Properties

Label 26.0.384...563.1
Degree $26$
Signature $[0, 13]$
Discriminant $-3.848\times 10^{47}$
Root discriminant $67.64$
Ramified primes $3, 53$
Class number $2809$ (GRH)
Class group $[53, 53]$ (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 + 25*x^24 - 14*x^23 + 405*x^22 - 192*x^21 + 3603*x^20 - 1111*x^19 + 22650*x^18 - 5414*x^17 + 87624*x^16 - 5096*x^15 + 234451*x^14 - 19756*x^13 + 367484*x^12 - 30562*x^11 + 404986*x^10 - 57146*x^9 + 232119*x^8 - 34742*x^7 + 91429*x^6 - 16226*x^5 + 7319*x^4 + 98*x^3 + 168*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^26 - x^25 + 25*x^24 - 14*x^23 + 405*x^22 - 192*x^21 + 3603*x^20 - 1111*x^19 + 22650*x^18 - 5414*x^17 + 87624*x^16 - 5096*x^15 + 234451*x^14 - 19756*x^13 + 367484*x^12 - 30562*x^11 + 404986*x^10 - 57146*x^9 + 232119*x^8 - 34742*x^7 + 91429*x^6 - 16226*x^5 + 7319*x^4 + 98*x^3 + 168*x^2 - 10*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 168, 98, 7319, -16226, 91429, -34742, 232119, -57146, 404986, -30562, 367484, -19756, 234451, -5096, 87624, -5414, 22650, -1111, 3603, -192, 405, -14, 25, -1, 1]);
 

\( x^{26} - x^{25} + 25 x^{24} - 14 x^{23} + 405 x^{22} - 192 x^{21} + 3603 x^{20} - 1111 x^{19} + 22650 x^{18} - 5414 x^{17} + 87624 x^{16} - 5096 x^{15} + 234451 x^{14} - 19756 x^{13} + 367484 x^{12} - 30562 x^{11} + 404986 x^{10} - 57146 x^{9} + 232119 x^{8} - 34742 x^{7} + 91429 x^{6} - 16226 x^{5} + 7319 x^{4} + 98 x^{3} + 168 x^{2} - 10 x + 1 \)

