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Properties

Label 25.25.2276114846...0625.2
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![2343310686055630999, -15239564267440452800, -180902906700178007650, -448031467429506142850, -481642554117131820875, -250276302703900127990, -50596670999271679375, 7170562108917464575, 4908918704216766225, 394840637488502375, -150186181700207005, -24981626726799450, 2222178011701800, 600140932816050, -16797346268375, -8111522824135, 59197696100, 66876014025, -77000125, -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 - 77000125*x^18 + 66876014025*x^17 + 59197696100*x^16 - 8111522824135*x^15 - 16797346268375*x^14 + 600140932816050*x^13 + 2222178011701800*x^12 - 24981626726799450*x^11 - 150186181700207005*x^10 + 394840637488502375*x^9 + 4908918704216766225*x^8 + 7170562108917464575*x^7 - 50596670999271679375*x^6 - 250276302703900127990*x^5 - 481642554117131820875*x^4 - 448031467429506142850*x^3 - 180902906700178007650*x^2 - 15239564267440452800*x + 2343310686055630999)
 
gp: K = bnfinit(x^25 - 1550*x^23 + 985025*x^21 - 335148750*x^19 - 77000125*x^18 + 66876014025*x^17 + 59197696100*x^16 - 8111522824135*x^15 - 16797346268375*x^14 + 600140932816050*x^13 + 2222178011701800*x^12 - 24981626726799450*x^11 - 150186181700207005*x^10 + 394840637488502375*x^9 + 4908918704216766225*x^8 + 7170562108917464575*x^7 - 50596670999271679375*x^6 - 250276302703900127990*x^5 - 481642554117131820875*x^4 - 448031467429506142850*x^3 - 180902906700178007650*x^2 - 15239564267440452800*x + 2343310686055630999, 1)
 

Normalized defining polynomial

\( x^{25} - 1550 x^{23} + 985025 x^{21} - 335148750 x^{19} - 77000125 x^{18} + 66876014025 x^{17} + 59197696100 x^{16} - 8111522824135 x^{15} - 16797346268375 x^{14} + 600140932816050 x^{13} + 2222178011701800 x^{12} - 24981626726799450 x^{11} - 150186181700207005 x^{10} + 394840637488502375 x^{9} + 4908918704216766225 x^{8} + 7170562108917464575 x^{7} - 50596670999271679375 x^{6} - 250276302703900127990 x^{5} - 481642554117131820875 x^{4} - 448031467429506142850 x^{3} - 180902906700178007650 x^{2} - 15239564267440452800 x + 2343310686055630999 \)

