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Properties

Label 26.26.2582123849...8432.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{26}\cdot 3^{13}\cdot 53^{24}$
Root discriminant $135.28$
Ramified primes $2, 3, 53$
Class number $1$ (GRH)
Class group Trivial (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![279841, -2695784, 1921305, 21937294, -26467591, -67862476, 87553846, 110464796, -136412460, -106283650, 118804702, 63396508, -62988943, -24022934, 21232303, 5839670, -4625769, -906778, 648879, 88060, -57267, -5094, 3024, 158, -86, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 86*x^24 + 158*x^23 + 3024*x^22 - 5094*x^21 - 57267*x^20 + 88060*x^19 + 648879*x^18 - 906778*x^17 - 4625769*x^16 + 5839670*x^15 + 21232303*x^14 - 24022934*x^13 - 62988943*x^12 + 63396508*x^11 + 118804702*x^10 - 106283650*x^9 - 136412460*x^8 + 110464796*x^7 + 87553846*x^6 - 67862476*x^5 - 26467591*x^4 + 21937294*x^3 + 1921305*x^2 - 2695784*x + 279841)
 
gp: K = bnfinit(x^26 - 2*x^25 - 86*x^24 + 158*x^23 + 3024*x^22 - 5094*x^21 - 57267*x^20 + 88060*x^19 + 648879*x^18 - 906778*x^17 - 4625769*x^16 + 5839670*x^15 + 21232303*x^14 - 24022934*x^13 - 62988943*x^12 + 63396508*x^11 + 118804702*x^10 - 106283650*x^9 - 136412460*x^8 + 110464796*x^7 + 87553846*x^6 - 67862476*x^5 - 26467591*x^4 + 21937294*x^3 + 1921305*x^2 - 2695784*x + 279841, 1)
 

Normalized defining polynomial

\( x^{26} - 2 x^{25} - 86 x^{24} + 158 x^{23} + 3024 x^{22} - 5094 x^{21} - 57267 x^{20} + 88060 x^{19} + 648879 x^{18} - 906778 x^{17} - 4625769 x^{16} + 5839670 x^{15} + 21232303 x^{14} - 24022934 x^{13} - 62988943 x^{12} + 63396508 x^{11} + 118804702 x^{10} - 106283650 x^{9} - 136412460 x^{8} + 110464796 x^{7} + 87553846 x^{6} - 67862476 x^{5} - 26467591 x^{4} + 21937294 x^{3} + 1921305 x^{2} - 2695784 x + 279841 \)

