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Properties

Label 26.26.1266871942...4341.1
Degree $26$
Signature $[26, 0]$
Discriminant $17^{13}\cdot 53^{25}$
Root discriminant $187.58$
Ramified primes $17, 53$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_{26}$ (as 26T1)

Related objects

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-253846469, -1531488649, 124126156759, -491134065712, -2570589441833, -367759090103, 3678187011560, 713094712440, -2083673460942, -342523579686, 626544673310, 80106834336, -112332798212, -10620996476, 12679275824, 847617156, -922097601, -41744121, 43336299, 1272480, -1295607, -23361, 23596, 236, -237, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 237*x^24 + 236*x^23 + 23596*x^22 - 23361*x^21 - 1295607*x^20 + 1272480*x^19 + 43336299*x^18 - 41744121*x^17 - 922097601*x^16 + 847617156*x^15 + 12679275824*x^14 - 10620996476*x^13 - 112332798212*x^12 + 80106834336*x^11 + 626544673310*x^10 - 342523579686*x^9 - 2083673460942*x^8 + 713094712440*x^7 + 3678187011560*x^6 - 367759090103*x^5 - 2570589441833*x^4 - 491134065712*x^3 + 124126156759*x^2 - 1531488649*x - 253846469)
 
gp: K = bnfinit(x^26 - x^25 - 237*x^24 + 236*x^23 + 23596*x^22 - 23361*x^21 - 1295607*x^20 + 1272480*x^19 + 43336299*x^18 - 41744121*x^17 - 922097601*x^16 + 847617156*x^15 + 12679275824*x^14 - 10620996476*x^13 - 112332798212*x^12 + 80106834336*x^11 + 626544673310*x^10 - 342523579686*x^9 - 2083673460942*x^8 + 713094712440*x^7 + 3678187011560*x^6 - 367759090103*x^5 - 2570589441833*x^4 - 491134065712*x^3 + 124126156759*x^2 - 1531488649*x - 253846469, 1)
 

