↑

Properties

Label 26.0.85463837848...3251.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,131^{25}$
Root discriminant $108.60$
Ramified prime $131$
Class number $7155$ (GRH)
Class group $[3, 3, 795]$ (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![4461913, 472321, 1868225, -4829444, 3599581, -2783945, 4162585, -3344770, 3177833, -410278, 163498, -1088095, 143672, 424503, -140905, 35200, 26193, -12939, -4713, -1660, 172, 444, 44, -11, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 + 3*x^24 - 11*x^23 + 44*x^22 + 444*x^21 + 172*x^20 - 1660*x^19 - 4713*x^18 - 12939*x^17 + 26193*x^16 + 35200*x^15 - 140905*x^14 + 424503*x^13 + 143672*x^12 - 1088095*x^11 + 163498*x^10 - 410278*x^9 + 3177833*x^8 - 3344770*x^7 + 4162585*x^6 - 2783945*x^5 + 3599581*x^4 - 4829444*x^3 + 1868225*x^2 + 472321*x + 4461913)
 
gp: K = bnfinit(x^26 - x^25 + 3*x^24 - 11*x^23 + 44*x^22 + 444*x^21 + 172*x^20 - 1660*x^19 - 4713*x^18 - 12939*x^17 + 26193*x^16 + 35200*x^15 - 140905*x^14 + 424503*x^13 + 143672*x^12 - 1088095*x^11 + 163498*x^10 - 410278*x^9 + 3177833*x^8 - 3344770*x^7 + 4162585*x^6 - 2783945*x^5 + 3599581*x^4 - 4829444*x^3 + 1868225*x^2 + 472321*x + 4461913, 1)
 

Normalized defining polynomial

\( x^{26} - x^{25} + 3 x^{24} - 11 x^{23} + 44 x^{22} + 444 x^{21} + 172 x^{20} - 1660 x^{19} - 4713 x^{18} - 12939 x^{17} + 26193 x^{16} + 35200 x^{15} - 140905 x^{14} + 424503 x^{13} + 143672 x^{12} - 1088095 x^{11} + 163498 x^{10} - 410278 x^{9} + 3177833 x^{8} - 3344770 x^{7} + 4162585 x^{6} - 2783945 x^{5} + 3599581 x^{4} - 4829444 x^{3} + 1868225 x^{2} + 472321 x + 4461913 \)

