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Properties

Label 25.25.1035996976...8001.1
Degree $25$
Signature $[25, 0]$
Discriminant $751^{24}$
Root discriminant $576.25$
Ramified prime $751$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-2175105944071, 4021161749229, 41570931977659, 66222925208094, 13790562028053, -49405455102305, -37245464805375, 4915687383185, 13615398610851, 2498218571582, -2002221575510, -701125573722, 134888109273, 76639409753, -3802678517, -4593891060, -11071180, 167015969, 3222735, -3762281, -71185, 50509, 561, -360, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - x^24 - 360*x^23 + 561*x^22 + 50509*x^21 - 71185*x^20 - 3762281*x^19 + 3222735*x^18 + 167015969*x^17 - 11071180*x^16 - 4593891060*x^15 - 3802678517*x^14 + 76639409753*x^13 + 134888109273*x^12 - 701125573722*x^11 - 2002221575510*x^10 + 2498218571582*x^9 + 13615398610851*x^8 + 4915687383185*x^7 - 37245464805375*x^6 - 49405455102305*x^5 + 13790562028053*x^4 + 66222925208094*x^3 + 41570931977659*x^2 + 4021161749229*x - 2175105944071)
 
gp: K = bnfinit(x^25 - x^24 - 360*x^23 + 561*x^22 + 50509*x^21 - 71185*x^20 - 3762281*x^19 + 3222735*x^18 + 167015969*x^17 - 11071180*x^16 - 4593891060*x^15 - 3802678517*x^14 + 76639409753*x^13 + 134888109273*x^12 - 701125573722*x^11 - 2002221575510*x^10 + 2498218571582*x^9 + 13615398610851*x^8 + 4915687383185*x^7 - 37245464805375*x^6 - 49405455102305*x^5 + 13790562028053*x^4 + 66222925208094*x^3 + 41570931977659*x^2 + 4021161749229*x - 2175105944071, 1)
 

Normalized defining polynomial

\( x^{25} - x^{24} - 360 x^{23} + 561 x^{22} + 50509 x^{21} - 71185 x^{20} - 3762281 x^{19} + 3222735 x^{18} + 167015969 x^{17} - 11071180 x^{16} - 4593891060 x^{15} - 3802678517 x^{14} + 76639409753 x^{13} + 134888109273 x^{12} - 701125573722 x^{11} - 2002221575510 x^{10} + 2498218571582 x^{9} + 13615398610851 x^{8} + 4915687383185 x^{7} - 37245464805375 x^{6} - 49405455102305 x^{5} + 13790562028053 x^{4} + 66222925208094 x^{3} + 41570931977659 x^{2} + 4021161749229 x - 2175105944071 \)

