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Properties

Label 26.0.19196410211...8128.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,2^{39}\cdot 79^{24}$
Root discriminant $159.66$
Ramified primes $2, 79$
Class number Not computed
Class group Not computed
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![8027125537, -16566954540, 21481183702, -20853231606, 17351590713, -12586842022, 8189022689, -4640948852, 2196377085, -834583714, 268110160, -71922706, 3723051, 11424264, -4196097, -1872300, 1833359, -349218, -134835, 65886, 801, -5448, 768, 178, -45, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537)
 
gp: K = bnfinit(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537, 1)
 

Normalized defining polynomial

\( x^{26} - 2 x^{25} - 45 x^{24} + 178 x^{23} + 768 x^{22} - 5448 x^{21} + 801 x^{20} + 65886 x^{19} - 134835 x^{18} - 349218 x^{17} + 1833359 x^{16} - 1872300 x^{15} - 4196097 x^{14} + 11424264 x^{13} + 3723051 x^{12} - 71922706 x^{11} + 268110160 x^{10} - 834583714 x^{9} + 2196377085 x^{8} - 4640948852 x^{7} + 8189022689 x^{6} - 12586842022 x^{5} + 17351590713 x^{4} - 20853231606 x^{3} + 21481183702 x^{2} - 16566954540 x + 8027125537 \)

