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Properties

Label 25.25.2276114846...0625.1
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![-3211621783659145751, 24592344442048304950, -7418028362553619150, -64048521279405896600, 7090953337926595375, 36464874055184939260, -1152117614947971625, -7291124562630257675, 55973138695881225, 621114013802522375, -2042542765396255, -25990984386144450, 36170557945800, 598759114519800, -288253992125, -8126381085385, 1015873100, 66971873775, -1321375, -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 - 1321375*x^18 + 66971873775*x^17 + 1015873100*x^16 - 8126381085385*x^15 - 288253992125*x^14 + 598759114519800*x^13 + 36170557945800*x^12 - 25990984386144450*x^11 - 2042542765396255*x^10 + 621114013802522375*x^9 + 55973138695881225*x^8 - 7291124562630257675*x^7 - 1152117614947971625*x^6 + 36464874055184939260*x^5 + 7090953337926595375*x^4 - 64048521279405896600*x^3 - 7418028362553619150*x^2 + 24592344442048304950*x - 3211621783659145751)
 
gp: K = bnfinit(x^25 - 1550*x^23 + 985025*x^21 - 335148750*x^19 - 1321375*x^18 + 66971873775*x^17 + 1015873100*x^16 - 8126381085385*x^15 - 288253992125*x^14 + 598759114519800*x^13 + 36170557945800*x^12 - 25990984386144450*x^11 - 2042542765396255*x^10 + 621114013802522375*x^9 + 55973138695881225*x^8 - 7291124562630257675*x^7 - 1152117614947971625*x^6 + 36464874055184939260*x^5 + 7090953337926595375*x^4 - 64048521279405896600*x^3 - 7418028362553619150*x^2 + 24592344442048304950*x - 3211621783659145751, 1)
 

Normalized defining polynomial

\( x^{25} - 1550 x^{23} + 985025 x^{21} - 335148750 x^{19} - 1321375 x^{18} + 66971873775 x^{17} + 1015873100 x^{16} - 8126381085385 x^{15} - 288253992125 x^{14} + 598759114519800 x^{13} + 36170557945800 x^{12} - 25990984386144450 x^{11} - 2042542765396255 x^{10} + 621114013802522375 x^{9} + 55973138695881225 x^{8} - 7291124562630257675 x^{7} - 1152117614947971625 x^{6} + 36464874055184939260 x^{5} + 7090953337926595375 x^{4} - 64048521279405896600 x^{3} - 7418028362553619150 x^{2} + 24592344442048304950 x - 3211621783659145751 \)