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:  \(-384766437057818380952237905666104641217782272563\)\(\medspace = -\,3^{13}\cdot 53^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $67.64$
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$$\chi_{159}(1,·)$, $\chi_{159}(130,·)$, $\chi_{159}(68,·)$, $\chi_{159}(134,·)$, $\chi_{159}(10,·)$, $\chi_{159}(13,·)$, $\chi_{159}(142,·)$, $\chi_{159}(77,·)$, $\chi_{159}(16,·)$, $\chi_{159}(148,·)$, $\chi_{159}(152,·)$, $\chi_{159}(89,·)$, $\chi_{159}(155,·)$, $\chi_{159}(28,·)$, $\chi_{159}(95,·)$, $\chi_{159}(97,·)$, $\chi_{159}(100,·)$, $\chi_{159}(107,·)$, $\chi_{159}(44,·)$, $\chi_{159}(46,·)$, $\chi_{159}(47,·)$, $\chi_{159}(49,·)$, $\chi_{159}(116,·)$, $\chi_{159}(119,·)$, $\chi_{159}(121,·)$, $\chi_{159}(122,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22} - \frac{3}{23} a^{21} - \frac{8}{23} a^{20} + \frac{5}{23} a^{19} + \frac{8}{23} a^{18} - \frac{2}{23} a^{17} - \frac{10}{23} a^{16} - \frac{10}{23} a^{14} - \frac{10}{23} a^{13} + \frac{4}{23} a^{12} - \frac{9}{23} a^{11} - \frac{7}{23} a^{10} - \frac{7}{23} a^{9} - \frac{9}{23} a^{8} - \frac{2}{23} a^{7} + \frac{1}{23} a^{6} + \frac{3}{23} a^{5} + \frac{10}{23} a^{4} + \frac{2}{23} a^{3} - \frac{2}{23} a^{2} - \frac{1}{23} a + \frac{4}{23}$, $\frac{1}{23} a^{23} + \frac{6}{23} a^{21} + \frac{4}{23} a^{20} - \frac{1}{23} a^{18} + \frac{7}{23} a^{17} - \frac{7}{23} a^{16} - \frac{10}{23} a^{15} + \frac{6}{23} a^{14} - \frac{3}{23} a^{13} + \frac{3}{23} a^{12} - \frac{11}{23} a^{11} - \frac{5}{23} a^{10} - \frac{7}{23} a^{9} - \frac{6}{23} a^{8} - \frac{5}{23} a^{7} + \frac{6}{23} a^{6} - \frac{4}{23} a^{5} + \frac{9}{23} a^{4} + \frac{4}{23} a^{3} - \frac{7}{23} a^{2} + \frac{1}{23} a - \frac{11}{23}$, $\frac{1}{435105007} a^{24} - \frac{6511450}{435105007} a^{23} + \frac{6511474}{435105007} a^{22} - \frac{21353795}{435105007} a^{21} - \frac{29492337}{435105007} a^{20} + \frac{7684633}{435105007} a^{19} + \frac{170977751}{435105007} a^{18} + \frac{23875413}{435105007} a^{17} - \frac{175808914}{435105007} a^{16} + \frac{163443991}{435105007} a^{15} - \frac{35916163}{435105007} a^{14} - \frac{533332}{435105007} a^{13} - \frac{15687192}{435105007} a^{12} + \frac{138182201}{435105007} a^{11} - \frac{189374544}{435105007} a^{10} - \frac{197759169}{435105007} a^{9} - \frac{53900542}{435105007} a^{8} - \frac{11497280}{435105007} a^{7} - \frac{7263133}{18917609} a^{6} + \frac{179207798}{435105007} a^{5} + \frac{25522905}{435105007} a^{4} + \frac{11250547}{435105007} a^{3} + \frac{131851964}{435105007} a^{2} + \frac{183386491}{435105007} a + \frac{16110395}{435105007}$, $\frac{1}{1365220377829828203115595168074635041613268941514172023597} a^{25} - \frac{406017783584370443327565549872701562558026047166}{1365220377829828203115595168074635041613268941514172023597} a^{24} - \frac{6904047626912328110876965887822271084240905796235622477}{1365220377829828203115595168074635041613268941514172023597} a^{23} + \frac{29323593610972396322021368460009649846832269138918548607}{1365220377829828203115595168074635041613268941514172023597} a^{22} - \frac{135515756698165995059639584286112323307037142623889494750}{1365220377829828203115595168074635041613268941514172023597} a^{21} - \frac{615107161698357627770812767700755334699834287636261025630}{1365220377829828203115595168074635041613268941514172023597} a^{20} - \frac{672829308607094425937406720507672630890146566540313355588}{1365220377829828203115595168074635041613268941514172023597} a^{19} - \frac{625358850504638576360859254186594649583369533651686212516}{1365220377829828203115595168074635041613268941514172023597} a^{18} - \frac{375407253104663712206007379032769987991042808218546838557}{1365220377829828203115595168074635041613268941514172023597} a^{17} + \frac{43394179261886025017011280587476349277189074768710654121}{1365220377829828203115595168074635041613268941514172023597} a^{16} - \frac{67072177062589755015214068849110262202459600684959330532}{1365220377829828203115595168074635041613268941514172023597} a^{15} - \frac{510707022786953772768524893150405047309188953767226847425}{1365220377829828203115595168074635041613268941514172023597} a^{14} + \frac{387223318185388271085633136514536336441982591853991530596}{1365220377829828203115595168074635041613268941514172023597} a^{13} + \frac{633558940259632069154825855282436871355948815855755359981}{1365220377829828203115595168074635041613268941514172023597} a^{12} - \frac{196836180870030280184716713104369669807561364932637929140}{1365220377829828203115595168074635041613268941514172023597} a^{11} + \frac{657151474320215300207387574591012630778368031730353478772}{1365220377829828203115595168074635041613268941514172023597} a^{10} - \frac{3060420176995503207016226300866757003443141618130264093}{16448438287106363892958977928610060742328541464026168959} a^{9} + \frac{68423659877116820840700539973282086681663002948591630840}{1365220377829828203115595168074635041613268941514172023597} a^{8} - \frac{668893522721584913009825686710269382742777797716759076844}{1365220377829828203115595168074635041613268941514172023597} a^{7} - \frac{525981852067395271039494859393273520073210674999992601111}{1365220377829828203115595168074635041613268941514172023597} a^{6} + \frac{159338131030312929103581753969277716380461935302785354290}{1365220377829828203115595168074635041613268941514172023597} a^{5} - \frac{437679221508859436159799048335632797726365118257863623622}{1365220377829828203115595168074635041613268941514172023597} a^{4} + \frac{501670577804556117310984972579246828435617680984835450807}{1365220377829828203115595168074635041613268941514172023597} a^{3} + \frac{450642196253928842624086071895904658178616299140713306799}{1365220377829828203115595168074635041613268941514172023597} a^{2} + \frac{378745931654771918228547160213929003447539766601602744316}{1365220377829828203115595168074635041613268941514172023597} a - \frac{325941543781279299332382160781623535381532700389688515136}{1365220377829828203115595168074635041613268941514172023597}$

