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Properties

Label 26.0.83315494722...1211.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,11^{13}\cdot 53^{24}$
Root discriminant $129.52$
Ramified primes $11, 53$
Class number $14446809$ (GRH)
Class group $[3, 3, 1605201]$ (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![4569738463, -5536163135, 9149067838, -7821265802, 7375230728, -4894096774, 3357872176, -1819021316, 1004077677, -466972712, 223682928, -94709797, 41351964, -16258486, 6343892, -2159628, 699117, -206429, 68365, -23643, 6796, -984, 34, -60, 45, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 + 45*x^24 - 60*x^23 + 34*x^22 - 984*x^21 + 6796*x^20 - 23643*x^19 + 68365*x^18 - 206429*x^17 + 699117*x^16 - 2159628*x^15 + 6343892*x^14 - 16258486*x^13 + 41351964*x^12 - 94709797*x^11 + 223682928*x^10 - 466972712*x^9 + 1004077677*x^8 - 1819021316*x^7 + 3357872176*x^6 - 4894096774*x^5 + 7375230728*x^4 - 7821265802*x^3 + 9149067838*x^2 - 5536163135*x + 4569738463)
 
gp: K = bnfinit(x^26 - 11*x^25 + 45*x^24 - 60*x^23 + 34*x^22 - 984*x^21 + 6796*x^20 - 23643*x^19 + 68365*x^18 - 206429*x^17 + 699117*x^16 - 2159628*x^15 + 6343892*x^14 - 16258486*x^13 + 41351964*x^12 - 94709797*x^11 + 223682928*x^10 - 466972712*x^9 + 1004077677*x^8 - 1819021316*x^7 + 3357872176*x^6 - 4894096774*x^5 + 7375230728*x^4 - 7821265802*x^3 + 9149067838*x^2 - 5536163135*x + 4569738463, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} + 45 x^{24} - 60 x^{23} + 34 x^{22} - 984 x^{21} + 6796 x^{20} - 23643 x^{19} + 68365 x^{18} - 206429 x^{17} + 699117 x^{16} - 2159628 x^{15} + 6343892 x^{14} - 16258486 x^{13} + 41351964 x^{12} - 94709797 x^{11} + 223682928 x^{10} - 466972712 x^{9} + 1004077677 x^{8} - 1819021316 x^{7} + 3357872176 x^{6} - 4894096774 x^{5} + 7375230728 x^{4} - 7821265802 x^{3} + 9149067838 x^{2} - 5536163135 x + 4569738463 \)