Normalized defining polynomial

\( x^{26} - x^{25} - 237 x^{24} + 236 x^{23} + 23596 x^{22} - 23361 x^{21} - 1295607 x^{20} + 1272480 x^{19} + 43336299 x^{18} - 41744121 x^{17} - 922097601 x^{16} + 847617156 x^{15} + 12679275824 x^{14} - 10620996476 x^{13} - 112332798212 x^{12} + 80106834336 x^{11} + 626544673310 x^{10} - 342523579686 x^{9} - 2083673460942 x^{8} + 713094712440 x^{7} + 3678187011560 x^{6} - 367759090103 x^{5} - 2570589441833 x^{4} - 491134065712 x^{3} + 124126156759 x^{2} - 1531488649 x - 253846469 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $26$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[26, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(126687194260488432068189542176205538728418933203425107834341=17^{13}\cdot 53^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $187.58$
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:  $17, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(901=17\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{901}(256,·)$, $\chi_{901}(1,·)$, $\chi_{901}(322,·)$, $\chi_{901}(579,·)$, $\chi_{901}(900,·)$, $\chi_{901}(645,·)$, $\chi_{901}(135,·)$, $\chi_{901}(460,·)$, $\chi_{901}(205,·)$, $\chi_{901}(271,·)$, $\chi_{901}(594,·)$, $\chi_{901}(596,·)$, $\chi_{901}(664,·)$, $\chi_{901}(222,·)$, $\chi_{901}(69,·)$, $\chi_{901}(832,·)$, $\chi_{901}(545,·)$, $\chi_{901}(356,·)$, $\chi_{901}(679,·)$, $\chi_{901}(237,·)$, $\chi_{901}(305,·)$, $\chi_{901}(307,·)$, $\chi_{901}(630,·)$, $\chi_{901}(696,·)$, $\chi_{901}(441,·)$, $\chi_{901}(766,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{1}{23} a^{16} + \frac{9}{23} a^{15} - \frac{4}{23} a^{14} - \frac{7}{23} a^{13} + \frac{3}{23} a^{12} + \frac{2}{23} a^{11} + \frac{11}{23} a^{10} + \frac{3}{23} a^{9} + \frac{5}{23} a^{8} - \frac{6}{23} a^{7} - \frac{4}{23} a^{6} + \frac{2}{23} a^{5} + \frac{3}{23} a^{4} + \frac{9}{23} a^{3} + \frac{7}{23} a^{2} + \frac{11}{23} a$, $\frac{1}{23} a^{19} + \frac{1}{23} a^{17} + \frac{9}{23} a^{16} - \frac{4}{23} a^{15} - \frac{7}{23} a^{14} + \frac{3}{23} a^{13} + \frac{2}{23} a^{12} + \frac{11}{23} a^{11} + \frac{3}{23} a^{10} + \frac{5}{23} a^{9} - \frac{6}{23} a^{8} - \frac{4}{23} a^{7} + \frac{2}{23} a^{6} + \frac{3}{23} a^{5} + \frac{9}{23} a^{4} + \frac{7}{23} a^{3} + \frac{11}{23} a^{2}$, $\frac{1}{23} a^{20} + \frac{9}{23} a^{17} - \frac{5}{23} a^{16} + \frac{7}{23} a^{15} + \frac{7}{23} a^{14} + \frac{9}{23} a^{13} + \frac{8}{23} a^{12} + \frac{1}{23} a^{11} - \frac{6}{23} a^{10} - \frac{9}{23} a^{9} - \frac{9}{23} a^{8} + \frac{8}{23} a^{7} + \frac{7}{23} a^{6} + \frac{7}{23} a^{5} + \frac{4}{23} a^{4} + \frac{2}{23} a^{3} - \frac{7}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{23} a^{21} - \frac{5}{23} a^{17} - \frac{2}{23} a^{16} - \frac{5}{23} a^{15} - \frac{1}{23} a^{14} + \frac{2}{23} a^{13} - \frac{3}{23} a^{12} - \frac{1}{23} a^{11} + \frac{7}{23} a^{10} + \frac{10}{23} a^{9} + \frac{9}{23} a^{8} - \frac{8}{23} a^{7} - \frac{3}{23} a^{6} + \frac{9}{23} a^{5} - \frac{2}{23} a^{4} + \frac{4}{23} a^{3} - \frac{5}{23} a^{2} - \frac{7}{23} a$, $\frac{1}{23} a^{22} - \frac{2}{23} a^{17} - \frac{2}{23} a^{15} + \frac{5}{23} a^{14} + \frac{8}{23} a^{13} - \frac{9}{23} a^{12} - \frac{6}{23} a^{11} - \frac{4}{23} a^{10} + \frac{1}{23} a^{9} - \frac{6}{23} a^{8} - \frac{10}{23} a^{7} - \frac{11}{23} a^{6} + \frac{8}{23} a^{5} - \frac{4}{23} a^{4} - \frac{6}{23} a^{3} + \frac{5}{23} a^{2} + \frac{9}{23} a$, $\frac{1}{23} a^{23} - \frac{1}{23} a$, $\frac{1}{529} a^{24} + \frac{10}{529} a^{23} - \frac{1}{529} a^{22} - \frac{10}{529} a^{21} - \frac{8}{529} a^{20} - \frac{7}{529} a^{19} + \frac{2}{529} a^{18} - \frac{27}{529} a^{17} + \frac{91}{529} a^{16} + \frac{19}{529} a^{15} + \frac{105}{529} a^{14} + \frac{118}{529} a^{13} + \frac{243}{529} a^{12} - \frac{88}{529} a^{11} - \frac{109}{529} a^{10} - \frac{173}{529} a^{9} - \frac{236}{529} a^{8} + \frac{226}{529} a^{7} - \frac{14}{529} a^{6} - \frac{56}{529} a^{5} - \frac{226}{529} a^{4} + \frac{126}{529} a^{3} + \frac{129}{529} a^{2} - \frac{7}{23} a$, $\frac{1}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{25} - \frac{141943313171704362044567191163772455486865926617203937139788527960214515408276315940504300532053973793551137760861866522609627015920012374}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{24} + \frac{2349019231938243733548091271992354927756736401533053879235671756838537071452216420912199979718808497981110518109663951439556621184818704788}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{23} + \frac{17096256599890313797743906142096135214354660678774892592382240776841219410353791418160259318297056975529592381269696450920578565297672423985}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{22} - \frac{18612211610682650763803286259766277074236873492543868962799019817607574596659117745315580260873222315851569466750149444334810486661631475040}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{21} + \frac{32281612881377433025680011540492966940486822137663755317259733450573119924926874702489845705281620872863279191611218333216607709900203855921}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{20} - \frac{23970212762627502180938805018057227024346210897427958165070825295159