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:  \(-1919641021107487828653877081110544587155912070949008048128=-\,2^{39}\cdot 79^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $159.66$
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:  $2, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(632=2^{3}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{632}(1,·)$, $\chi_{632}(259,·)$, $\chi_{632}(97,·)$, $\chi_{632}(179,·)$, $\chi_{632}(65,·)$, $\chi_{632}(275,·)$, $\chi_{632}(457,·)$, $\chi_{632}(459,·)$, $\chi_{632}(337,·)$, $\chi_{632}(403,·)$, $\chi_{632}(225,·)$, $\chi_{632}(89,·)$, $\chi_{632}(539,·)$, $\chi_{632}(289,·)$, $\chi_{632}(67,·)$, $\chi_{632}(283,·)$, $\chi_{632}(131,·)$, $\chi_{632}(561,·)$, $\chi_{632}(617,·)$, $\chi_{632}(299,·)$, $\chi_{632}(433,·)$, $\chi_{632}(563,·)$, $\chi_{632}(417,·)$, $\chi_{632}(441,·)$, $\chi_{632}(571,·)$, $\chi_{632}(475,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23} a^{20} + \frac{1}{23} a^{19} + \frac{6}{23} a^{18} + \frac{1}{23} a^{17} + \frac{8}{23} a^{16} - \frac{5}{23} a^{15} + \frac{2}{23} a^{14} - \frac{10}{23} a^{13} + \frac{10}{23} a^{12} + \frac{10}{23} a^{11} - \frac{5}{23} a^{10} + \frac{5}{23} a^{8} - \frac{11}{23} a^{7} + \frac{2}{23} a^{6} + \frac{7}{23} a^{5} - \frac{11}{23} a^{4} + \frac{6}{23} a^{3} - \frac{9}{23} a^{2} + \frac{3}{23} a - \frac{6}{23}$, $\frac{1}{23} a^{21} + \frac{5}{23} a^{19} - \frac{5}{23} a^{18} + \frac{7}{23} a^{17} + \frac{10}{23} a^{16} + \frac{7}{23} a^{15} + \frac{11}{23} a^{14} - \frac{3}{23} a^{13} + \frac{8}{23} a^{11} + \frac{5}{23} a^{10} + \frac{5}{23} a^{9} + \frac{7}{23} a^{8} - \frac{10}{23} a^{7} + \frac{5}{23} a^{6} + \frac{5}{23} a^{5} - \frac{6}{23} a^{4} + \frac{8}{23} a^{3} - \frac{11}{23} a^{2} - \frac{9}{23} a + \frac{6}{23}$, $\frac{1}{23} a^{22} - \frac{10}{23} a^{19} + \frac{5}{23} a^{17} - \frac{10}{23} a^{16} - \frac{10}{23} a^{15} + \frac{10}{23} a^{14} + \frac{4}{23} a^{13} + \frac{4}{23} a^{12} + \frac{1}{23} a^{11} + \frac{7}{23} a^{10} + \frac{7}{23} a^{9} + \frac{11}{23} a^{8} - \frac{9}{23} a^{7} - \frac{5}{23} a^{6} + \frac{5}{23} a^{5} - \frac{6}{23} a^{4} + \frac{5}{23} a^{3} - \frac{10}{23} a^{2} - \frac{9}{23} a + \frac{7}{23}$, $\frac{1}{23} a^{23} + \frac{10}{23} a^{19} - \frac{4}{23} a^{18} + \frac{1}{23} a^{16} + \frac{6}{23} a^{15} + \frac{1}{23} a^{14} - \frac{4}{23} a^{13} + \frac{9}{23} a^{12} - \frac{8}{23} a^{11} + \frac{3}{23} a^{10} + \frac{11}{23} a^{9} - \frac{5}{23} a^{8} + \frac{2}{23} a^{6} - \frac{5}{23} a^{5} + \frac{10}{23} a^{4} + \frac{4}{23} a^{3} - \frac{7}{23} a^{2} - \frac{9}{23} a + \frac{9}{23}$, $\frac{1}{1253201} a^{24} - \frac{22033}{1253201} a^{23} + \frac{26467}{1253201} a^{22} + \frac{16710}{1253201} a^{21} - \frac{22941}{1253201} a^{20} + \frac{272495}{1253201} a^{19} - \frac{637}{2369} a^{18} + \frac{469626}{1253201} a^{17} + \frac{303}{1253201} a^{16} - \frac{80824}{1253201} a^{15} + \frac{334265}{1253201} a^{14} - \frac{260376}{1253201} a^{13} + \frac{515943}{1253201} a^{12} + \frac{14986}{54487} a^{11} - \frac{383738}{1253201} a^{10} - \frac{461126}{1253201} a^{9} + \frac{130323}{1253201} a^{8} - \frac{46281}{1253201} a^{7} + \frac{430331}{1253201} a^{6} - \frac{104341}{1253201} a^{5} + \frac{89829}{1253201} a^{4} + \frac{518892}{1253201} a^{3} - \frac{10219}{54487} a^{2} - \frac{339893}{1253201} a - \frac{150560}{1253201}$, $\frac{1}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{25} - \frac{7595324819163346999347961278885364799280535574780897176577810649786865785042171778620046582321}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{24} - \frac{306811838768828407593008916301873461138992102232276790556851929333316039599646347432690669181327443}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{23} + \frac{657897696436867790071813261089445435019464831692841922672125314324307994512403424900886621162509}{231034347677722538678921771967486109974054116642348765610620798810585613028870637691357341541966151} a^{22} - \frac{6763386909401044960574464769488344766170371205181466427290440349873139129787681436178041230816262}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{21} - \frac{421633780081353399987483014247799061897828474123819733122311538413399669313522971545850255269801553}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{20} - \frac{8051125443715959060484314157391684200865915444574504032452439557748312393369449225505196562977074555}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{19} + \frac{6156520901523779262523408290999680382665148194578306787183235407803935133577964843627250257317915897}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{18} + \frac{6040660933165420472188292330694107528465814674388496927996407477863710634062692988619246587720941673}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{17} + \frac{1759328217310563034458389471279761462423871555186393584636564064134899403140005120778245576127995983}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{16} + \frac{755880517418436774345676881659567125106368266984653819879631850069732583449506877068679785738098191}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{15} + \frac{136301170708048701309533348643510530741317142846177977763649599996816239875233817109709726666219515}{1034632078730670499301258370115263883796851044093996645995388794673492093129290247052600268644457111} a^{14} - \frac{8276516153227278492088993392950204927888270798758575898302490177261921287210111783288274624080715594}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{13} - \frac{8331589040941780442372897403483367353099185603271810593070864355443795571583382642542079354782369958}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{12} + \frac{5698176803745766066732286944882793640768604200591317656591189390343144774207934647146979581883258000}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{11} - \frac{3912093318032735142843757042744540121732354428415567605727871439289228173838877434548929897772595318}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{10} + \frac{437244015442974661475191564072100364651253993735976634521874802728936872799399407853912845354345083}{1034632078730670499301258370115263883796851044093996645995388794673492093129290247052600268644457111} a^{9} + \frac{10612850118582638646785248602186444428071179747517755001336221129530839080531365663937446064589537844}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{8} + \frac{10774061566850416545037027238733968166309039479544432446049840219239328636533466208705521268794149713}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{7} - \frac{8809975910152536424667301426327703818082253728279106493651893781449835507317211665794173406268359944}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{6} - \frac{5330791028826560831209176804870309283151448973100207516507085101484306526844517083130924700902192573}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{5} - \frac{1688251195649566851744788070934932133093241752614663172413356199173886607820120562737936538468588971}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{4} + \frac{2827802203567039868058336595023459574105417129328695799317538793022543681314530203798640256655837558}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{3} + \frac{9888251975599311491227605029214175636226320892770838447855658274237291806093875727026179354763113241}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a^{2} - \frac{2413120765709046985955024639480014294956930503689959934629198863330199580369511566563977016638444766}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553} a + \frac{9955508482215288316647776146341113092970184716292992119977139764533462828981839175975054646669413124}{23796537810805421483928942512651069327327574014161922857893942277490318141973675682209806178822513553}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{-2}) \), 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 R ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ $26$ $26$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
2Data not computed
79Data not computed