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Properties

Label 25.25.1295303893...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 181^{20}$
Root discriminant $840.42$
Ramified primes $5, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5^2$ (as 25T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-187339360201, -1596471019745, 999608868745, 21125601745505, -18090480890620, -45133153562130, 34441335142480, 24640171762500, -19407576480120, -5158532346840, 4911284705363, 371426734895, -640862104660, 15017327870, 45027904315, -3601148553, -1712570110, 195237720, 35508745, -4886245, -398086, 62750, 2250, -400, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 400*x^23 + 2250*x^22 + 62750*x^21 - 398086*x^20 - 4886245*x^19 + 35508745*x^18 + 195237720*x^17 - 1712570110*x^16 - 3601148553*x^15 + 45027904315*x^14 + 15017327870*x^13 - 640862104660*x^12 + 371426734895*x^11 + 4911284705363*x^10 - 5158532346840*x^9 - 19407576480120*x^8 + 24640171762500*x^7 + 34441335142480*x^6 - 45133153562130*x^5 - 18090480890620*x^4 + 21125601745505*x^3 + 999608868745*x^2 - 1596471019745*x - 187339360201)
 
gp: K = bnfinit(x^25 - 5*x^24 - 400*x^23 + 2250*x^22 + 62750*x^21 - 398086*x^20 - 4886245*x^19 + 35508745*x^18 + 195237720*x^17 - 1712570110*x^16 - 3601148553*x^15 + 45027904315*x^14 + 15017327870*x^13 - 640862104660*x^12 + 371426734895*x^11 + 4911284705363*x^10 - 5158532346840*x^9 - 19407576480120*x^8 + 24640171762500*x^7 + 34441335142480*x^6 - 45133153562130*x^5 - 18090480890620*x^4 + 21125601745505*x^3 + 999608868745*x^2 - 1596471019745*x - 187339360201, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 400 x^{23} + 2250 x^{22} + 62750 x^{21} - 398086 x^{20} - 4886245 x^{19} + 35508745 x^{18} + 195237720 x^{17} - 1712570110 x^{16} - 3601148553 x^{15} + 45027904315 x^{14} + 15017327870 x^{13} - 640862104660 x^{12} + 371426734895 x^{11} + 4911284705363 x^{10} - 5158532346840 x^{9} - 19407576480120 x^{8} + 24640171762500 x^{7} + 34441335142480 x^{6} - 45133153562130 x^{5} - 18090480890620 x^{4} + 21125601745505 x^{3} + 999608868745 x^{2} - 1596471019745 x - 187339360201 \)

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:  \(12953038931091714779183434289245559553901690387647249735891819000244140625=5^{40}\cdot 181^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $840.42$
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:  $5, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4525=5^{2}\cdot 181\)
Dirichlet character group:    $\lbrace$$\chi_{4525}(3136,·)$, $\chi_{4525}(1,·)$, $\chi_{4525}(2116,·)$, $\chi_{4525}(1221,·)$, $\chi_{4525}(1671,·)$, $\chi_{4525}(4041,·)$, $\chi_{4525}(906,·)$, $\chi_{4525}(3021,·)$, $\chi_{4525}(2126,·)$, $\chi_{4525}(2576,·)$, $\chi_{4525}(1811,·)$, $\chi_{4525}(3926,·)$, $\chi_{4525}(3031,·)$, $\chi_{4525}(3481,·)$, $\chi_{4525}(2716,·)$, $\chi_{4525}(421,·)$, $\chi_{4525}(3936,·)$, $\chi_{4525}(4386,·)$, $\chi_{4525}(3621,·)$, $\chi_{4525}(1326,·)$, $\chi_{4525}(306,·)$, $\chi_{4525}(2231,·)$, $\chi_{4525}(1211,·)$, $\chi_{4525}(316,·)$, $\chi_{4525}(766,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{14} a^{9} - \frac{1}{2} a^{6} - \frac{1}{14} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{2}$, $\frac{1}{196} a^{12} - \frac{1}{98} a^{11} - \frac{5}{98} a^{10} + \frac{13}{196} a^{9} - \frac{3}{49} a^{8} + \frac{3}{196} a^{7} + \frac{12}{49} a^{6} + \frac{65}{196} a^{5} - \frac{9}{98} a^{4} - \frac{3}{98} a^{3} + \frac{24}{49} a^{2} - \frac{73}{196} a + \frac{13}{28}$, $\frac{1}{196} a^{13} + \frac{1}{28} a^{10} + \frac{1}{28} a^{8} - \frac{1}{98} a^{7} + \frac{9}{28} a^{6} - \frac{1}{2} a^{5} + \frac{3}{14} a^{4} + \frac{13}{28} a^{2} + \frac{1}{196} a + \frac{3}{7}$, $\frac{1}{196} a^{14} - \frac{1}{28} a^{11} - \frac{1}{14} a^{10} - \frac{1}{28} a^{9} - \frac{1}{98} a^{8} + \frac{1}{28} a^{7} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{28} a^{3} + \frac{1}{196} a^{2} - \frac{2}{7} a - \frac{1}{2}$, $\frac{1}{196} a^{15} + \frac{1}{28} a^{10} + \frac{5}{196} a^{9} + \frac{1}{28} a^{8} - \frac{1}{28} a^{7} - \frac{1}{4} a^{5} - \frac{1}{28} a^{4} + \frac{43}{196} a^{3} - \frac{2}{7} a^{2} + \frac{1}{28} a + \frac{1}{4}$, $\frac{1}{196} a^{16} - \frac{1}{28} a^{11} - \frac{9}{196} a^{10} - \frac{1}{28} a^{9} - \frac{1}{28} a^{8} + \frac{1}{4} a^{6} + \frac{1}{28} a^{5} - \frac{41}{196} a^{4} + \frac{2}{7} a^{3} + \frac{1}{28} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{196} a^{17} + \frac{5}{196} a^{11} + \frac{1}{28} a^{10} - \frac{1}{14} a^{7} - \frac{1}{4} a^{6} - \frac{3}{98} a^{5} + \frac{3}{14} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{9}{28} a + \frac{1}{4}$, $\frac{1}{1372} a^{18} + \frac{3}{1372} a^{17} - \frac{1}{686} a^{16} + \frac{3}{1372} a^{15} + \frac{3}{1372} a^{14} - \frac{3}{1372} a^{13} - \frac{3}{1372} a^{12} - \frac{11}{1372} a^{11} - \frac{3}{196} a^{10} - \frac{41}{686} a^{9} + \frac{3}{686} a^{8} - \frac{81}{1372} a^{7} - \frac{69}{686} a^{6} - \frac{153}{1372} a^{5} + \frac{197}{686} a^{4} - \frac{209}{686} a^{3} - \frac{267}{686} a^{2} - \frac{45}{98} a + \frac{3}{14}$, $\frac{1}{204428} a^{19} - \frac{53}{204428} a^{18} - \frac{1}{1372} a^{17} + \frac{101}{204428} a^{16} + \frac{3}{204428} a^{15} - \frac{159}{102214} a^{14} + \frac{403}{204428} a^{13} - \frac{9}{102214} a^{12} - \frac{529}{29204} a^{11} - \frac{8839}{204428} a^{10} - \frac{11}{102214} a^{9} + \frac{3937}{204428} a^{8} + \frac{4075}{102214} a^{7} + \frac{44643}{102214} a^{6} + \frac{25195}{204428} a^{5} + \frac{8810}{51107} a^{4} + \frac{16901}{51107} a^{3} + \frac{1926}{7301} a^{2} + \frac{2864}{7301} a + \frac{45}{2086}$, $\frac{1}{1430996} a^{20} + \frac{80}{357749} a^{18} - \frac{12}{357749} a^{17} - \frac{300}{357749} a^{16} - \frac{755}{1430996} a^{15} - \frac{2445}{1430996} a^{14} + \frac{17}{715498} a^{13} - \frac{233}{357749} a^{12} - \frac{3967}{357749} a^{11} + \frac{47}{1043} a^{10} + \frac{15630}{357749} a^{9} - \frac{43947}{715498} a^{8} - \frac{74913}{1430996} a^{7} + \frac{61669}{357749} a^{6} + \frac{119479}{357749} a^{5} + \frac{75003}{715498} a^{4} - \frac{102587}{715498} a^{3} - \frac{521553}{1430996} a^{2} + \frac{18117}{102214} a + \frac{4919}{29204}$, $\frac{1}{4292988} a^{21} + \frac{5}{4292988} a^{19} + \frac{167}{715498} a^{18} - \frac{8501}{4292988} a^{17} - \frac{8581}{4292988} a^{16} + \frac{8083}{4292988} a^{15} + \frac{1081}{2146494} a^{14} + \frac{3335}{2146494} a^{13} + \frac{58}{1073247} a^{12} + \frac{15821}{613284} a^{11} - \frac{62351}{2146494} a^{10} + \frac{99475}{4292988} a^{9} + \frac{204583}{4292988} a^{8} + \frac{82083}{1430996} a^{7} + \frac{229795}{2146494} a^{6} + \frac{559879}{2146494} a^{5} - \frac{87149}{1430996} a^{4} + \frac{1252261}{4292988} a^{3} - \frac{69553}{613284} a^{2} + \frac{6865}{43806} a + \frac{1849}{6258}$, $\frac{1}{3876568164} a^{22} - \frac{41}{969142041} a^{21} - \frac{811}{3876568164} a^{20} - \frac{115}{138448863} a^{19} + \frac{634993}{3876568164} a^{18} + \frac{414825}{215364898} a^{17} + \frac{43023}{30766414} a^{16} + \frac{412449}{215364898} a^{15} - \frac{111759}{107682449} a^{14} - \frac{9779839}{3876568164} a^{13} - \frac{2885}{8672412} a^{12} - \frac{114766}{6504309} a^{11} + \frac{6113099}{215364898} a^{10} + \frac{89680543}{1938284082} a^{9} - \frac{184733177}{3876568164} a^{8} - \frac{17336650}{969142041} a^{7} + \frac{1726046095}{3876568164} a^{6} + \frac{882069091}{1938284082} a^{5} + \frac{589989041}{1938284082} a^{4} - \frac{34641683}{92299242} a^{3} - \frac{437481035}{3876568164} a^{2} - \frac{218014133}{553795452} a + \frac{129641}{1839852}$, $\frac{1}{23259408984} a^{23} - \frac{1}{23259408984} a^{22} - \frac{151}{7753136328} a^{21} + \frac{5455}{23259408984} a^{20} + \frac{6217}{2584378776} a^{19} + \frac{6760897}{23259408984} a^{18} - \frac{377957}{430729796} a^{17} - \frac{373733}{323047347} a^{16} + \frac{395573}{1292189388} a^{15} - \frac{120635}{415346589} a^{14} + \frac{2687219}{23259408984} a^{13} + \frac{40580695}{23259408984} a^{12} + \frac{449793499}{23259408984} a^{11} + \frac{912325667}{23259408984} a^{10} + \frac{46940057}{1292189388} a^{9} - \frac{240848483}{3876568164} a^{8} + \frac{44554261}{969142041} a^{7} + \frac{132473207}{1292189388} a^{6} - \frac{129725165}{276897726} a^{5} - \frac{1220664589}{11629704492} a^{4} - \frac{49118005}{11629704492} a^{3} + \frac{632204647}{2907426123} a^{2} - \frac{72650119}{1107590904} a + \frac{1115879}{11039112}$, $\frac{1}{3501678242643447318260243639209207510016664474327808149137234264824862855345117374461553278956024166840328272} a^{24} + \frac{10587704297388683705851390407460227467555781782612896012826475517632577917961551592458800157092915}{583613040440574553043373939868201251669444079054634691522872377470810475890852895743592213159337361140054712} a^{23} - \frac{73863979412713665857258492722314383535359572962638690390367404488735223055796149063664607197252421}{1750839121321723659130121819604603755008332237163904074568617132412431427672558687230776639478012083420164136} a^{22} + \frac{18832221227967994484432862098196525932393189589363387263965557681112530801080501655284113373959268512}{218854890165215457391265227450575469376041529645488009321077141551553928459069835903847079934751510427520517} a^{21} + \frac{164465018313757692174659650945085758612638383630304019294507156921162554400981293150374947797002560469}{1750839121321723659130121819604603755008332237163904074568617132412431427672558687230776639478012083420164136} a^{20} + \frac{40137229464262755396019250045802576789282529902340212680883046172291220306832848977990461162916899597}{17865705319609425093164508363312283214370737113917388516006297269514606404822027420722210606918490647144532} a^{19} - \frac{149440551841504831509269610965037066648953579835656644330547791877969270537597201975947705780658249617379}{500239748949063902608606234172743930002380639189686878448176323546408979335016767780221896993717738120046896} a^{18} - \frac{420342339301007455990185152128043434139904190736803377390213106928813664534918049760987219536187886436091}{194537680146858184347791313289400417223148026351544897174290792490270158630284298581197404386445787046684904} a^{17} + \frac{5118955698572425499466737376229119763502801862125613938980977816917238457654692508618912896735287695}{27007868963884240503650050435846233128300434034644578255489489447489956772217728527168874689219184651768} a^{16} + \frac{800755287399359468495673414542163597525351724735539959372896266678584909076424706106274114449511898060303}{875419560660861829565060909802301877504166118581952037284308566206215713836279343615388319739006041710082068} a^{15} + \frac{4339256365637883766407467768867515373953433684424303045180114112550114326884784651543729740917983455264847}{3501678242643447318260243639209207510016664474327808149137234264824862855345117374461553278956024166840328272} a^{14} - \frac{5490009316327792252226613603082520623487680306740561237951151353465439054753359739259297985656681858023}{72951630055071819130421742483525156458680509881829336440359047183851309486356611967949026644917170142506839} a^{13} + \frac{2282568