465106466705880338905225998814325410107538760278850453489102919663882994}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{19} - \frac{29237227612367385485634142130750339441943137154147962713206727087688059825181500076208347385989880728808847576920335583699763176451680939453}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{18} - \frac{697531301403841629423518407184679375583615053746310466490468365599109990442224044872625012833739081817648754060638880315512402335151085816738}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{17} - \frac{237046686325823823496253520971717351689078171575752939636320849655652599048794957436167257983774991202507988587103948593353013065827854495648}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{16} - \frac{473142832364761088564642284166158053720349672397188881538867387242842181114556690641654625433004654291909486541990230385537086511550678989382}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{15} + \frac{507953124438844163277128628864255206253178991174969451534125878352739128839701365770169385894619831977517815598772170545544498692192693355232}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{14} + \frac{568367313278338355439236847793903766369522176896511238175166692398798863823499321257077482342802592040616536998009919537641074684999465846380}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{13} - \frac{487075864627487335361477786380820192723426287662355298640990645963960346835740537942621731871733139018373334915708892541619671137501873740}{88898273585285780423363241280049880426199240816788442512452567194651057305283136440834595527459131346195628552417523855941477467179937780699} a^{12} - \frac{532790198680271161836710057476856474145320510169886516022020878846260510348297566126203904697022178246701279167201725118234928354733157032668}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{11} + \frac{50491332583163691212517460230613805522008851148294059369350165777819063689213038311681710061647113032851364969763116746177531947079454512324}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{10} + \frac{896931893936080077739812514798839838397856488857083558976765056792512364691257869167288103287146885112245150554557977735462068124967205593486}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{9} - \frac{780685398910414624193808465013961426044766501350696039681656166202001698929713617105721211577237219617172201235129833276487510439665314979008}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{8} + \frac{192566064804534927459240064237134183785963963838733638398545367356436836051588532711240393350927781497675939835778432213043010607257375996493}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{7} + \frac{229875927523085216682527146161076245144465599676376057517304728705616054674361326610497190426985188926901092938652707054303838060850457914050}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{6} + \frac{286045138374340825851391856898749307001520665229072508960425317819661578008867559116462367638833751110570503790873099522985116656804167432}{88898273585285780423363241280049880426199240816788442512452567194651057305283136440834595527459131346195628552417523855941477467179937780699} a^{5} - \frac{841166961883893263471994448278909699037303454553453932074254371547130020103166582811473540232909285018948569707981332323540521994938253854318}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{4} + \frac{8815013701033335315351235078537728645698318676285655708309803420348941119981356716305224193371331477629537680669652567290640995500466153976}{88898273585285780423363241280049880426199240816788442512452567194651057305283136440834595527459131346195628552417523855941477467179937780699} a^{3} + \frac{56464293049797376967089021529362625715396262708638548507902113066457168779280404878786362766330008132979071264905360003563522489023885381748}{2044660292461572949737354549441147249802582538786134177786409045476974318021512138139195697131560020962499456705603048686653981745138568956077} a^{2} + \frac{33272664814592290050401719028994237967273293548912135640249616269757515168998162218543905538984707460524957752298913776171121138549101282608}{88898273585285780423363241280049880426199240816788442512452567194651057305283136440834595527459131346195628552417523855941477467179937780699} a + \frac{857125878962681588780589910911398450213648013362341120568056790209726002109724241288520279334832894707213977986773823899712859698464774454}{3865142329795033931450575707828255670704314818121236630976198573680480752403614627862373718585179623747636024018153211127890324659997294813}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (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:  $25$
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:  \( 788899536890369100000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{26}$ (as 26T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{901}) \), 13.13.491258904256726154641.1

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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ R $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ R ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
17Data not computed
53Data not computed