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Properties

Label 26.0.46968559211...9375.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,3^{13}\cdot 5^{13}\cdot 53^{24}$
Root discriminant $151.25$
Ramified primes $3, 5, 53$
Class number $208891462$ (GRH)
Class group $[208891462]$ (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![52746287941, -54641860909, 86147056237, -65397167421, 58711220016, -35392358120, 23269915520, -11704874804, 6241296158, -2731752221, 1253687550, -492849091, 200174256, -71057510, 25499556, -8058692, 2553145, -732225, 214432, -57343, 14666, -3120, 694, -192, 58, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 + 58*x^24 - 192*x^23 + 694*x^22 - 3120*x^21 + 14666*x^20 - 57343*x^19 + 214432*x^18 - 732225*x^17 + 2553145*x^16 - 8058692*x^15 + 25499556*x^14 - 71057510*x^13 + 200174256*x^12 - 492849091*x^11 + 1253687550*x^10 - 2731752221*x^9 + 6241296158*x^8 - 11704874804*x^7 + 23269915520*x^6 - 35392358120*x^5 + 58711220016*x^4 - 65397167421*x^3 + 86147056237*x^2 - 54641860909*x + 52746287941)
 
gp: K = bnfinit(x^26 - 11*x^25 + 58*x^24 - 192*x^23 + 694*x^22 - 3120*x^21 + 14666*x^20 - 57343*x^19 + 214432*x^18 - 732225*x^17 + 2553145*x^16 - 8058692*x^15 + 25499556*x^14 - 71057510*x^13 + 200174256*x^12 - 492849091*x^11 + 1253687550*x^10 - 2731752221*x^9 + 6241296158*x^8 - 11704874804*x^7 + 23269915520*x^6 - 35392358120*x^5 + 58711220016*x^4 - 65397167421*x^3 + 86147056237*x^2 - 54641860909*x + 52746287941, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} + 58 x^{24} - 192 x^{23} + 694 x^{22} - 3120 x^{21} + 14666 x^{20} - 57343 x^{19} + 214432 x^{18} - 732225 x^{17} + 2553145 x^{16} - 8058692 x^{15} + 25499556 x^{14} - 71057510 x^{13} + 200174256 x^{12} - 492849091 x^{11} + 1253687550 x^{10} - 2731752221 x^{9} + 6241296158 x^{8} - 11704874804 x^{7} + 23269915520 x^{6} - 35392358120 x^{5} + 58711220016 x^{4} - 65397167421 x^{3} + 86147056237 x^{2} - 54641860909 x + 52746287941 \)

