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Properties

Label 25.25.1154816935...0625.4
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 101^{24}$
Root discriminant $1102.81$
Ramified primes $5, 101$
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![42821375557427150093, -54470348529213734710, -3546244457438956345, 32931106903916290285, -13215853303684560625, -2551473066213323146, 2459705659955183385, -151479667998984250, -181562185323651575, 28657059461272800, 6999683430290308, -1631967413736815, -154430289182685, 50484244881730, 1991258743100, -960015631260, -14569481290, 11627188680, 55116205, -88874950, -82214, 408545, 0, -1010, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 1010*x^23 + 408545*x^21 - 82214*x^20 - 88874950*x^19 + 55116205*x^18 + 11627188680*x^17 - 14569481290*x^16 - 960015631260*x^15 + 1991258743100*x^14 + 50484244881730*x^13 - 154430289182685*x^12 - 1631967413736815*x^11 + 6999683430290308*x^10 + 28657059461272800*x^9 - 181562185323651575*x^8 - 151479667998984250*x^7 + 2459705659955183385*x^6 - 2551473066213323146*x^5 - 13215853303684560625*x^4 + 32931106903916290285*x^3 - 3546244457438956345*x^2 - 54470348529213734710*x + 42821375557427150093)
 
gp: K = bnfinit(x^25 - 1010*x^23 + 408545*x^21 - 82214*x^20 - 88874950*x^19 + 55116205*x^18 + 11627188680*x^17 - 14569481290*x^16 - 960015631260*x^15 + 1991258743100*x^14 + 50484244881730*x^13 - 154430289182685*x^12 - 1631967413736815*x^11 + 6999683430290308*x^10 + 28657059461272800*x^9 - 181562185323651575*x^8 - 151479667998984250*x^7 + 2459705659955183385*x^6 - 2551473066213323146*x^5 - 13215853303684560625*x^4 + 32931106903916290285*x^3 - 3546244457438956345*x^2 - 54470348529213734710*x + 42821375557427150093, 1)
 

Normalized defining polynomial

\( x^{25} - 1010 x^{23} + 408545 x^{21} - 82214 x^{20} - 88874950 x^{19} + 55116205 x^{18} + 11627188680 x^{17} - 14569481290 x^{16} - 960015631260 x^{15} + 1991258743100 x^{14} + 50484244881730 x^{13} - 154430289182685 x^{12} - 1631967413736815 x^{11} + 6999683430290308 x^{10} + 28657059461272800 x^{9} - 181562185323651575 x^{8} - 151479667998984250 x^{7} + 2459705659955183385 x^{6} - 2551473066213323146 x^{5} - 13215853303684560625 x^{4} + 32931106903916290285 x^{3} - 3546244457438956345 x^{2} - 54470348529213734710 x + 42821375557427150093 \)