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

Class group and class number

$C_{53}\times C_{53}$, which has order $2809$ (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:  \( -\frac{229987870023954017059236497189737894134743541980}{3137680228602444474088975878125518885635965438371} a^{25} + \frac{217136480836551540404588632151115363363948328202}{3137680228602444474088975878125518885635965438371} a^{24} - \frac{5734368197620153028358493469558145698063203294598}{3137680228602444474088975878125518885635965438371} a^{23} + \frac{2895769853953695503445872483572537182694702551756}{3137680228602444474088975878125518885635965438371} a^{22} - \frac{92903090782795538430449179718308485758584153051060}{3137680228602444474088975878125518885635965438371} a^{21} + \frac{38910900127714861412854367362364439244898433859460}{3137680228602444474088975878125518885635965438371} a^{20} - \frac{825175169192090249025432598882797587021843700983579}{3137680228602444474088975878125518885635965438371} a^{19} + \frac{208619354791426775899170622718184951223205376339973}{3137680228602444474088975878125518885635965438371} a^{18} - \frac{5186025624562473117779830913522560002009908665030605}{3137680228602444474088975878125518885635965438371} a^{17} + \frac{950284046221967557819071670013626109187374876093148}{3137680228602444474088975878125518885635965438371} a^{16} - \frac{20026979216450597144491354405047755726381392519412945}{3137680228602444474088975878125518885635965438371} a^{15} + \frac{26026733398959552629061003677239193487412451502180}{3137680228602444474088975878125518885635965438371} a^{14} - \frac{53640081111470508555505879268919239920346609308326833}{3137680228602444474088975878125518885635965438371} a^{13} + \frac{1493283954000585215753084023395936297151511603120853}{3137680228602444474088975878125518885635965438371} a^{12} - \frac{83693528184162061027106813090510590280448512245797904}{3137680228602444474088975878125518885635965438371} a^{11} + \frac{2190262566064004435436005680569916912366060961235047}{3137680228602444474088975878125518885635965438371} a^{10} - \frac{91866436702243517298271969914984914726430188109122519}{3137680228602444474088975878125518885635965438371} a^{9} + \frac{7761435268729161722573189589323550986192528410571847}{3137680228602444474088975878125518885635965438371} a^{8} - \frac{51689186418254482677926859052944291289597257812419914}{3137680228602444474088975878125518885635965438371} a^{7} + \frac{4755838253545939984437862860116971870622966564403172}{3137680228602444474088975878125518885635965438371} a^{6} - \frac{20044193060199090632308192233670562915612723850599729}{3137680228602444474088975878125518885635965438371} a^{5} + \frac{2415825135601289194505318683440073804435597153774499}{3137680228602444474088975878125518885635965438371} a^{4} - \frac{1267780444146117601025600216051378277239936331305850}{3137680228602444474088975878125518885635965438371} a^{3} - \frac{180775132780258128332551968251888391460806926920950}{3137680228602444474088975878125518885635965438371} a^{2} - \frac{28118311719579089033381263797378756317127766600308}{3137680228602444474088975878125518885635965438371} a + \frac{1646897395182689836224362555359664278287504292319}{3137680228602444474088975878125518885635965438371} \) (order $6$)
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

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 2809}{6\sqrt{384766437057818380952237905666104641217782272563}}\approx 0.0966370230307511$ (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

\(\Q(\sqrt{-3}) \), 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 $26$ R $26$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ R $26$

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

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