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Properties

Label 26.26.7309445804...8893.1
Degree $26$
Signature $[26, 0]$
Discriminant $13^{13}\cdot 53^{24}$
Root discriminant $140.80$
Ramified primes $13, 53$
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![529, -57707, -1480706, -5737562, 33037124, 61867022, -136303988, -131092304, 236602449, 114942028, -218101734, -43965313, 117022908, 2064770, -37710724, 4114140, 7250271, -1483913, -794597, 233157, 43660, -18660, -650, 732, -33, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 - 33*x^24 + 732*x^23 - 650*x^22 - 18660*x^21 + 43660*x^20 + 233157*x^19 - 794597*x^18 - 1483913*x^17 + 7250271*x^16 + 4114140*x^15 - 37710724*x^14 + 2064770*x^13 + 117022908*x^12 - 43965313*x^11 - 218101734*x^10 + 114942028*x^9 + 236602449*x^8 - 131092304*x^7 - 136303988*x^6 + 61867022*x^5 + 33037124*x^4 - 5737562*x^3 - 1480706*x^2 - 57707*x + 529)
 
gp: K = bnfinit(x^26 - 11*x^25 - 33*x^24 + 732*x^23 - 650*x^22 - 18660*x^21 + 43660*x^20 + 233157*x^19 - 794597*x^18 - 1483913*x^17 + 7250271*x^16 + 4114140*x^15 - 37710724*x^14 + 2064770*x^13 + 117022908*x^12 - 43965313*x^11 - 218101734*x^10 + 114942028*x^9 + 236602449*x^8 - 131092304*x^7 - 136303988*x^6 + 61867022*x^5 + 33037124*x^4 - 5737562*x^3 - 1480706*x^2 - 57707*x + 529, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} - 33 x^{24} + 732 x^{23} - 650 x^{22} - 18660 x^{21} + 43660 x^{20} + 233157 x^{19} - 794597 x^{18} - 1483913 x^{17} + 7250271 x^{16} + 4114140 x^{15} - 37710724 x^{14} + 2064770 x^{13} + 117022908 x^{12} - 43965313 x^{11} - 218101734 x^{10} + 114942028 x^{9} + 236602449 x^{8} - 131092304 x^{7} - 136303988 x^{6} + 61867022 x^{5} + 33037124 x^{4} - 5737562 x^{3} - 1480706 x^{2} - 57707 x + 529 \)

