↑

Properties

Label 26.0.12054663673...6811.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,11^{13}\cdot 79^{24}$
Root discriminant $187.22$
Ramified primes $11, 79$
Class number $298678381$ (GRH)
Class group $[298678381]$ (GRH)
Galois group $C_{26}$ (as 26T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25840556879, 29028971434, 36604214346, 23592642699, 18829589581, 9320051345, 6020127942, 2360295077, 1365115456, 425393325, 209322522, 26865868, 16867730, 494653, 2458481, -678796, -90833, -78793, 77627, 8273, -690, -2824, 282, 88, 21, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 + 21*x^24 + 88*x^23 + 282*x^22 - 2824*x^21 - 690*x^20 + 8273*x^19 + 77627*x^18 - 78793*x^17 - 90833*x^16 - 678796*x^15 + 2458481*x^14 + 494653*x^13 + 16867730*x^12 + 26865868*x^11 + 209322522*x^10 + 425393325*x^9 + 1365115456*x^8 + 2360295077*x^7 + 6020127942*x^6 + 9320051345*x^5 + 18829589581*x^4 + 23592642699*x^3 + 36604214346*x^2 + 29028971434*x + 25840556879)
 
gp: K = bnfinit(x^26 - 11*x^25 + 21*x^24 + 88*x^23 + 282*x^22 - 2824*x^21 - 690*x^20 + 8273*x^19 + 77627*x^18 - 78793*x^17 - 90833*x^16 - 678796*x^15 + 2458481*x^14 + 494653*x^13 + 16867730*x^12 + 26865868*x^11 + 209322522*x^10 + 425393325*x^9 + 1365115456*x^8 + 2360295077*x^7 + 6020127942*x^6 + 9320051345*x^5 + 18829589581*x^4 + 23592642699*x^3 + 36604214346*x^2 + 29028971434*x + 25840556879, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} + 21 x^{24} + 88 x^{23} + 282 x^{22} - 2824 x^{21} - 690 x^{20} + 8273 x^{19} + 77627 x^{18} - 78793 x^{17} - 90833 x^{16} - 678796 x^{15} + 2458481 x^{14} + 494653 x^{13} + 16867730 x^{12} + 26865868 x^{11} + 209322522 x^{10} + 425393325 x^{9} + 1365115456 x^{8} + 2360295077 x^{7} + 6020127942 x^{6} + 9320051345 x^{5} + 18829589581 x^{4} + 23592642699 x^{3} + 36604214346 x^{2} + 29028971434 x + 25840556879 \)

