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Properties

Label 25.25.2276114846...0625.4
Degree $25$
Signature $[25, 0]$
Discriminant $5^{68}\cdot 31^{20}$
Root discriminant $1242.48$
Ramified primes $5, 31$
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![11183922944644501751, 62045773461582216700, -50566067906613262850, -182032030329568110350, 30828560929555647125, 62279523875038372510, -5260769512829156375, -8867062293830230925, 368288011589757525, 647652419463437375, -11487780086685245, -26022499540269450, 172020628126200, 598125348233550, -1284040510375, -8133195776635, 4525252900, 67015839525, -5886125, -335148750, 0, 985025, 0, -1550, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 1550*x^23 + 985025*x^21 - 335148750*x^19 - 5886125*x^18 + 67015839525*x^17 + 4525252900*x^16 - 8133195776635*x^15 - 1284040510375*x^14 + 598125348233550*x^13 + 172020628126200*x^12 - 26022499540269450*x^11 - 11487780086685245*x^10 + 647652419463437375*x^9 + 368288011589757525*x^8 - 8867062293830230925*x^7 - 5260769512829156375*x^6 + 62279523875038372510*x^5 + 30828560929555647125*x^4 - 182032030329568110350*x^3 - 50566067906613262850*x^2 + 62045773461582216700*x + 11183922944644501751)
 
gp: K = bnfinit(x^25 - 1550*x^23 + 985025*x^21 - 335148750*x^19 - 5886125*x^18 + 67015839525*x^17 + 4525252900*x^16 - 8133195776635*x^15 - 1284040510375*x^14 + 598125348233550*x^13 + 172020628126200*x^12 - 26022499540269450*x^11 - 11487780086685245*x^10 + 647652419463437375*x^9 + 368288011589757525*x^8 - 8867062293830230925*x^7 - 5260769512829156375*x^6 + 62279523875038372510*x^5 + 30828560929555647125*x^4 - 182032030329568110350*x^3 - 50566067906613262850*x^2 + 62045773461582216700*x + 11183922944644501751, 1)
 

Normalized defining polynomial

\( x^{25} - 1550 x^{23} + 985025 x^{21} - 335148750 x^{19} - 5886125 x^{18} + 67015839525 x^{17} + 4525252900 x^{16} - 8133195776635 x^{15} - 1284040510375 x^{14} + 598125348233550 x^{13} + 172020628126200 x^{12} - 26022499540269450 x^{11} - 11487780086685245 x^{10} + 647652419463437375 x^{9} + 368288011589757525 x^{8} - 8867062293830230925 x^{7} - 5260769512829156375 x^{6} + 62279523875038372510 x^{5} + 30828560929555647125 x^{4} - 182032030329568110350 x^{3} - 50566067906613262850 x^{2} + 62045773461582216700 x + 11183922944644501751 \)

