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Properties

Label 25.25.1154816935...0625.2
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![-16510160157919896943, -41158441525542510360, 15474193591179457270, 45789263205923977660, -27984454640013654375, -1829877645315965021, 4274886158200428415, -422037681808798000, -276485257390184925, 47457692171150925, 9743726823879892, -2266218619182215, -204409768833865, 62777145784355, 2606214708525, -1103386742210, -19563201060, 12625715080, 77648295, -92851825, -119786, 416120, 0, -1010, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 1010*x^23 + 416120*x^21 - 119786*x^20 - 92851825*x^19 + 77648295*x^18 + 12625715080*x^17 - 19563201060*x^16 - 1103386742210*x^15 + 2606214708525*x^14 + 62777145784355*x^13 - 204409768833865*x^12 - 2266218619182215*x^11 + 9743726823879892*x^10 + 47457692171150925*x^9 - 276485257390184925*x^8 - 422037681808798000*x^7 + 4274886158200428415*x^6 - 1829877645315965021*x^5 - 27984454640013654375*x^4 + 45789263205923977660*x^3 + 15474193591179457270*x^2 - 41158441525542510360*x - 16510160157919896943)
 
gp: K = bnfinit(x^25 - 1010*x^23 + 416120*x^21 - 119786*x^20 - 92851825*x^19 + 77648295*x^18 + 12625715080*x^17 - 19563201060*x^16 - 1103386742210*x^15 + 2606214708525*x^14 + 62777145784355*x^13 - 204409768833865*x^12 - 2266218619182215*x^11 + 9743726823879892*x^10 + 47457692171150925*x^9 - 276485257390184925*x^8 - 422037681808798000*x^7 + 4274886158200428415*x^6 - 1829877645315965021*x^5 - 27984454640013654375*x^4 + 45789263205923977660*x^3 + 15474193591179457270*x^2 - 41158441525542510360*x - 16510160157919896943, 1)
 

Normalized defining polynomial

\( x^{25} - 1010 x^{23} + 416120 x^{21} - 119786 x^{20} - 92851825 x^{19} + 77648295 x^{18} + 12625715080 x^{17} - 19563201060 x^{16} - 1103386742210 x^{15} + 2606214708525 x^{14} + 62777145784355 x^{13} - 204409768833865 x^{12} - 2266218619182215 x^{11} + 9743726823879892 x^{10} + 47457692171150925 x^{9} - 276485257390184925 x^{8} - 422037681808798000 x^{7} + 4274886158200428415 x^{6} - 1829877645315965021 x^{5} - 27984454640013654375 x^{4} + 45789263205923977660 x^{3} + 15474193591179457270 x^{2} - 41158441525542510360 x - 16510160157919896943 \)