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:  \(73094458047088417407552639076238127506405695058400788893=13^{13}\cdot 53^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.80$
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:  $13, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(689=13\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{689}(1,·)$, $\chi_{689}(66,·)$, $\chi_{689}(259,·)$, $\chi_{689}(261,·)$, $\chi_{689}(311,·)$, $\chi_{689}(519,·)$, $\chi_{689}(584,·)$, $\chi_{689}(521,·)$, $\chi_{689}(651,·)$, $\chi_{689}(77,·)$, $\chi_{689}(142,·)$, $\chi_{689}(365,·)$, $\chi_{689}(599,·)$, $\chi_{689}(664,·)$, $\chi_{689}(155,·)$, $\chi_{689}(222,·)$, $\chi_{689}(415,·)$, $\chi_{689}(545,·)$, $\chi_{689}(625,·)$, $\chi_{689}(493,·)$, $\chi_{689}(558,·)$, $\chi_{689}(417,·)$, $\chi_{689}(116,·)$, $\chi_{689}(649,·)$, $\chi_{689}(248,·)$, $\chi_{689}(183,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23} a^{15} - \frac{10}{23} a^{14} - \frac{2}{23} a^{13} - \frac{5}{23} a^{12} - \frac{3}{23} a^{11} + \frac{9}{23} a^{10} - \frac{11}{23} a^{9} - \frac{11}{23} a^{8} + \frac{4}{23} a^{7} + \frac{3}{23} a^{6} + \frac{6}{23} a^{5} + \frac{11}{23} a^{4} + \frac{4}{23} a^{3} - \frac{2}{23} a^{2} - \frac{4}{23} a$, $\frac{1}{23} a^{16} - \frac{10}{23} a^{14} - \frac{2}{23} a^{13} - \frac{7}{23} a^{12} + \frac{2}{23} a^{11} + \frac{10}{23} a^{10} - \frac{6}{23} a^{9} + \frac{9}{23} a^{8} - \frac{3}{23} a^{7} - \frac{10}{23} a^{6} + \frac{2}{23} a^{5} - \frac{1}{23} a^{4} - \frac{8}{23} a^{3} - \frac{1}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{23} a^{17} - \frac{10}{23} a^{14} - \frac{4}{23} a^{13} - \frac{2}{23} a^{12} + \frac{3}{23} a^{11} - \frac{8}{23} a^{10} - \frac{9}{23} a^{9} + \frac{2}{23} a^{8} + \frac{7}{23} a^{7} + \frac{9}{23} a^{6} - \frac{10}{23} a^{5} + \frac{10}{23} a^{4} - \frac{7}{23} a^{3} + \frac{9}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{23} a^{18} + \frac{11}{23} a^{14} + \frac{1}{23} a^{13} - \frac{1}{23} a^{12} + \frac{8}{23} a^{11} - \frac{11}{23} a^{10} + \frac{7}{23} a^{9} - \frac{11}{23} a^{8} + \frac{3}{23} a^{7} - \frac{3}{23} a^{6} + \frac{1}{23} a^{5} + \frac{11}{23} a^{4} + \frac{3}{23} a^{3} + \frac{9}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{23} a^{19} - \frac{4}{23} a^{14} - \frac{2}{23} a^{13} - \frac{6}{23} a^{12} - \frac{1}{23} a^{11} - \frac{5}{23} a^{9} + \frac{9}{23} a^{8} - \frac{1}{23} a^{7} - \frac{9}{23} a^{6} - \frac{9}{23} a^{5} - \frac{3}{23} a^{4} + \frac{11}{23} a^{3} + \frac{5}{23} a^{2} - \frac{2}{23} a$, $\frac{1}{23} a^{20} + \frac{4}{23} a^{14} + \frac{9}{23} a^{13} + \frac{2}{23} a^{12} + \frac{11}{23} a^{11} + \frac{8}{23} a^{10} + \frac{11}{23} a^{9} + \frac{1}{23} a^{8} + \frac{7}{23} a^{7} + \frac{3}{23} a^{6} - \frac{2}{23} a^{5} + \frac{9}{23} a^{4} - \frac{2}{23} a^{3} - \frac{10}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{21} + \frac{3}{23} a^{14} + \frac{10}{23} a^{13} + \frac{8}{23} a^{12} - \frac{3}{23} a^{11} - \frac{2}{23} a^{10} - \frac{1}{23} a^{9} + \frac{5}{23} a^{8} + \frac{10}{23} a^{7} + \frac{9}{23} a^{6} + \frac{8}{23} a^{5} - \frac{3}{23} a^{3} - \frac{8}{23} a^{2} - \frac{7}{23} a$, $\frac{1}{23} a^{22} - \frac{6}{23} a^{14} - \frac{9}{23} a^{13} - \frac{11}{23} a^{12} + \frac{7}{23} a^{11} - \frac{5}{23} a^{10} - \frac{8}{23} a^{9} - \frac{3}{23} a^{8} - \frac{3}{23} a^{7} - \frac{1}{23} a^{6} + \frac{5}{23} a^{5} + \frac{10}{23} a^{4} + \frac{3}{23} a^{3} - \frac{1}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{529} a^{23} - \frac{9}{529} a^{22} - \frac{1}{529} a^{21} + \frac{5}{529} a^{20} - \frac{9}{529} a^{19} - \frac{6}{529} a^{18} + \frac{9}{529} a^{17} - \frac{9}{529} a^{16} - \frac{4}{529} a^{15} + \frac{196}{529} a^{14} + \frac{72}{529} a^{13} - \frac{188}{529} a^{12} - \frac{7}{23} a^{11} + \frac{116}{529} a^{10} + \frac{10}{529} a^{9} - \frac{168}{529} a^{8} + \frac{255}{529} a^{7} - \frac{26}{529} a^{6} + \frac{64}{529} a^{5} + \frac{17}{529} a^{4} + \frac{141}{529} a^{3} - \frac{218}{529} a^{2} - \frac{6}{23} a$, $\frac{1}{2111564598971} a^{24} + \frac{1053491383}{2111564598971} a^{23} - \frac{28207289703}{2111564598971} a^{22} + \frac{30363100346}{2111564598971} a^{21} - \frac{1098020680}{2111564598971} a^{20} - \frac{43038266870}{2111564598971} a^{19} - \frac{1776702498}{91807156477} a^{18} + \frac{43740281342}{2111564598971} a^{17} - \frac{41682817068}{2111564598971} a^{16} + \frac{1380885690}{2111564598971} a^{15} - \frac{854969523887}{2111564598971} a^{14} - \frac{550247058553}{2111564598971} a^{13} - \frac{988872795792}{2111564598971} a^{12} + \frac{524897678302}{2111564598971} a^{11} - \frac{31520058047}{91807156477} a^{10} + \frac{382729795616}{2111564598971} a^{9} + \frac{959003614202}{2111564598971} a^{8} - \frac{12917763996}{91807156477} a^{7} - \frac{1024134721407}{2111564598971} a^{6} + \frac{689061777213}{2111564598971} a^{5} - \frac{434773468952}{2111564598971} a^{4} - \frac{360724619318}{2111564598971} a^{3} - \frac{649562827086}{2111564598971} a^{2} + \frac{42272953455}{91807156477} a - \frac{266597606}{3991615499}$, $\frac{1}{119965369551431639360592667068915186067799950058520858184799} a^{25} + \frac{22357031479443531470538872744169623791593683816}{119965369551431639360592667068915186067799950058520858184799} a^{24} + \frac{58239774368824946635770649636972129500784945716317372716}{119965369551431639360592667068915186067799950058520858184799} a^{23} + \frac{102013629059568232566022445797506578477444342154317959596}{119965369551431639360592667068915186067799950058520858184799} a^{22} - \frac{964638863056285893856652280707335363699982566347632311672}{119965369551431639360592667068915186067799950058520858184799} a^{21} - \frac{1887166426384767400131005320030347540886840439349580512488}{119965369551431639360592667068915186067799950058520858184799} a^{20} + \frac{706780303322175618710176816778636795989102769962208491376}{119965369551431639360592667068915186067799950058520858184799} a^{19} + \frac{2322878184300948911675856437691108249906623556613503502810}{119965369551431639360592667068915186067799950058520858184799} a^{18} + \frac{1310578157357527142256504210382308786294454374813889378833}{119965369551431639360592667068915186067799950058520858184799} a^{17} - \frac{847720212309869126798402904834152569670652988386758649911}{119965369551431639360592667068915186067799950058520858184799} a^{16} - \frac{838797245600377769884557156585726572394387721566610420606}{119965369551431639360592667068915186067799950058520858184799} a^{15} - \frac{20455017110426597722811789186718348310679505569898802510464}{119965369551431639360592667068915186067799950058520858184799} a^{14} + \frac{25749973914242473185880855662011378591283669934374759977840}{119965369551431639360592667068915186067799950058520858184799} a^{13} + \frac{53333475241515195249459156502382024140028371169922528271140}{119965369551431639360592667068915186067799950058520858184799} a^{12} + \frac{27416538454327470359395802245887244043052343774567930557664}{119965369551431639360592667068915186067799950058520858184799} a^{11} - \frac{19169066743810813228156872474865675825796488027028386427567}{119965369551431639360592667068915186067799950058520858184799} a^{10} + \frac{56521137888358064153109434274418571174302008647431017131068}{119965369551431639360592667068915186067799950058520858184799} a^{9} - \frac{56650990579658954447725279634528561926564771528149580502810}{119965369551431639360592667068915186067799950058520858184799} a^{8} + \frac{43845782002337848918755639365540540022441122645463083120393}{119965369551431639360592667068915186067799950058520858184799} a^{7} - \frac{1488188064058087236188725837928429450572459522433961069034}{119965369551431639360592667068915186067799950058520858184799} a^{6} + \frac{11315300611766829469477210390700569819904460320500997426646}{119965369551431639360592667068915186067799950058520858184799} a^{5} - \frac{35117687087269348462567715598150562486966122534968889060739}{119965369551431639360592667068915186067799950058520858184799} a^{4} + \frac{44571312955255280953926902601886248175748466196205797858433}{119965369551431639360592667068915186067799950058520858184799} a^{3} + \frac{33727197867559896088078885882512451312813101408569258221776}{119965369551431639360592667068915186067799950058520858184799} a^{2} - \frac{2076912428541775367251864532498465727652990307784525805760}{5215885632670940841764898568213703742078258698196559051513} a + \frac{107969964778025863290703429305627231435268110149554072065}{226777636203084384424560807313639293133837334704198219631}$

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:  \( 104723787342670040000 \) (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

\(\Q(\sqrt{13}) \), 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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ $26$ $26$ R ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ 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
13Data 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}$