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

Invariants

Degree:  $26$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 13]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8331549472216739873971116493728468253329428095507981211=-\,11^{13}\cdot 53^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.52$
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:  $11, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(583=11\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{583}(384,·)$, $\chi_{583}(1,·)$, $\chi_{583}(386,·)$, $\chi_{583}(331,·)$, $\chi_{583}(452,·)$, $\chi_{583}(10,·)$, $\chi_{583}(395,·)$, $\chi_{583}(142,·)$, $\chi_{583}(208,·)$, $\chi_{583}(342,·)$, $\chi_{583}(89,·)$, $\chi_{583}(153,·)$, $\chi_{583}(155,·)$, $\chi_{583}(540,·)$, $\chi_{583}(417,·)$, $\chi_{583}(100,·)$, $\chi_{583}(362,·)$, $\chi_{583}(364,·)$, $\chi_{583}(175,·)$, $\chi_{583}(307,·)$, $\chi_{583}(309,·)$, $\chi_{583}(54,·)$, $\chi_{583}(439,·)$, $\chi_{583}(505,·)$, $\chi_{583}(122,·)$, $\chi_{583}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{9}{23} a^{16} - \frac{1}{23} a^{15} - \frac{8}{23} a^{14} - \frac{11}{23} a^{13} - \frac{5}{23} a^{12} + \frac{4}{23} a^{11} - \frac{7}{23} a^{10} + \frac{10}{23} a^{9} + \frac{9}{23} a^{8} - \frac{6}{23} a^{7} - \frac{9}{23} a^{6} + \frac{10}{23} a^{5} - \frac{9}{23} a^{4} + \frac{8}{23} a^{3} + \frac{3}{23} a^{2} - \frac{3}{23} a$, $\frac{1}{23} a^{18} + \frac{10}{23} a^{16} + \frac{6}{23} a^{15} + \frac{9}{23} a^{14} + \frac{11}{23} a^{13} + \frac{5}{23} a^{12} + \frac{6}{23} a^{11} - \frac{7}{23} a^{10} + \frac{7}{23} a^{9} + \frac{6}{23} a^{8} + \frac{6}{23} a^{7} - \frac{2}{23} a^{6} - \frac{11}{23} a^{5} - \frac{4}{23} a^{4} + \frac{6}{23} a^{3} + \frac{1}{23} a^{2} - \frac{4}{23} a$, $\frac{1}{23} a^{19} + \frac{4}{23} a^{16} - \frac{4}{23} a^{15} - \frac{1}{23} a^{14} + \frac{10}{23} a^{12} - \frac{1}{23} a^{11} + \frac{8}{23} a^{10} - \frac{2}{23} a^{9} + \frac{8}{23} a^{8} - \frac{11}{23} a^{7} + \frac{10}{23} a^{6} + \frac{11}{23} a^{5} + \frac{4}{23} a^{4} - \frac{10}{23} a^{3} - \frac{11}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{20} + \frac{9}{23} a^{16} + \frac{3}{23} a^{15} + \frac{9}{23} a^{14} + \frac{8}{23} a^{13} - \frac{4}{23} a^{12} - \frac{8}{23} a^{11} + \frac{3}{23} a^{10} - \frac{9}{23} a^{9} - \frac{1}{23} a^{8} + \frac{11}{23} a^{7} + \frac{1}{23} a^{6} + \frac{10}{23} a^{5} + \frac{3}{23} a^{4} + \frac{3}{23} a^{3} - \frac{5}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{23} a^{21} - \frac{8}{23} a^{16} - \frac{5}{23} a^{15} + \frac{11}{23} a^{14} + \frac{3}{23} a^{13} - \frac{9}{23} a^{12} - \frac{10}{23} a^{11} + \frac{8}{23} a^{10} + \frac{1}{23} a^{9} - \frac{1}{23} a^{8} + \frac{9}{23} a^{7} - \frac{1}{23} a^{6} + \frac{5}{23} a^{5} - \frac{8}{23} a^{4} - \frac{8}{23} a^{3} + \frac{8}{23} a^{2} + \frac{4}{23} a$, $\frac{1}{23} a^{22} - \frac{8}{23} a^{16} + \frac{3}{23} a^{15} + \frac{8}{23} a^{14} - \frac{5}{23} a^{13} - \frac{4}{23} a^{12} - \frac{6}{23} a^{11} - \frac{9}{23} a^{10} + \frac{10}{23} a^{9} - \frac{11}{23} a^{8} - \frac{3}{23} a^{7} + \frac{2}{23} a^{6} + \frac{3}{23} a^{5} - \frac{11}{23} a^{4} + \frac{3}{23} a^{3} + \frac{5}{23} a^{2} - \frac{1}{23} a$, $\frac{1}{16537} a^{23} - \frac{191}{16537} a^{22} + \frac{237}{16537} a^{21} + \frac{275}{16537} a^{20} + \frac{170}{16537} a^{19} - \frac{10}{719} a^{18} + \frac{218}{16537} a^{17} + \frac{1032}{16537} a^{16} - \frac{4062}{16537} a^{15} + \frac{2192}{16537} a^{14} + \frac{1284}{16537} a^{13} - \frac{1169}{16537} a^{12} + \frac{3488}{16537} a^{11} - \frac{3799}{16537} a^{10} + \frac{5374}{16537} a^{9} - \frac{3231}{16537} a^{8} + \frac{41}{719} a^{7} + \frac{5489}{16537} a^{6} - \frac{8205}{16537} a^{5} + \frac{6257}{16537} a^{4} - \frac{4544}{16537} a^{3} - \frac{3076}{16537} a^{2} - \frac{4019}{16537} a + \frac{342}{719}$, $\frac{1}{435105007} a^{24} - \frac{1057}{435105007} a^{23} + \frac{8232104}{435105007} a^{22} + \frac{5451406}{435105007} a^{21} + \frac{165379}{435105007} a^{20} - \frac{3142804}{435105007} a^{19} + \frac{217961}{18917609} a^{18} - \frac{1414370}{435105007} a^{17} - \frac{201544071}{435105007} a^{16} + \frac{71082157}{435105007} a^{15} - \frac{135126250}{435105007} a^{14} - \frac{124111724}{435105007} a^{13} + \frac{61291769}{435105007} a^{12} + \frac{99698404}{435105007} a^{11} - \frac{64406451}{435105007} a^{10} + \frac{181276285}{435105007} a^{9} + \frac{96660844}{435105007} a^{8} + \frac{108653006}{435105007} a^{7} + \frac{69935950}{435105007} a^{6} + \frac{204238579}{435105007} a^{5} - \frac{16289353}{435105007} a^{4} - \frac{214247084}{435105007} a^{3} + \frac{52110460}{435105007} a^{2} + \frac{204964343}{435105007} a + \frac{4007762}{18917609}$, $\frac{1}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{25} - \frac{319182743172443362033130336881748783209782477641590792349534113161895254195620621478693}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{24} + \frac{3109555799349931120232588962952867584576856266953112667445808886256704559594685312966734928}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{23} - \frac{11476676979445366611036052028078646143457046009374131523575692305454959851143248959278182631759}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{22} + \frac{6853901823947665482351156803464147707693535564513668764550137756772381377878616968225632133446}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{21} + \frac{9511537420353229170153053408644961316415883707033998125254238844732966613662984658496001780572}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{20} + \frac{8522073811868994794736734803434266326310696884599551311176409419315800311282035851360992580885}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{19} - \frac{11008210862899506630492658462338518299363238824644964591909800674957984847842310699568755021994}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{18} - \frac{501893375265498107955976803419633871903992794714083527931993831562438774106741025272887203043}{23838853616142596257487659630266166921047344888605488557402914040049298657306290581771323573891} a^{17} + \frac{188291704633449880138574352295618144620689968135011945179312768542602397199569128424773875515550}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{16} - \frac{164976584892238028456512472982684666278594926792201192830636244071374561259589995894055103326980}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{15} - \frac{88599888078802478410187033921969692143478068118297757269703932599715447449464540637913461291965}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{14} - \frac{121880221563466804079151431699009701427063245648341093069235222532976214526336697700700253054069}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{13} - \frac{81941318271658608571458668496131201727945701354398952948919292818474994546686453576129244680158}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{12} - \frac{249383017418352786772522592061053429461113148923679743231296981207693363076888158150127193016695}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{11} + \frac{234906305989263046597176384147253828329952555540654967027898876067049673005303954655683760637370}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{10} - \frac{209515848779099754033182658546322636730145809803884062189254443692886878565461436750503140611989}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{9} - \frac{97716521427489870539061009004835078212155916080169524631792950154850823044542628939329328299359}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{8} - \frac{1121073026060671750122592918449987125960429865091436014424948881296961114742459009654570364302}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{7} + \frac{11061511801650860558681468460025843260242993546563838922442353246185814672470204281124715075628}{23838853616142596257487659630266166921047344888605488557402914040049298657306290581771323573891} a^{6} + \frac{213468302573303362750604511183498598921250321911925026974683218983525567747801167805195240411433}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{5} - \frac{220125319915873732414402963608251889916144867614651706537315796466324708688478066142409000230800}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{4} + \frac{51240627549225975503782271682917750590699813429490132540151736239972864874025985043610305491766}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{3} + \frac{138029568234854447502698016752915264644031258965867694444324810974649310813369068514094719614862}{548293633171279713922216171496121839184088932437926236820267022921133869118044683380740442199493} a^{2} - \frac{5318030137288119934605120486171553462199434277074716446041158701283359859295925322430888960629}{23838853616142596257487659630266166921047344888605488557402914040049298657306290581771323573891} a + \frac{333485127903861786471416838552965212637906794022510596437930992165078636418121628328909346036}{1036471896354025924238593896968094213958580212548064719887083219132578202491577851381361894517}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{1605201}$, which has order $14446809$ (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:  $12$
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:  \( 5382739421.971964 \) (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

\(\Q(\sqrt{-11}) \), 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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ R $26$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ 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
11Data 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}$