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Properties

Label 25.25.2279365643...0625.3
Degree $25$
Signature $[25, 0]$
Discriminant $5^{68}\cdot 11^{20}$
Root discriminant $542.39$
Ramified primes $5, 11$
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![-6753826844501, 48016300765950, 164764650851350, -1243323318639600, -51166647560875, 1547329331945760, -22665313001375, -719226536302175, 9127440610475, 159208867549875, -678182906005, -18421283506450, 22243405800, 1194622146300, -415438375, -45731609385, 4126100, 1062050275, -15125, -14973750, 0, 124025, 0, -550, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 550*x^23 + 124025*x^21 - 14973750*x^19 - 15125*x^18 + 1062050275*x^17 + 4126100*x^16 - 45731609385*x^15 - 415438375*x^14 + 1194622146300*x^13 + 22243405800*x^12 - 18421283506450*x^11 - 678182906005*x^10 + 159208867549875*x^9 + 9127440610475*x^8 - 719226536302175*x^7 - 22665313001375*x^6 + 1547329331945760*x^5 - 51166647560875*x^4 - 1243323318639600*x^3 + 164764650851350*x^2 + 48016300765950*x - 6753826844501)
 
gp: K = bnfinit(x^25 - 550*x^23 + 124025*x^21 - 14973750*x^19 - 15125*x^18 + 1062050275*x^17 + 4126100*x^16 - 45731609385*x^15 - 415438375*x^14 + 1194622146300*x^13 + 22243405800*x^12 - 18421283506450*x^11 - 678182906005*x^10 + 159208867549875*x^9 + 9127440610475*x^8 - 719226536302175*x^7 - 22665313001375*x^6 + 1547329331945760*x^5 - 51166647560875*x^4 - 1243323318639600*x^3 + 164764650851350*x^2 + 48016300765950*x - 6753826844501, 1)
 

Normalized defining polynomial

\( x^{25} - 550 x^{23} + 124025 x^{21} - 14973750 x^{19} - 15125 x^{18} + 1062050275 x^{17} + 4126100 x^{16} - 45731609385 x^{15} - 415438375 x^{14} + 1194622146300 x^{13} + 22243405800 x^{12} - 18421283506450 x^{11} - 678182906005 x^{10} + 159208867549875 x^{9} + 9127440610475 x^{8} - 719226536302175 x^{7} - 22665313001375 x^{6} + 1547329331945760 x^{5} - 51166647560875 x^{4} - 1243323318639600 x^{3} + 164764650851350 x^{2} + 48016300765950 x - 6753826844501 \)