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:  \(-469685592111594703310837287190069142111550625687255859375=-\,3^{13}\cdot 5^{13}\cdot 53^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $151.25$
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:  $3, 5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(795=3\cdot 5\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{795}(256,·)$, $\chi_{795}(1,·)$, $\chi_{795}(646,·)$, $\chi_{795}(584,·)$, $\chi_{795}(331,·)$, $\chi_{795}(524,·)$, $\chi_{795}(526,·)$, $\chi_{795}(16,·)$, $\chi_{795}(466,·)$, $\chi_{795}(599,·)$, $\chi_{795}(89,·)$, $\chi_{795}(346,·)$, $\chi_{795}(704,·)$, $\chi_{795}(736,·)$, $\chi_{795}(766,·)$, $\chi_{795}(134,·)$, $\chi_{795}(554,·)$, $\chi_{795}(44,·)$, $\chi_{795}(301,·)$, $\chi_{795}(46,·)$, $\chi_{795}(434,·)$, $\chi_{795}(629,·)$, $\chi_{795}(119,·)$, $\chi_{795}(121,·)$, $\chi_{795}(314,·)$, $\chi_{795}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{6}{23} a^{17} - \frac{5}{23} a^{16} + \frac{9}{23} a^{15} + \frac{1}{23} a^{14} - \frac{9}{23} a^{13} - \frac{6}{23} a^{11} + \frac{7}{23} a^{10} + \frac{6}{23} a^{9} + \frac{4}{23} a^{8} + \frac{1}{23} a^{7} - \frac{5}{23} a^{6} - \frac{4}{23} a^{5} - \frac{9}{23} a^{4} - \frac{8}{23} a^{3} + \frac{6}{23} a^{2} - \frac{6}{23} a$, $\frac{1}{23} a^{19} + \frac{5}{23} a^{17} + \frac{2}{23} a^{16} + \frac{9}{23} a^{15} - \frac{3}{23} a^{14} - \frac{8}{23} a^{13} - \frac{6}{23} a^{12} - \frac{6}{23} a^{11} + \frac{2}{23} a^{10} - \frac{6}{23} a^{9} + \frac{2}{23} a^{8} + \frac{1}{23} a^{7} - \frac{11}{23} a^{6} - \frac{10}{23} a^{5} + \frac{7}{23} a^{4} + \frac{4}{23} a^{3} + \frac{7}{23} a^{2} + \frac{10}{23} a$, $\frac{1}{23} a^{20} + \frac{9}{23} a^{17} + \frac{11}{23} a^{16} - \frac{2}{23} a^{15} + \frac{10}{23} a^{14} - \frac{7}{23} a^{13} - \frac{6}{23} a^{12} + \frac{9}{23} a^{11} + \frac{5}{23} a^{10} - \frac{5}{23} a^{9} + \frac{4}{23} a^{8} + \frac{7}{23} a^{7} - \frac{8}{23} a^{6} + \frac{4}{23} a^{5} + \frac{3}{23} a^{4} + \frac{1}{23} a^{3} + \frac{3}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{21} - \frac{4}{23} a^{17} - \frac{3}{23} a^{16} - \frac{2}{23} a^{15} + \frac{7}{23} a^{14} + \frac{6}{23} a^{13} + \frac{9}{23} a^{12} - \frac{10}{23} a^{11} + \frac{1}{23} a^{10} - \frac{4}{23} a^{9} - \frac{6}{23} a^{8} + \frac{6}{23} a^{7} + \frac{3}{23} a^{6} - \frac{7}{23} a^{5} - \frac{10}{23} a^{4} + \frac{6}{23} a^{3} - \frac{1}{23} a^{2} + \frac{8}{23} a$, $\frac{1}{23} a^{22} - \frac{4}{23} a^{17} + \frac{1}{23} a^{16} - \frac{3}{23} a^{15} + \frac{10}{23} a^{14} - \frac{4}{23} a^{13} - \frac{10}{23} a^{12} + \frac{1}{23} a^{10} - \frac{5}{23} a^{9} - \frac{1}{23} a^{8} + \frac{7}{23} a^{7} - \frac{4}{23} a^{6} - \frac{3}{23} a^{5} - \frac{7}{23} a^{4} - \frac{10}{23} a^{3} + \frac{9}{23} a^{2} - \frac{1}{23} a$, $\frac{1}{1909} a^{23} - \frac{19}{1909} a^{22} - \frac{41}{1909} a^{21} - \frac{20}{1909} a^{20} - \frac{26}{1909} a^{19} + \frac{19}{1909} a^{18} - \frac{28}{83} a^{17} - \frac{539}{1909} a^{16} - \frac{367}{1909} a^{15} - \frac{695}{1909} a^{14} + \frac{950}{1909} a^{13} + \frac{856}{1909} a^{12} - \frac{280}{1909} a^{11} + \frac{772}{1909} a^{10} + \frac{215}{1909} a^{9} + \frac{853}{1909} a^{8} - \frac{710}{1909} a^{7} + \frac{580}{1909} a^{6} - \frac{357}{1909} a^{5} + \frac{245}{1909} a^{4} - \frac{309}{1909} a^{3} - \frac{327}{1909} a^{2} - \frac{341}{1909} a + \frac{8}{83}$, $\frac{1}{10007415161} a^{24} + \frac{2082584}{10007415161} a^{23} + \frac{112143131}{10007415161} a^{22} + \frac{203090547}{10007415161} a^{21} - \frac{66907326}{10007415161} a^{20} + \frac{80585494}{10007415161} a^{19} + \frac{99407165}{10007415161} a^{18} + \frac{2162721667}{10007415161} a^{17} + \frac{3658624054}{10007415161} a^{16} + \frac{4434673445}{10007415161} a^{15} + \frac{2486255519}{10007415161} a^{14} + \frac{2203648556}{10007415161} a^{13} + \frac{3991320209}{10007415161} a^{12} - \frac{3879209626}{10007415161} a^{11} + \frac{1039733999}{10007415161} a^{10} + \frac{4703293836}{10007415161} a^{9} - \frac{684545546}{10007415161} a^{8} + \frac{3496587337}{10007415161} a^{7} - \frac{2660694721}{10007415161} a^{6} + \frac{6992656}{18917609} a^{5} - \frac{3029504327}{10007415161} a^{4} + \frac{1860691669}{10007415161} a^{3} + \frac{231916434}{10007415161} a^{2} + \frac{170715702}{435105007} a - \frac{9185672}{18917609}$, $\frac{1}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{25} - \frac{5866536595199432667213687872397351844080026689414377910963035357423052483335228553738035277190}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{24} - \frac{9015868838543774013201832371222032512795610403066589688123525573979470551459958773688039947600884783}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{23} + \frac{2226671110754783623634902836622093274671054538127773123495104796602551066253826683116949800838167325193}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{22} + \frac{1556253359024493624717584369905445859533108837716493628090817778280502333258728322479627427024011127775}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{21} - \frac{743363109328744554552397589491329430766897532783924604583886692572982075993192915181327140538957267028}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{20} + \frac{125919278604164652606950778472802660513878121305491984490989585864869197822750531343623862865042386437}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{19} + \frac{456661820343183497948961014826888900640839536286946847453104601596341718211807923991259010306028465266}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{18} + \frac{16537752260832878318387902313226905422062236175140332732080920055536451155056786242027944003569565215076}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{17} + \frac{22567728695479606117390294975037808407292259641834447845248779061127554299771827016338284503497247667651}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{16} - \frac{57884406001079385921254141307215358314784454012010558190079804455197403891597244325747095495633414649889}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{15} - \frac{29805871901775180016505696742929961656741614227900041464694330723893750749625595888837750959571978291186}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{14} - \frac{19446460771690707822940933241008013084397458630666533039784448264699628059404214274755905765530599689989}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{13} + \frac{358533506430403705851954976657835792030776808877974914286387257282122623137925707491021783518512920484}{5336184723050341098282125336451948626448813975724123725959279128032617891912183338171805504549359291331} a^{12} - \frac{52767821512693265580627132685649047966069907545668648671718677679810900139672137333643225514436202713096}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{11} - \frac{24215128676703212171186958853603224248519152483553897468612595701845479321890633346921373295205429901338}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{10} + \frac{51037191887047641773189324703816279670084039513304366783871339704430054483747491002701592156919861020677}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{9} + \frac{33700961347580143440621174333442856030162101222181867237046565414556204922950324926430709329597905814179}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{8} + \frac{33748403370675180797390178805175163398026444621944387322603694359578552256739117942393538419423393196672}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{7} + \frac{42335100925904318447779381328028064369369509004115923149861208823249720807371577845193473578815364787022}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{6} - \frac{30197456654260977951769208551734272613185876615495728081986345019900307621027187256131957266404924808634}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{5} - \frac{54873874339796663005377240294993180415596460409950958822007584043796445876980124448963254584724185296747}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{4} + \frac{26852570048552326148578937227158673904273684444906059059641624324378642551818432795866152361185104609977}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{3} - \frac{29908998510529270790325191650203836207388405619509232092187648550961903172858875400331169217601898301158}{122732248630157845260488882738394818408322721441654845697063419944750211513980216777951526604635263700613} a^{2} + \frac{1290514961753812124243207149375654682370744918342486570116191550427966448684470119884187275719417056090}{5336184723050341098282125336451948626448813975724123725959279128032617891912183338171805504549359291331} a - \frac{13307587497698189035253443009832699003298199646283436750074908935713448706167518754661139468271181219}{232008031436971352099222840715302114193426694596701031563446918610113821387486232094426326284754751797}$

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

Class group and class number

$C_{208891462}$, which has order $208891462$ (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

\(\Q(\sqrt{-15}) \), 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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ R R $26$ $26$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ 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
3Data not computed
5Data 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}$