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$$\chi_{3875}(2496,·)$, $\chi_{3875}(1,·)$, $\chi_{3875}(3011,·)$, $\chi_{3875}(2116,·)$, $\chi_{3875}(3271,·)$, $\chi_{3875}(776,·)$, $\chi_{3875}(3786,·)$, $\chi_{3875}(2891,·)$, $\chi_{3875}(1551,·)$, $\chi_{3875}(3666,·)$, $\chi_{3875}(686,·)$, $\chi_{3875}(2326,·)$, $\chi_{3875}(281,·)$, $\chi_{3875}(3101,·)$, $\chi_{3875}(1056,·)$, $\chi_{3875}(1831,·)$, $\chi_{3875}(171,·)$, $\chi_{3875}(2606,·)$, $\chi_{3875}(946,·)$, $\chi_{3875}(3381,·)$, $\chi_{3875}(566,·)$, $\chi_{3875}(1721,·)$, $\chi_{3875}(2236,·)$, $\chi_{3875}(1341,·)$, $\chi_{3875}(1461,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{31} a^{5}$, $\frac{1}{31} a^{6}$, $\frac{1}{31} a^{7}$, $\frac{1}{31} a^{8}$, $\frac{1}{31} a^{9}$, $\frac{1}{961} a^{10}$, $\frac{1}{961} a^{11}$, $\frac{1}{961} a^{12}$, $\frac{1}{961} a^{13}$, $\frac{1}{961} a^{14}$, $\frac{1}{29791} a^{15}$, $\frac{1}{29791} a^{16}$, $\frac{1}{29791} a^{17}$, $\frac{1}{29791} a^{18}$, $\frac{1}{29791} a^{19}$, $\frac{1}{923521} a^{20}$, $\frac{1}{923521} a^{21}$, $\frac{1}{923521} a^{22}$, $\frac{1}{923521} a^{23}$, $\frac{1}{173317867054206421143788732717271296671584755689319700486473715357291895118004583962742912516260124390404904539334295134939894682697184521654652692527705931205327758478180381664484436059529} a^{24} + \frac{54654829315239527557148140681434932183287668477886442682088836436613760203751618795906467919898425678869277663403600879428286995084138086709099768355812959702679846730979777126750180}{173317867054206421143788732717271296671584755689319700486473715357291895118004583962742912516260124390404904539334295134939894682697184521654652692527705931205327758478180381664484436059529} a^{23} - \frac{39098980482066000802578548016641926909087764209314187401977384231628047646752051437804628474711176507994553034047439189882418871977993322672853348222787706404916503229370488015563215}{173317867054206421143788732717271296671584755689319700486473715357291895118004583962742912516260124390404904539334295134939894682697184521654652692527705931205327758478180381664484436059529} a^{22} + \frac{52737577625510833453153415696861486723846604670336624825477215004680296462383434640136294243249882676183022916665647465828165750138502682732612460407244953608869236560967286890178935}{173317867054206421143788732717271296671584755689319700486473715357291895118004583962742912516260124390404904539334295134939894682697184521654652692527705931205327758478180381664484436059529} a^{21} + \frac{71732477233308249590543470894624980195155664594923239597028199787239330489405366996550687701227486176675484202838709520488863646006616075630035048297719540775348989544919682189606848}{173317867054206421143788732717271296671584755689319700486473715357291895118004583962742912516260124390404904539334295134939894682697184521654652692527705931205327758478180381664484436059529} a^{20} - \frac{82062249446457668571245572669022930266912259401620347674141759148472745047898997509033783633378518823436325681281983936455839840191163167165389267964825433367254994838506336088162214}{5590898937232465198186733313460364408760798570623216144724958559912641778000147869765900403750326593238867888365622423707738538151522081343698473952506642942107347047683238118209175356759} a^{19} - \frac{16595110062394606063712712840445218927256845682944840726342449545916298950401489739243700140880557964193004672707021080269035744086204748502429568634816030034115489485632470136427545}{5590898937232465198186733313460364408760798570623216144724958559912641778000147869765900403750326593238867888365622423707738538151522081343698473952506642942107347047683238118209175356759} a^{18} - \frac{12359518734760112540266298262031638604570414386156351765174935258951053448091094398859018571904814796223408665667138647119926866300444629084102902414794044663871514316483120562732314}{5590898937232465198186733313460364408760798570623216144724958559912641778000147869765900403750326593238867888365622423707738538151522081343698473952506642942107347047683238118209175356759} a^{17} - \frac{91061135665335039688414480741283833192233830734087134954459516530237965351042010712085456871093300828306994399708075563039456163474664249917153575510989938706234312310676743227978354}{5590898937232465198186733313460364408760798570623216144724958559912641778000147869765900403750326593238867888365622423707738538151522081343698473952506642942107347047683238118209175356759} a^{16} + \frac{74273508985536777807223007772708914105280581724082804541552907368203247020206401432433442703516312162754282556876602855840620564394411632522233873323629516117188248060432956332260883}{5590898937232465198186733313460364408760798570623216144724958559912641778000147869765900403750326593238867888365622423707738538151522081343698473952506642942107347047683238118209175356759} a^{15} - \frac{17090836185973650096439493281446374179542264827869686738304232709277554404605829944952080323364822504018447176721702145905268593550754496580753289739696561460135710365369935871867839}{180351578620402103167313977853560142218090276471716649829837372900407799290327350637609690443558922362544125431149110442185114133920067140119305611371182030390559582183330261877715334089} a^{14} + \frac{326744616309795056093344352170441776887350892236961998213683030349578627729501126440364233722723590653913653102002093890540201757550308748449642236406709775960