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:  \(11548169354972874033626020541245956538701891644459465169347822666168212890625=5^{40}\cdot 101^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1102.81$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2525=5^{2}\cdot 101\)
Dirichlet character group:    $\lbrace$$\chi_{2525}(1,·)$, $\chi_{2525}(1091,·)$, $\chi_{2525}(391,·)$, $\chi_{2525}(1801,·)$, $\chi_{2525}(2381,·)$, $\chi_{2525}(1551,·)$, $\chi_{2525}(786,·)$, $\chi_{2525}(536,·)$, $\chi_{2525}(731,·)$, $\chi_{2525}(1501,·)$, $\chi_{2525}(1696,·)$, $\chi_{2525}(2146,·)$, $\chi_{2525}(611,·)$, $\chi_{2525}(1381,·)$, $\chi_{2525}(2241,·)$, $\chi_{2525}(1771,·)$, $\chi_{2525}(1006,·)$, $\chi_{2525}(1391,·)$, $\chi_{2525}(496,·)$, $\chi_{2525}(1586,·)$, $\chi_{2525}(1971,·)$, $\chi_{2525}(2036,·)$, $\chi_{2525}(56,·)$, $\chi_{2525}(441,·)$, $\chi_{2525}(701,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{41} a^{17} - \frac{12}{41} a^{16} + \frac{9}{41} a^{15} + \frac{2}{41} a^{14} - \frac{17}{41} a^{13} + \frac{15}{41} a^{12} + \frac{20}{41} a^{11} + \frac{13}{41} a^{10} + \frac{20}{41} a^{9} - \frac{18}{41} a^{7} - \frac{12}{41} a^{6} - \frac{7}{41} a^{5} - \frac{1}{41} a^{4} + \frac{9}{41} a^{3} - \frac{2}{41} a^{2} - \frac{16}{41} a + \frac{17}{41}$, $\frac{1}{41} a^{18} - \frac{12}{41} a^{16} - \frac{13}{41} a^{15} + \frac{7}{41} a^{14} + \frac{16}{41} a^{13} - \frac{5}{41} a^{12} + \frac{7}{41} a^{11} + \frac{12}{41} a^{10} - \frac{6}{41} a^{9} - \frac{18}{41} a^{8} + \frac{18}{41} a^{7} + \frac{13}{41} a^{6} - \frac{3}{41} a^{5} - \frac{3}{41} a^{4} - \frac{17}{41} a^{3} + \frac{1}{41} a^{2} - \frac{11}{41} a - \frac{1}{41}$, $\frac{1}{41} a^{19} + \frac{7}{41} a^{16} - \frac{8}{41} a^{15} - \frac{1}{41} a^{14} - \frac{4}{41} a^{13} - \frac{18}{41} a^{12} + \frac{6}{41} a^{11} - \frac{14}{41} a^{10} + \frac{17}{41} a^{9} + \frac{18}{41} a^{8} + \frac{2}{41} a^{7} + \frac{17}{41} a^{6} - \frac{5}{41} a^{5} + \frac{12}{41} a^{4} - \frac{14}{41} a^{3} + \frac{6}{41} a^{2} + \frac{12}{41} a - \frac{1}{41}$, $\frac{1}{41} a^{20} - \frac{6}{41} a^{16} + \frac{18}{41} a^{15} - \frac{18}{41} a^{14} + \frac{19}{41} a^{13} - \frac{17}{41} a^{12} + \frac{10}{41} a^{11} + \frac{8}{41} a^{10} + \frac{1}{41} a^{9} + \frac{2}{41} a^{8} + \frac{20}{41} a^{7} - \frac{3}{41} a^{6} + \frac{20}{41} a^{5} - \frac{7}{41} a^{4} - \frac{16}{41} a^{3} - \frac{15}{41} a^{2} - \frac{12}{41} a + \frac{4}{41}$, $\frac{1}{41} a^{21} - \frac{13}{41} a^{16} - \frac{5}{41} a^{15} - \frac{10}{41} a^{14} + \frac{4}{41} a^{13} + \frac{18}{41} a^{12} + \frac{5}{41} a^{11} - \frac{3}{41} a^{10} - \frac{1}{41} a^{9} + \frac{20}{41} a^{8} + \frac{12}{41} a^{7} - \frac{11}{41} a^{6} - \frac{8}{41} a^{5} + \frac{19}{41} a^{4} - \frac{2}{41} a^{3} + \frac{17}{41} a^{2} - \frac{10}{41} a + \frac{20}{41}$, $\frac{1}{41} a^{22} + \frac{3}{41} a^{16} - \frac{16}{41} a^{15} - \frac{11}{41} a^{14} + \frac{2}{41} a^{13} - \frac{5}{41} a^{12} + \frac{11}{41} a^{11} + \frac{4}{41} a^{10} - \frac{7}{41} a^{9} + \frac{12}{41} a^{8} + \frac{1}{41} a^{7} + \frac{10}{41} a^{5} - \frac{15}{41} a^{4} + \frac{11}{41} a^{3} + \frac{5}{41} a^{2} + \frac{17}{41} a + \frac{16}{41}$, $\frac{1}{231937} a^{23} - \frac{2157}{231937} a^{22} - \frac{2674}{231937} a^{21} - \frac{2627}{231937} a^{20} + \frac{391}{231937} a^{19} + \frac{994}{231937} a^{18} + \frac{11}{231937} a^{17} - \frac{20026}{231937} a^{16} + \frac{93995}{231937} a^{15} - \frac{47690}{231937} a^{14} - \frac{53348}{231937} a^{13} - \frac{30108}{231937} a^{12} + \frac{114570}{231937} a^{11} - \frac{114004}{231937} a^{10} - \frac{7943}{231937} a^{9} + \frac{57869}{231937} a^{8} - \frac{13602}{231937} a^{7} - \frac{14152}{231937} a^{6} - \frac{86877}{231937} a^{5} - \frac{13601}{231937} a^{4} + \frac{76347}{231937} a^{3} - \frac{81352}{231937} a^{2} + \frac{4909}{231937} a - \frac{38331}{231937}$, $\frac{1}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{24} - \frac{1210060399427537617649846433111138875095874509702629454245994914161663358415262271947259222893972746999192819898274111609667991102466432701428026406073819502258400829753847672962365129631707474323}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{23} + \frac{7776908423699322644244194990548496481506469479789836494426969570767763894965363490412109731044213398744827150362717225255196051924058369954825877566260105436124822305548250189754617579116825290311250}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{22} - \frac{8053009868166964355710597793055426238416369316530117686431686497779818620228670893369463707828374279306764007444299166381881380551832900893749594390559177461217051332134931574227675141049119139855780}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{21} + \frac{7535369869166533854958207391023439005266023622867758314819901737822395051121281323380213378194642087647394657797829212600164190816699598045814753077040439178428792088242171982481382973833208517031504}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{20} + \frac{3048753093197058144911278252644784812987279627448908268200532633051991358021429339762463635312173806472411377790768468743592526295380782454217826880215508297112255802046246009469069560198366038223078}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{19} + \frac{8843254988717545266253487738749629210428779217879648318698032904439499478499708098421384416385343528278073087911927675