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Properties

Label 25.25.1795754272...0625.3
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 151^{24}$
Root discriminant $1622.44$
Ramified primes $5, 151$
Class number Not computed
Class group Not computed
Galois group $C_{25}$ (as 25T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2276256034653585107, -7863433649974415910, 190576219504785438330, 231433895175806702610, 3656876625295283000, -87579212713502803421, -29327372321687245415, 4098328317611888825, 2900196984390428225, 48336836091231675, -122615074701281442, -8366656352727740, 2854195063152865, 272562107887855, -39654613098025, -4572304527685, 333293260385, 45005546980, -1628643720, -264336825, 4101311, 886370, -3775, -1510, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 1510*x^23 - 3775*x^22 + 886370*x^21 + 4101311*x^20 - 264336825*x^19 - 1628643720*x^18 + 45005546980*x^17 + 333293260385*x^16 - 4572304527685*x^15 - 39654613098025*x^14 + 272562107887855*x^13 + 2854195063152865*x^12 - 8366656352727740*x^11 - 122615074701281442*x^10 + 48336836091231675*x^9 + 2900196984390428225*x^8 + 4098328317611888825*x^7 - 29327372321687245415*x^6 - 87579212713502803421*x^5 + 3656876625295283000*x^4 + 231433895175806702610*x^3 + 190576219504785438330*x^2 - 7863433649974415910*x - 2276256034653585107)
 
gp: K = bnfinit(x^25 - 1510*x^23 - 3775*x^22 + 886370*x^21 + 4101311*x^20 - 264336825*x^19 - 1628643720*x^18 + 45005546980*x^17 + 333293260385*x^16 - 4572304527685*x^15 - 39654613098025*x^14 + 272562107887855*x^13 + 2854195063152865*x^12 - 8366656352727740*x^11 - 122615074701281442*x^10 + 48336836091231675*x^9 + 2900196984390428225*x^8 + 4098328317611888825*x^7 - 29327372321687245415*x^6 - 87579212713502803421*x^5 + 3656876625295283000*x^4 + 231433895175806702610*x^3 + 190576219504785438330*x^2 - 7863433649974415910*x - 2276256034653585107, 1)
 