570475946604551383228}{5817792858722648489268192833985811039293234724894085478381850738722832235171850020568054530437384592340133078424164852973713359158711843229655019721651033238405147812365492318635978519} a^{13} - \frac{68670666357451984507643032898396393899516955703255016508323427666352385365005368551392608635918936487822204594367938148880685895368344866062430272601897970048007342993456575448115580}{180351578620402103167313977853560142218090276471716649829837372900407799290327350637609690443558922362544125431149110442185114133920067140119305611371182030390559582183330261877715334089} a^{12} + \frac{68707815729708865308840155634516007003142422604780522671703264133947114079288597748828758565485198411709458358115524127704476065828553003806212960754212082034715059532978723136231926}{180351578620402103167313977853560142218090276471716649829837372900407799290327350637609690443558922362544125431149110442185114133920067140119305611371182030390559582183330261877715334089} a^{11} + \frac{11153831208311440025681046852990718822479894382688658818349554370524260319724106612153861345193818409223802278039022966413587381233335934666202588160453837255019519303858591389191867}{180351578620402103167313977853560142218090276471716649829837372900407799290327350637609690443558922362544125431149110442185114133920067140119305611371182030390559582183330261877715334089} a^{10} + \frac{52086975426185054402580753736590453903101497990244286351886094361236113499486319125086418907413718231317620518142526289102029195025494734079966669517492566193632670705246140245125197}{5817792858722648489268192833985811039293234724894085478381850738722832235171850020568054530437384592340133078424164852973713359158711843229655019721651033238405147812365492318635978519} a^{9} + \frac{93269858038034605227210518610026126148413628801501171059771760979045240176001289514772958073079127470683770812440388682760557657982766942976817985633946825093213403722467300980945796}{5817792858722648489268192833985811039293234724894085478381850738722832235171850020568054530437384592340133078424164852973713359158711843229655019721651033238405147812365492318635978519} a^{8} + \frac{86180913394732243943528787241943853661331094512520267928357383824048865860786028869900599522671735059353180418659217089096678137538661872204863436978741813675212410933386669414204380}{5817792858722648489268192833985811039293234724894085478381850738722832235171850020568054530437384592340133078424164852973713359158711843229655019721651033238405147812365492318635978519} a^{7} - \frac{2800572175552035842105533516984682129544845869239016500026953088787820053240567325647957479746355927783896567662416408800911148127093494237017332599138997217661518571425003348516824}{187670737378149951266715897870510033525588216932067273496188733507188136618446774857034017110883373946455905755618221063668172876087478813859839345859710749625972510076306203826967049} a^{6} - \frac{36685152448549516198878318467847984620737733587452701999497878355191767830879019821797682665650452098878855595148703484079238700691060039821427552993160212096591844909421751981235641}{5817792858722648489268192833985811039293234724894085478381850738722832235171850020568054530437384592340133078424164852973713359158711843229655019721651033238405147812365492318635978519} a^{5} - \frac{71324126627286538998098387759880760355772875025826771243297974925675039225943694611331528294096369062572874439955253836661134332068372587058391290256900956485517973389594835819529662}{187670737378149951266715897870510033525588216932067273496188733507188136618446774857034017110883373946455905755618221063668172876087478813859839345859710749625972510076306203826967049} a^{4} + \frac{24734150828435177122496109744784370513087921420906668885837174285000177997218327440792369293734263051713710241315438552809290737228554479124672234469898151863048894576254757699245236}{187670737378149951266715897870510033525588216932067273496188733507188136618446774857034017110883373946455905755618221063668172876087478813859839345859710749625972510076306203826967049} a^{3} + \frac{61419577128379502335433553215755868605128451873617108589177934508745135061405591782725965082916272754247108669318094475587486420187374997994234193059754655146947843779741646374920145}{187670737378149951266715897870510033525588216932067273496188733507188136618446774857034017110883373946455905755618221063668172876087478813859839345859710749625972510076306203826967049} a^{2} + \frac{67818727408405426665712665240896682031867495499741050005056167258993081286088909679444435959426235116512448828120068020262741012361285381958389636354983938318456351401537139912322977}{187670737378149951266715897870510033525588216932067273496188733507188136618446774857034017110883373946455905755618221063668172876087478813859839345859710749625972510076306203826967049} a + \frac{15212847032888652370215776374806434520038896486148629702507140948423094877102738772863534319218575016560295105587467211308913425864595003011676411285996142245688742446728985573268663}{187670737378149951266715897870510033525588216932067273496188733507188136618446774857034017110883373946455905755618221063668172876087478813859839345859710749625972510076306203826967049}$

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 $25$ $25$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ $25$ $25$ $25$

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