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:  \(227936564389216769066972959924266550757465665810741484165191650390625=5^{68}\cdot 11^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $542.39$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1375=5^{3}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1375}(1,·)$, $\chi_{1375}(196,·)$, $\chi_{1375}(1281,·)$, $\chi_{1375}(456,·)$, $\chi_{1375}(586,·)$, $\chi_{1375}(1291,·)$, $\chi_{1375}(1101,·)$, $\chi_{1375}(1296,·)$, $\chi_{1375}(466,·)$, $\chi_{1375}(276,·)$, $\chi_{1375}(471,·)$, $\chi_{1375}(731,·)$, $\chi_{1375}(861,·)$, $\chi_{1375}(36,·)$, $\chi_{1375}(741,·)$, $\chi_{1375}(551,·)$, $\chi_{1375}(746,·)$, $\chi_{1375}(1006,·)$, $\chi_{1375}(1136,·)$, $\chi_{1375}(181,·)$, $\chi_{1375}(311,·)$, $\chi_{1375}(1016,·)$, $\chi_{1375}(826,·)$, $\chi_{1375}(1021,·)$, $\chi_{1375}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{11} a^{9}$, $\frac{1}{121} a^{10}$, $\frac{1}{121} a^{11}$, $\frac{1}{121} a^{12}$, $\frac{1}{121} a^{13}$, $\frac{1}{121} a^{14}$, $\frac{1}{1331} a^{15}$, $\frac{1}{1331} a^{16}$, $\frac{1}{1331} a^{17}$, $\frac{1}{1331} a^{18}$, $\frac{1}{1331} a^{19}$, $\frac{1}{14641} a^{20}$, $\frac{1}{14641} a^{21}$, $\frac{1}{14641} a^{22}$, $\frac{1}{14641} a^{23}$, $\frac{1}{53635025962612926577049342176353483306465579292548412823404002454819930045723942484759227174424450689973002724367706979344333372405515593091322591} a^{24} - \frac{680465661023210142783513658297520532643856694210202294828598241645122117683961583980463712414172945451675507050448983719474698122678948699929}{53635025962612926577049342176353483306465579292548412823404002454819930045723942484759227174424450689973002724367706979344333372405515593091322591} a^{23} - \frac{927076182029345419638132934858923112360366799137836498546284272151173375212054863032884312675892076847467711264503530330346208880779766676156}{53635025962612926577049342176353483306465579292548412823404002454819930045723942484759227174424450689973002724367706979344333372405515593091322591} a^{22} - \frac{112045007232856648857293945632526736759392016010194717540397789845944235780854720949167820963646678878262354536228089342896911291053238609179}{53635025962612926577049342176353483306465579292548412823404002454819930045723942484759227174424450689973002724367706979344333372405515593091322591} a^{21} - \frac{570234724165931682593688908762259384104566886101073790336782807789304179275399790248449634856595916297516478863991897676703541407682499960577}{53635025962612926577049342176353483306465579292548412823404002454819930045723942484759227174424450689973002724367706979344333372405515593091322591} a^{20} + \frac{1219381865811163819541301974670027773553815453400215542104864375032717812160580016978077581255713668827380349952585524292743174857786303183721}{4875911451146629688822667470577589391496870844777128438491272950438175458702176589523566106765859153633909338578882452667666670218683235735574781} a^{19} - \frac{1323246790194389380175349154957477559823962907131822497137546320028250417361990443235052722506198837230016330152511963358429929552453434705756}{4875911451146629688822667470577589391496870844777128438491272950438175458702176589523566106765859153633909338578882452667666670218683235735574781} a^{18} - \frac{1357346108074826001742136161869664982796563030829711044897008597853062300942661525412415138701159156236591036569371203275012511156678121732385}{4875911451146629688822667470577589391496870844777128438491272950438175458702176589523566106765859153633909338578882452667666670218683235735574781} a^{17} - \frac{450061634627687094813041168357355543116183947489873901452155591917883746903606453756049002009354985856921369226843575214544433215787464510442}{4875911451146629688822667470577589391496870844777128438491272950438175458702176589523566106765859153633909338578882452667666670218683235735574781} a^{16} + \frac{180357815188656919265245864594173700296056671115144161869255810347233360418067718253724519248336364744905161009890653446673953108983218347549}{4875911451146629688822667470577589391496870844777128438491272950438175458702176589523566106765859153633909338578882452667666670218683235735574781} a^{15} - \frac{898486021428908488167140586501338173627458656483075789296287130854709865945290280094104733012769392875986974543518265401907807586677221900231}{443264677376966335347515224597962671954260985888829858044661177312561405336561508138506009705987195784900848961716586606151515474425748703234071} a^{14} - \frac{830590647398471326190518766787711115087610163817551163131713471381672612446023818545477645570542582757692823738916345683169657748989003605853}{443264677376966335347515224597962671954260985888829858044661177312561405336561508138506009705987195784900848961716586606151515474425748703234071} a^{13} - \frac{751948564198579775091006062260698405018376152888681886471724100786281873433879542357811937260788746420747817930752457089383425904900207100476}{443264677376966335347515224597962671954260985888829858044661177312561405336561508138506009705987195784900848961716586606151515474425748703234071} a^{12} + \frac{1575652407162942068365401902940141863794027967426812353829968864770679527966976046029546816755746334296100973075944268422071751478703212722537}{443264677376966335347515224597962671954260985888829858044661177312561405336561508138506009705987195784900848961716586606151515474425748703234071} a^{11} - \frac{1054117931030763989997650171195856991743245217410230038674238531337632135234409647801707040119504672826112214980774162686869306310250638671873}{443264677376966335347515224597962671954260985888829858044661177312561405336561508138506009705987195784900848961716586606151515474425748703234071} a^{10} - \frac{562349720643299336418780388192227683398812011030003918896633566609681310338245707941006878173652486630904374533465493036920624293415823570732}{40296788852451485031592293145269333814023725989893623458605561573869218666960137103500546336907926889536440814701507873286501406765977154839461} a^{9} + \frac{1749683088934572046679472293545016284276223804957688834050322170347788140471105243593221314330432670098487255591901770931653440828703060538116}{40296788852451485031592293145269333814023725989893623458605561573869218666960137103500546336907926889536440814701507873286501406765977154839461} a^{8} + \frac{801431713932377074740052451005323727743701933982309901452323688245308657036246980102301867624338909297507376743132638925180470757539259338076}{40296788852451485031592293145269333814023725989893623458605561573869218666960137103500546336907926889536440814701507873286501406765977154839461} a^{7} + \frac{218739728791385986331500656940092516447744452349738295610523626110075733455253155312797497159537395162158521788958369200846214720628292252372}{40296788852451485031592293145269333814023725989893623458605561573869218666960137103500546336907926889536440814701507873286501406765977154839461} a^{6} - \frac{674217534989593092043609621503371082966122833897481842434238362837293855839731838380727105420963447132247125797089167239036211658572268099870}{40296788852451485031592293145269333814023725989893623458605561573869218666960137103500546336907926889536440814701507873286501406765977154839461} a^{5} + \frac{682517087088369338432807467138205131844419930823437784935336917255400433255739665385135197490784378822529299274439558001227703291684309048453}{3663344441131953184690208467751757619456702362717602132600505597624474424269103373045504212446175171776040074063773443026045582433270650439951} a^{4} + \frac{395472854871985581759809886191030837184651693564961588748648296983182326304846639902620866993144741986498398360775746178474873884663085164202}{3663344441131953184690208467751757619456702362717602132600505597624474424269103373045504212446175171776040074063773443026045582433270650439951} a^{3} - \frac{1485136753433760036215525239743724813957149455072673522962761451446213525585938276875837041001938908929817859296788815764019132185338291154125}{3663344441131953184690208467751757619456702362717602132600505597624474424269103373045504212446175171776040074063773443026045582433270650439951} a^{2} + \frac{519712385702919837113739595248640374219863148105833146320819144307086434681850670040779677287869818476369912251570428894395937663952064419641}{3663344441131953184690208467751757619456702362717602132600505597624474424269103373045504212446175171776040074063773443026045582433270650439951} a + \frac{1493304819969952542599118729654639288217863915972313084377666333963202003615774455918250189308822614637954889556785857812776800225085093549}{3485579867870554885528266857994060532308946111053855501998578113819671193405426615647482599853639554496707967710536101832583808214339343901}$

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 ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ R $25$ $25$ $25$ $25$ $25$ $25$ $25$ $25$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ $25$ $25$ $25$

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
11Data not computed