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:  \(1035996976866061687572354401200188865607516483497915359292674342768001=751^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $576.25$
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:  $751$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(751\)
Dirichlet character group:    $\lbrace$$\chi_{751}(1,·)$, $\chi_{751}(450,·)$, $\chi_{751}(179,·)$, $\chi_{751}(710,·)$, $\chi_{751}(193,·)$, $\chi_{751}(392,·)$, $\chi_{751}(460,·)$, $\chi_{751}(499,·)$, $\chi_{751}(80,·)$, $\chi_{751}(466,·)$, $\chi_{751}(666,·)$, $\chi_{751}(475,·)$, $\chi_{751}(348,·)$, $\chi_{751}(325,·)$, $\chi_{751}(481,·)$, $\chi_{751}(162,·)$, $\chi_{751}(420,·)$, $\chi_{751}(485,·)$, $\chi_{751}(171,·)$, $\chi_{751}(556,·)$, $\chi_{751}(51,·)$, $\chi_{751}(53,·)$, $\chi_{751}(569,·)$, $\chi_{751}(117,·)$, $\chi_{751}(703,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{73} a^{21} + \frac{18}{73} a^{20} + \frac{2}{73} a^{19} - \frac{9}{73} a^{18} - \frac{13}{73} a^{17} - \frac{26}{73} a^{16} - \frac{21}{73} a^{15} + \frac{20}{73} a^{14} + \frac{3}{73} a^{13} + \frac{29}{73} a^{12} + \frac{22}{73} a^{11} + \frac{22}{73} a^{10} - \frac{35}{73} a^{9} - \frac{35}{73} a^{8} + \frac{23}{73} a^{7} + \frac{3}{73} a^{6} - \frac{19}{73} a^{5} - \frac{24}{73} a^{4} + \frac{15}{73} a^{3} + \frac{36}{73} a^{2} + \frac{16}{73} a + \frac{4}{73}$, $\frac{1}{73} a^{22} - \frac{30}{73} a^{20} + \frac{28}{73} a^{19} + \frac{3}{73} a^{18} - \frac{11}{73} a^{17} + \frac{9}{73} a^{16} + \frac{33}{73} a^{15} + \frac{8}{73} a^{14} - \frac{25}{73} a^{13} + \frac{11}{73} a^{12} - \frac{9}{73} a^{11} + \frac{7}{73} a^{10} + \frac{11}{73} a^{9} - \frac{4}{73} a^{8} + \frac{27}{73} a^{7} + \frac{26}{73} a^{5} + \frac{9}{73} a^{4} - \frac{15}{73} a^{3} + \frac{25}{73} a^{2} + \frac{8}{73} a + \frac{1}{73}$, $\frac{1}{73} a^{23} - \frac{16}{73} a^{20} - \frac{10}{73} a^{19} + \frac{11}{73} a^{18} - \frac{16}{73} a^{17} - \frac{17}{73} a^{16} + \frac{35}{73} a^{15} - \frac{9}{73} a^{14} + \frac{28}{73} a^{13} - \frac{15}{73} a^{12} + \frac{10}{73} a^{11} + \frac{14}{73} a^{10} - \frac{32}{73} a^{9} - \frac{1}{73} a^{8} + \frac{33}{73} a^{7} - \frac{30}{73} a^{6} + \frac{23}{73} a^{5} - \frac{5}{73} a^{4} - \frac{36}{73} a^{3} - \frac{7}{73} a^{2} - \frac{30}{73} a - \frac{26}{73}$, $\frac{1}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{24} + \frac{169061521982828018888876149797502453835524837317731165027966767275292704168199960963363274044765092578902421380874567739346456873669323142550292}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{23} - \frac{1360553996384048883263448100409768516779425862326312505116207664379038707804694136037256463480450956410544474075102501702938700958890747173154361}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{22} + \frac{1434076012174694367642668385601037538301399422347761108425445136555925228520852212115710264250770169803754876859386288672584378096266915195727512}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{21} + \frac{32443048747362347961194081563688465758334274669444413183183952944437514257207476029270504879989233752122663224260271488483144166316749361917617469}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{20} + \frac{4039113819932412757940468445045096003320313952937137876934766063141953605101675657273754043594548429784631865328696798341652058987736162326205623}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{19} + \frac{69631162272475950163036098563383267778486975665194597341622425923024883294974288975046170208405914256947195822083493679667972913226836528045221926}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{18} + \frac{66562122874601122228461135828442057774711823213435614892661446084672661093175836897931462383620950869981665391044171627355731264947326470364280327}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{17} - \frac{64635135886194078067534692430835363544865841713370807833901539238582822374767855788076718638209044482400815319769964895223118836065092894271840874}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{16} - \frac{39076225368355549588383321966809243680694740032416166443909759887838490768138614792276178441638163357373900350535503847631076626328788132924217843}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{15} + \frac{36534197445346444872892423097330681622678647753786326988224689131621325719258704426019094612393005655155609063245481749890626182882358628627424662}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{14} - \frac{27432783179309035052711446202156743761685772454431703637620570405983785943173033414191286551878511137942900577584359573450867727858257555906866889}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{13} - \frac{101396186594361488764616676130244506280482051237423920290018524545288399665305709956371584675880284147312716304183449556616638667281702676859961560}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{12} - \frac{4394200208206665598144416418057568452874901963459897093163650456689029797076280244676040014946147456379062700939100777875997362527678678373372324}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{11} + \frac{44995272586902276975206940266512362250125328840667913122853317894124528307812714609690655962896719114287395312658258576841598215222218542212664769}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{10} + \frac{69435547651819004828067789014882308191051614031079296609735164696738807234746822531488057458972131367739078206966272705113008950424984359527760683}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{9} + \frac{58236687436643242607994111823753053961377582437302939548579547705604151534061942100790095033497360169241034264826925507114961637667283478929253612}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{8} - \frac{90223438375078085313496619577088953292234031205188256897412949810748398371345749327074957733838034511174264246061769100072628937708666766544955495}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{7} + \frac{79269973866073090800463561101663962403947073097604364121701794015358532291660641345127889258908164159565729040775302545210187796011450885420484745}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{6} + \frac{1224120565613055613557627156509514132807241955827992445414316571997513091085248633794531544483977909329548017991070837040620077073046257404594405}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{5} - \frac{89989358313114459522160895874568339145473272418361597154833798483632251760277532613501355687891929145063075794345295617150195202287611387939046286}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{4} - \frac{85307167554023014939068058066919375936417660284193790691542515381906783323949911864997429501316386313456517805345900852854387208370989272773622596}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{3} + \frac{22111378289558033336747734612042453980558788827422016082305859399366954302411696560341143655025869388276825835319813950642949802484805442622329482}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a^{2} - \frac{50885411670243319424443181370862636298430278003636268951844005363337213715433417844539559453826791231254636670496104672695268404763223732205543388}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081} a + \frac{75342039097290790063023747689924888066597588297020718943061905659059628829911424969871648313509127495486864913186087991168559264258554766986311142}{230367483464878429769285235159061131552350727705412259678963025567032129596758719154010991680521844525325836677292085395207161427078241485378307081}$

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:  \( 295898715808607820000000000 \) (assuming GRH)
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

5.5.318097128001.1

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 $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ $25$ $25$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ $25$

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
751Data not computed