833495835264730533271681439270673322544660644956467613955276841927731271082851831}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{18} + \frac{5230494714268792516517036144266141189368464606134064702274584303663004576362284618315580025998615892112863207877393214439987795213373566288089123796485921547792316711510800871628816597775853095901266}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{17} + \frac{466902832339117949415539674374254838025405720904935768835434515602114937564381267333427479179200560346186458252592601896496276126180357400998783742175143617514509412070348441225885266468874904574072748}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{16} + \frac{368288230753039845517097857804161212431146010918990273132160787047388362131372010750783246149778679641639794470281446182370231640697874032333549551177111256523641943118934459943109507112343123968937634}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{15} - \frac{198962683068677697610929577977474894130100056548597967137674975426780810862885180051063238064401004458521470373065403510624163073924147275642646051917421967908982198841005159462311404522372776080910210}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{14} + \frac{75055799414420039992444008806234246019256463367438099994736855441528476269837881206658193495397202396376195452817926595366359268237115551547371658954260587287208584429031644494819275499557126219558798}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{13} - \frac{229392660804662980384500641304011087891544611451818537799613258783504695168123193801577531276002420283212543950439080076432521494302553841574584812062929060105931307157912471548274114233222884382308842}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{12} + \frac{338680474686713820290643084447242295288397897615987383986133527900170267076446373234661513038091852445728157393920287786832056585317582876701442117913458053925676995247648700480579076169234631203075602}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{11} - \frac{75029196549796398479652032754967033120205854111641479116978535616491919233791166111756353887086348535762758666316358232735925945066160731861911631122425058873876746549490894477519214117311431490396300}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{10} - \frac{206018146518953183385661917822846664776019538731526888853414192655505606170740622903682128002090463555347535994152450491885884106138968725505920576968608819685353376985390181125404413742548121115773822}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{9} + \frac{266215318430505875362243215139534868326900681224079562579008196707773121879111372063518030502681016252062590794479450334172890202974105554092714767211675706573964237978373979015707670748719594825568150}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{8} + \frac{54785557533899360307700034227726789528865180378141232051647013055996371704653885450797004145566203583120306940843094298365671030252673554442048997578358950594158988973993312870415476154069735514513043}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{7} - \frac{251394994459687135352708823845875405429400268335673039648892021799987375176639190232469619236648105899051003404495851683028173481797642260110425978967444923409016297399928420805664205765083125691847655}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{6} + \frac{401579518023134386829902858129761163336651235634069372366875569105396639557457119765989028718337792379555675610596400151006630506364579513307815335480043421164237664034416640350614224564715862302475598}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{5} + \frac{77639530583330112394609466747053615807251975528137845158890803096705292506628211634500287974045134962926617123611387666768712071147249131170809804638521141662489500967845036826456482742286793210083964}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{4} + \frac{441514623725908264693709587122189762658706793243053069750885147532334629659652616087027011479500726800899343443876425450843979243696633175175147063135573024375129739691340085190371521729186900687563904}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{3} - \frac{410464548428982166396526942199718454081016210292765926258398570698355921451760240914794822627646640558397422708556027696856438562942767230797584151481964915732975191228748327828181198279685320950434805}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a^{2} - \frac{168483396945395795201008569589615909789718399562386467727777163910223679520406674886974366996936099592412494504904210352081305115169309949857535597451958114486297087791576456549944154051827544149883372}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427} a - \frac{93605052752570083651297411285957808560638299161784808739341055673025345427234624926538893949900026775929470663681093588601552928594941240768018546100166475272137608913086135400746881499520045578519246}{968778025909839409525056126156386456521654773370165448480807201246482610712420610758852772788827054439419965532945361106991028335293177522849247639064236182733242720471248573284555990480870765234692427}$

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

5.5.104060401.1

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 $25$ $25$ R $25$ $25$ $25$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{25}$ $25$ $25$ $25$ $25$

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
101Data not computed