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Properties

Label 26.0.23382739460...4567.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,7^{13}\cdot 53^{24}$
Root discriminant $103.32$
Ramified primes $7, 53$
Class number $1152931$ (GRH)
Class group $[1152931]$ (GRH)
Galois group $C_{26}$ (as 26T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![183561919, -275771917, 493165779, -494051417, 505907604, -383212608, 281189852, -167065424, 94651964, -45568357, 21707936, -9349303, 4404928, -2174910, 1145956, -489500, 126811, -1493, 2138, -11543, 5790, -300, -470, 72, 32, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 + 32*x^24 + 72*x^23 - 470*x^22 - 300*x^21 + 5790*x^20 - 11543*x^19 + 2138*x^18 - 1493*x^17 + 126811*x^16 - 489500*x^15 + 1145956*x^14 - 2174910*x^13 + 4404928*x^12 - 9349303*x^11 + 21707936*x^10 - 45568357*x^9 + 94651964*x^8 - 167065424*x^7 + 281189852*x^6 - 383212608*x^5 + 505907604*x^4 - 494051417*x^3 + 493165779*x^2 - 275771917*x + 183561919)
 
gp: K = bnfinit(x^26 - 11*x^25 + 32*x^24 + 72*x^23 - 470*x^22 - 300*x^21 + 5790*x^20 - 11543*x^19 + 2138*x^18 - 1493*x^17 + 126811*x^16 - 489500*x^15 + 1145956*x^14 - 2174910*x^13 + 4404928*x^12 - 9349303*x^11 + 21707936*x^10 - 45568357*x^9 + 94651964*x^8 - 167065424*x^7 + 281189852*x^6 - 383212608*x^5 + 505907604*x^4 - 494051417*x^3 + 493165779*x^2 - 275771917*x + 183561919, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} + 32 x^{24} + 72 x^{23} - 470 x^{22} - 300 x^{21} + 5790 x^{20} - 11543 x^{19} + 2138 x^{18} - 1493 x^{17} + 126811 x^{16} - 489500 x^{15} + 1145956 x^{14} - 2174910 x^{13} + 4404928 x^{12} - 9349303 x^{11} + 21707936 x^{10} - 45568357 x^{9} + 94651964 x^{8} - 167065424 x^{7} + 281189852 x^{6} - 383212608 x^{5} + 505907604 x^{4} - 494051417 x^{3} + 493165779 x^{2} - 275771917 x + 183561919 \)

