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Properties

Label 26.26.2672709777...3593.1
Degree $26$
Signature $[26, 0]$
Discriminant $7^{13}\cdot 79^{25}$
Root discriminant $176.68$
Ramified primes $7, 79$
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![-65884397, -1573526488, -13595597644, -53539363068, -102426530893, -78629976783, 33139423840, 101906597908, 50701543916, -22839488411, -30521065603, -4639134330, 6019633502, 2536241039, -383501239, -402354875, -25391827, 30844380, 5310580, -1208507, -329551, 22195, 10071, -110, -156, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 156*x^24 - 110*x^23 + 10071*x^22 + 22195*x^21 - 329551*x^20 - 1208507*x^19 + 5310580*x^18 + 30844380*x^17 - 25391827*x^16 - 402354875*x^15 - 383501239*x^14 + 2536241039*x^13 + 6019633502*x^12 - 4639134330*x^11 - 30521065603*x^10 - 22839488411*x^9 + 50701543916*x^8 + 101906597908*x^7 + 33139423840*x^6 - 78629976783*x^5 - 102426530893*x^4 - 53539363068*x^3 - 13595597644*x^2 - 1573526488*x - 65884397)
 
gp: K = bnfinit(x^26 - x^25 - 156*x^24 - 110*x^23 + 10071*x^22 + 22195*x^21 - 329551*x^20 - 1208507*x^19 + 5310580*x^18 + 30844380*x^17 - 25391827*x^16 - 402354875*x^15 - 383501239*x^14 + 2536241039*x^13 + 6019633502*x^12 - 4639134330*x^11 - 30521065603*x^10 - 22839488411*x^9 + 50701543916*x^8 + 101906597908*x^7 + 33139423840*x^6 - 78629976783*x^5 - 102426530893*x^4 - 53539363068*x^3 - 13595597644*x^2 - 1573526488*x - 65884397, 1)
 

Normalized defining polynomial

\( x^{26} - x^{25} - 156 x^{24} - 110 x^{23} + 10071 x^{22} + 22195 x^{21} - 329551 x^{20} - 1208507 x^{19} + 5310580 x^{18} + 30844380 x^{17} - 25391827 x^{16} - 402354875 x^{15} - 383501239 x^{14} + 2536241039 x^{13} + 6019633502 x^{12} - 4639134330 x^{11} - 30521065603 x^{10} - 22839488411 x^{9} + 50701543916 x^{8} + 101906597908 x^{7} + 33139423840 x^{6} - 78629976783 x^{5} - 102426530893 x^{4} - 53539363068 x^{3} - 13595597644 x^{2} - 1573526488 x - 65884397 \)

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:  \(26727097776294998942448064582453023235154520026711313563593=7^{13}\cdot 79^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.68$
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:  $7, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(553=7\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{553}(512,·)$, $\chi_{553}(1,·)$, $\chi_{553}(69,·)$, $\chi_{553}(64,·)$, $\chi_{553}(8,·)$, $\chi_{553}(204,·)$, $\chi_{553}(141,·)$, $\chi_{553}(526,·)$, $\chi_{553}(337,·)$, $\chi_{553}(531,·)$, $\chi_{553}(22,·)$, $\chi_{553}(216,·)$, $\chi_{553}(27,·)$, $\chi_{553}(412,·)$, $\chi_{553}(349,·)$, $\chi_{553}(545,·)$, $\chi_{553}(484,·)$, $\chi_{553}(552,·)$, $\chi_{553}(41,·)$, $\chi_{553}(225,·)$, $\chi_{553}(302,·)$, $\chi_{553}(176,·)$, $\chi_{553}(328,·)$, $\chi_{553}(489,·)$, $\chi_{553}(377,·)$, $\chi_{553}(251,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23} a^{15} + \frac{6}{23} a^{14} + \frac{2}{23} a^{13} - \frac{5}{23} a^{12} + \frac{2}{23} a^{11} + \frac{9}{23} a^{10} + \frac{3}{23} a^{9} - \frac{3}{23} a^{8} - \frac{5}{23} a^{7} - \frac{5}{23} a^{6} - \frac{6}{23} a^{5} - \frac{2}{23} a^{4} + \frac{7}{23} a^{3} - \frac{7}{23} a^{2} + \frac{3}{23} a$, $\frac{1}{23} a^{16} - \frac{11}{23} a^{14} + \frac{6}{23} a^{13} + \frac{9}{23} a^{12} - \frac{3}{23} a^{11} - \frac{5}{23} a^{10} + \frac{2}{23} a^{9} - \frac{10}{23} a^{8} + \frac{2}{23} a^{7} + \frac{1}{23} a^{6} + \frac{11}{23} a^{5} - \frac{4}{23} a^{4} - \frac{3}{23} a^{3} - \frac{1}{23} a^{2} + \frac{5}{23} a$, $\frac{1}{23} a^{17} + \frac{3}{23} a^{14} + \frac{8}{23} a^{13} + \frac{11}{23} a^{12} - \frac{6}{23} a^{11} + \frac{9}{23} a^{10} - \frac{8}{23} a^{8} - \frac{8}{23} a^{7} + \frac{2}{23} a^{6} - \frac{1}{23} a^{5} - \frac{2}{23} a^{4} + \frac{7}{23} a^{3} - \frac{3}{23} a^{2} + \frac{10}{23} a$, $\frac{1}{23} a^{18} - \frac{10}{23} a^{14} + \frac{5}{23} a^{13} + \frac{9}{23} a^{12} + \frac{3}{23} a^{11} - \frac{4}{23} a^{10} + \frac{6}{23} a^{9} + \frac{1}{23} a^{8} - \frac{6}{23} a^{7} - \frac{9}{23} a^{6} - \frac{7}{23} a^{5} - \frac{10}{23} a^{4} - \frac{1}{23} a^{3} + \frac{8}{23} a^{2} - \frac{9}{23} a$, $\frac{1}{23} a^{19} - \frac{4}{23} a^{14} + \frac{6}{23} a^{13} - \frac{1}{23} a^{12} - \frac{7}{23} a^{11} + \frac{4}{23} a^{10} + \frac{8}{23} a^{9} + \frac{10}{23} a^{8} + \frac{10}{23} a^{7} - \frac{11}{23} a^{6} - \frac{1}{23} a^{5} + \frac{2}{23} a^{4} + \frac{9}{23} a^{3} - \frac{10}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{20} + \frac{7}{23} a^{14} + \frac{7}{23} a^{13} - \frac{4}{23} a^{12} - \frac{11}{23} a^{11} - \frac{2}{23} a^{10} - \frac{1}{23} a^{9} - \frac{2}{23} a^{8} - \frac{8}{23} a^{7} + \frac{2}{23} a^{6} + \frac{1}{23} a^{5} + \frac{1}{23} a^{4} - \frac{5}{23} a^{3} + \frac{2}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{23} a^{21} + \frac{11}{23} a^{14} + \frac{5}{23} a^{13} + \frac{1}{23} a^{12} + \frac{7}{23} a^{11} + \frac{5}{23} a^{10} - \frac{10}{23} a^{8} - \frac{9}{23} a^{7} - \frac{10}{23} a^{6} - \frac{3}{23} a^{5} + \frac{9}{23} a^{4} - \frac{1}{23} a^{3} - \frac{8}{23} a^{2} + \frac{2}{23} a$, $\frac{1}{6739} a^{22} + \frac{83}{6739} a^{21} + \frac{62}{6739} a^{20} + \frac{135}{6739} a^{19} + \frac{39}{6739} a^{18} - \frac{14}{6739} a^{17} + \frac{4}{293} a^{16} - \frac{33}{6739} a^{15} + \frac{2646}{6739} a^{14} + \frac{413}{6739} a^{13} - \frac{1486}{6739} a^{12} + \frac{2292}{6739} a^{11} - \frac{2469}{6739} a^{10} + \frac{1915}{6739} a^{9} - \frac{1768}{6739} a^{8} - \frac{1553}{6739} a^{7} - \frac{3250}{6739} a^{6} - \frac{1895}{6739} a^{5} + \frac{1241}{6739} a^{4} + \frac{553}{6739} a^{3} - \frac{2100}{6739} a^{2} + \frac{404}{6739} a + \frac{85}{293}$, $\frac{1}{220035089} a^{23} - \frac{8622}{220035089} a^{22} - \frac{4029544}{220035089} a^{21} - \frac{140991}{9566743} a^{20} + \frac{953509}{220035089} a^{19} - \frac{4393750}{220035089} a^{18} + \frac{191696}{220035089} a^{17} - \frac{1970549}{220035089} a^{16} - \frac{824368}{220035089} a^{15} + \frac{103126044}{220035089} a^{14} - \frac{109275891}{220035089} a^{13} - \frac{36803223}{220035089} a^{12} + \frac{14786974}{220035089} a^{11} - \frac{57976416}{220035089} a^{10} + \frac{106660354}{220035089} a^{9} - \frac{7728227}{220035089} a^{8} - \frac{91148673}{220035089} a^{7} - \frac{79798052}{220035089} a^{6} - \frac{69546336}{220035089} a^{5} + \frac{68740116}{220035089} a^{4} - \frac{59627475}{220035089} a^{3} - \frac{88848686}{220035089} a^{2} + \frac{1742342}{9566743} a - \frac{4248893}{9566743}$, $\frac{1}{794546706379} a^{24} + \frac{1309}{794546706379} a^{23} + \frac{25342196}{794546706379} a^{22} + \frac{433577451}{34545508973} a^{21} - \frac{8442340}{7714045693} a^{20} - \frac{7451350414}{794546706379} a^{19} + \frac{3671478172}{794546706379} a^{18} - \frac{3305288449}{794546706379} a^{17} - \frac{11678893646}{794546706379} a^{16} + \frac{11234683398}{794546706379} a^{15} + \frac{338895796796}{794546706379} a^{14} + \frac{269150771972}{794546706379} a^{13} - \frac{38701327271}{794546706379} a^{12} + \frac{329936570001}{794546706379} a^{11} - \frac{255520993954}{794546706379} a^{10} - \frac{129018143441}{794546706379} a^{9} + \frac{257132868060}{794546706379} a^{8} + \frac{3118541524}{7714045693} a^{7} - \frac{174384624522}{794546706379} a^{6} - \frac{161629707637}{794546706379} a^{5} - \frac{27996121262}{794546706379} a^{4} + \frac{231383404033}{794546706379} a^{3} - \frac{73897775722}{794546706379} a^{2} + \frac{54365409886}{794546706379} a - \frac{7470960769}{34545508973}$, $\frac{1}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{25} + \frac{66129419272794891995942049961865914855983181980660057085737}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{24} - \frac{73252829238671985081161954161696136091786606985694354874173404}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{23} - \frac{5984749732412135711868095623694421806034147940189006603500117546239}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{22} - \frac{2483562048106235066535357219480389287737066645219317238134688437402443}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{21} - \frac{6046384747567323001085883352019891764798098395589014893953993977010305}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{20} + \frac{1638701429291327043274983129405221840484893227243540482355377897653049}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{19} + \frac{875185934665211505828838501271750155542931352396180853200860557462337}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{18} + \frac{4173404588538161166379600963068400215815052067867923981169957732977445}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{17} - \frac{1824696599234455435747961121932635114696415988019599124312202452470838}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{16} - \frac{6051092520328326783061982483883257743633892157221762252506162193488610}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{15} - \frac{146973574147931968556863946618218607114806224932549062048330053647957584}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{14} + \frac{57676493156461588676769844991480268861470614525887004797072394348353135}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{13} - \frac{145517033751862747828092271702329266818385767844041431003765066903349380}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{12} - \frac{139322862611470145671273663822172878347495155479599197313747269058650689}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{11} + \frac{102165237601191759800669381614170876750576032183704479066813462289415246}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{10} - \frac{120916645718858925493148857945359797881480066130979226144597485218039789}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{9} + \frac{76759441739728739415306306675649468944815150413336026610293375158556417}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{8} - \frac{5615013877377022725378637608877725382305811391379645464933796384674434}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{7} - \frac{81492228529165754769433142751887243374336363601357005997670265889124345}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{6} + \frac{59262562135964774042063833226021873509755893843199649527761215103044974}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{5} + \frac{115537437282727921268990275843607606645514640981262955817475812942472111}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{4} + \frac{6237047065040035738934540911742713329938147365351940508665845064217245}{14207774815384270418403111419666770836991976062643121918304603267973999} a^{3} - \frac{29437766736779721481897009716231533341409192887917045704579075931160236}{326778820753838219623271562652335729250815449440791804121005875163401977} a^{2} + \frac{134680524404638993672885443473302068182763458931292850862605934748795776}{326778820753838219623271562652335729250815449440791804121005875163401977} a - \frac{5014170131296875823904267927373442450768819550447006137898001805933472}{14207774815384270418403111419666770836991976062643121918304603267973999}$

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:  \( 2411631151251373700000 \) (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

\(\Q(\sqrt{553}) \), 13.13.59091511031674153381441.1

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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ R ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ $26$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
7Data not computed
79Data not computed