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$$\chi_{2525}(1,·)$, $\chi_{2525}(1796,·)$, $\chi_{2525}(71,·)$, $\chi_{2525}(521,·)$, $\chi_{2525}(1546,·)$, $\chi_{2525}(1551,·)$, $\chi_{2525}(81,·)$, $\chi_{2525}(1236,·)$, $\chi_{2525}(1621,·)$, $\chi_{2525}(1231,·)$, $\chi_{2525}(1501,·)$, $\chi_{2525}(1886,·)$, $\chi_{2525}(361,·)$, $\chi_{2525}(1956,·)$, $\chi_{2525}(1191,·)$, $\chi_{2525}(1641,·)$, $\chi_{2525}(1511,·)$, $\chi_{2525}(1906,·)$, $\chi_{2525}(381,·)$, $\chi_{2525}(561,·)$, $\chi_{2525}(1266,·)$, $\chi_{2525}(1801,·)$, $\chi_{2525}(2516,·)$, $\chi_{2525}(1466,·)$, $\chi_{2525}(701,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{2}{17} a^{13} + \frac{3}{17} a^{12} + \frac{6}{17} a^{11} - \frac{4}{17} a^{10} - \frac{8}{17} a^{9} + \frac{1}{17} a^{6} + \frac{2}{17} a^{5} + \frac{3}{17} a^{4} + \frac{6}{17} a^{3} - \frac{4}{17} a^{2} - \frac{8}{17} a$, $\frac{1}{17} a^{15} - \frac{1}{17} a^{13} + \frac{1}{17} a^{11} - \frac{1}{17} a^{9} + \frac{1}{17} a^{7} - \frac{1}{17} a^{5} + \frac{1}{17} a^{3} - \frac{1}{17} a$, $\frac{1}{17} a^{16} + \frac{2}{17} a^{13} + \frac{4}{17} a^{12} + \frac{6}{17} a^{11} - \frac{5}{17} a^{10} - \frac{8}{17} a^{9} + \frac{1}{17} a^{8} + \frac{2}{17} a^{5} + \frac{4}{17} a^{4} + \frac{6}{17} a^{3} - \frac{5}{17} a^{2} - \frac{8}{17} a$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{289} a^{21} + \frac{2}{289} a^{20} + \frac{5}{289} a^{19} + \frac{8}{289} a^{18} + \frac{6}{289} a^{17} - \frac{1}{289} a^{15} - \frac{8}{289} a^{14} - \frac{83}{289} a^{13} + \frac{44}{289} a^{12} + \frac{104}{289} a^{11} + \frac{117}{289} a^{10} - \frac{3}{289} a^{9} + \frac{6}{17} a^{8} - \frac{1}{289} a^{7} - \frac{110}{289} a^{6} + \frac{52}{289} a^{5} - \frac{111}{289} a^{4} + \frac{116}{289} a^{3} + \frac{75}{289} a^{2} + \frac{93}{289} a - \frac{7}{17}$, $\frac{1}{289} a^{22} + \frac{1}{289} a^{20} - \frac{2}{289} a^{19} + \frac{7}{289} a^{18} + \frac{5}{289} a^{17} - \frac{1}{289} a^{16} - \frac{6}{289} a^{15} + \frac{1}{289} a^{14} + \frac{57}{289} a^{13} - \frac{69}{289} a^{12} + \frac{28}{289} a^{11} + \frac{69}{289} a^{10} + \frac{142}{289} a^{9} + \frac{84}{289} a^{8} - \frac{108}{289} a^{7} + \frac{3}{17} a^{6} - \frac{79}{289} a^{5} - \frac{36}{289} a^{4} - \frac{38}{289} a^{3} - \frac{57}{289} a^{2} + \frac{1}{289} a - \frac{3}{17}$, $\frac{1}{42590797} a^{23} + \frac{15026}{42590797} a^{22} - \frac{2879}{2505341} a^{21} - \frac{94713}{42590797} a^{20} + \frac{131229}{42590797} a^{19} + \frac{61896}{2505341} a^{18} + \frac{63239}{2505341} a^{17} + \frac{4773}{42590797} a^{16} + \frac{259315}{42590797} a^{15} - \frac{855020}{42590797} a^{14} - \frac{6013816}{42590797} a^{13} + \frac{3148726}{42590797} a^{12} - \frac{12152014}{42590797} a^{11} - \frac{2874966}{42590797} a^{10} - \frac{16811429}{42590797} a^{9} - \frac{16628996}{42590797} a^{8} - \frac{19506297}{42590797} a^{7} - \frac{6380545}{42590797} a^{6} - \frac{2294403}{42590797} a^{5} - \frac{9683803}{42590797} a^{4} + \frac{8002245}{42590797} a^{3} + \frac{8426821}{42590797} a^{2} + \frac{18501243}{42590797} a - \frac{1043464}{2505341}$, $\frac{1}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{24} - \frac{68419159657025838870618125183018122573590923085995575999697385883677424629707133647195034108318105270466107911907132621930828764956436468455544287742985010934963129991184387048973912743}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{23} - \frac{883485638776231730952326282579509657824726311055507728853043109384997796412595681752399133208844819720306212576291113598027274230937253867535353747372070886302092360474084635307540378872060}{791736720668290007383079632632841317507511412284565927020872666013722696367797617855445959466915755590359501171961492180970389556206056518395126351163784110022830596068586456536789434692351213} a^{22} - \frac{10944272831519758529626912562177634149605529908290878410349688023904507715077432795277545895420452576260781938543687797139788757314793686375813990914057616048912364092074971361752283634064801}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{21} - \frac{266755021349965971689660207771699145817869767930562303220399309593093220760719061903104516267364243119081722486164034917904122059910545473842264473028167058719031518298535320244003500198132785}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{20} + \frac{372183550976831137136130172233111098837552667232574524287268636392943487670635136614211709268740863605804824015269006694920177807830271068740407514610469971557290136830222901298575223285746}{791736720668290007383079632632841317507511412284565927020872666013722696367797617855445959466915755590359501171961492180970389556206056518395126351163784110022830596068586456536789434692351213} a^{19} + \frac{22361399116412351613686886455697046185582102571372732550223919249623111526103545118186465954332597289090700497758048214871688401189718814895642824381066004688756592197685894496392248858734410}{791736720668290007383079632632841317507511412284565927020872666013722696367797617855445959466915755590359501171961492180970389556206056518395126351163784110022830596068586456536789434692351213} a^{18} - \frac{56046523278753126525246560061990157786695675383324131403881791468443643167228