Normalized defining polynomial
\( x^{26} - x^{25} + 37 x^{24} - 118 x^{23} + 1008 x^{22} - 3235 x^{21} + 16642 x^{20} - 51633 x^{19} + 196206 x^{18} - 526339 x^{17} + 1528709 x^{16} - 3470974 x^{15} + 8121237 x^{14} - 15101678 x^{13} + 27024309 x^{12} - 37166826 x^{11} + 48531861 x^{10} - 47702921 x^{9} + 49613581 x^{8} - 37639084 x^{7} + 35389248 x^{6} - 18788771 x^{5} + 15390312 x^{4} - 4606662 x^{3} + 4701993 x^{2} - 643428 x + 85849 \)
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: | \(-5567067695110660347277981181082226208360544116097363=-\,3^{13}\cdot 79^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $97.77$ | 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: | $3, 79$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(237=3\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{237}(64,·)$, $\chi_{237}(1,·)$, $\chi_{237}(131,·)$, $\chi_{237}(196,·)$, $\chi_{237}(65,·)$, $\chi_{237}(8,·)$, $\chi_{237}(10,·)$, $\chi_{237}(143,·)$, $\chi_{237}(80,·)$, $\chi_{237}(146,·)$, $\chi_{237}(67,·)$, $\chi_{237}(22,·)$, $\chi_{237}(89,·)$, $\chi_{237}(220,·)$, $\chi_{237}(223,·)$, $\chi_{237}(97,·)$, $\chi_{237}(100,·)$, $\chi_{237}(101,·)$, $\chi_{237}(38,·)$, $\chi_{237}(46,·)$, $\chi_{237}(176,·)$, $\chi_{237}(179,·)$, $\chi_{237}(52,·)$, $\chi_{237}(125,·)$, $\chi_{237}(62,·)$, $\chi_{237}(166,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23} a^{20} - \frac{11}{23} a^{19} - \frac{7}{23} a^{18} + \frac{2}{23} a^{17} - \frac{11}{23} a^{16} + \frac{11}{23} a^{15} + \frac{6}{23} a^{14} - \frac{9}{23} a^{13} - \frac{10}{23} a^{12} - \frac{5}{23} a^{11} + \frac{10}{23} a^{10} + \frac{6}{23} a^{9} - \frac{9}{23} a^{8} - \frac{7}{23} a^{7} + \frac{11}{23} a^{6} + \frac{2}{23} a^{5} + \frac{6}{23} a^{4} - \frac{11}{23} a^{3} - \frac{5}{23} a^{2} - \frac{8}{23} a + \frac{6}{23}$, $\frac{1}{23} a^{21} + \frac{10}{23} a^{19} - \frac{6}{23} a^{18} + \frac{11}{23} a^{17} + \frac{5}{23} a^{16} - \frac{11}{23} a^{15} + \frac{11}{23} a^{14} + \frac{6}{23} a^{13} + \frac{1}{23} a^{11} + \frac{1}{23} a^{10} + \frac{11}{23} a^{9} + \frac{9}{23} a^{8} + \frac{3}{23} a^{7} + \frac{8}{23} a^{6} + \frac{5}{23} a^{5} + \frac{9}{23} a^{4} - \frac{11}{23} a^{3} + \frac{6}{23} a^{2} + \frac{10}{23} a - \frac{3}{23}$, $\frac{1}{23} a^{22} - \frac{11}{23} a^{19} - \frac{11}{23} a^{18} + \frac{8}{23} a^{17} + \frac{7}{23} a^{16} - \frac{7}{23} a^{15} - \frac{8}{23} a^{14} - \frac{2}{23} a^{13} + \frac{9}{23} a^{12} + \frac{5}{23} a^{11} + \frac{3}{23} a^{10} - \frac{5}{23} a^{9} + \frac{1}{23} a^{8} + \frac{9}{23} a^{7} + \frac{10}{23} a^{6} - \frac{11}{23} a^{5} - \frac{2}{23} a^{4} + \frac{1}{23} a^{3} - \frac{9}{23} a^{2} + \frac{8}{23} a + \frac{9}{23}$, $\frac{1}{428789} a^{23} - \frac{5069}{428789} a^{22} + \frac{8906}{428789} a^{21} - \frac{2061}{428789} a^{20} - \frac{117957}{428789} a^{19} - \frac{124539}{428789} a^{18} - \frac{35045}{428789} a^{17} + \frac{168026}{428789} a^{16} - \frac{102429}{428789} a^{15} - \frac{171887}{428789} a^{14} + \frac{157703}{428789} a^{13} + \frac{8136}{18643} a^{12} + \frac{204425}{428789} a^{11} + \frac{39296}{428789} a^{10} + \frac{91347}{428789} a^{9} + \frac{88070}{428789} a^{8} - \frac{179757}{428789} a^{7} - \frac{126778}{428789} a^{6} + \frac{190188}{428789} a^{5} - \frac{7532}{18643} a^{4} + \frac{26249}{428789} a^{3} + \frac{54161}{428789} a^{2} - \frac{176238}{428789} a + \frac{94462}{428789}$, $\frac{1}{226829381} a^{24} - \frac{172}{226829381} a^{23} + \frac{4213164}{226829381} a^{22} + \frac{4106104}{226829381} a^{21} - \frac{1858610}{226829381} a^{20} - \frac{47869979}{226829381} a^{19} + \frac{33616546}{226829381} a^{18} - \frac{4581589}{9862147} a^{17} + \frac{111770088}{226829381} a^{16} - \frac{1873298}{226829381} a^{15} + \frac{14662768}{226829381} a^{14} + \frac{74613943}{226829381} a^{13} - \frac{82731845}{226829381} a^{12} + \frac{58182867}{226829381} a^{11} + \frac{95505687}{226829381} a^{10} + \frac{90643238}{226829381} a^{9} - \frac{96218222}{226829381} a^{8} + \frac{86317315}{226829381} a^{7} + \frac{105801460}{226829381} a^{6} + \frac{31000145}{226829381} a^{5} + \frac{25263251}{226829381} a^{4} - \frac{90571594}{226829381} a^{3} - \frac{76881084}{226829381} a^{2} - \frac{109374321}{226829381} a + \frac{26296928}{226829381}$, $\frac{1}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{25} + \frac{5028663798190928254528323920936572584182730421797642648353564}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{24} - \frac{5434653987004154132623952505350183698315702147712834306868790608}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{23} + \frac{318197472441256836753020362878167400882060374350039664878918213148}{27458753166746220460456279498267102003093043285507133203113331064067} a^{22} - \frac{89172295766774851059824672145468410746853159869900697711356451658694}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{21} + \frac{94449048245117941852355170631684093869025745253987485613157410436300}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{20} + \frac{1899320740136436344805470928540974779806955814195038244027010200501926}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{19} - \frac{396087458497819336827905513861593729087823818500121558313149009777938}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{18} - \frac{1483796104834721733698746233505438041699316146812671190626186898588789}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{17} - \frac{46701144102496773509147559495115035687422013909564120907497860707576}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{16} - \frac{77016878389198076225514896114731634286808499467582650674785766305352}{216088448833959387101851590834188933154775688464208309120152735765049} a^{15} - \frac{943056410165870923319829573935699984321340129132480408054925644120555}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{14} - \frac{1718370952544416449922801099810467006121155899499387558937005488138274}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{13} + \frac{2016659838655940485263762863568968408054030925043247927180243093250843}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{12} - \frac{46285145085367410492753972208270692341521720389307185755073869684731}{216088448833959387101851590834188933154775688464208309120152735765049} a^{11} - \frac{1941966838676569509425823732968292537268964529062631078000734997137317}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{10} - \frac{111668386770699229751448684534340909430538867758413491535107859816359}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{9} + \frac{634981441447736744110235049039056222458520509185300759752867931821670}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{8} - \frac{368587450744057366517166518444708878834309563288865737220747696780087}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{7} + \frac{1796292615976580076137062033458352636289965547356300749880518507011697}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{6} - \frac{2092833474493745750954311974239731017396754771090460468897756559482172}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{5} + \frac{2343291326028156135756712671682988525232457641768199690460086394941000}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{4} + \frac{1202210273977095535591824423266761629803354423829264025525037271656856}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{3} - \frac{540068367185817302694686993260825844932075378850671857914166333213366}{4970034323181065903342586589186345462559840834676791109763512922596127} a^{2} + \frac{1995885664860806506326164508622745816281628883640993479309286529785190}{4970034323181065903342586589186345462559840834676791109763512922596127} a + \frac{1896651770403063920128459921086845598832055047492501558517552914462}{16962574481846641308336473000635991339794678616644338258578542397939}$
Class group and class number
$C_{57473}$, which has order $57473$ (assuming GRH)
Unit group