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$$\chi_{3875}(1,·)$, $\chi_{3875}(1411,·)$, $\chi_{3875}(1496,·)$, $\chi_{3875}(1366,·)$, $\chi_{3875}(776,·)$, $\chi_{3875}(2186,·)$, $\chi_{3875}(1551,·)$, $\chi_{3875}(2961,·)$, $\chi_{3875}(531,·)$, $\chi_{3875}(2326,·)$, $\chi_{3875}(3736,·)$, $\chi_{3875}(1306,·)$, $\chi_{3875}(591,·)$, $\chi_{3875}(3101,·)$, $\chi_{3875}(2271,·)$, $\chi_{3875}(2081,·)$, $\chi_{3875}(2916,·)$, $\chi_{3875}(3046,·)$, $\chi_{3875}(721,·)$, $\chi_{3875}(2856,·)$, $\chi_{3875}(3691,·)$, $\chi_{3875}(3821,·)$, $\chi_{3875}(2141,·)$, $\chi_{3875}(3631,·)$, $\chi_{3875}(636,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{31} a^{5}$, $\frac{1}{31} a^{6}$, $\frac{1}{31} a^{7}$, $\frac{1}{31} a^{8}$, $\frac{1}{31} a^{9}$, $\frac{1}{961} a^{10}$, $\frac{1}{961} a^{11}$, $\frac{1}{961} a^{12}$, $\frac{1}{961} a^{13}$, $\frac{1}{961} a^{14}$, $\frac{1}{29791} a^{15}$, $\frac{1}{29791} a^{16}$, $\frac{1}{29791} a^{17}$, $\frac{1}{29791} a^{18}$, $\frac{1}{29791} a^{19}$, $\frac{1}{39711403} a^{20} + \frac{17}{1281013} a^{19} - \frac{5}{1281013} a^{18} - \frac{13}{1281013} a^{17} - \frac{13}{1281013} a^{16} - \frac{14}{1281013} a^{15} - \frac{12}{41323} a^{14} - \frac{6}{41323} a^{13} - \frac{2}{41323} a^{12} + \frac{18}{41323} a^{11} + \frac{13}{41323} a^{10} - \frac{11}{1333} a^{9} + \frac{21}{1333} a^{8} - \frac{1}{1333} a^{7} - \frac{17}{1333} a^{6} + \frac{13}{1333} a^{5} + \frac{20}{43} a^{4} + \frac{4}{43} a^{3} - \frac{10}{43} a^{2} - \frac{16}{43} a + \frac{16}{43}$, $\frac{1}{39711403} a^{21} - \frac{20}{1281013} a^{19} - \frac{1}{1281013} a^{18} + \frac{1}{1281013} a^{17} - \frac{3}{1281013} a^{15} - \frac{3}{41323} a^{14} + \frac{21}{41323} a^{13} - \frac{3}{41323} a^{12} - \frac{13}{41323} a^{11} - \frac{11}{41323} a^{10} + \frac{13}{1333} a^{9} - \frac{17}{1333} a^{8} - \frac{6}{1333} a^{7} - \frac{15}{1333} a^{6} + \frac{4}{1333} a^{5} - \frac{1}{43} a^{4} - \frac{11}{43} a^{3} + \frac{8}{43} a^{2} + \frac{20}{43} a - \frac{4}{43}$, $\frac{1}{39711403} a^{22} + \frac{4}{1281013} a^{19} - \frac{3}{1281013} a^{18} - \frac{19}{1281013} a^{17} + \frac{21}{1281013} a^{16} - \frac{1}{1281013} a^{15} + \frac{20}{41323} a^{14} + \frac{18}{41323} a^{13} - \frac{6}{41323} a^{12} + \frac{12}{41323} a^{11} - \frac{8}{41323} a^{10} - \frac{15}{1333} a^{8} + \frac{10}{1333} a^{7} - \frac{1}{1333} a^{6} - \frac{12}{1333} a^{5} + \frac{5}{43} a^{4} - \frac{6}{43} a^{3} + \frac{12}{43} a^{2} + \frac{9}{43} a - \frac{13}{43}$, $\frac{1}{306675002827979} a^{23} + \frac{2299327}{306675002827979} a^{22} - \frac{2724352}{306675002827979} a^{21} - \frac{2742189}{306675002827979} a^{20} - \frac{417617}{38493159637} a^{19} + \frac{24626187}{9892742026709} a^{18} - \frac{138220656}{9892742026709} a^{17} + \frac{137298634}{9892742026709} a^{16} + \frac{86145377}{9892742026709} a^{15} - \frac{9898490}{319120710539} a^{14} + \frac{3531498}{10294216469} a^{13} - \frac{114978239}{319120710539} a^{12} + \frac{132409504}{319120710539} a^{11} + \frac{18582627}{319120710539} a^{10} - \frac{95582108}{10294216469} a^{9} - \frac{101986468}{10294216469} a^{8} - \frac{81117649}{10294216469} a^{7} + \frac{52086184}{10294216469} a^{6} - \frac{111255212}{10294216469} a^{5} - \frac{47505064}{332071499} a^{4} + \frac{91075949}{332071499} a^{3} - \frac{164029095}{332071499} a^{2} - \frac{109988055}{332071499} a + \frac{35481071}{332071499}$, $\frac{1}{3629970585070113015725477195364489110732553954097802518063959960572253413759426316871802644361915275643209579157166071058779250661261945464226042927117112647} a^{24} + \frac{4142265777780846334463366207450367823879316129974102253500909775964330546701968907863607984759553346114123889275904355865308349592782073499678}{3629970585070113015725477195364489110732553954097802518063959960572253413759426316871802644361915275643209579157166071058779250661261945464226042927117112647} a^{23} - \frac{41498649803953832294776035410216283256050563265091156426043906617082810305959313955186986151248910707994054770926278623897288214266400906691141482420}{3629970585070113015725477195364489110732553954097802518063959960572253413759426316871802644361915275643209579157166071058779250661261945464226042927117112647} a^{22} - \frac{28276021543316875658939598719311221008989627282285300091811849783288392199828507184908904782617375736140795959869270808757851098577906835782342582507}{3629970585070113015725477195364489110732553954097802518063959960572253413759426316871802644361915275643209579157166071058779250661261945464226042927117112647} a^{21} - \frac{2162446305818724489482706399850042987940540508503047734717353622612651561269051346458899988594453104372515255272358058032070888264395614141357217678}{3629970585070113015725477195364489110732553954097802518063959960572253413759426316871802644361915275643209579157166071058779250661261945464226042927117112647} a^{20} + \frac{1795356903378877176305693004082823313922064328799771654720268764599526884379537231242933288178270095422156680331657130329113540550209560066922672066343}{117095825324842355345983135334338358410727546906380726389159998728137206895465365060380730463287589536877728359908582937379975827782643402071807836358616537} a^{19} - \frac{304328352356234389266270168700764319841351240081746430518196084011608988302686624763321372205256349687914979116248922072011500506598286357999093770960}{117095825324842355345983135334338358410727546906380726389159998728137206895465365060380730463287589536877728359908582937379975827782643402071807836358616537} a^{18} + \frac{1384725958730065402892788996644960518903236356423190230833956120127823464363442015120000621035240819938494126340328211375669859172411944757012000994817}{117095825324842355345983135334338358410727546906380726389159998728137206895465365060380730463287589