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

Invariants

Degree:  $26$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 13]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-23382739460171668835485199054409361747964940454234567=-\,7^{13}\cdot 53^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.32$
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:  $7, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(371=7\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{371}(1,·)$, $\chi_{371}(258,·)$, $\chi_{371}(195,·)$, $\chi_{371}(69,·)$, $\chi_{371}(134,·)$, $\chi_{371}(97,·)$, $\chi_{371}(328,·)$, $\chi_{371}(13,·)$, $\chi_{371}(15,·)$, $\chi_{371}(148,·)$, $\chi_{371}(342,·)$, $\chi_{371}(281,·)$, $\chi_{371}(153,·)$, $\chi_{371}(155,·)$, $\chi_{371}(160,·)$, $\chi_{371}(225,·)$, $\chi_{371}(99,·)$, $\chi_{371}(36,·)$, $\chi_{371}(293,·)$, $\chi_{371}(169,·)$, $\chi_{371}(365,·)$, $\chi_{371}(174,·)$, $\chi_{371}(307,·)$, $\chi_{371}(309,·)$, $\chi_{371}(183,·)$, $\chi_{371}(314,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{6}{23} a^{17} + \frac{11}{23} a^{16} - \frac{11}{23} a^{15} - \frac{2}{23} a^{14} + \frac{2}{23} a^{13} + \frac{9}{23} a^{12} + \frac{4}{23} a^{11} - \frac{8}{23} a^{10} + \frac{7}{23} a^{9} + \frac{10}{23} a^{7} + \frac{3}{23} a^{6} + \frac{2}{23} a^{5} + \frac{8}{23} a^{4} - \frac{11}{23} a^{3} + \frac{1}{23} a^{2} + \frac{3}{23} a$, $\frac{1}{23} a^{19} - \frac{2}{23} a^{17} + \frac{9}{23} a^{16} + \frac{1}{23} a^{15} - \frac{10}{23} a^{14} - \frac{2}{23} a^{13} - \frac{11}{23} a^{12} - \frac{7}{23} a^{11} + \frac{5}{23} a^{10} - \frac{4}{23} a^{9} + \frac{10}{23} a^{8} - \frac{6}{23} a^{7} - \frac{3}{23} a^{6} - \frac{3}{23} a^{5} - \frac{9}{23} a^{4} + \frac{4}{23} a^{3} + \frac{9}{23} a^{2} - \frac{5}{23} a$, $\frac{1}{23} a^{20} - \frac{3}{23} a^{17} - \frac{9}{23} a^{15} - \frac{6}{23} a^{14} - \frac{7}{23} a^{13} + \frac{11}{23} a^{12} - \frac{10}{23} a^{11} + \frac{3}{23} a^{10} + \frac{1}{23} a^{9} - \frac{6}{23} a^{8} - \frac{6}{23} a^{7} + \frac{3}{23} a^{6} - \frac{5}{23} a^{5} - \frac{3}{23} a^{4} + \frac{10}{23} a^{3} - \frac{3}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{23} a^{21} + \frac{5}{23} a^{17} + \frac{1}{23} a^{16} + \frac{7}{23} a^{15} + \frac{10}{23} a^{14} - \frac{6}{23} a^{13} - \frac{6}{23} a^{12} - \frac{8}{23} a^{11} - \frac{8}{23} a^{9} - \frac{6}{23} a^{8} + \frac{10}{23} a^{7} + \frac{4}{23} a^{6} + \frac{3}{23} a^{5} + \frac{11}{23} a^{4} + \frac{10}{23} a^{3} + \frac{9}{23} a^{2} + \frac{9}{23} a$, $\frac{1}{23} a^{22} + \frac{8}{23} a^{17} - \frac{2}{23} a^{16} - \frac{4}{23} a^{15} + \frac{4}{23} a^{14} + \frac{7}{23} a^{13} - \frac{7}{23} a^{12} + \frac{3}{23} a^{11} + \frac{9}{23} a^{10} + \frac{5}{23} a^{9} + \frac{10}{23} a^{8} + \frac{11}{23} a^{6} + \frac{1}{23} a^{5} - \frac{7}{23} a^{4} - \frac{5}{23} a^{3} + \frac{4}{23} a^{2} + \frac{8}{23} a$, $\frac{1}{23} a^{23} - \frac{1}{23} a$, $\frac{1}{10007415161} a^{24} - \frac{176153202}{10007415161} a^{23} + \frac{17782985}{10007415161} a^{22} + \frac{73398035}{10007415161} a^{21} - \frac{23241509}{10007415161} a^{20} + \frac{137024165}{10007415161} a^{19} - \frac{142825772}{10007415161} a^{18} - \frac{1470907843}{10007415161} a^{17} + \frac{35599056}{435105007} a^{16} - \frac{1987021286}{10007415161} a^{15} + \frac{140652819}{10007415161} a^{14} - \frac{1817617091}{10007415161} a^{13} + \frac{3134224801}{10007415161} a^{12} + \frac{1337899672}{10007415161} a^{11} - \frac{4196757732}{10007415161} a^{10} - \frac{3723761276}{10007415161} a^{9} + \frac{3673186742}{10007415161} a^{8} + \frac{80441129}{10007415161} a^{7} - \frac{1765275883}{10007415161} a^{6} - \frac{2270336062}{10007415161} a^{5} - \frac{1492477418}{10007415161} a^{4} - \frac{1889174221}{10007415161} a^{3} - \frac{4728644963}{10007415161} a^{2} + \frac{3488671863}{10007415161} a + \frac{211781356}{435105007}$, $\frac{1}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{25} + \frac{1618438531006262138344972397430812124250337590019727547414740032003507246247695}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{24} - \frac{988095858420635087185602640499418217689770090153769016054325990356369126379299787963603}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{23} - \frac{732391859158590375348608521013329951779527013286871407883700321068484465618275042111496}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{22} + \frac{106785913165579852596568541912053918770654558675088199846545613895244682163854131694128}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{21} + \frac{315974244233303107217533223255260223702046083247068851759191985662424516057368233786286}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{20} + \frac{814981930712276083527719405750974491392056761327219413538962131589031599616707229217850}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{19} - \frac{40621616869172404401181478354860657974705508549465425415563654610577006441409198314007}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{18} + \frac{16519381965096977677405711551499712267949871015831285036372645033672867255158983870771078}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{17} + \frac{18349467717481646808767570719220302367835160171145522304802718425876799810835581962612732}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{16} + \frac{21258017475613101491425182276324238867099274456351346798785721312686877640511379952860092}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{15} - \frac{12156956807118011249844374787844263005352034383994212623139920110027452218707786584398133}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{14} + \frac{1550231394714123020220144034277947497085106241113407887532602897322949920659276188867270}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{13} + \frac{23042846064769950194998317282882564622305066932668231877843711108080040577033425684763633}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{12} - \frac{188637600964374701383373624049212816845198315509660559375122014074286610395379644755162}{527630129176623885293522607960860251146531139405444054763633035336342394633803241415013} a^{11} - \frac{12023168557766686920063984091847295151242801248272272400081766563896032707255459972537301}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{10} + \frac{16732136476962778854335566946604577203535675001982859862453539819791427490487376526438076}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{9} + \frac{11431881769485595956252423337964985291617176672754650556229436348060307934772241603893054}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{8} - \frac{5377036597380342213862687987989970095710181275605559629370560592583167594937823689591489}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{7} - \frac{2218877949700904901936049883839065372321970531540919438087087574934219626651196718166602}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{6} - \frac{28175916348020930992722639304368948633688597996729180649208626105938075990360076112849252}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{5} - \frac{24886809815982359942713458454273820864907302921858337942839727930647371619628104769402713}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{4} - \frac{8636182415663500031675965075497822795488237495997391386522363133789157577565033001600252}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{3} + \frac{26944017569550136187012482443025679278036883208509695574639754343385570919516612198237886}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a^{2} + \frac{17249227555146007877338438360496697454669709931126077267482525645411721742902341915879372}{56456423821898755726406919051812046872678831916382513859708734780988636225816946831406391} a - \frac{692340192213149494968539226766917975892355004699935026232325422342064654285368947745943}{2454627122691250248974213871817915081420818778973152776509075425260375488078997688322017}$

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

Class group and class number

$C_{1152931}$, which has order $1152931$ (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:  $12$
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:  \( 5382739421.971964 \) (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

\(\Q(\sqrt{-7}) \), 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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ $26$ $26$ R ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ R $26$

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
7Data not computed
$53$53.13.12.1$x^{13} - 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$
53.13.12.1$x^{13} - 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$