222381986729929306263178510681104309207247615862877530521569665204383494306239720635817440238173975587578834259491}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{17} - \frac{248188809818612478858638963962425545040528310665256723460934378404504400390637709908201588925398241910987542009474528373009725347317162125421140809711055905542051754853877901975905166824870710}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{16} + \frac{75853476856171297939239687542347126605561776992045790236204275281059432942948821939312367993850808582755544343804075884788637397648690025203313850110179725079477522704840320014422334211394830}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{15} - \frac{73461482688721917148334617222947793946817645077764979122211073657432638494507262691348124427900984318405617061111494439131906461313449499331082973205262557704215789417046636207578009414028932}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{14} + \frac{5870673523773254624915519916518888097341611033878847365103648250431271154549442134133428224372339783866738753748455049812506572452480804055941218405029434521829390891255285329486155319574098098}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{13} - \frac{5542814903997389436223459185242659629162106146359359191197595344865732123123121751096858279951689133922920429860992683927252217703508839088779301166001522807205115636546805316294912197750021308}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{12} - \frac{638715652840871315931641732768262248131170835410046454640595778332039357625302758890723570641788312586637152292121064157157337014308157903445502352761049340976543198316321031115474360020688140}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{11} - \frac{544622714027657180025100186279032420742650052615264547124805317290765789379911513580832923716145471279896138838975952765702809839548603085742004537352203523640956101450679628416449906095057249}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{10} + \frac{3212494398707064354716911893393154928302326394905108392742644370712135526054224275875690132298926722453465639537201366721352285958724812491996962885929831099712209212893380988413848470973179938}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{9} + \frac{5262881667631332478627987635076635724941824979642814259767016887772030967580661835780445965918229886061602173379286812599735354904969341674673947063233333267656658111359185554410250949850611615}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{8} + \frac{1866260034461491123495114223945536849366970507009862950206170674846785067815425738070974852332855433641919993947368398551248454902901850008663491415308364938652720054392565959622717639368471386}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{7} - \frac{6236896605062652786462603862900440196462323585968662916017829769800729095771239079124221633108447954279490373634244841092193916033163188785991332589803309559106843297708247300650032036823804481}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{6} + \frac{4696845458215436235755650061815109360748031784259016701341220067313495666387707288494449810773500842597912647036296647579813798730486801008893903341401764950452931769580844422221477280477011048}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{5} - \frac{895021821966240374615062623089597546859735296027235599278191964691690697016732579814835027284052210885039299838080146860100858820448943979149559432198760415068331524474213684548140575681251192}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{4} + \frac{4274976137243605236189922750009449747641762598900521638599681087643851242905815835586142658466142197576700989451463398177722585290809638740263934961090915173382417481977875884109775267851449564}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{3} + \frac{2424513966548648802969599294372772622396388903643382803600994101978690286053552707613115395509817280307065219195912786694111600629406201221186412103139231696660665785456807274224595036577841691}{13459524251360930125512353754758302397627694008837620759354835322233285838252559503542581310937567845036111519923345367076496622455502960812717147969784329870388120133165969761125420389769970621} a^{2} + \frac{2243160857390208997567009086444971070234231713459835209384556429295059642802312120439971170604292175385111978046500620642539686264329112290036080036675795816090319110951590543983544545991509}{46572748274605294551945860743108312794559494840268583942404274471395452727517506932673291733347985622962323598350676010645317032718003324611478020656693182942519446827563909208046437334844189} a + \frac{20763903672363802092164994567152732844344675584139030921456463024041752089019822536754159592599085190008574241580792107478617738062580459176462462643921210439679491677448990556977724917335336}{46572748274605294551945860743108312794559494840268583942404274471395452727517506932673291733347985622962323598350676010645317032718003324611478020656693182942519446827563909208046437334844189}$

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 $25$ $25$ $25$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{25}$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$

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