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:  \(-120546636738027837347808557425307577626634601068205941946811=-\,11^{13}\cdot 79^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $187.22$
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:  $11, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(869=11\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{869}(1,·)$, $\chi_{869}(131,·)$, $\chi_{869}(65,·)$, $\chi_{869}(10,·)$, $\chi_{869}(87,·)$, $\chi_{869}(89,·)$, $\chi_{869}(144,·)$, $\chi_{869}(210,·)$, $\chi_{869}(67,·)$, $\chi_{869}(21,·)$, $\chi_{869}(791,·)$, $\chi_{869}(857,·)$, $\chi_{869}(538,·)$, $\chi_{869}(670,·)$, $\chi_{869}(417,·)$, $\chi_{869}(100,·)$, $\chi_{869}(166,·)$, $\chi_{869}(615,·)$, $\chi_{869}(617,·)$, $\chi_{869}(362,·)$, $\chi_{869}(749,·)$, $\chi_{869}(496,·)$, $\chi_{869}(694,·)$, $\chi_{869}(441,·)$, $\chi_{869}(571,·)$, $\chi_{869}(650,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{5}{23} a^{15} + \frac{5}{23} a^{14} + \frac{1}{23} a^{13} + \frac{4}{23} a^{12} - \frac{3}{23} a^{11} - \frac{7}{23} a^{10} - \frac{3}{23} a^{9} - \frac{3}{23} a^{8} + \frac{7}{23} a^{7} - \frac{4}{23} a^{6} - \frac{2}{23} a^{5} + \frac{7}{23} a^{4} - \frac{8}{23} a^{3} - \frac{1}{23} a^{2} - \frac{2}{23} a$, $\frac{1}{23} a^{17} + \frac{3}{23} a^{15} + \frac{3}{23} a^{14} + \frac{9}{23} a^{13} - \frac{6}{23} a^{12} + \frac{1}{23} a^{11} + \frac{8}{23} a^{10} + \frac{5}{23} a^{9} - \frac{8}{23} a^{8} + \frac{8}{23} a^{7} + \frac{1}{23} a^{6} - \frac{3}{23} a^{5} + \frac{4}{23} a^{4} + \frac{5}{23} a^{3} - \frac{7}{23} a^{2} - \frac{10}{23} a$, $\frac{1}{23} a^{18} - \frac{5}{23} a^{15} - \frac{6}{23} a^{14} - \frac{9}{23} a^{13} - \frac{11}{23} a^{12} - \frac{6}{23} a^{11} + \frac{3}{23} a^{10} + \frac{1}{23} a^{9} - \frac{6}{23} a^{8} + \frac{3}{23} a^{7} + \frac{9}{23} a^{6} + \frac{10}{23} a^{5} + \frac{7}{23} a^{4} - \frac{6}{23} a^{3} - \frac{7}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{23} a^{19} - \frac{8}{23} a^{15} - \frac{7}{23} a^{14} - \frac{6}{23} a^{13} - \frac{9}{23} a^{12} + \frac{11}{23} a^{11} - \frac{11}{23} a^{10} + \frac{2}{23} a^{9} + \frac{11}{23} a^{8} - \frac{2}{23} a^{7} - \frac{10}{23} a^{6} - \frac{3}{23} a^{5} + \frac{6}{23} a^{4} - \frac{1}{23} a^{3} + \frac{1}{23} a^{2} - \frac{10}{23} a$, $\frac{1}{23} a^{20} - \frac{1}{23} a^{15} + \frac{11}{23} a^{14} - \frac{1}{23} a^{13} - \frac{3}{23} a^{12} + \frac{11}{23} a^{11} - \frac{8}{23} a^{10} + \frac{10}{23} a^{9} - \frac{3}{23} a^{8} + \frac{11}{23} a^{6} - \frac{10}{23} a^{5} + \frac{9}{23} a^{4} + \frac{6}{23} a^{3} + \frac{5}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{21} + \frac{6}{23} a^{15} + \frac{4}{23} a^{14} - \frac{2}{23} a^{13} - \frac{8}{23} a^{12} - \frac{11}{23} a^{11} + \frac{3}{23} a^{10} - \frac{6}{23} a^{9} - \frac{3}{23} a^{8} - \frac{5}{23} a^{7} + \frac{9}{23} a^{6} + \frac{7}{23} a^{5} - \frac{10}{23} a^{4} - \frac{3}{23} a^{3} + \frac{6}{23} a^{2} - \frac{2}{23} a$, $\frac{1}{529} a^{22} - \frac{9}{529} a^{21} - \frac{11}{529} a^{20} - \frac{2}{529} a^{19} - \frac{5}{529} a^{18} - \frac{1}{529} a^{16} - \frac{101}{529} a^{15} - \frac{127}{529} a^{14} + \frac{117}{529} a^{13} - \frac{206}{529} a^{12} - \frac{128}{529} a^{11} + \frac{42}{529} a^{10} - \frac{162}{529} a^{9} + \frac{15}{529} a^{8} + \frac{224}{529} a^{7} + \frac{15}{529} a^{6} - \frac{62}{529} a^{5} - \frac{39}{529} a^{4} - \frac{129}{529} a^{3} + \frac{44}{529} a^{2} + \frac{175}{529} a - \frac{7}{23}$, $\frac{1}{54487} a^{23} + \frac{20}{54487} a^{22} + \frac{740}{54487} a^{21} + \frac{576}{54487} a^{20} + \frac{29}{54487} a^{19} + \frac{522}{54487} a^{18} - \frac{737}{54487} a^{17} + \frac{445}{54487} a^{16} - \frac{2803}{54487} a^{15} + \frac{20124}{54487} a^{14} - \frac{17099}{54487} a^{13} - \frac{20454}{54487} a^{12} + \frac{19238}{54487} a^{11} + \frac{25551}{54487} a^{10} - \frac{17448}{54487} a^{9} - \frac{25055}{54487} a^{8} - \frac{13407}{54487} a^{7} + \frac{24569}{54487} a^{6} - \frac{17960}{54487} a^{5} + \frac{2995}{54487} a^{4} + \frac{17486}{54487} a^{3} - \frac{11268}{54487} a^{2} + \frac{3120}{54487} a - \frac{8}{23}$, $\frac{1}{1253201} a^{24} + \frac{10}{1253201} a^{23} - \frac{181}{1253201} a^{22} + \frac{9141}{1253201} a^{21} - \frac{14383}{1253201} a^{20} + \frac{18257}{1253201} a^{19} - \frac{26042}{1253201} a^{18} - \frac{6399}{1253201} a^{17} + \frac{1055}{54487} a^{16} - \frac{343349}{1253201} a^{15} - \frac{24905}{1253201} a^{14} + \frac{407315}{1253201} a^{13} + \frac{339138}{1253201} a^{12} + \frac{129193}{1253201} a^{11} + \frac{419305}{1253201} a^{10} + \frac{270965}{1253201} a^{9} + \frac{539036}{1253201} a^{8} + \frac{165334}{1253201} a^{7} - \frac{615601}{1253201} a^{6} - \frac{9603}{1253201} a^{5} + \frac{572370}{1253201} a^{4} - \frac{194986}{1253201} a^{3} - \frac{126765}{1253201} a^{2} - \frac{157375}{1253201} a - \frac{193}{529}$, $\frac{1}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{25} + \frac{231867383479269068716528256211213792741290081788173941984924972481707727798586933778225314}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{24} + \frac{50897839966715653562748489612969856346221218988949293240972846137325381076105200578863391012}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{23} + \frac{5680835146765187684949519923548461810275257867998898401141778435838622340351157750351303872759}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{22} - \frac{3401300660013743657234208049648524639527237608658439362846404932783743067808742191461228214812}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{21} + \frac{66675784711936767694585806785236474496646949047273978313288906175087955939736185825167369343211}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{20} - \frac{88398761818921762699958496799796098380727114002458431900800398887829794500401659092807357421647}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{19} - \frac{85107981468501357368416449208733112759876023855388358704518387755447042762650582079819061954583}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{18} - \frac{3126492927662998173119158390796993537015808392014162928587390269490311042311854898020220950142}{261699772885660788216604723604910559452382308903713094190745553988178345439289117728994484311337} a^{17} - \frac{38925594975436790002675478384665022099909887276366747094317398506083394597655725304185467844736}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{16} - \frac{2258965352328110968646593706655331874642372219158519859776114895229150495380093579486137056718082}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{15} - \frac{1641131795885079793631289494795092059170231376879837132364659525234653088033246678556891185685256}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{14} - \frac{655018763501898760858475934658834268576205746020233948370831888671771250346770548273112780674327}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{13} - \frac{847530248932909243847101804998782210088944519421467224757397111217091328419991774408692459531762}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{12} + \frac{1313353173458857928806049577108154470941653769742275959264242396906262452749899574987779084431116}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{11} + \frac{162612237349351723438609546692370434024356846611551865993475344954419207569430091466617608211071}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{10} + \frac{1042332561145148490732017026946797268940644684352475515325485028051286506615389835169887875228490}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{9} + \frac{999897358339924330505898831991955349573241348561894593913840552932808000141945560150701882908474}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{8} + \frac{960073893651948591476105060958266524220131881660589707206408501113790011684168938545247164355793}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{7} - \frac{2419115108175612491117342221022929849704273409680437220517761533561085249234464326518057658081321}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{6} + \frac{121074679028883030530268542279012068110218947876972898110248685157057344511693740031184097275786}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{5} + \frac{2467251067094782443324543957215363944973198044413424062030373837603323559432334091877242821402095}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{4} + \frac{862833811184580577169254089285255812454855015237924703342223193331614406065234425464539331834497}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{3} + \frac{633725004675319370034647890304001627420428517346383099068764549726327333099630007070062848933271}{6019094776370198128981908642912942867404793104785401166387147741728101945103649707766873139160751} a^{2} - \frac{18522683521638845051272540989035900173452566303787931617871297928566536423658056850469143732955}{261699772885660788216604723604910559452382308903713094190745553988178345439289117728994484311337} a + \frac{2523171319051493620489476138673778531324835969743772410848581494894114075835869133315962319}{110468456262414853616126941158679003567911485396248667872834763186229778572937576078089693673}$

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

Class group and class number

$C_{298678381}$, which has order $298678381$ (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:  \( 57529828940.82975 \) (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

\(\Q(\sqrt{-11}) \), 13.13.59091511031674153381441.1

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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ R $26$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
79Data not computed