099206686957854755454262047767371570641233571420867007235891319786913856520613262061983904128729491}{1750839121321723659130121819604603755008332237163904074568617132412431427672558687230776639478012083420164136} a^{12} - \frac{19878016336533233387020381473031158432584421557863594827287776480343040942596922828522621364117467738562153}{583613040440574553043373939868201251669444079054634691522872377470810475890852895743592213159337361140054712} a^{11} - \frac{6407940013887530123462783058879825033678171669562293366861219528070959051436192377618646263860461209210787}{3501678242643447318260243639209207510016664474327808149137234264824862855345117374461553278956024166840328272} a^{10} - \frac{3541004492416861116875932935015350545246515779055471604202573081891688857382707750231519104495369147505627}{83373291491510650434767705695457321667063439864947813074696053924401496555836127963370316165619623020007816} a^{9} + \frac{2920189986633512772251511379139833078590114666915142077097320329176317239168022016488147108030764886417149}{583613040440574553043373939868201251669444079054634691522872377470810475890852895743592213159337361140054712} a^{8} + \frac{62021515870740718382765348339146360361955982831161254090280821086684696540213307314551885317337712524181}{583613040440574553043373939868201251669444079054634691522872377470810475890852895743592213159337361140054712} a^{7} + \frac{49898206043406293803414281099804897732178003936740039287713827693546236157775378181979492557177335030144673}{583613040440574553043373939868201251669444079054634691522872377470810475890852895743592213159337361140054712} a^{6} - \frac{759942207632363566510476131640292571887139791621122310457350125029227409895719404285888986227313446043668563}{1750839121321723659130121819604603755008332237163904074568617132412431427672558687230776639478012083420164136} a^{5} - \frac{294307100281608485250124065817587670247459819820842660637162086005453760531965670264707663313079941827156537}{875419560660861829565060909802301877504166118581952037284308566206215713836279343615388319739006041710082068} a^{4} + \frac{48363727122123229291919332469019850575560132766944564468963646658162288155846206227491919006593290431936189}{291806520220287276521686969934100625834722039527317345761436188735405237945426447871796106579668680570027356} a^{3} - \frac{173998025058994184944705790745202156460595451577188775277687688274679060775609323477957371601888380078548339}{3501678242643447318260243639209207510016664474327808149137234264824862855345117374461553278956024166840328272} a^{2} - \frac{18810840438095780648417504906623098808266489758086444004795021064839441126284149796667684566572612850926421}{125059937237265975652151558543185982500595159797421719612044080886602244833754191945055474248429434530011724} a + \frac{213314688267860202912445330015039346116215439743952802526842529315562453513501277437375980453322411730177}{1661926076242737217968791475656956578080998801294640792186632304140893619053211853090438195992417734618096}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  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:  \( 2970119090537361000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2$ (as 25T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 25
The 25 conjugacy class representatives for $C_5^2$
Character table for $C_5^2$ is not computed

Intermediate fields

5.5.390625.1, 5.5.419251219140625.1, 5.5.1073283121.1, 5.5.419251219140625.2, 5.5.419251219140625.4, 5.5.419251219140625.3

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{25}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{25}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
181Data not computed