Properties

Label 26.0.203...839.1
Degree $26$
Signature $[0, 13]$
Discriminant $-2.039\times 10^{49}$
Root discriminant $78.80$
Ramified primes $3, 53$
Class number $32510$ (GRH)
Class group $[32510]$ (GRH)
Galois group $C_{26}$ (as 26T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 + 28*x^24 - 29*x^23 + 276*x^22 - 306*x^21 + 1038*x^20 - 1375*x^19 + 54*x^18 - 1586*x^17 - 8106*x^16 - 449*x^15 - 5336*x^14 - 49021*x^13 + 97058*x^12 - 111859*x^11 + 338650*x^10 + 82909*x^9 + 16023*x^8 + 521830*x^7 - 124280*x^6 - 1299383*x^5 + 3115247*x^4 - 2524927*x^3 + 1035234*x^2 + 2787416*x + 1009861)
 
gp: K = bnfinit(x^26 - x^25 + 28*x^24 - 29*x^23 + 276*x^22 - 306*x^21 + 1038*x^20 - 1375*x^19 + 54*x^18 - 1586*x^17 - 8106*x^16 - 449*x^15 - 5336*x^14 - 49021*x^13 + 97058*x^12 - 111859*x^11 + 338650*x^10 + 82909*x^9 + 16023*x^8 + 521830*x^7 - 124280*x^6 - 1299383*x^5 + 3115247*x^4 - 2524927*x^3 + 1035234*x^2 + 2787416*x + 1009861, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1009861, 2787416, 1035234, -2524927, 3115247, -1299383, -124280, 521830, 16023, 82909, 338650, -111859, 97058, -49021, -5336, -449, -8106, -1586, 54, -1375, 1038, -306, 276, -29, 28, -1, 1]);
 

\( x^{26} - x^{25} + 28 x^{24} - 29 x^{23} + 276 x^{22} - 306 x^{21} + 1038 x^{20} - 1375 x^{19} + 54 x^{18} - 1586 x^{17} - 8106 x^{16} - 449 x^{15} - 5336 x^{14} - 49021 x^{13} + 97058 x^{12} - 111859 x^{11} + 338650 x^{10} + 82909 x^{9} + 16023 x^{8} + 521830 x^{7} - 124280 x^{6} - 1299383 x^{5} + 3115247 x^{4} - 2524927 x^{3} + 1035234 x^{2} + 2787416 x + 1009861 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-20392621164064374190468609000303545984542460445839\)\(\medspace = -\,3^{13}\cdot 53^{25}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $78.80$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 53$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $26$
This field is Galois and abelian over $\Q$.
Conductor:  \(159=3\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{159}(1,·)$, $\chi_{159}(130,·)$, $\chi_{159}(131,·)$, $\chi_{159}(10,·)$, $\chi_{159}(11,·)$, $\chi_{159}(13,·)$, $\chi_{159}(142,·)$, $\chi_{159}(143,·)$, $\chi_{159}(16,·)$, $\chi_{159}(17,·)$, $\chi_{159}(146,·)$, $\chi_{159}(148,·)$, $\chi_{159}(149,·)$, $\chi_{159}(28,·)$, $\chi_{159}(29,·)$, $\chi_{159}(158,·)$, $\chi_{159}(97,·)$, $\chi_{159}(100,·)$, $\chi_{159}(38,·)$, $\chi_{159}(113,·)$, $\chi_{159}(110,·)$, $\chi_{159}(46,·)$, $\chi_{159}(49,·)$, $\chi_{159}(121,·)$, $\chi_{159}(59,·)$, $\chi_{159}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{7}{23} a^{16} - \frac{9}{23} a^{14} + \frac{9}{23} a^{13} + \frac{2}{23} a^{12} + \frac{1}{23} a^{11} + \frac{11}{23} a^{10} - \frac{3}{23} a^{9} + \frac{3}{23} a^{7} + \frac{11}{23} a^{6} + \frac{6}{23} a^{5} + \frac{7}{23} a^{4} - \frac{8}{23} a^{3} + \frac{8}{23} a^{2} - \frac{9}{23} a$, $\frac{1}{23} a^{18} - \frac{3}{23} a^{16} - \frac{9}{23} a^{15} - \frac{8}{23} a^{14} - \frac{4}{23} a^{13} - \frac{8}{23} a^{12} - \frac{5}{23} a^{11} + \frac{5}{23} a^{10} + \frac{2}{23} a^{9} + \frac{3}{23} a^{8} + \frac{9}{23} a^{7} - \frac{9}{23} a^{6} + \frac{3}{23} a^{5} - \frac{5}{23} a^{4} - \frac{2}{23} a^{3} + \frac{1}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{23} a^{19} - \frac{7}{23} a^{16} - \frac{8}{23} a^{15} - \frac{8}{23} a^{14} - \frac{4}{23} a^{13} + \frac{1}{23} a^{12} + \frac{8}{23} a^{11} - \frac{11}{23} a^{10} - \frac{6}{23} a^{9} + \frac{9}{23} a^{8} - \frac{10}{23} a^{6} - \frac{10}{23} a^{5} - \frac{4}{23} a^{4} + \frac{7}{23} a^{2} - \frac{4}{23} a$, $\frac{1}{23} a^{20} - \frac{11}{23} a^{16} - \frac{8}{23} a^{15} + \frac{2}{23} a^{14} - \frac{5}{23} a^{13} - \frac{1}{23} a^{12} - \frac{4}{23} a^{11} + \frac{2}{23} a^{10} + \frac{11}{23} a^{9} + \frac{11}{23} a^{7} - \frac{2}{23} a^{6} - \frac{8}{23} a^{5} + \frac{3}{23} a^{4} - \frac{3}{23} a^{3} + \frac{6}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{23} a^{21} + \frac{7}{23} a^{16} + \frac{2}{23} a^{15} + \frac{11}{23} a^{14} + \frac{6}{23} a^{13} - \frac{5}{23} a^{12} - \frac{10}{23} a^{11} - \frac{6}{23} a^{10} - \frac{10}{23} a^{9} + \frac{11}{23} a^{8} + \frac{8}{23} a^{7} - \frac{2}{23} a^{6} + \frac{5}{23} a^{4} + \frac{10}{23} a^{3} + \frac{2}{23} a^{2} - \frac{7}{23} a$, $\frac{1}{529} a^{22} - \frac{6}{529} a^{21} - \frac{3}{529} a^{20} - \frac{2}{529} a^{19} - \frac{10}{529} a^{18} + \frac{9}{529} a^{17} + \frac{10}{23} a^{16} + \frac{129}{529} a^{15} - \frac{57}{529} a^{14} - \frac{52}{529} a^{13} - \frac{125}{529} a^{12} + \frac{240}{529} a^{11} + \frac{106}{529} a^{10} - \frac{252}{529} a^{9} - \frac{152}{529} a^{8} + \frac{224}{529} a^{7} - \frac{218}{529} a^{6} - \frac{61}{529} a^{5} - \frac{49}{529} a^{4} + \frac{162}{529} a^{3} + \frac{24}{529} a^{2} - \frac{6}{23} a$, $\frac{1}{213080671} a^{23} - \frac{189827}{213080671} a^{22} - \frac{3711800}{213080671} a^{21} + \frac{4193456}{213080671} a^{20} + \frac{2873476}{213080671} a^{19} + \frac{1942448}{213080671} a^{18} - \frac{4544956}{213080671} a^{17} + \frac{14678292}{213080671} a^{16} + \frac{81751046}{213080671} a^{15} - \frac{350141}{2567237} a^{14} + \frac{26757052}{213080671} a^{13} - \frac{86861356}{213080671} a^{12} - \frac{72063031}{213080671} a^{11} + \frac{1371946}{213080671} a^{10} + \frac{50412350}{213080671} a^{9} + \frac{51744157}{213080671} a^{8} - \frac{104107314}{213080671} a^{7} - \frac{92193906}{213080671} a^{6} + \frac{33461439}{213080671} a^{5} + \frac{3295056}{213080671} a^{4} + \frac{45858411}{213080671} a^{3} - \frac{72830905}{213080671} a^{2} + \frac{3328155}{9264377} a - \frac{15}{211}$, $\frac{1}{153205002449} a^{24} + \frac{71}{153205002449} a^{23} + \frac{129135131}{153205002449} a^{22} - \frac{248551430}{153205002449} a^{21} - \frac{1322217158}{153205002449} a^{20} + \frac{2788879666}{153205002449} a^{19} + \frac{2776157404}{153205002449} a^{18} - \frac{289266543}{153205002449} a^{17} + \frac{8739865390}{153205002449} a^{16} - \frac{34061054298}{153205002449} a^{15} + \frac{26248864633}{153205002449} a^{14} - \frac{67858340302}{153205002449} a^{13} - \frac{69170912016}{153205002449} a^{12} + \frac{29650695668}{153205002449} a^{11} - \frac{48573255029}{153205002449} a^{10} + \frac{2352697363}{6661087063} a^{9} + \frac{36258471776}{153205002449} a^{8} - \frac{52323385590}{153205002449} a^{7} + \frac{55798811353}{153205002449} a^{6} - \frac{1638331583}{153205002449} a^{5} - \frac{2171559363}{6661087063} a^{4} - \frac{26205620182}{153205002449} a^{3} - \frac{286849956}{726090059} a^{2} + \frac{2671328397}{6661087063} a + \frac{6782}{151709}$, $\frac{1}{15231797105456104875423888713872769022305639001007425847666699} a^{25} - \frac{46544898421568630390550972902105534199177150622371}{15231797105456104875423888713872769022305639001007425847666699} a^{24} - \frac{23104546994512473114775046026986035258753252327424641}{15231797105456104875423888713872769022305639001007425847666699} a^{23} + \frac{2740402821755227593995639384624220147031682027514116992623}{15231797105456104875423888713872769022305639001007425847666699} a^{22} - \frac{238960698280809411900407753447879712888599946471841350142308}{15231797105456104875423888713872769022305639001007425847666699} a^{21} - \frac{120149735643332029121083935585527982566206111582921727265073}{15231797105456104875423888713872769022305639001007425847666699} a^{20} - \frac{51843160939304540261822254388732025580731315908264544759265}{15231797105456104875423888713872769022305639001007425847666699} a^{19} + \frac{71153256749970884708274762885818174168130497657166908856614}{15231797105456104875423888713872769022305639001007425847666699} a^{18} + \frac{10203300454692560391589348892224319064518529346534446954252}{662252048063308907627125596255337783578506043522061993376813} a^{17} - \frac{6067206024896050939767606576392054557467143874138194450265356}{15231797105456104875423888713872769022305639001007425847666699} a^{16} + \frac{5638772821864327146186834799971076147580363699041945524874901}{15231797105456104875423888713872769022305639001007425847666699} a^{15} - \frac{1381010253631074934919935830310293806451988752064867540687910}{15231797105456104875423888713872769022305639001007425847666699} a^{14} + \frac{7600913793141229004471425311473117131591282711480220806840890}{15231797105456104875423888713872769022305639001007425847666699} a^{13} + \frac{3209316491364858584771535528433525081340128830772528228447066}{15231797105456104875423888713872769022305639001007425847666699} a^{12} - \frac{6375257478983057065209469069200932648100965097042042928745942}{15231797105456104875423888713872769022305639001007425847666699} a^{11} - \frac{4083497493748572066573593146090133647586051374185928541711058}{15231797105456104875423888713872769022305639001007425847666699} a^{10} - \frac{2780707981138620193338460683112454022832844568770437935099398}{15231797105456104875423888713872769022305639001007425847666699} a^{9} - \frac{6344650733660253693748057219928878209960257601246913839550107}{15231797105456104875423888713872769022305639001007425847666699} a^{8} + \frac{7224558374699197549023977309190560975907177181027873607230358}{15231797105456104875423888713872769022305639001007425847666699} a^{7} + \frac{6439822713280447118217241625761318287254829509262308521867383}{15231797105456104875423888713872769022305639001007425847666699} a^{6} - \frac{340259318439859409919314837416901991663905104244218875727608}{15231797105456104875423888713872769022305639001007425847666699} a^{5} - \frac{4425051638002939190380793423142761136815867515979329434379234}{15231797105456104875423888713872769022305639001007425847666699} a^{4} - \frac{5826976341257916410225734615968250899435208349865910954996756}{15231797105456104875423888713872769022305639001007425847666699} a^{3} - \frac{2294628538210365205157874423479492307815647687418506758248165}{15231797105456104875423888713872769022305639001007425847666699} a^{2} - \frac{28914241997847419216883179817754312221487454219730851279394}{662252048063308907627125596255337783578506043522061993376813} a - \frac{6712370705134500962820419094284859748318092869609596089}{15083063021005965054026136977141179847826224600224610959}$

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

Class group and class number

$C_{32510}$, which has order $32510$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5382739421.971964 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 32510}{2\sqrt{20392621164064374190468609000303545984542460445839}}\approx 0.460884489749922$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-159}) \), 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 ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ R $26$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
53Data not computed