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:  \(25821238496277693864023924106991445777252944911041298432=2^{26}\cdot 3^{13}\cdot 53^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.28$
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:  $2, 3, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(636=2^{2}\cdot 3\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{636}(1,·)$, $\chi_{636}(203,·)$, $\chi_{636}(577,·)$, $\chi_{636}(493,·)$, $\chi_{636}(599,·)$, $\chi_{636}(395,·)$, $\chi_{636}(13,·)$, $\chi_{636}(205,·)$, $\chi_{636}(275,·)$, $\chi_{636}(121,·)$, $\chi_{636}(407,·)$, $\chi_{636}(155,·)$, $\chi_{636}(95,·)$, $\chi_{636}(289,·)$, $\chi_{636}(227,·)$, $\chi_{636}(97,·)$, $\chi_{636}(169,·)$, $\chi_{636}(107,·)$, $\chi_{636}(301,·)$, $\chi_{636}(47,·)$, $\chi_{636}(49,·)$, $\chi_{636}(611,·)$, $\chi_{636}(625,·)$, $\chi_{636}(311,·)$, $\chi_{636}(505,·)$, $\chi_{636}(119,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{11}{23} a^{16} + \frac{7}{23} a^{15} + \frac{5}{23} a^{14} - \frac{11}{23} a^{13} - \frac{10}{23} a^{12} - \frac{10}{23} a^{11} + \frac{6}{23} a^{10} - \frac{1}{23} a^{9} + \frac{3}{23} a^{8} + \frac{7}{23} a^{7} + \frac{5}{23} a^{6} - \frac{6}{23} a^{5} + \frac{7}{23} a^{4} - \frac{3}{23} a^{3} - \frac{5}{23} a^{2} - \frac{7}{23} a$, $\frac{1}{23} a^{18} + \frac{1}{23} a^{16} - \frac{10}{23} a^{15} - \frac{2}{23} a^{14} + \frac{7}{23} a^{13} - \frac{5}{23} a^{12} + \frac{11}{23} a^{11} - \frac{4}{23} a^{10} - \frac{8}{23} a^{9} - \frac{6}{23} a^{8} - \frac{10}{23} a^{7} + \frac{3}{23} a^{6} + \frac{10}{23} a^{5} + \frac{5}{23} a^{4} + \frac{8}{23} a^{3} + \frac{7}{23} a^{2} - \frac{8}{23} a$, $\frac{1}{23} a^{19} + \frac{1}{23} a^{16} - \frac{9}{23} a^{15} + \frac{2}{23} a^{14} + \frac{6}{23} a^{13} - \frac{2}{23} a^{12} + \frac{6}{23} a^{11} + \frac{9}{23} a^{10} - \frac{5}{23} a^{9} + \frac{10}{23} a^{8} - \frac{4}{23} a^{7} + \frac{5}{23} a^{6} + \frac{11}{23} a^{5} + \frac{1}{23} a^{4} + \frac{10}{23} a^{3} - \frac{3}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{20} + \frac{2}{23} a^{16} - \frac{5}{23} a^{15} + \frac{1}{23} a^{14} + \frac{9}{23} a^{13} - \frac{7}{23} a^{12} - \frac{4}{23} a^{11} - \frac{11}{23} a^{10} + \frac{11}{23} a^{9} - \frac{7}{23} a^{8} - \frac{2}{23} a^{7} + \frac{6}{23} a^{6} + \frac{7}{23} a^{5} + \frac{3}{23} a^{4} - \frac{11}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{21} - \frac{6}{23} a^{16} + \frac{10}{23} a^{15} - \frac{1}{23} a^{14} - \frac{8}{23} a^{13} - \frac{7}{23} a^{12} + \frac{9}{23} a^{11} - \frac{1}{23} a^{10} - \frac{5}{23} a^{9} - \frac{8}{23} a^{8} - \frac{8}{23} a^{7} - \frac{3}{23} a^{6} - \frac{8}{23} a^{5} + \frac{9}{23} a^{4} - \frac{5}{23} a^{3} - \frac{6}{23} a^{2} - \frac{9}{23} a$, $\frac{1}{1909} a^{22} - \frac{36}{1909} a^{21} - \frac{6}{1909} a^{20} - \frac{32}{1909} a^{19} + \frac{18}{1909} a^{18} - \frac{21}{1909} a^{17} - \frac{187}{1909} a^{16} + \frac{454}{1909} a^{15} + \frac{882}{1909} a^{14} + \frac{234}{1909} a^{13} + \frac{795}{1909} a^{12} - \frac{720}{1909} a^{11} + \frac{153}{1909} a^{10} + \frac{827}{1909} a^{9} + \frac{585}{1909} a^{8} + \frac{347}{1909} a^{7} - \frac{945}{1909} a^{6} - \frac{310}{1909} a^{5} - \frac{256}{1909} a^{4} - \frac{578}{1909} a^{3} - \frac{534}{1909} a^{2} - \frac{418}{1909} a - \frac{11}{83}$, $\frac{1}{1909} a^{23} + \frac{26}{1909} a^{21} + \frac{1}{1909} a^{20} + \frac{28}{1909} a^{19} - \frac{37}{1909} a^{18} - \frac{30}{1909} a^{17} - \frac{385}{1909} a^{16} - \frac{619}{1909} a^{15} + \frac{944}{1909} a^{14} + \frac{753}{1909} a^{13} - \frac{818}{1909} a^{12} + \frac{544}{1909} a^{11} - \frac{139}{1909} a^{10} + \frac{228}{1909} a^{9} + \frac{657}{1909} a^{8} - \frac{737}{1909} a^{7} + \frac{198}{1909} a^{6} - \frac{543}{1909} a^{5} - \frac{6}{23} a^{4} + \frac{404}{1909} a^{3} + \frac{859}{1909} a^{2} + \frac{635}{1909} a + \frac{19}{83}$, $\frac{1}{435105007} a^{24} - \frac{31289}{435105007} a^{23} + \frac{22804}{435105007} a^{22} - \frac{2684633}{435105007} a^{21} + \frac{134571}{18917609} a^{20} + \frac{9205974}{435105007} a^{19} + \frac{4855048}{435105007} a^{18} - \frac{6896592}{435105007} a^{17} - \frac{191990358}{435105007} a^{16} + \frac{6532206}{18917609} a^{15} + \frac{217137410}{435105007} a^{14} - \frac{72461010}{435105007} a^{13} - \frac{212427738}{435105007} a^{12} - \frac{44316344}{435105007} a^{11} + \frac{72656437}{435105007} a^{10} - \frac{19798562}{435105007} a^{9} - \frac{59734960}{435105007} a^{8} - \frac{190827530}{435105007} a^{7} - \frac{93694853}{435105007} a^{6} + \frac{4749821}{18917609} a^{5} - \frac{146537593}{435105007} a^{4} + \frac{196980363}{435105007} a^{3} - \frac{77977692}{435105007} a^{2} - \frac{208536067}{435105007} a + \frac{1518791}{18917609}$, $\frac{1}{110673434272822919814930802451352839384075604977081320504267579} a^{25} + \frac{124308993940754990415087028895035326575088114754801331}{110673434272822919814930802451352839384075604977081320504267579} a^{24} + \frac{23501820873119495191172519261616319739791331909531078994542}{110673434272822919814930802451352839384075604977081320504267579} a^{23} - \frac{19759884161172162123165797828646000952579548587676798052544}{110673434272822919814930802451352839384075604977081320504267579} a^{22} - \frac{1841427177022575992816102016695602182200590092226652966806112}{110673434272822919814930802451352839384075604977081320504267579} a^{21} + \frac{320021295207148595252010924003648305036085881039859211919631}{110673434272822919814930802451352839384075604977081320504267579} a^{20} + \frac{1945521300353106807559250912899944254951709458276069000834653}{110673434272822919814930802451352839384075604977081320504267579} a^{19} + \frac{422519682102841596749438424053430329747703464101624465974759}{110673434272822919814930802451352839384075604977081320504267579} a^{18} + \frac{124416892200055612864512773083522598415125196144552177951066}{110673434272822919814930802451352839384075604977081320504267579} a^{17} - \frac{34013913148889571668590668411387427494916560420436215968794749}{110673434272822919814930802451352839384075604977081320504267579} a^{16} - \frac{14297044570883067563164276704320138045969851766468624651645072}{110673434272822919814930802451352839384075604977081320504267579} a^{15} - \frac{39151693749634890488588355453995933282935997460995375758542199}{110673434272822919814930802451352839384075604977081320504267579} a^{14} + \frac{47358646754041668210735012174911771652644973208666036576849433}{110673434272822919814930802451352839384075604977081320504267579} a^{13} - \frac{4750780233505350314989550337423152985578412708192641175000320}{110673434272822919814930802451352839384075604977081320504267579} a^{12} + \frac{32073323544212467090138037897747829326289681681317528952028773}{110673434272822919814930802451352839384075604977081320504267579} a^{11} + \frac{19373940742646486584189676949819535145847034929359976966336429}{110673434272822919814930802451352839384075604977081320504267579} a^{10} + \frac{47690613658065235026216501957569537287815839472621242345922565}{110673434272822919814930802451352839384075604977081320504267579} a^{9} - \frac{17165947523991155113498873929493981724817848325116049230622594}{110673434272822919814930802451352839384075604977081320504267579} a^{8} - \frac{8822662405157110531061563576684283830666120295556125416585236}{110673434272822919814930802451352839384075604977081320504267579} a^{7} - \frac{23496729016769156100476153854338086982393266997553891206908700}{110673434272822919814930802451352839384075604977081320504267579} a^{6} - \frac{41260234501538649759430325204794818784475374532262523510433502}{110673434272822919814930802451352839384075604977081320504267579} a^{5} + \frac{36544431443233244800330995019505384866719770826196142686669617}{110673434272822919814930802451352839384075604977081320504267579} a^{4} - \frac{41441033485422469292884093435113667790663187939934162476900243}{110673434272822919814930802451352839384075604977081320504267579} a^{3} - \frac{54082915867649795108925747722559085070430017943335234290037067}{110673434272822919814930802451352839384075604977081320504267579} a^{2} + \frac{17985201686619058569704019024976841859318216293370329649765}{44970920062097895089366437404044225674146934163787614995639} a - \frac{24621172726602380927094568994536902230835208237539388846629}{209212541158455424980965600097075310744944432848924991501451}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 74976425731025760000 \) (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

\(\Q(\sqrt{3}) \), 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 R R $26$ $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ 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
2Data not computed
3Data not computed
53Data not computed