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:  \(227611484625260901780055732479538161718013444811958834179677069187164306640625=5^{68}\cdot 31^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1242.48$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3875=5^{3}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3875}(256,·)$, $\chi_{3875}(1,·)$, $\chi_{3875}(1986,·)$, $\chi_{3875}(1031,·)$, $\chi_{3875}(776,·)$, $\chi_{3875}(2761,·)$, $\chi_{3875}(1806,·)$, $\chi_{3875}(1551,·)$, $\chi_{3875}(16,·)$, $\chi_{3875}(2581,·)$, $\chi_{3875}(2326,·)$, $\chi_{3875}(791,·)$, $\chi_{3875}(3356,·)$, $\chi_{3875}(3101,·)$, $\chi_{3875}(1566,·)$, $\chi_{3875}(3536,·)$, $\chi_{3875}(996,·)$, $\chi_{3875}(2341,·)$, $\chi_{3875}(1771,·)$, $\chi_{3875}(3116,·)$, $\chi_{3875}(221,·)$, $\chi_{3875}(2546,·)$, $\chi_{3875}(436,·)$, $\chi_{3875}(3321,·)$, $\chi_{3875}(1211,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{31} a^{5}$, $\frac{1}{31} a^{6}$, $\frac{1}{217} a^{7} + \frac{2}{7} a$, $\frac{1}{217} a^{8} + \frac{2}{7} a^{2}$, $\frac{1}{217} a^{9} + \frac{2}{7} a^{3}$, $\frac{1}{6727} a^{10} + \frac{3}{7} a^{4}$, $\frac{1}{6727} a^{11} + \frac{2}{217} a^{5}$, $\frac{1}{6727} a^{12} + \frac{2}{217} a^{6}$, $\frac{1}{47089} a^{13} - \frac{3}{1519} a^{7} - \frac{10}{49} a$, $\frac{1}{47089} a^{14} - \frac{3}{1519} a^{8} - \frac{10}{49} a^{2}$, $\frac{1}{1459759} a^{15} - \frac{1}{1519} a^{9} + \frac{20}{49} a^{3}$, $\frac{1}{1459759} a^{16} - \frac{3}{47089} a^{10} + \frac{6}{49} a^{4}$, $\frac{1}{10218313} a^{17} - \frac{3}{10218313} a^{15} + \frac{3}{329623} a^{13} - \frac{3}{329623} a^{11} + \frac{24}{10633} a^{9} - \frac{9}{10633} a^{7} + \frac{2}{217} a^{6} - \frac{59}{10633} a^{5} + \frac{3}{7} a^{4} + \frac{31}{343} a^{3} - \frac{2}{7} a^{2} + \frac{166}{343} a + \frac{3}{7}$, $\frac{1}{10218313} a^{18} - \frac{3}{10218313} a^{16} + \frac{3}{329623} a^{14} - \frac{3}{329623} a^{12} + \frac{9}{329623} a^{10} - \frac{9}{10633} a^{8} - \frac{59}{10633} a^{6} + \frac{2}{217} a^{5} - \frac{116}{343} a^{4} - \frac{2}{7} a^{3} + \frac{166}{343} a^{2} - \frac{1}{7} a$, $\frac{1}{10218313} a^{19} - \frac{1}{329623} a^{13} - \frac{16}{10633} a^{7} + \frac{1}{217} a^{6} - \frac{20}{343} a + \frac{2}{7}$, $\frac{1}{2217373921} a^{20} - \frac{2}{71528191} a^{19} + \frac{3}{71528191} a^{18} + \frac{3}{71528191} a^{17} - \frac{9}{71528191} a^{16} - \frac{9}{71528191} a^{15} + \frac{11}{2307361} a^{14} - \frac{17}{2307361} a^{13} - \frac{58}{2307361} a^{12} - \frac{9}{2307361} a^{11} + \frac{76}{2307361} a^{10} + \frac{23}{74431} a^{9} - \frac{51}{74431} a^{8} - \frac{9}{74431} a^{7} + \frac{264}{74431} a^{6} - \frac{569}{74431} a^{5} + \frac{926}{2401} a^{4} - \frac{985}{2401} a^{3} - \frac{925}{2401} a^{2} - \frac{554}{2401} a + \frac{19}{49}$, $\frac{1}{2217373921} a^{21} - \frac{2}{71528191} a^{19} + \frac{2}{71528191} a^{17} + \frac{2}{10218313} a^{15} + \frac{8}{2307361} a^{13} + \frac{1}{47089} a^{12} + \frac{43}{2307361} a^{11} - \frac{1}{47089} a^{10} - \frac{130}{74431} a^{9} - \frac{3}{1519} a^{8} - \frac{12}{10633} a^{7} + \frac{4}{1519} a^{6} - \frac{363}{74431} a^{5} + \frac{10}{49} a^{4} + \frac{445}{2401} a^{3} + \frac{11}{49} a^{2} + \frac{862}{2401} a + \frac{9}{49}$, $\frac{1}{2217373921} a^{22} + \frac{2}{71528191} a^{19} - \frac{1}{71528191} a^{18} - \frac{3}{71528191} a^{17} + \frac{23}{71528191} a^{16} + \frac{9}{71528191} a^{15} - \frac{24}{2307361} a^{14} + \frac{17}{2307361} a^{13} + \frac{101}{2307361} a^{12} - \frac{40}{2307361} a^{11} + \frac{10}{2307361} a^{10} - \frac{170}{74431} a^{9} - \frac{75}{74431} a^{8} + \frac{9}{74431} a^{7} - \frac{627}{74431} a^{6} - \frac{1048}{74431} a^{5} - \frac{138}{2401} a^{4} - \frac{877}{2401} a^{3} - \frac{985}{2401} a^{2} + \frac{799}{2401} a - \frac{19}{49}$, $\frac{1}{108651322129} a^{23} + \frac{3}{15521617447} a^{22} + \frac{13}{108651322129} a^{21} + \frac{3}{15521617447} a^{20} + \frac{13}{3504881359} a^{19} - \frac{1}{500697337} a^{18} + \frac{143}{3504881359} a^{17} + \frac{5}{71528191} a^{16} - \frac{60}{3504881359} a^{15} - \frac{67}{16151527} a^{14} + \frac{1105}{113060689} a^{13} + \frac{1060}{16151527} a^{12} - \frac{841}{16151527} a^{11} - \frac{64}{16151527} a^{10} - \frac{8231}{3647119} a^{9} + \frac{6}{74431} a^{8} - \frac{2876}{3647119} a^{7} + \frac{6240}{521017} a^{6} + \frac{24853}{3647119} a^{5} - \frac{926}{16807} a^{4} - \frac{2741}{16807} a^{3} + \frac{4133}{16807} a^{2} - \frac{8941}{117649} a - \frac{95}{2401}$, $\frac{1}{789666360318374481279335899864940804955082190326479143230769868472173541149490323089466078777272016109681088435376454544728411390506447825580100647} a^{24} - \frac{3498230364408414722957276606908210372290512239298498348207641538571326322498374794898455435161925807953018894241443669876813585745841697}{789666360318374481279335899864940804955082190326479143230769868472173541149490323089466078777272016109681088435376454544728411390506447825580100647} a^{23} - \frac{82379557360451767624805638023677998052051248175458342346448008671845624099783823579271524967575502879155274796059509435172965445592646569}{789666360318374481279335899864940804955082190326479143230769868472173541149490323089466078777272016109681088435376454544728411390506447825580100647} a^{22} - \frac{65357951680053244876835527744869168909573740491629503373128008476623518073729110372546022497403434697035692170444356772357077292569839221}{789666360318374481279335899864940804955082190326479143230769868472173541149490323089466078777272016109681088435376454544728411390506447825580100647} a^{21} - \frac{123933500917485922087500544121088774142406927766406159957290286391999149733722989034076182588876032145666543174628540451599094968817440735}{789666360318374481279335899864940804955082190326479143230769868472173541149490323089466078777272016109681088435376454544728411390506447825580100647} a^{20} - \frac{1115482072296674037560693982071505217320997694772134348492025812408846312448277514559381894303537160063235345786219645582390366436538747740}{25473108397366918750946319350481961450163941623434811071960318337812049714499687841595679960557161809989712530173434017571884238403433800825164537} a^{19} + \frac{108553825957738385577003717901568703236725838913682521680519700326934127874195724137574145338869635589115277015522407465701316997638991524}{25473108397366918750946319350481961450163941623434811071960318337812049714499687841595679960557161809989712530173434017571884238403433800825164537} a^{18} + \frac{931578036527963758588502903999101562879346513611743921660906962405305788209691347044457720646267084871387956172772145420648462445504640976}{25473108397366918750946319350481961450163941623434811071960318337812049714499687841595679960557161809989712530173434017571884238403433800825164537} a^{17} + \frac{7715024499312120667179234026671494593360231443685301180161740552093351911056817491558463325648803600429665836112239347052154764780700635083}{25473108397366918750946319350481961450163941623434811071960318337812049714499687841595679960557161809989712530173434017571884238403433800825164537} a^{16} - \frac{5319232414569315805233346532719580612327040666105689714356717263203494838434392264004229543086385440153962046502825127859668714553449090002}{25473108397366918750946319350481961450163941623434811071960318337812049714499687841595679960557161809989712530173434017571884238403433800825164537} a^{15} - \frac{7283187258236733416328383216576278047500310711622563657543248473500399681173167860033724614706124390570613278200403095638024124068638304564}{821713174108610282288590946789740691940772310433381002321300591542324184338699607793409030985714897096442339683014000566834975432368832284682727} a^{14} - \frac{1121040160630023968881950357588389005700171675180653086302399803033139981010592675870243487497014770186231045070454328281395865818109300400}{821713174108610282288590946789740691940772310433381002321300591542324184338699607793409030985714897096442339683014000566834975432368832284682727} a^{13} + \frac{90649657007405769822660622119256246815007355230179707011290763430542350057011054108617489993228750557344187866280569249702822047785257490}{2395665230637347761774