536877728359908582937379975827782643402071807836358616537} a^{17} + \frac{998216932694798199015864099503773306099469460337274243535455423073594466861881674408464065856808626997793943557653286957488586885937475004992045023297}{117095825324842355345983135334338358410727546906380726389159998728137206895465365060380730463287589536877728359908582937379975827782643402071807836358616537} a^{16} + \frac{60592005555123759672365045133008392577206332748204934210128149887611509495978340715653488496632354486155907627070647317418210709266145943741744007915}{3777284687898140495031714043043172851958953126012281496424516088004426028885979518076797756880244823770249301932534933463870187992988496841026059237374727} a^{15} - \frac{6381159738429666764250583185413307336082782277352740954448825273019696498707172752923083822437800460040697363935226931819631288600654475474683476409}{18981330089940404497646804236397853527431925256343123097610633608062442356210952352144712346131883536533916090113240871677739638155721089653397282599873} a^{14} - \frac{176968058325929745130171155882390935113487947736780254522540930199388445273933303167992631329293434152554721818948381444128270351818906282965131410214}{3777284687898140495031714043043172851958953126012281496424516088004426028885979518076797756880244823770249301932534933463870187992988496841026059237374727} a^{13} + \frac{1279915667191807535837378143837657174508267137719077370988335947422372025989447997389260718203350232008022165208871144335885820065373554569494595174400}{3777284687898140495031714043043172851958953126012281496424516088004426028885979518076797756880244823770249301932534933463870187992988496841026059237374727} a^{12} + \frac{112927420196169825779305349563536170378624767715712783610522764251762832422978414058386579334131454455393807461695209645814434435679731138737849905719}{3777284687898140495031714043043172851958953126012281496424516088004426028885979518076797756880244823770249301932534933463870187992988496841026059237374727} a^{11} + \frac{519183648359249430513255301124333298016702478253959676172433571376871068616140205761205214215318998704825176015906221299938095736606011382721897138575}{3777284687898140495031714043043172851958953126012281496424516088004426028885979518076797756880244823770249301932534933463870187992988496841026059237374727} a^{10} + \frac{97378904565181913725127589785834317007363806519649142649361175174819878183859935627927570122584949083167283482813017311405501679971611386964456771219}{121847893158004532097797227194941059740611391161686499884661809290465355770515468325057992157427252379685461352662417208511941548160919252936324491528217} a^{9} + \frac{1723084587567636397811820086369615413206095954781256474802731502130076174656028590575859907159604209641832433215774785980395351110282679065251631251093}{121847893158004532097797227194941059740611391161686499884661809290465355770515468325057992157427252379685461352662417208511941548160919252936324491528217} a^{8} - \frac{975673141404783833569904807370248628567914141278731955587814540428790005884278783063048389077739788449055219449031568812062630133812954962561659879135}{121847893158004532097797227194941059740611391161686499884661809290465355770515468325057992157427252379685461352662417208511941548160919252936324491528217} a^{7} - \frac{1514539858906228635955435148233073252124475612991915591851368401229994589495057573104818992298952697331270165400507286916529669536902768816976324057991}{121847893158004532097797227194941059740611391161686499884661809290465355770515468325057992157427252379685461352662417208511941548160919252936324491528217} a^{6} + \frac{1464626022717707286988281009259996357171405232609729270016430181827871440629810335163625725801311722669359733926724415719715129736529619938926793654490}{121847893158004532097797227194941059740611391161686499884661809290465355770515468325057992157427252379685461352662417208511941548160919252936324491528217} a^{5} + \frac{1106691058921995148043787861030172381961265587835725238361127321726789428120461446898543104234988017551626369132346117320579367413928564313887571197106}{3930577198645307487025717006288421281955206166506016125311671267434366315177918333066386843787975883215660043634271522855223920908416750094720144888007} a^{4} - \frac{844300505502949804665268832963644980282035694996002980114726939031275888048073066761342773528432199845322892959775797065576835003666525024808195952593}{3930577198645307487025717006288421281955206166506016125311671267434366315177918333066386843787975883215660043634271522855223920908416750094720144888007} a^{3} + \frac{7578152931652727091747959520495398460363567589069871909231094161652618828085348267022591435331528895394522606426072772330393709362661887041714720789}{26030312573809983357786205339658419085796067327854411425905107731353419305814028695803886382701827041163311547246831277186913383499448676123974469457} a^{2} + \frac{1763519058356424399753726446129828032257182387414771412987450962714968958450227660069563193994531358507895173690101188328285216274776254752262107716151}{3930577198645307487025717006288421281955206166506016125311671267434366315177918333066386843787975883215660043634271522855223920908416750094720144888007} a - \frac{1175964782820065112035011883507879007358463458758911619330816711716505254862947748371992479223129225918652872291883968665860436961971491914927036580166}{3930577198645307487025717006288421281955206166506016125311671267434366315177918333066386843787975883215660043634271522855223920908416750094720144888007}$

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 ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ R $25$ $25$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{25}$ $25$ $25$ $25$

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
31Data not computed