Normalized defining polynomial
\( x^{26} - x^{25} + 61 x^{24} + 6 x^{23} + 2428 x^{22} + 745 x^{21} + 53416 x^{20} + 34671 x^{19} + 835048 x^{18} + 606289 x^{17} + 7656274 x^{16} + 7911027 x^{15} + 50259437 x^{14} + 53321755 x^{13} + 222049643 x^{12} + 235929538 x^{11} + 711364146 x^{10} + 658848433 x^{9} + 1489046291 x^{8} + 1147597018 x^{7} + 2167670685 x^{6} + 1324856027 x^{5} + 1912484318 x^{4} + 753653983 x^{3} + 998748064 x^{2} + 333252757 x + 245454889 \)
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: | \(-1040129483587052483222994327325975449420803964347689607283=-\,3^{13}\cdot 131^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $155.94$ | 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: | $3, 131$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(393=3\cdot 131\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{393}(1,·)$, $\chi_{393}(322,·)$, $\chi_{393}(325,·)$, $\chi_{393}(211,·)$, $\chi_{393}(263,·)$, $\chi_{393}(194,·)$, $\chi_{393}(80,·)$, $\chi_{393}(193,·)$, $\chi_{393}(215,·)$, $\chi_{393}(346,·)$, $\chi_{393}(112,·)$, $\chi_{393}(230,·)$, $\chi_{393}(361,·)$, $\chi_{393}(170,·)$, $\chi_{393}(107,·)$, $\chi_{393}(301,·)$, $\chi_{393}(238,·)$, $\chi_{393}(176,·)$, $\chi_{393}(113,·)$, $\chi_{393}(307,·)$, $\chi_{393}(52,·)$, $\chi_{393}(374,·)$, $\chi_{393}(244,·)$, $\chi_{393}(314,·)$, $\chi_{393}(62,·)$, $\chi_{393}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4453} a^{21} - \frac{196}{4453} a^{20} + \frac{1765}{4453} a^{19} + \frac{1800}{4453} a^{18} - \frac{711}{4453} a^{17} + \frac{1638}{4453} a^{16} + \frac{790}{4453} a^{15} - \frac{1552}{4453} a^{14} + \frac{1980}{4453} a^{13} + \frac{246}{4453} a^{12} - \frac{12}{73} a^{11} + \frac{856}{4453} a^{10} + \frac{1682}{4453} a^{9} - \frac{1339}{4453} a^{8} + \frac{15}{4453} a^{7} - \frac{1475}{4453} a^{6} + \frac{675}{4453} a^{5} - \frac{841}{4453} a^{4} + \frac{847}{4453} a^{3} + \frac{1886}{4453} a^{2} + \frac{1326}{4453} a - \frac{1559}{4453}$, $\frac{1}{236009} a^{22} + \frac{18}{236009} a^{21} + \frac{35522}{236009} a^{20} - \frac{21260}{236009} a^{19} + \frac{77232}{236009} a^{18} + \frac{49869}{236009} a^{17} - \frac{107337}{236009} a^{16} + \frac{29465}{236009} a^{15} + \frac{26092}{236009} a^{14} - \frac{70317}{236009} a^{13} - \frac{54960}{236009} a^{12} + \frac{98029}{236009} a^{11} - \frac{37784}{236009} a^{10} + \frac{91429}{236009} a^{9} + \frac{83068}{236009} a^{8} + \frac{95248}{236009} a^{7} - \frac{52248}{236009} a^{6} - \frac{70135}{236009} a^{5} - \frac{103426}{236009} a^{4} - \frac{110754}{236009} a^{3} + \frac{21972}{236009} a^{2} - \frac{29505}{236009} a + \frac{22614}{236009}$, $\frac{1}{236009} a^{23} + \frac{6}{236009} a^{21} + \frac{100742}{236009} a^{20} - \frac{55619}{236009} a^{19} - \frac{19441}{236009} a^{18} - \frac{56385}{236009} a^{17} + \frac{15159}{236009} a^{16} + \frac{15122}{236009} a^{15} + \frac{31950}{236009} a^{14} - \frac{26804}{236009} a^{13} - \frac{17635}{236009} a^{12} - \frac{1880}{3869} a^{11} - \frac{87695}{236009} a^{10} - \frac{101285}{236009} a^{9} - \frac{63634}{236009} a^{8} + \frac{65498}{236009} a^{7} - \frac{87487}{236009} a^{6} + \frac{1156}{4453} a^{5} - \frac{41811}{236009} a^{4} + \frac{56982}{236009} a^{3} - \frac{6566}{236009} a^{2} - \frac{1221}{3233} a - \frac{60803}{236009}$, $\frac{1}{13730499269153818751} a^{24} - \frac{3747429131818}{13730499269153818751} a^{23} + \frac{275060234507}{154275272687121559} a^{22} + \frac{6900471573866}{188089031084298887} a^{21} - \frac{2855149411838162947}{13730499269153818751} a^{20} + \frac{5112407930213874374}{13730499269153818751} a^{19} + \frac{909770710000391278}{13730499269153818751} a^{18} + \frac{5858616068937589550}{13730499269153818751} a^{17} + \frac{2666046729070930422}{13730499269153818751} a^{16} + \frac{6851999546016251271}{13730499269153818751} a^{15} + \frac{6542925776016779517}{13730499269153818751} a^{14} - \frac{4571552389994434954}{13730499269153818751} a^{13} - \frac{5546596843808230164}{13730499269153818751} a^{12} - \frac{1643556979475409344}{13730499269153818751} a^{11} - \frac{4159380323100470955}{13730499269153818751} a^{10} + \frac{2050597109837804313}{13730499269153818751} a^{9} + \frac{6684968419535818673}{13730499269153818751} a^{8} - \frac{91848836420757}{486155835752357} a^{7} - \frac{2347262925327371431}{13730499269153818751} a^{6} - \frac{5274526781317891977}{13730499269153818751} a^{5} - \frac{2823384799589261808}{13730499269153818751} a^{4} - \frac{4221927354987652917}{13730499269153818751} a^{3} - \frac{6254083239000294910}{13730499269153818751} a^{2} - \frac{1892898010458645288}{13730499269153818751} a + \frac{2720341375463877856}{13730499269153818751}$, $\frac{1}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{25} - \frac{22930153308141064392363884759460796573858576162685854859132798553051443522}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{24} + \frac{574972100238851892852615466325282010975845852703018701911870680851015082330514489852024}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{23} - \frac{156888795456040615468643080367232870799323702421790815694093258567652003165965651079879}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{22} - \frac{39184817614966996931619583259732087649072691800135740318494756867548063096303269349778872}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{21} - \frac{168087253317474320859350264457615172886485415131417623653763674384761632568922522679992593942}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{20} - \frac{11586181041618185981935751926709063679935119609117921641222298213352837177465341564228489199}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{19} - \frac{294617874333674647259234514682673384768226074335031567313590520123332674898698318636201657583}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{18} - \frac{15576313856889616810364296429780446390887353993317059826081302257155024767012713688359839046}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{17} + \frac{251381282827512327775996411101409141312491596676338714423585607158760921325871569314268121371}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{16} - \frac{194009602777028709062413862172988565719423279103843156520908852788856449233570859670031721328}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{15} - \frac{40553331734266521072040706351520332503060130167165464643758482128870778390213784770014207735}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{14} - \frac{77313777318364429779458402811277887622372818799528585654960076790622843725205422410886953334}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{13} - \frac{248133557013255207171114183565510318186969915978226379161262226945059705811243590281530022691}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{12} + \frac{195501611024859382031813865444413121987917160007435918785840854093271512348868765638507443503}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{11} - \frac{204602228707202208567869841666434928599531600493604390193850623915489589559624744423034495974}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{10} + \frac{87055794118533211351912025616688257382274442691377788986909275293282063187826665361521172539}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{9} - \frac{143976915150920311018749076916637662230148509033224257866333156584442696348184937411123242901}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{8} - \frac{65912769881578894415764024390603422511823783657995619767728251465922392803675407938260106469}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{7} + \frac{227636714116865457384905772820973462039628799143967526982577675962780289978526541268346089430}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{6} + \frac{204195961313266117175792769763332884841611603998590790211497490919741425179132722266637955398}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{5} - \frac{3783351202016823995556333049660241578669184250039507045147371849179070206714974472774116994}{7666114994191451484351933324102302416523967334384029424594347601376957352476235656656444849} a^{4} + \frac{207833360530695331879147809650606339180954303349016616444661775872370602646924713741149610545}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{3} - \frac{243113121066587210693244188861608016058014073552320722146394952010850055382168582820004685993}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a^{2} + \frac{284877841444063163342729537715680216139656061677633866391899871305321293524655086301417932856}{682284234483039182107322065845104915070633092760178618788896936522549204370384973442423591561} a + \frac{5878094733155952618315448594833526074931634887940527767197935172093407289009693487852504}{43549130942939885243334529000134353422520782074435349383346967289369324335889766607673683}$