| Rank: | $12$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{745437435626525585791097888709341304681484807006785254875549}{408484780404460089039416995905839193109216802389807767712954131881} a^{25} - \frac{800600851644091915550693456619893048232672919112354031509062}{408484780404460089039416995905839193109216802389807767712954131881} a^{24} + \frac{27633491585445404582125825140468867200147918128306953923210841}{408484780404460089039416995905839193109216802389807767712954131881} a^{23} - \frac{89932332821032303795495171336158580850423032047539225217894220}{408484780404460089039416995905839193109216802389807767712954131881} a^{22} + \frac{757736344994103711379521133711857773162434426200308472994013920}{408484780404460089039416995905839193109216802389807767712954131881} a^{21} - \frac{2464288908392365058113282363130840345871255333259680285988693784}{408484780404460089039416995905839193109216802389807767712954131881} a^{20} + \frac{12573204623928256474854921990373761147483149692735488756348734413}{408484780404460089039416995905839193109216802389807767712954131881} a^{19} - \frac{39330911361288242705889494852756800570596591189199619922913689537}{408484780404460089039416995905839193109216802389807767712954131881} a^{18} + \frac{148845080043023163759577860857899614387193792271760207280706212906}{408484780404460089039416995905839193109216802389807767712954131881} a^{17} - \frac{401933652602234683585264079828767046313057346437865059333368878931}{408484780404460089039416995905839193109216802389807767712954131881} a^{16} + \frac{1164644218294350376610152560579646099693911031526036638568843250948}{408484780404460089039416995905839193109216802389807767712954131881} a^{15} - \frac{2657455733490793958085602735923700346711360631808299696136786022297}{408484780404460089039416995905839193109216802389807767712954131881} a^{14} + \frac{6207029985588047906671405250161930715896574910644884226087144376576}{408484780404460089039416995905839193109216802389807767712954131881} a^{13} - \frac{11598064215401636134133795453151717443426255953600115214049227112301}{408484780404460089039416995905839193109216802389807767712954131881} a^{12} + \frac{20736838298104688526927598480458816125218529837050164805240642217353}{408484780404460089039416995905839193109216802389807767712954131881} a^{11} - \frac{28650958448430114553819787935661239455313835538399976808195141005716}{408484780404460089039416995905839193109216802389807767712954131881} a^{10} + \frac{37238537460543477723099146238779884229080542315073926124361974436810}{408484780404460089039416995905839193109216802389807767712954131881} a^{9} - \frac{36549407897392657125357430970714790943501781202717326421796785248527}{408484780404460089039416995905839193109216802389807767712954131881} a^{8} + \frac{37429046557079935182304712656079883099486240822683688346665522654115}{408484780404460089039416995905839193109216802389807767712954131881} a^{7} - \frac{28167755699276684561756024859865494506498191349179582440485315490428}{408484780404460089039416995905839193109216802389807767712954131881} a^{6} + \frac{26158145806174808843074478835580681763373827382739869584454294313980}{408484780404460089039416995905839193109216802389807767712954131881} a^{5} - \frac{13699740916400386158370648641827002546266938032398234891827527703172}{408484780404460089039416995905839193109216802389807767712954131881} a^{4} + \frac{10905024500702360702073152572174989223827794283152320479011396627999}{408484780404460089039416995905839193109216802389807767712954131881} a^{3} - \frac{2604055139933440516041199089854875479500442801564892131076838478333}{408484780404460089039416995905839193109216802389807767712954131881} a^{2} + \frac{3161861817590699445201304561941655417888284129745114459011791636626}{408484780404460089039416995905839193109216802389807767712954131881} a - \frac{78298508111038596112801991594433972609517842594379667909711265}{1394146008206348426755689405821976768290842328975453132126123317} \) (order $6$) | 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
| \(\Q(\sqrt{-3}) \), 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 | $26$ | R | $26$ | ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | $26$ | ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ |
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 | ||||||
| $79$ | 79.13.12.1 | $x^{13} - 79$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |
| 79.13.12.1 | $x^{13} - 79$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ | |