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

\( x^{26} - x^{25} + 28 x^{24} - 29 x^{23} + 276 x^{22} - 306 x^{21} + 1038 x^{20} - 1375 x^{19} + 54 x^{18} - 1586 x^{17} - 8106 x^{16} - 449 x^{15} - 5336 x^{14} - 49021 x^{13} + \cdots + 1009861 \) Copy content Toggle raw display

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-20392621164064374190468609000303545984542460445839\) \(\medspace = -\,3^{13}\cdot 53^{25}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(78.80\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}53^{25/26}\approx 78.79854425324184$
Ramified primes:   \(3\), \(53\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-159}) \)
$\card{ \Gal(K/\Q) }$:  $26$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23}a^{17}-\frac{7}{23}a^{16}-\frac{9}{23}a^{14}+\frac{9}{23}a^{13}+\frac{2}{23}a^{12}+\frac{1}{23}a^{11}+\frac{11}{23}a^{10}-\frac{3}{23}a^{9}+\frac{3}{23}a^{7}+\frac{11}{23}a^{6}+\frac{6}{23}a^{5}+\frac{7}{23}a^{4}-\frac{8}{23}a^{3}+\frac{8}{23}a^{2}-\frac{9}{23}a$, $\frac{1}{23}a^{18}-\frac{3}{23}a^{16}-\frac{9}{23}a^{15}-\frac{8}{23}a^{14}-\frac{4}{23}a^{13}-\frac{8}{23}a^{12}-\frac{5}{23}a^{11}+\frac{5}{23}a^{10}+\frac{2}{23}a^{9}+\frac{3}{23}a^{8}+\frac{9}{23}a^{7}-\frac{9}{23}a^{6}+\frac{3}{23}a^{5}-\frac{5}{23}a^{4}-\frac{2}{23}a^{3}+\frac{1}{23}a^{2}+\frac{6}{23}a$, $\frac{1}{23}a^{19}-\frac{7}{23}a^{16}-\frac{8}{23}a^{15}-\frac{8}{23}a^{14}-\frac{4}{23}a^{13}+\frac{1}{23}a^{12}+\frac{8}{23}a^{11}-\frac{11}{23}a^{10}-\frac{6}{23}a^{9}+\frac{9}{23}a^{8}-\frac{10}{23}a^{6}-\frac{10}{23}a^{5}-\frac{4}{23}a^{4}+\frac{7}{23}a^{2}-\frac{4}{23}a$, $\frac{1}{23}a^{20}-\frac{11}{23}a^{16}-\frac{8}{23}a^{15}+\frac{2}{23}a^{14}-\frac{5}{23}a^{13}-\frac{1}{23}a^{12}-\frac{4}{23}a^{11}+\frac{2}{23}a^{10}+\frac{11}{23}a^{9}+\frac{11}{23}a^{7}-\frac{2}{23}a^{6}-\frac{8}{23}a^{5}+\frac{3}{23}a^{4}-\frac{3}{23}a^{3}+\frac{6}{23}a^{2}+\frac{6}{23}a$, $\frac{1}{23}a^{21}+\frac{7}{23}a^{16}+\frac{2}{23}a^{15}+\frac{11}{23}a^{14}+\frac{6}{23}a^{13}-\frac{5}{23}a^{12}-\frac{10}{23}a^{11}-\frac{6}{23}a^{10}-\frac{10}{23}a^{9}+\frac{11}{23}a^{8}+\frac{8}{23}a^{7}-\frac{2}{23}a^{6}+\frac{5}{23}a^{4}+\frac{10}{23}a^{3}+\frac{2}{23}a^{2}-\frac{7}{23}a$, $\frac{1}{529}a^{22}-\frac{6}{529}a^{21}-\frac{3}{529}a^{20}-\frac{2}{529}a^{19}-\frac{10}{529}a^{18}+\frac{9}{529}a^{17}+\frac{10}{23}a^{16}+\frac{129}{529}a^{15}-\frac{57}{529}a^{14}-\frac{52}{529}a^{13}-\frac{125}{529}a^{12}+\frac{240}{529}a^{11}+\frac{106}{529}a^{10}-\frac{252}{529}a^{9}-\frac{152}{529}a^{8}+\frac{224}{529}a^{7}-\frac{218}{529}a^{6}-\frac{61}{529}a^{5}-\frac{49}{529}a^{4}+\frac{162}{529}a^{3}+\frac{24}{529}a^{2}-\frac{6}{23}a$, $\frac{1}{213080671}a^{23}-\frac{189827}{213080671}a^{22}-\frac{3711800}{213080671}a^{21}+\frac{4193456}{213080671}a^{20}+\frac{2873476}{213080671}a^{19}+\frac{1942448}{213080671}a^{18}-\frac{4544956}{213080671}a^{17}+\frac{14678292}{213080671}a^{16}+\frac{81751046}{213080671}a^{15}-\frac{350141}{2567237}a^{14}+\frac{26757052}{213080671}a^{13}-\frac{86861356}{213080671}a^{12}-\frac{72063031}{213080671}a^{11}+\frac{1371946}{213080671}a^{10}+\frac{50412350}{213080671}a^{9}+\frac{51744157}{213080671}a^{8}-\frac{104107314}{213080671}a^{7}-\frac{92193906}{213080671}a^{6}+\frac{33461439}{213080671}a^{5}+\frac{3295056}{213080671}a^{4}+\frac{45858411}{213080671}a^{3}-\frac{72830905}{213080671}a^{2}+\frac{3328155}{9264377}a-\frac{15}{211}$, $\frac{1}{153205002449}a^{24}+\frac{71}{153205002449}a^{23}+\frac{129135131}{153205002449}a^{22}-\frac{248551430}{153205002449}a^{21}-\frac{1322217158}{153205002449}a^{20}+\frac{2788879666}{153205002449}a^{19}+\frac{2776157404}{153205002449}a^{18}-\frac{289266543}{153205002449}a^{17}+\frac{8739865390}{153205002449}a^{16}-\frac{34061054298}{153205002449}a^{15}+\frac{26248864633}{153205002449}a^{14}-\frac{67858340302}{153205002449}a^{13}-\frac{69170912016}{153205002449}a^{12}+\frac{29650695668}{153205002449}a^{11}-\frac{48573255029}{153205002449}a^{10}+\frac{2352697363}{6661087063}a^{9}+\frac{36258471776}{153205002449}a^{8}-\frac{52323385590}{153205002449}a^{7}+\frac{55798811353}{153205002449}a^{6}-\frac{1638331583}{153205002449}a^{5}-\frac{2171559363}{6661087063}a^{4}-\frac{26205620182}{153205002449}a^{3}-\frac{286849956}{726090059}a^{2}+\frac{2671328397}{6661087063}a+\frac{6782}{151709}$, $\frac{1}{15\!\cdots\!99}a^{25}-\frac{46\!\cdots\!71}{15\!\cdots\!99}a^{24}-\frac{23\!\cdots\!41}{15\!\cdots\!99}a^{23}+\frac{27\!\cdots\!23}{15\!\cdots\!99}a^{22}-\frac{23\!\cdots\!08}{15\!\cdots\!99}a^{21}-\frac{12\!\cdots\!73}{15\!\cdots\!99}a^{20}-\frac{51\!\cdots\!65}{15\!\cdots\!99}a^{19}+\frac{71\!\cdots\!14}{15\!\cdots\!99}a^{18}+\frac{10\!\cdots\!52}{66\!\cdots\!13}a^{17}-\frac{60\!\cdots\!56}{15\!\cdots\!99}a^{16}+\frac{56\!\cdots\!01}{15\!\cdots\!99}a^{15}-\frac{13\!\cdots\!10}{15\!\cdots\!99}a^{14}+\frac{76\!\cdots\!90}{15\!\cdots\!99}a^{13}+\frac{32\!\cdots\!66}{15\!\cdots\!99}a^{12}-\frac{63\!\cdots\!42}{15\!\cdots\!99}a^{11}-\frac{40\!\cdots\!58}{15\!\cdots\!99}a^{10}-\frac{27\!\cdots\!98}{15\!\cdots\!99}a^{9}-\frac{63\!\cdots\!07}{15\!\cdots\!99}a^{8}+\frac{72\!\cdots\!58}{15\!\cdots\!99}a^{7}+\frac{64\!\cdots\!83}{15\!\cdots\!99}a^{6}-\frac{34\!\cdots\!08}{15\!\cdots\!99}a^{5}-\frac{44\!\cdots\!34}{15\!\cdots\!99}a^{4}-\frac{58\!\cdots\!56}{15\!\cdots\!99}a^{3}-\frac{22\!\cdots\!65}{15\!\cdots\!99}a^{2}-\frac{28\!\cdots\!94}{66\!\cdots\!13}a-\frac{67\!\cdots\!89}{15\!\cdots\!59}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $23$

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{51\!\cdots\!60}{15\!\cdots\!99}a^{25}-\frac{17\!\cdots\!80}{15\!\cdots\!99}a^{24}+\frac{13\!\cdots\!02}{15\!\cdots\!99}a^{23}-\frac{65\!\cdots\!14}{15\!\cdots\!99}a^{22}+\frac{12\!\cdots\!70}{15\!\cdots\!99}a^{21}-\frac{10\!\cdots\!13}{18\!\cdots\!53}a^{20}+\frac{34\!\cdots\!47}{15\!\cdots\!99}a^{19}-\frac{56\!\cdots\!28}{15\!\cdots\!99}a^{18}-\frac{24\!\cdots\!98}{66\!\cdots\!13}a^{17}-\frac{11\!\cdots\!93}{15\!\cdots\!99}a^{16}-\frac{33\!\cdots\!98}{15\!\cdots\!99}a^{15}-\frac{60\!\cdots\!69}{15\!\cdots\!99}a^{14}+\frac{13\!\cdots\!69}{15\!\cdots\!99}a^{13}-\frac{16\!\cdots\!50}{15\!\cdots\!99}a^{12}+\frac{34\!\cdots\!50}{15\!\cdots\!99}a^{11}-\frac{61\!\cdots\!32}{15\!\cdots\!99}a^{10}+\frac{15\!\cdots\!07}{15\!\cdots\!99}a^{9}+\frac{10\!\cdots\!87}{15\!\cdots\!99}a^{8}-\frac{67\!\cdots\!78}{15\!\cdots\!99}a^{7}-\frac{10\!\cdots\!03}{15\!\cdots\!99}a^{6}-\frac{59\!\cdots\!87}{15\!\cdots\!99}a^{5}-\frac{42\!\cdots\!84}{15\!\cdots\!99}a^{4}+\frac{12\!\cdots\!21}{15\!\cdots\!99}a^{3}+\frac{24\!\cdots\!33}{15\!\cdots\!99}a^{2}-\frac{20\!\cdots\!50}{66\!\cdots\!13}a+\frac{27\!\cdots\!16}{15\!\cdots\!59}$, $\frac{76\!\cdots\!64}{15\!\cdots\!99}a^{25}-\frac{30\!\cdots\!82}{15\!\cdots\!99}a^{24}+\frac{20\!\cdots\!56}{15\!\cdots\!99}a^{23}-\frac{83\!\cdots\!25}{15\!\cdots\!99}a^{22}+\frac{18\!\cdots\!92}{15\!\cdots\!99}a^{21}-\frac{79\!\cdots\!77}{15\!\cdots\!99}a^{20}+\frac{54\!\cdots\!88}{15\!\cdots\!99}a^{19}-\frac{29\!\cdots\!94}{15\!\cdots\!99}a^{18}-\frac{49\!\cdots\!37}{66\!\cdots\!13}a^{17}-\frac{87\!\cdots\!10}{15\!\cdots\!99}a^{16}-\frac{80\!\cdots\!91}{15\!\cdots\!99}a^{15}-\frac{37\!\cdots\!64}{15\!\cdots\!99}a^{14}-\frac{35\!\cdots\!50}{15\!\cdots\!99}a^{13}-\frac{45\!\cdots\!96}{15\!\cdots\!99}a^{12}+\frac{54\!\cdots\!29}{15\!\cdots\!99}a^{11}-\frac{31\!\cdots\!20}{15\!\cdots\!99}a^{10}+\frac{15\!\cdots\!60}{15\!\cdots\!99}a^{9}+\frac{33\!\cdots\!69}{15\!\cdots\!99}a^{8}-\frac{87\!\cdots\!21}{15\!\cdots\!99}a^{7}+\frac{71\!\cdots\!84}{15\!\cdots\!99}a^{6}+\frac{11\!\cdots\!38}{15\!\cdots\!99}a^{5}-\frac{12\!\cdots\!85}{15\!\cdots\!99}a^{4}+\frac{19\!\cdots\!77}{15\!\cdots\!99}a^{3}-\frac{89\!\cdots\!37}{15\!\cdots\!99}a^{2}-\frac{83\!\cdots\!29}{66\!\cdots\!13}a+\frac{33\!\cdots\!05}{15\!\cdots\!59}$, $\frac{10\!\cdots\!93}{15\!\cdots\!99}a^{25}-\frac{54\!\cdots\!99}{15\!\cdots\!99}a^{24}+\frac{28\!\cdots\!50}{15\!\cdots\!99}a^{23}-\frac{16\!\cdots\!88}{15\!\cdots\!99}a^{22}+\frac{27\!\cdots\!98}{15\!\cdots\!99}a^{21}-\frac{18\!\cdots\!58}{15\!\cdots\!99}a^{20}+\frac{89\!\cdots\!17}{15\!\cdots\!99}a^{19}-\frac{93\!\cdots\!57}{15\!\cdots\!99}a^{18}-\frac{32\!\cdots\!27}{66\!\cdots\!13}a^{17}-\frac{16\!\cdots\!76}{15\!\cdots\!99}a^{16}-\frac{91\!\cdots\!79}{15\!\cdots\!99}a^{15}-\frac{45\!\cdots\!04}{15\!\cdots\!99}a^{14}-\frac{52\!\cdots\!47}{15\!\cdots\!99}a^{13}-\frac{52\!\cdots\!93}{15\!\cdots\!99}a^{12}+\frac{74\!\cdots\!39}{15\!\cdots\!99}a^{11}-\frac{54\!\cdots\!30}{15\!\cdots\!99}a^{10}+\frac{29\!\cdots\!13}{15\!\cdots\!99}a^{9}+\frac{28\!\cdots\!71}{15\!\cdots\!99}a^{8}+\frac{66\!\cdots\!87}{15\!\cdots\!99}a^{7}+\frac{51\!\cdots\!56}{15\!\cdots\!99}a^{6}+\frac{23\!\cdots\!77}{15\!\cdots\!99}a^{5}-\frac{14\!\cdots\!49}{15\!\cdots\!99}a^{4}+\frac{23\!\cdots\!07}{15\!\cdots\!99}a^{3}-\frac{80\!\cdots\!94}{15\!\cdots\!99}a^{2}-\frac{91\!\cdots\!08}{66\!\cdots\!13}a+\frac{27\!\cdots\!30}{15\!\cdots\!59}$, $\frac{38\!\cdots\!22}{15\!\cdots\!99}a^{25}-\frac{95\!\cdots\!09}{18\!\cdots\!53}a^{24}+\frac{10\!\cdots\!42}{15\!\cdots\!99}a^{23}-\frac{21\!\cdots\!33}{15\!\cdots\!99}a^{22}+\frac{97\!\cdots\!74}{15\!\cdots\!99}a^{21}-\frac{19\!\cdots\!90}{15\!\cdots\!99}a^{20}+\frac{34\!\cdots\!32}{18\!\cdots\!53}a^{19}-\frac{61\!\cdots\!24}{15\!\cdots\!99}a^{18}-\frac{27\!\cdots\!44}{66\!\cdots\!13}a^{17}+\frac{52\!\cdots\!26}{15\!\cdots\!99}a^{16}-\frac{46\!\cdots\!53}{15\!\cdots\!99}a^{15}-\frac{20\!\cdots\!22}{15\!\cdots\!99}a^{14}-\frac{31\!\cdots\!31}{15\!\cdots\!99}a^{13}-\frac{23\!\cdots\!11}{15\!\cdots\!99}a^{12}+\frac{23\!\cdots\!88}{15\!\cdots\!99}a^{11}-\frac{12\!\cdots\!59}{15\!\cdots\!99}a^{10}+\frac{66\!\cdots\!95}{15\!\cdots\!99}a^{9}+\frac{19\!\cdots\!09}{15\!\cdots\!99}a^{8}-\frac{86\!\cdots\!15}{15\!\cdots\!99}a^{7}+\frac{37\!\cdots\!87}{15\!\cdots\!99}a^{6}+\frac{25\!\cdots\!57}{15\!\cdots\!99}a^{5}-\frac{59\!\cdots\!22}{15\!\cdots\!99}a^{4}+\frac{92\!\cdots\!74}{15\!\cdots\!99}a^{3}-\frac{43\!\cdots\!78}{15\!\cdots\!99}a^{2}-\frac{39\!\cdots\!96}{66\!\cdots\!13}a+\frac{51\!\cdots\!37}{15\!\cdots\!59}$, $\frac{59\!\cdots\!98}{15\!\cdots\!99}a^{25}-\frac{26\!\cdots\!37}{15\!\cdots\!99}a^{24}+\frac{16\!\cdots\!91}{15\!\cdots\!99}a^{23}-\frac{81\!\cdots\!04}{15\!\cdots\!99}a^{22}+\frac{15\!\cdots\!43}{15\!\cdots\!99}a^{21}-\frac{96\!\cdots\!42}{15\!\cdots\!99}a^{20}+\frac{54\!\cdots\!23}{15\!\cdots\!99}a^{19}-\frac{54\!\cdots\!56}{15\!\cdots\!99}a^{18}-\frac{37\!\cdots\!29}{28\!\cdots\!31}a^{17}-\frac{13\!\cdots\!97}{15\!\cdots\!99}a^{16}-\frac{47\!\cdots\!82}{15\!\cdots\!99}a^{15}-\frac{36\!\cdots\!84}{15\!\cdots\!99}a^{14}-\frac{34\!\cdots\!72}{15\!\cdots\!99}a^{13}-\frac{29\!\cdots\!77}{15\!\cdots\!99}a^{12}+\frac{38\!\cdots\!77}{15\!\cdots\!99}a^{11}-\frac{35\!\cdots\!86}{15\!\cdots\!99}a^{10}+\frac{17\!\cdots\!76}{15\!\cdots\!99}a^{9}+\frac{11\!\cdots\!03}{15\!\cdots\!99}a^{8}+\frac{11\!\cdots\!09}{15\!\cdots\!99}a^{7}+\frac{20\!\cdots\!76}{15\!\cdots\!99}a^{6}+\frac{17\!\cdots\!37}{15\!\cdots\!99}a^{5}-\frac{76\!\cdots\!21}{15\!\cdots\!99}a^{4}+\frac{12\!\cdots\!69}{15\!\cdots\!99}a^{3}-\frac{45\!\cdots\!01}{15\!\cdots\!99}a^{2}-\frac{48\!\cdots\!55}{66\!\cdots\!13}a+\frac{85\!\cdots\!49}{15\!\cdots\!59}$, $\frac{79\!\cdots\!05}{15\!\cdots\!99}a^{25}-\frac{26\!\cdots\!18}{15\!\cdots\!99}a^{24}+\frac{21\!\cdots\!43}{15\!\cdots\!99}a^{23}-\frac{73\!\cdots\!46}{15\!\cdots\!99}a^{22}+\frac{18\!\cdots\!70}{15\!\cdots\!99}a^{21}-\frac{72\!\cdots\!16}{15\!\cdots\!99}a^{20}+\frac{51\!\cdots\!52}{15\!\cdots\!99}a^{19}-\frac{29\!\cdots\!10}{15\!\cdots\!99}a^{18}-\frac{57\!\cdots\!16}{66\!\cdots\!13}a^{17}-\frac{14\!\cdots\!73}{15\!\cdots\!99}a^{16}-\frac{78\!\cdots\!21}{15\!\cdots\!99}a^{15}-\frac{33\!\cdots\!07}{15\!\cdots\!99}a^{14}-\frac{13\!\cdots\!18}{15\!\cdots\!99}a^{13}-\frac{43\!\cdots\!37}{15\!\cdots\!99}a^{12}+\frac{54\!\cdots\!26}{15\!\cdots\!99}a^{11}-\frac{21\!\cdots\!28}{15\!\cdots\!99}a^{10}+\frac{15\!\cdots\!89}{15\!\cdots\!99}a^{9}+\frac{31\!\cdots\!25}{15\!\cdots\!99}a^{8}-\frac{16\!\cdots\!50}{15\!\cdots\!99}a^{7}+\frac{54\!\cdots\!56}{15\!\cdots\!99}a^{6}-\frac{34\!\cdots\!84}{15\!\cdots\!99}a^{5}-\frac{11\!\cdots\!77}{15\!\cdots\!99}a^{4}+\frac{21\!\cdots\!74}{15\!\cdots\!99}a^{3}-\frac{56\!\cdots\!62}{15\!\cdots\!99}a^{2}-\frac{74\!\cdots\!39}{66\!\cdots\!13}a+\frac{45\!\cdots\!11}{15\!\cdots\!59}$, $\frac{60\!\cdots\!27}{15\!\cdots\!99}a^{25}-\frac{22\!\cdots\!29}{15\!\cdots\!99}a^{24}+\frac{16\!\cdots\!13}{15\!\cdots\!99}a^{23}-\frac{62\!\cdots\!31}{15\!\cdots\!99}a^{22}+\frac{15\!\cdots\!78}{15\!\cdots\!99}a^{21}-\frac{62\!\cdots\!81}{15\!\cdots\!99}a^{20}+\frac{43\!\cdots\!15}{15\!\cdots\!99}a^{19}-\frac{25\!\cdots\!68}{15\!\cdots\!99}a^{18}-\frac{43\!\cdots\!64}{66\!\cdots\!13}a^{17}-\frac{15\!\cdots\!94}{15\!\cdots\!99}a^{16}-\frac{67\!\cdots\!85}{15\!\cdots\!99}a^{15}-\frac{26\!\cdots\!17}{15\!\cdots\!99}a^{14}-\frac{21\!\cdots\!55}{15\!\cdots\!99}a^{13}-\frac{34\!\cdots\!41}{15\!\cdots\!99}a^{12}+\frac{48\!\cdots\!45}{15\!\cdots\!99}a^{11}-\frac{19\!\cdots\!01}{15\!\cdots\!99}a^{10}+\frac{12\!\cdots\!74}{15\!\cdots\!99}a^{9}+\frac{27\!\cdots\!71}{15\!\cdots\!99}a^{8}-\frac{13\!\cdots\!36}{15\!\cdots\!99}a^{7}+\frac{47\!\cdots\!92}{15\!\cdots\!99}a^{6}+\frac{63\!\cdots\!43}{15\!\cdots\!99}a^{5}-\frac{98\!\cdots\!77}{15\!\cdots\!99}a^{4}+\frac{16\!\cdots\!96}{15\!\cdots\!99}a^{3}-\frac{48\!\cdots\!16}{15\!\cdots\!99}a^{2}-\frac{71\!\cdots\!31}{79\!\cdots\!11}a+\frac{44\!\cdots\!30}{15\!\cdots\!59}$, $\frac{56\!\cdots\!61}{15\!\cdots\!99}a^{25}+\frac{18\!\cdots\!06}{15\!\cdots\!99}a^{24}+\frac{15\!\cdots\!04}{15\!\cdots\!99}a^{23}+\frac{50\!\cdots\!72}{15\!\cdots\!99}a^{22}+\frac{13\!\cdots\!28}{15\!\cdots\!99}a^{21}+\frac{48\!\cdots\!11}{15\!\cdots\!99}a^{20}+\frac{23\!\cdots\!62}{15\!\cdots\!99}a^{19}+\frac{16\!\cdots\!09}{15\!\cdots\!99}a^{18}-\frac{10\!\cdots\!41}{66\!\cdots\!13}a^{17}-\frac{39\!\cdots\!80}{15\!\cdots\!99}a^{16}-\frac{12\!\cdots\!71}{15\!\cdots\!99}a^{15}-\frac{16\!\cdots\!78}{15\!\cdots\!99}a^{14}-\frac{24\!\cdots\!12}{15\!\cdots\!99}a^{13}-\frac{45\!\cdots\!04}{15\!\cdots\!99}a^{12}-\frac{66\!\cdots\!08}{15\!\cdots\!99}a^{11}+\frac{77\!\cdots\!65}{15\!\cdots\!99}a^{10}+\frac{15\!\cdots\!74}{15\!\cdots\!99}a^{9}+\frac{10\!\cdots\!38}{18\!\cdots\!53}a^{8}+\frac{19\!\cdots\!49}{15\!\cdots\!99}a^{7}+\frac{12\!\cdots\!10}{15\!\cdots\!99}a^{6}+\frac{25\!\cdots\!44}{15\!\cdots\!99}a^{5}+\frac{24\!\cdots\!15}{15\!\cdots\!99}a^{4}-\frac{11\!\cdots\!71}{15\!\cdots\!99}a^{3}-\frac{22\!\cdots\!80}{15\!\cdots\!99}a^{2}-\frac{56\!\cdots\!23}{66\!\cdots\!13}a-\frac{25\!\cdots\!01}{15\!\cdots\!59}$, $\frac{32\!\cdots\!37}{15\!\cdots\!99}a^{25}-\frac{32\!\cdots\!53}{15\!\cdots\!99}a^{24}+\frac{91\!\cdots\!56}{15\!\cdots\!99}a^{23}-\frac{82\!\cdots\!18}{15\!\cdots\!99}a^{22}+\frac{87\!\cdots\!27}{15\!\cdots\!99}a^{21}-\frac{71\!\cdots\!87}{15\!\cdots\!99}a^{20}+\frac{30\!\cdots\!78}{15\!\cdots\!99}a^{19}-\frac{22\!\cdots\!00}{15\!\cdots\!99}a^{18}-\frac{10\!\cdots\!03}{66\!\cdots\!13}a^{17}-\frac{64\!\cdots\!20}{15\!\cdots\!99}a^{16}-\frac{35\!\cdots\!42}{15\!\cdots\!99}a^{15}-\frac{17\!\cdots\!78}{15\!\cdots\!99}a^{14}-\frac{29\!\cdots\!60}{15\!\cdots\!99}a^{13}-\frac{19\!\cdots\!67}{15\!\cdots\!99}a^{12}+\frac{29\!\cdots\!60}{15\!\cdots\!99}a^{11}-\frac{22\!\cdots\!09}{15\!\cdots\!99}a^{10}+\frac{77\!\cdots\!86}{15\!\cdots\!99}a^{9}+\frac{10\!\cdots\!01}{15\!\cdots\!99}a^{8}+\frac{26\!\cdots\!50}{15\!\cdots\!99}a^{7}+\frac{43\!\cdots\!32}{15\!\cdots\!99}a^{6}+\frac{86\!\cdots\!94}{15\!\cdots\!99}a^{5}-\frac{68\!\cdots\!81}{15\!\cdots\!99}a^{4}+\frac{92\!\cdots\!21}{15\!\cdots\!99}a^{3}-\frac{57\!\cdots\!76}{15\!\cdots\!99}a^{2}-\frac{45\!\cdots\!71}{66\!\cdots\!13}a+\frac{14\!\cdots\!77}{15\!\cdots\!59}$, $\frac{10\!\cdots\!68}{66\!\cdots\!13}a^{25}+\frac{10\!\cdots\!50}{66\!\cdots\!13}a^{24}+\frac{24\!\cdots\!41}{66\!\cdots\!13}a^{23}+\frac{25\!\cdots\!84}{66\!\cdots\!13}a^{22}+\frac{17\!\cdots\!10}{66\!\cdots\!13}a^{21}+\frac{21\!\cdots\!62}{66\!\cdots\!13}a^{20}-\frac{79\!\cdots\!41}{66\!\cdots\!13}a^{19}+\frac{57\!\cdots\!60}{66\!\cdots\!13}a^{18}-\frac{45\!\cdots\!07}{66\!\cdots\!13}a^{17}+\frac{64\!\cdots\!71}{66\!\cdots\!13}a^{16}-\frac{10\!\cdots\!20}{66\!\cdots\!13}a^{15}-\frac{27\!\cdots\!46}{28\!\cdots\!31}a^{14}+\frac{12\!\cdots\!07}{66\!\cdots\!13}a^{13}-\frac{45\!\cdots\!70}{66\!\cdots\!13}a^{12}+\frac{38\!\cdots\!55}{66\!\cdots\!13}a^{11}+\frac{47\!\cdots\!75}{28\!\cdots\!31}a^{10}+\frac{14\!\cdots\!93}{66\!\cdots\!13}a^{9}+\frac{74\!\cdots\!56}{66\!\cdots\!13}a^{8}-\frac{29\!\cdots\!23}{28\!\cdots\!31}a^{7}+\frac{65\!\cdots\!36}{66\!\cdots\!13}a^{6}-\frac{52\!\cdots\!08}{66\!\cdots\!13}a^{5}-\frac{55\!\cdots\!97}{66\!\cdots\!13}a^{4}+\frac{27\!\cdots\!83}{66\!\cdots\!13}a^{3}+\frac{68\!\cdots\!69}{66\!\cdots\!13}a^{2}-\frac{35\!\cdots\!57}{28\!\cdots\!31}a-\frac{84\!\cdots\!16}{65\!\cdots\!33}$, $\frac{22\!\cdots\!23}{15\!\cdots\!99}a^{25}-\frac{13\!\cdots\!80}{15\!\cdots\!99}a^{24}+\frac{65\!\cdots\!16}{15\!\cdots\!99}a^{23}-\frac{34\!\cdots\!53}{15\!\cdots\!99}a^{22}+\frac{70\!\cdots\!08}{15\!\cdots\!99}a^{21}-\frac{29\!\cdots\!22}{15\!\cdots\!99}a^{20}+\frac{33\!\cdots\!09}{15\!\cdots\!99}a^{19}-\frac{77\!\cdots\!57}{15\!\cdots\!99}a^{18}+\frac{27\!\cdots\!83}{66\!\cdots\!13}a^{17}+\frac{14\!\cdots\!75}{15\!\cdots\!99}a^{16}-\frac{78\!\cdots\!48}{15\!\cdots\!99}a^{15}+\frac{61\!\cdots\!19}{15\!\cdots\!99}a^{14}+\frac{37\!\cdots\!90}{15\!\cdots\!99}a^{13}-\frac{14\!\cdots\!31}{15\!\cdots\!99}a^{12}+\frac{66\!\cdots\!70}{15\!\cdots\!99}a^{11}-\frac{79\!\cdots\!98}{15\!\cdots\!99}a^{10}+\frac{21\!\cdots\!60}{15\!\cdots\!99}a^{9}-\frac{19\!\cdots\!02}{15\!\cdots\!99}a^{8}-\frac{22\!\cdots\!24}{15\!\cdots\!99}a^{7}+\frac{39\!\cdots\!21}{15\!\cdots\!99}a^{6}-\frac{11\!\cdots\!31}{15\!\cdots\!99}a^{5}-\frac{13\!\cdots\!55}{15\!\cdots\!99}a^{4}+\frac{19\!\cdots\!22}{15\!\cdots\!99}a^{3}-\frac{12\!\cdots\!09}{15\!\cdots\!99}a^{2}-\frac{56\!\cdots\!73}{66\!\cdots\!13}a+\frac{46\!\cdots\!09}{15\!\cdots\!59}$, $\frac{53\!\cdots\!39}{15\!\cdots\!99}a^{25}-\frac{87\!\cdots\!73}{15\!\cdots\!99}a^{24}+\frac{15\!\cdots\!50}{15\!\cdots\!99}a^{23}-\frac{24\!\cdots\!07}{15\!\cdots\!99}a^{22}+\frac{16\!\cdots\!17}{15\!\cdots\!99}a^{21}-\frac{24\!\cdots\!90}{15\!\cdots\!99}a^{20}+\frac{78\!\cdots\!23}{15\!\cdots\!99}a^{19}-\frac{10\!\cdots\!69}{15\!\cdots\!99}a^{18}+\frac{45\!\cdots\!41}{66\!\cdots\!13}a^{17}-\frac{15\!\cdots\!35}{15\!\cdots\!99}a^{16}-\frac{36\!\cdots\!76}{15\!\cdots\!99}a^{15}-\frac{20\!\cdots\!31}{15\!\cdots\!99}a^{14}-\frac{89\!\cdots\!13}{15\!\cdots\!99}a^{13}-\frac{27\!\cdots\!09}{15\!\cdots\!99}a^{12}+\frac{46\!\cdots\!89}{15\!\cdots\!99}a^{11}-\frac{82\!\cdots\!13}{15\!\cdots\!99}a^{10}+\frac{20\!\cdots\!86}{15\!\cdots\!99}a^{9}-\frac{58\!\cdots\!87}{15\!\cdots\!99}a^{8}+\frac{38\!\cdots\!40}{15\!\cdots\!99}a^{7}+\frac{45\!\cdots\!68}{15\!\cdots\!99}a^{6}+\frac{37\!\cdots\!12}{15\!\cdots\!99}a^{5}-\frac{10\!\cdots\!22}{15\!\cdots\!99}a^{4}+\frac{10\!\cdots\!23}{15\!\cdots\!99}a^{3}-\frac{10\!\cdots\!41}{15\!\cdots\!99}a^{2}-\frac{69\!\cdots\!95}{66\!\cdots\!13}a-\frac{15\!\cdots\!14}{15\!\cdots\!59}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 5382739421.971964 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 32510}{2\cdot\sqrt{20392621164064374190468609000303545984542460445839}}\cr\approx \mathstrut & 0.460884489749922 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.13.0.1}{13} }^{2}$ R ${\href{/padicField/5.13.0.1}{13} }^{2}$ ${\href{/padicField/7.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/13.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/padicField/23.1.0.1}{1} }^{26}$ $26$ $26$ ${\href{/padicField/37.13.0.1}{13} }^{2}$ ${\href{/padicField/41.13.0.1}{13} }^{2}$ ${\href{/padicField/43.13.0.1}{13} }^{2}$ $26$ R $26$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $26$$2$$13$$13$
\(53\) Copy content Toggle raw display Deg $26$$26$$1$$25$