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:  \(-85463837848355618843993505004759251339989023133673251=-\,131^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.60$
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:  $131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(131\)
Dirichlet character group:    $\lbrace$$\chi_{131}(1,·)$, $\chi_{131}(130,·)$, $\chi_{131}(68,·)$, $\chi_{131}(69,·)$, $\chi_{131}(71,·)$, $\chi_{131}(79,·)$, $\chi_{131}(80,·)$, $\chi_{131}(18,·)$, $\chi_{131}(19,·)$, $\chi_{131}(84,·)$, $\chi_{131}(86,·)$, $\chi_{131}(24,·)$, $\chi_{131}(92,·)$, $\chi_{131}(32,·)$, $\chi_{131}(99,·)$, $\chi_{131}(39,·)$, $\chi_{131}(107,·)$, $\chi_{131}(45,·)$, $\chi_{131}(47,·)$, $\chi_{131}(112,·)$, $\chi_{131}(113,·)$, $\chi_{131}(51,·)$, $\chi_{131}(52,·)$, $\chi_{131}(60,·)$, $\chi_{131}(62,·)$, $\chi_{131}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{89} a^{21} + \frac{40}{89} a^{20} - \frac{36}{89} a^{19} + \frac{31}{89} a^{18} + \frac{7}{89} a^{17} - \frac{18}{89} a^{16} + \frac{13}{89} a^{15} - \frac{5}{89} a^{14} + \frac{22}{89} a^{13} - \frac{24}{89} a^{12} + \frac{44}{89} a^{11} + \frac{35}{89} a^{10} - \frac{20}{89} a^{9} + \frac{30}{89} a^{8} + \frac{14}{89} a^{7} - \frac{41}{89} a^{6} - \frac{5}{89} a^{5} - \frac{21}{89} a^{4} - \frac{2}{89} a^{3} + \frac{10}{89} a^{2} + \frac{10}{89} a - \frac{32}{89}$, $\frac{1}{89} a^{22} - \frac{34}{89} a^{20} - \frac{42}{89} a^{19} + \frac{13}{89} a^{18} - \frac{31}{89} a^{17} + \frac{21}{89} a^{16} + \frac{9}{89} a^{15} + \frac{44}{89} a^{14} - \frac{14}{89} a^{13} + \frac{25}{89} a^{12} - \frac{34}{89} a^{11} + \frac{4}{89} a^{10} + \frac{29}{89} a^{9} - \frac{29}{89} a^{8} + \frac{22}{89} a^{7} + \frac{33}{89} a^{6} + \frac{1}{89} a^{5} + \frac{37}{89} a^{4} + \frac{1}{89} a^{3} - \frac{34}{89} a^{2} + \frac{13}{89} a + \frac{34}{89}$, $\frac{1}{287737} a^{23} + \frac{1093}{287737} a^{22} + \frac{529}{287737} a^{21} + \frac{2938}{287737} a^{20} + \frac{56036}{287737} a^{19} + \frac{30474}{287737} a^{18} + \frac{126808}{287737} a^{17} - \frac{71544}{287737} a^{16} - \frac{2825}{287737} a^{15} + \frac{79617}{287737} a^{14} + \frac{25411}{287737} a^{13} - \frac{88838}{287737} a^{12} - \frac{50389}{287737} a^{11} + \frac{138382}{287737} a^{10} + \frac{113947}{287737} a^{9} + \frac{33364}{287737} a^{8} - \frac{51699}{287737} a^{7} - \frac{71563}{287737} a^{6} + \frac{125496}{287737} a^{5} + \frac{109876}{287737} a^{4} + \frac{90624}{287737} a^{3} + \frac{432}{287737} a^{2} - \frac{110512}{287737} a - \frac{15386}{287737}$, $\frac{1}{287737} a^{24} - \frac{1143}{287737} a^{22} + \frac{215}{287737} a^{21} + \frac{19608}{287737} a^{20} + \frac{32311}{287737} a^{19} - \frac{120616}{287737} a^{18} - \frac{135405}{287737} a^{17} - \frac{50299}{287737} a^{16} + \frac{92759}{287737} a^{15} + \frac{23458}{287737} a^{14} + \frac{34496}{287737} a^{13} - \frac{18047}{287737} a^{12} - \frac{22246}{287737} a^{11} + \frac{92462}{287737} a^{10} - \frac{140430}{287737} a^{9} - \frac{43845}{287737} a^{8} + \frac{100419}{287737} a^{7} + \frac{27663}{287737} a^{6} - \frac{52411}{287737} a^{5} + \frac{98873}{287737} a^{4} - \frac{28043}{287737} a^{3} + \frac{2485}{287737} a^{2} - \frac{104407}{287737} a + \frac{118453}{287737}$, $\frac{1}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{25} - \frac{830336118910176547974896943481024020797143984452104209275792799008211334073325875}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{24} - \frac{1040116654196085023778025074489759777851153529798379848763418606851245486476197016}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{23} + \frac{110499240603326547332169129543623846620452606725396469661987831509074399727814349233}{29856193517504457330916796573210200384342778112417370913815524578222872367446671217223} a^{22} - \frac{3534921753616895957746736737782538700508700842487218870693634778127187484942303694427}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{21} + \frac{322622592716428249525163795612010984792599820070786482628536451209121430099626510795680}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{20} + \frac{673772882677606494714141236960939931199089442321546710083979293953845343382790365669542}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{19} - \frac{322418973640401738678883482386201398037573653697883812081602255472569716411986144090040}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{18} - \frac{767650876464099465520480387536209388138692169915966600669441229799327700093238981807990}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{17} - \frac{283171544982565953709137682547982492588258687572630051166323898867612818588555230191942}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{16} - \frac{677102188986653336996530712688291902973924917471037589786664480532222402473615819598656}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{15} + \frac{405163668414011024616487828130735074551998183286976850586202649757156688031374864190126}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{14} - \frac{681965262105319699742932454593262607657444231336245861263943201455201549770510673820113}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{13} + \frac{14312074953919946228483924327742231953186644364451005800623691747365563885521358248228}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{12} - \frac{587652649351326966378374201292007110984193297161961074266956875130810840272849383995500}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{11} + \frac{228442824194248902404407550271863691516777368325287264351598421195921140499441236728783}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{10} - \frac{688560577868228366843066407948867804989039601528676786620572282219415056750326246440058}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{9} - \frac{620266157712702662119117241416542824042642737018581753012941500015677555907964221488032}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{8} + \frac{20481660829403521846278207193469275205364269451501095211250541969859462866697671333132}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{7} - \frac{616769151923867251064156578102745901355224866609521742279089489804828185201576092226813}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{6} - \frac{765113862545824097129448507974889572448444974773886152810608183575406597357451761602828}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{5} - \frac{276933321088833388370406330099776138692187821237367356280338205594956071177875006262542}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{4} + \frac{785948314035410867072039876897910213109600743559079222020427444573593847222553985219075}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{3} - \frac{196334676936740903247795974730043804576270558720318149305792562645944208696050175310343}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a^{2} + \frac{208612945965109222998506986116553413744047130793139596800494760924170947902585213557738}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819} a + \frac{448563092173536161928247211797240664391676292844424371124720776692476558298731425449855}{1582378256427736238538590218380140620370167239958120658432222802645812235474673574512819}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{795}$, which has order $7155$ (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:  \( 1197545162478.713 \) (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

\(\Q(\sqrt{-131}) \), 13.13.25542038069936263923006961.1

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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ $26$ $26$ $26$ $26$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
131Data not computed