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Properties

Label 25.25.1154816935...0625.3
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![13895971453380343, 207711763316129890, 54511774675262230, -260320879713631715, -57844425068056750, 125386209929929004, 21327357544787485, -30976450128926475, -3761130316961825, 4348657471082675, 344954657261958, -359838520343665, -16758299601935, 17626167204980, 425568926225, -504670619810, -5536293790, 8294519455, 35193955, -77133700, -82214, 393395, 0, -1010, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 1010*x^23 + 393395*x^21 - 82214*x^20 - 77133700*x^19 + 35193955*x^18 + 8294519455*x^17 - 5536293790*x^16 - 504670619810*x^15 + 425568926225*x^14 + 17626167204980*x^13 - 16758299601935*x^12 - 359838520343665*x^11 + 344954657261958*x^10 + 4348657471082675*x^9 - 3761130316961825*x^8 - 30976450128926475*x^7 + 21327357544787485*x^6 + 125386209929929004*x^5 - 57844425068056750*x^4 - 260320879713631715*x^3 + 54511774675262230*x^2 + 207711763316129890*x + 13895971453380343)
 
gp: K = bnfinit(x^25 - 1010*x^23 + 393395*x^21 - 82214*x^20 - 77133700*x^19 + 35193955*x^18 + 8294519455*x^17 - 5536293790*x^16 - 504670619810*x^15 + 425568926225*x^14 + 17626167204980*x^13 - 16758299601935*x^12 - 359838520343665*x^11 + 344954657261958*x^10 + 4348657471082675*x^9 - 3761130316961825*x^8 - 30976450128926475*x^7 + 21327357544787485*x^6 + 125386209929929004*x^5 - 57844425068056750*x^4 - 260320879713631715*x^3 + 54511774675262230*x^2 + 207711763316129890*x + 13895971453380343, 1)
 

Normalized defining polynomial

\( x^{25} - 1010 x^{23} + 393395 x^{21} - 82214 x^{20} - 77133700 x^{19} + 35193955 x^{18} + 8294519455 x^{17} - 5536293790 x^{16} - 504670619810 x^{15} + 425568926225 x^{14} + 17626167204980 x^{13} - 16758299601935 x^{12} - 359838520343665 x^{11} + 344954657261958 x^{10} + 4348657471082675 x^{9} - 3761130316961825 x^{8} - 30976450128926475 x^{7} + 21327357544787485 x^{6} + 125386209929929004 x^{5} - 57844425068056750 x^{4} - 260320879713631715 x^{3} + 54511774675262230 x^{2} + 207711763316129890 x + 13895971453380343 \)