Class group and class number
$C_{1526201}$, which has order $1526201$ (assuming GRH)
Unit group
| Rank: | $12$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{329455162218607639739474271052372157327819327858696196457822372925527031}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{25} + \frac{382449285773061352463202330882995926698545269843782583663589300161797655}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{24} - \frac{19983683173891931514585932513630961718442961966576507509969433762436379456}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{23} + \frac{1165589220255354624096719616327954128879680767135675425801989544434727262}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{22} - \frac{789706791481582489415186542262774512693506680462926882306558212455972698286}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{21} - \frac{111130524336663289154019221199491539181803414385385831347841487128188179419}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{20} - \frac{17164617811514879173062612363501643693664044544466405840600079905064341630782}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{19} - \frac{8293242361556784603078760535909340166533007300370878053534082644817517067331}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{18} - \frac{264731642166341478063313211428140506319787881532295929419868501942888087570463}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{17} - \frac{146117688808442953374305561621791319039952308885812836168817518586509232242817}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{16} - \frac{2357344570311560407208212280110391701122999796421500163914884138364727283470060}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{15} - \frac{2048621336164522232391846032828886421377669210973551287407675503279044786635435}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{14} - \frac{14953225719560610482689392540751167214551206299215415566267820840249955322238547}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{13} - \frac{13199576623317117881518301943862238713561150199987085583277333506146560691852509}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{12} - \frac{62569136242436531205558597918618384898422325789182303432864959454153718918132351}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{11} - \frac{55278324283455900389816154739547882696947081801392358192013700611560220251340986}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{10} - \frac{188883231328563928240803940515919241680709000200648261795238142016401535877554456}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{9} - \frac{135575882010519618560984161506203462265935583971602061940371836934535266992865286}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{8} - \frac{353689201463328621977640836742151937080699613326922236440982519948358945493045567}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{7} - \frac{184154079468388344060491509726956554255348513236845838484147596851913195582986500}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{6} - \frac{462717645718179826089245748752953207559493593690151246774502142557073199958391887}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{5} - \frac{143408955154797596318343523597246568501916067770696204101598586294928690227442695}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{4} - \frac{310801080868168250222301057390399736570630474219810512857724677341784569046128288}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{3} + \frac{74586161094996602381070130838344093059409782503408382929567624990616373544514465}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{2} - \frac{129814807172290856042757032532144446524983494881260299501248507615593557319424538}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a + \frac{5028970955693085496145270277585125484195767844010597163009494336588022449286}{6539233564716493582174147215585745099958684369723063405446103094890853033487} \) (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: | \( 1197545162478.713 \) (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
| \(\Q(\sqrt{-3}) \), 13.13.25542038069936263923006961.1 |
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 | $26$ | ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ | $26$ |
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 | ||||||
| 131 | Data not computed | ||||||