Normalized defining polynomial

\( x^{25} - 1510 x^{23} - 3775 x^{22} + 886370 x^{21} + 4101311 x^{20} - 264336825 x^{19} - 1628643720 x^{18} + 45005546980 x^{17} + 333293260385 x^{16} - 4572304527685 x^{15} - 39654613098025 x^{14} + 272562107887855 x^{13} + 2854195063152865 x^{12} - 8366656352727740 x^{11} - 122615074701281442 x^{10} + 48336836091231675 x^{9} + 2900196984390428225 x^{8} + 4098328317611888825 x^{7} - 29327372321687245415 x^{6} - 87579212713502803421 x^{5} + 3656876625295283000 x^{4} + 231433895175806702610 x^{3} + 190576219504785438330 x^{2} - 7863433649974415910 x - 2276256034653585107 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $25$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[25, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(179575427285891034426038718238151945439156258996386904982500709593296051025390625=5^{40}\cdot 151^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1622.44$
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:  $5, 151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3775=5^{2}\cdot 151\)
Dirichlet character group:    $\lbrace$$\chi_{3775}(2816,·)$, $\chi_{3775}(1,·)$, $\chi_{3775}(2626,·)$, $\chi_{3775}(261,·)$, $\chi_{3775}(3596,·)$, $\chi_{3775}(2466,·)$, $\chi_{3775}(3406,·)$, $\chi_{3775}(2726,·)$, $\chi_{3775}(1876,·)$, $\chi_{3775}(1486,·)$, $\chi_{3775}(2031,·)$, $\chi_{3775}(1181,·)$, $\chi_{3775}(3106,·)$, $\chi_{3775}(2661,·)$, $\chi_{3775}(3366,·)$, $\chi_{3775}(171,·)$, $\chi_{3775}(2796,·)$, $\chi_{3775}(1821,·)$, $\chi_{3775}(3696,·)$, $\chi_{3775}(1841,·)$, $\chi_{3775}(1076,·)$, $\chi_{3775}(1591,·)$, $\chi_{3775}(2356,·)$, $\chi_{3775}(1786,·)$, $\chi_{3775}(2111,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{38} a^{16} - \frac{4}{19} a^{15} - \frac{3}{19} a^{14} + \frac{6}{19} a^{13} - \frac{4}{19} a^{10} - \frac{6}{19} a^{9} - \frac{9}{38} a^{8} + \frac{9}{19} a^{7} + \frac{7}{38} a^{4} - \frac{9}{19} a^{3} + \frac{15}{38} a^{2} - \frac{11}{38} a$, $\frac{1}{38} a^{17} + \frac{3}{19} a^{15} + \frac{1}{19} a^{14} - \frac{9}{19} a^{13} - \frac{4}{19} a^{11} + \frac{9}{38} a^{9} - \frac{8}{19} a^{8} - \frac{4}{19} a^{7} + \frac{7}{38} a^{5} - \frac{15}{38} a^{3} - \frac{5}{38} a^{2} - \frac{6}{19} a$, $\frac{1}{38} a^{18} - \frac{7}{38} a^{15} + \frac{9}{19} a^{14} + \frac{2}{19} a^{13} - \frac{4}{19} a^{12} - \frac{1}{2} a^{10} + \frac{9}{19} a^{9} + \frac{4}{19} a^{8} - \frac{13}{38} a^{7} + \frac{7}{38} a^{6} - \frac{1}{2} a^{4} + \frac{4}{19} a^{3} + \frac{6}{19} a^{2} + \frac{9}{38} a - \frac{1}{2}$, $\frac{1}{38} a^{19} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{9}{19} a$, $\frac{1}{38} a^{20} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{9}{19} a^{2}$, $\frac{1}{38} a^{21} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{9}{19} a^{3}$, $\frac{1}{7486} a^{22} + \frac{28}{3743} a^{21} - \frac{63}{7486} a^{20} - \frac{79}{7486} a^{19} + \frac{36}{3743} a^{18} - \frac{53}{7486} a^{17} - \frac{43}{7486} a^{16} - \frac{277}{3743} a^{15} + \frac{1657}{7486} a^{14} - \frac{1271}{3743} a^{13} - \frac{785}{7486} a^{12} + \frac{2533}{7486} a^{11} + \frac{2187}{7486} a^{10} + \frac{2133}{7486} a^{9} + \frac{3711}{7486} a^{8} + \frac{1561}{3743} a^{7} - \frac{1035}{7486} a^{6} - \frac{3259}{7486} a^{5} - \frac{1063}{3743} a^{4} - \frac{2699}{7486} a^{3} + \frac{1271}{3743} a^{2} - \frac{1793}{7486} a - \frac{22}{197}$, $\frac{1}{35542712026} a^{23} - \frac{835548}{17771356013} a^{22} + \frac{125537895}{17771356013} a^{21} + \frac{37900034}{17771356013} a^{20} + \frac{102072178}{17771356013} a^{19} + \frac{13761107}{1870669054} a^{18} + \frac{103006971}{17771356013} a^{17} - \frac{222500231}{17771356013} a^{16} - \frac{2635877905}{17771356013} a^{15} + \frac{4878782587}{17771356013} a^{14} - \frac{8879227348}{17771356013} a^{13} + \frac{368005450}{935334527} a^{12} - \frac{11461730039}{35542712026} a^{11} - \frac{8182394069}{35542712026} a^{10} - \frac{16902111241}{35542712026} a^{9} + \frac{12673110669}{35542712026} a^{8} + \frac{17000999313}{35542712026} a^{7} + \frac{287036483}{1870669054} a^{6} - \frac{2178059506}{17771356013} a^{5} + \frac{4651928665}{35542712026} a^{4} - \frac{152048822}{935334527} a^{3} + \frac{8068681182}{17771356013} a^{2} - \frac{636109217}{1870669054} a + \frac{13295833}{98456266}$, $\frac{1}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{24} + \frac{6502025276221382208188281409237535607000804712343845860569252849928805637429255539719502584411646146387148492699648471119086396304436663363550446818108847478195867214019908083241458406547072244960611}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{23} - \frac{103545094325434353652351956858810388253764664965808595317858326021356274182170487681903070762321132132774024966322403441831430355314427962256230889288393427967911741746797627393814202555703473762120911322529}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{22} + \frac{7705973230848192468227688266895176114790595925139625774387657615061936308412664701681312118624050610556733593507975542183255868207312533905407049748991237657144983959627673335243010121443591749752230396018333}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{21} - \frac{864291111630986641780189472186930674160254718282735852815136308419837292549298026577464900116626527032989162559497913764977746562703433944951394792982185695639102243469943289782209777108518459115227260237951}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{20} + \frac{12535931562358033450984934415288827560248511158562792076018063258729912988182111313497872271318272446058334875297264270386337704576226397701941463338301785535906059958163078075466856255204711497248096143315897}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{19} - \frac{4878968798459955434480132192429621680410968802072982156172155900246603985167789496392657993658257413284110124617604779180294339641063968982558930013770125902291919537525274099514688491139560184398536188108636}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{18} - \frac{2966671950316984654305566292161710195983604700