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$$\chi_{2525}(256,·)$, $\chi_{2525}(1,·)$, $\chi_{2525}(2411,·)$, $\chi_{2525}(2246,·)$, $\chi_{2525}(1736,·)$, $\chi_{2525}(1801,·)$, $\chi_{2525}(586,·)$, $\chi_{2525}(1291,·)$, $\chi_{2525}(1551,·)$, $\chi_{2525}(16,·)$, $\chi_{2525}(1041,·)$, $\chi_{2525}(2521,·)$, $\chi_{2525}(1371,·)$, $\chi_{2525}(1116,·)$, $\chi_{2525}(2461,·)$, $\chi_{2525}(1506,·)$, $\chi_{2525}(1571,·)$, $\chi_{2525}(2091,·)$, $\chi_{2525}(1501,·)$, $\chi_{2525}(456,·)$, $\chi_{2525}(371,·)$, $\chi_{2525}(181,·)$, $\chi_{2525}(886,·)$, $\chi_{2525}(631,·)$, $\chi_{2525}(701,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{317} a^{23} - \frac{48}{317} a^{22} + \frac{126}{317} a^{21} - \frac{70}{317} a^{20} + \frac{108}{317} a^{19} + \frac{68}{317} a^{18} - \frac{64}{317} a^{17} + \frac{26}{317} a^{16} + \frac{123}{317} a^{15} + \frac{126}{317} a^{14} + \frac{13}{317} a^{13} + \frac{13}{317} a^{12} - \frac{18}{317} a^{11} + \frac{3}{317} a^{10} - \frac{157}{317} a^{9} - \frac{8}{317} a^{8} - \frac{107}{317} a^{7} + \frac{13}{317} a^{6} - \frac{56}{317} a^{5} - \frac{97}{317} a^{4} - \frac{49}{317} a^{3} - \frac{117}{317} a^{2} - \frac{112}{317} a - \frac{11}{317}$, $\frac{1}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{24} - \frac{85879832164081466126830726944604328531918681867655069710326578723613035339539377032248173748383611390070817781308769265943938358421061549787471930887117731147095077453887665253495867193357702254361598549819439445}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{23} - \frac{45376321922020657566662468821204169832161962845413311115805381666399235058272883643164667966288219748611599028877286689792708748567961731971919764005860952754432827514428937872698060596552864367948395212342699423485}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{22} - \frac{8425262647306933288165389682412535726026631075386296800611325728986641714235078927274885150361709947605925695265081105591016844252581189579532825751666125819090909963685495760935314185432424769096021981248288571256}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{21} + \frac{41733161237918744128293986060459172557200745808682816750370058378601317826281150699662245723504302537654680216593419806220471168254604064356270371632534809878058825181830237914016383438479665646075792996827540329172}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{20} + \frac{39779425735121075022464776839176497306889786382910816361558465471038160930517615078559954621420601586378352714009073099255217074786913772345407935440907141254171843665924450359269257301256730892657132057618713893869}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{19} - \frac{34379705002135786714136963428812317939369294562179148855458972164548291818811395146309848640040993341045958370827651878498024140093390956675954831532261155822418759626760862285232424645290127124597051622715202159324}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{18} - \frac{32453207625209385135383871989646340004391841875013571237849462012494913712529197327802182423705008768211754691893224656243536968330187667809128955654533822962115992131934610500911068999508948426284136544018562642726}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{17} - \frac{18872759783997051179388439274819186706383106243017259737544010270518924245033496269168744872448772253122744673112446107531101811981501880025902502320179783809325699538493134353034936758829092982722921116627352879510}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{16} - \frac{16570428695828117461678249622815370757046697941088600918625944455523638692289079215866314964065020854230532596799296992990635798394918999718042175103714734622321982459955492728695236386028918153741317146992569246884}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{15} - \frac{34898173051696857662389704249798870151764184731242260424989862732093752054034516342074728550118659973153712112874000898789234315110738675767454267704083968068777007893640439954010629549007906801143125473988560942713}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{14} + \frac{27350637651928830181948614077591663437167673699826307125838856469479052799264579290800577251871928689310270997318220053281548128239256189819746854753026179783628663154452058358925561472310783462634876122947104164215}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{13} + \frac{44517340661497514814920053883826106268827403048110475222999049402022399324660777063737909282754363143776372272897414291684325763432940673020004543543778666053514808938297535475244503231243285723881826635448927649737}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{12} + \frac{17012663072713559445613072738488429802398061818780350283568555763219867506919962724196480769867874988960818061971076078925903754658195935732108393165984732894838354313947559715507464095773570033777082662214487256882}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{11} + \frac{31032470010329886436331325167960974636752522505641052516696725883896656509364697271596237824376450924490240267652127949832279421091979093165843059313373814652824553745520222601194326811599178305059230769868408860218}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{10} - \frac{47019275372535536448256547995225673905839534935328319659746000229941457343463595575675063883158592949590048376195201684131682101218892967620860580362241486415913951207966049914556366466940761060184156357591490650953}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{9} - \frac{45978575993968762222569251093358272078940231740467623155313026193930479882370080611628400905787600013989188535732271091230399482509442452371139748335002496627072775538836324033404807055848675399698477173980895489661}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{8} - \frac{11161903079644637929421564340498062259233393473666608761008820835384900223457110712510068356377490232613969320854056385419284526395474262089054519198975703732443987263009587012210558117416381717221208059639923280759}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{7} - \frac{6690014753134592041720148054546686827894098975029573512262470659026401764037990358497636964544507647152580471750353557717998792244182245190070993675913923549270851463919749174215119154680242484830525697628683621676}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{6} + \frac{41452144110485627415674841514123426335924223761262437140285084638265073725487880639629564870323163470327988845659384571930639445935584220358156916026353190049941780729862156779364271443783186579978046135669522543763}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{5} - \frac{38568157402072037606644688561170906760232471585461750835773180338153806422686328694601687975557010425108439271202248242695594680383724990385661781743670492427499891908819665402486267229923227038023327508445398947061}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{4} - \frac{28829842973857106301508649988097449074250338372178541524687623443511770789710369430260144871651144277187683915845737597961559946212020345198641282218649257332941072846184241538410170219118831894227113942424995869450}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{3} + \frac{26625519108628458952809915286508007186119859183608114733167849142653616573208487550550271646903384846824739886948588461118605055651166471681056640032297623602703980979141944320227217868155282954616516543720229867142}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a^{2} + \frac{10720864639685007365220190205198742051113688074360471959582035493412967430347505482529709219677965242162796414375236043070467712168660610902943402006967974678723047220848782952579513841848940468409958721202900717795}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417} a + \frac{37941561667645095735047418963545043091127797778437448807491609173730349049882873263324982461662020743334731470009204081469680956564368749117146289453008003724348954094456655504628938108442840697269595439176236485553}{95387060223015775720559831181628659423334792422565555256604553633086377012602603214066435116263448777646812985155181965348863655040413930743550457130979006059909053240936737200628039606969031086209816793029576131417}$

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 $25$ $25$ $25$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$

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