317629124608431314204986686241989274928838315813948509328302604691052436486580456100115693918368999518878811570939605489} a^{12} - \frac{6351046318084981669819914929152906615786719166180387207580343298329619079886399945751403979869150172838886022899641748009555388257821716507}{117387596301230040326941563827105813134396044347625857474471513077474883476957086827629861569387842442348905669002000080976425061766976040668961} a^{11} + \frac{52861903624656267604615955569587827612353559958275611903143233840320729193665697130177707361232380056972546264379841464203609982342180101653}{821713174108610282288590946789740691940772310433381002321300591542324184338699607793409030985714897096442339683014000566834975432368832284682727} a^{10} - \frac{43763545800295006282311207341166172889012696550591266844484793098850953456852507539726338270089655826957919445671847455418004483569373164821}{26506876584148718783502933767410990062605558401076806526493567469107231752861277670755130031797254745046527086548838727962418562334478460796217} a^{9} + \frac{9847219810677796426272813724201710309233779587847484222446113148380641398054227833312426272004559547675229089681864298941071308100104667550}{26506876584148718783502933767410990062605558401076806526493567469107231752861277670755130031797254745046527086548838727962418562334478460796217} a^{8} + \frac{46335468310265549016901063250682093490355652910133088404031485456596214742135967024212843624042416497154336993719497917798405389663572725938}{26506876584148718783502933767410990062605558401076806526493567469107231752861277670755130031797254745046527086548838727962418562334478460796217} a^{7} - \frac{129279136577442139294964784947759247643394140177313672123793265484951072946579557810353656076223510369192750445729287081731985119764559201077}{26506876584148718783502933767410990062605558401076806526493567469107231752861277670755130031797254745046527086548838727962418562334478460796217} a^{6} - \frac{369644132516683185184672906051772722324243589913634157458011837057161050885206457373241324833712412738145402934569227915145933613707196879716}{26506876584148718783502933767410990062605558401076806526493567469107231752861277670755130031797254745046527086548838727962418562334478460796217} a^{5} + \frac{6218263780459035660183930420488490084495378772689901660063807902721705687500678654163846373403875260580447154838400510938127622140596189993}{17450214999439577869323853698098084307179432785435685665894382797305616690494587011688696531795427745257753184034785206031875287909465741143} a^{4} - \frac{31659282581264805386081031525146641988104367712695125796343737319157752033980309262342605221285567607154296266660716115289026482426605481206}{122151504996077045085266975886686590150256029498049799661260679581139316833462109081820875722567994216804272288243496442223127015366260188001} a^{3} - \frac{26277398906808861444465961416875174132588148148388580161286268123357062748955143375573204795079457867229696875206299944262027271497631255507}{855060534972539315596868831206806131051792206486348597628824757067975217834234763572746130057975959517629906017704475095561889107563821316007} a^{2} + \frac{229500550229819980245748975453413396301725419377923021128100660368645837879068197727336689477360165742028857140901547871851739454424572950419}{855060534972539315596868831206806131051792206486348597628824757067975217834234763572746130057975959517629906017704475095561889107563821316007} a - \frac{6265598342553771572327686559396603882636514603237752061685132892362986968545805684547451577971149574125000973593250177291377743150470575241}{17450214999439577869323853698098084307179432785435685665894382797305616690494587011688696531795427745257753184034785206031875287909465741143}$

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

5.5.390625.1

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 ${\href{/LocalNumberField/7.1.0.1}{1} }^{25}$ $25$ $25$ $25$ $25$ $25$ $25$ R $25$ $25$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ $25$ $25$ $25$

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
31Data not computed