048200600337155101887794900215605575137403758985313483777694398300410910727690242359009293319859655080101815920460060363427554076452240414644170324874644016700517}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{17} + \frac{5962317235081912022040670053716032395895561091548263938015975697660362983888275314335241330761561119751247711620187832042767165364772795231195322469889390655689824643770747452185351327972110916448597297394818}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{16} + \frac{168397674520065320045058337722304593830381873969474343922568844993292397015991445956916635498550578278740480371012438505550256316939645354806796142977087872605109246009281362180975488780352003374581977203926443}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{15} - \frac{696157791621786016793047224826025824090445140285469271966120259653154698353054532259434899357206549276709436009306510050761884653750483679610912154458874190994946666889206581549754931523725078850303588571894761}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{14} + \frac{750149488724901967950413975082843383396932182200920699710808773067172362064426345218145610190197544869108453639822949081024788443528311646667222872497567310576524729239731340059722444065668106288409259604651693}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{13} - \frac{201143705749319963757954869629920638022572710772842437828654483800951725795428890158128223310782978233783276887054760146591814478853843533329688822200452830567409249839342052462919938244713782717842419834310709}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{12} - \frac{156459544146489786281745701881419855906067414817783593323586615065023669451413546998129530832984555264931338143755844489380468327325166843031518631690320764678814704283707624212026269870176212273868005770172345}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{11} - \frac{126111197513887209780600972140541480610736695135013457765473404703359790934372757055212384149393818151638724797554088316977104927215765258290690710370232581886117451751621177736675420338124414029532242625902513}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{10} - \frac{578309756674953117457168205274484042103462478786563842750923960405172999017195763136959062433735846905821971675732868232580672909954651762705386480713914106052285813912040744982371721724321274828066027998826139}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{9} - \frac{136265866194136028701306022914799769950907582268171482755827795047026688299438590588944211295654217969093587567443088088158052502671582515272462739141869678139705516017963458595581529029511701506652967746873227}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{8} - \frac{218871141818908460689928544839847306733807080471117687110806261061095607449750520658033461271418189742099535752830924260410611494784690320161811672006872066999626935664724949735316080328602621620494991243593261}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{7} + \frac{264721246743144021049994810294788118121928581727828509231602904234938924464754222536499452587582980160398816170152421932519110918609261393641255278743593445381516546436400650368683515151981412510923463680770815}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{6} - \frac{605385663303608531723779383203299467414048517419754286225596157678119132762629308820298026767223687333760928606129226106607137739847606520207812209553544091015029570888917362782928295048701010484077917348651889}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{5} + \frac{350846018143230938923449550853955884150296131468086011781386372737551372450040577177375271374109434315771672325287692025290209846002960100942134253348272836689960556732991411101151770110859308437417933101447342}{776954545555076931144825695290120305118361965827703076825395648160770820307320346482542387031511280507086331869640176527579526224943085502362040345272130302917702574589865540906537750032726404782304480259025799} a^{4} - \frac{546511849619562384803276030207684125317738880801881497992407016258222927752180364563734749547256052454567816312041145778160285661908250235201266106254475595999877301850768178085372887571702383848635112290670215}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{3} - \frac{249855079841600773462794516601938285626143647029454192341983431960190344811557675849270928952285801806687099844642330940548762010474208724363125546784192858323931258040996249084506433296749873922489648172863279}{1553909091110153862289651390580240610236723931655406153650791296321541640614640692965084774063022561014172663739280353055159052449886171004724080690544260605835405149179731081813075500065452809564608960518051598} a^{2} - \frac{13474176570593194851957984524203941658241529062448662546251760807404078695222685889622388393226672036878779829014190299371832412658559920642604652164154654159374373576061283870433097600809236838554998551793328}{40892344502898785849727668173164226585176945569879109306599770955830043174069491920133809843763751605636122729981061922504185590786478184334844228698533173837773819715256081100344092106985600251700235803106621} a + \frac{787481285944769487682910823718284071373145054752811832212025869840036560516154018258728661861719212566592374058332845349942316399517491134857295788615286161380332592337878526662492103944553622043179559341209}{4304457316094609036813438755069918587913362691566222032273660100613688755165209675803558930922500169014328708419059149737282693766997703614194129336687702509239349443711166431615167590209010552810551137169118}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $24$
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

$C_{25}$ (as 25T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 25
The 25 conjugacy class representatives for $C_{25}$
Character table for $C_{25}$ is not computed

Intermediate fields

5.5.519885601.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ $25$ R $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{25}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
151Data not computed