195097520804251654912803923561603123817722534249008267446229846530027072460357172870352}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{22} - \frac{4580395026337550682032666486576520405741201632383414064604735238151497912375733036400707502622426487746173826335421757852286782894217164481225244064820233210876875}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{21} + \frac{988656222400945520771312644657995105782664862975588026549395781704276866043916843626496675514694296097424306951853007073045220437923969677738462503984041006629685}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{20} + \frac{1969834036512208396966677767954456060469595986210721200491256273952231105790024671996890680878164961995767730865228287710261004651528551779524655091127634831228222}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{19} + \frac{4591932384826304169477283385588406348206859730960990449226057615030819003747286823052075191769628276633849999835854021507908913007690530204984736373233979912639272}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{18} + \frac{1627782718300733230752395394642214749943838264425590902473263779090254668513981463789483859646156990690967511361286052441168345900810495395538798134495949946760339}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{17} + \frac{4318433877634809785669281733476325646258736073079730852316079305125549063788126275351655040736910206063474428836927230118001283121557685799750162472943812386648144}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{16} - \frac{982460151993559808975612031160736337159474244831574459447134003872093941483517876669938852335227501431182593937249219924049654276069556830136695476566099278781936}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{15} - \frac{895868623310436801018428317855464908802442549359383923217635509158190787104969713649447338187620282575880992558639254109526601194579262325801627584701184291529192}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{14} - \frac{3861362142216993738827367600713856348679829650773810233032932899620779121449615807239386522589253511844556870389025607114159777460238631209320209126982290273888414}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{13} + \frac{1998661385324355742787666712938492626670366090157857746728687511362355363792402889828753412914182029441840819199917416933997535543262403807816670188531151951896656}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{12} + \frac{4571889323538876068023217400501020668945646256013157651744554177329848729987345977875379017096835163011434167550090309977172345872377108292540777888593520206603949}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{11} - \frac{7198749724999725849117650096311641385903583809347533813533174716277693217998045860331802634653209902788781716223499110947125919945289408819420920590477759773438}{89049209688276406235645588614623794643660761861223303094273259420172764184198450533178377462102408212018841138184855288894229078950326650665144643130654528812741} a^{10} - \frac{3483981979272935770116518727176787486430495310600400231478071281339782777662949453785853029363644726124772085170582566780095958844397294505397238798963426042548}{30057619673960805890265230226387211441235651479971272653272046555074087595297268791956108480898920122037905368409399103822342307405946219625143459984164146949411} a^{9} + \frac{2234490668135939299286961484481533073401539666924269185119990433436765862730814125609988352664832448038808830588426066407269288838362729043998393706935671362468574}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{8} + \frac{20176855465433162579355157886714609322972472933116151610721591500686929344700290951545836461099892038586865930994543679396673213046071574783036238755657398871138}{45157656097846329228502739250069886383278206251899968867712032028239269041275991502607044494999799425052208539269097231808921855202298348915499890118388789492717} a^{7} - \frac{812865810209013405067414897540797205809636928566070740793126783012946430131990638488565486294604605386323169619277568661962446374423098531597099363087003664360832}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{6} + \frac{15829497873670461328916942195563666106936898047004468549945434390363966916844243306322923916992320157170636025063228634711650461750008151490590090596990863868590}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{5} - \frac{3954229119384957719061330274769925481245151142910403696976578711110642763546070217244556178143377303794238412656881842408124823908116064568507332920425599707843085}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{4} - \frac{4218681292505097405839961639970115584373619316595498070396916464334698504455501434632002942168162948017917360024034292400076788736294090614883953487766653784310197}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{3} + \frac{1017887691011820681770457815148771360544453058430179792456624106930999780482058493858331261702403343821345625318759511599511387545511735956359090774707910998547232}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a^{2} + \frac{2632070253989474809476039104625004072825280840040860109396991442782989043926628390955811252002394464085979885641187364727347041539514658122403270682047075258043787}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287} a + \frac{746596073669161788540344065907303513625464837958515916782123131141084673587412666600349239279837379001545845407422911599861137410114343167918887413578031809938683}{9528265436645575467214077981764746026871701519150893431087238757958485767709234207050086388444957678686016001785779515911682511447684951621170476814980034582963287}$
Class group and class number
Not computed
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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | 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
| \(\Q(\sqrt{-795}) \), 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 | $26$ | R | R | $26$ | $26$ | $26$ | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | $26$ | $26$ | $26$ | ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ | 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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 53 | Data not computed | ||||||