Properties

Label 26.0.275...999.1
Degree $26$
Signature $[0, 13]$
Discriminant $-2.759\times 10^{47}$
Root discriminant $66.78$
Ramified prime $79$
Class number $265$ (GRH)
Class group $[265]$ (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 + 2*x^24 + 48*x^23 - 41*x^22 + 75*x^21 + 827*x^20 - 597*x^19 + 990*x^18 + 6414*x^17 - 3755*x^16 + 5397*x^15 + 23537*x^14 - 10353*x^13 + 15676*x^12 + 35758*x^11 - 6997*x^10 + 39165*x^9 + 14468*x^8 + 8338*x^7 + 43270*x^6 - 25107*x^5 + 25775*x^4 - 10788*x^3 - 8746*x^2 + 10048*x + 4289)
 
gp: K = bnfinit(x^26 - x^25 + 2*x^24 + 48*x^23 - 41*x^22 + 75*x^21 + 827*x^20 - 597*x^19 + 990*x^18 + 6414*x^17 - 3755*x^16 + 5397*x^15 + 23537*x^14 - 10353*x^13 + 15676*x^12 + 35758*x^11 - 6997*x^10 + 39165*x^9 + 14468*x^8 + 8338*x^7 + 43270*x^6 - 25107*x^5 + 25775*x^4 - 10788*x^3 - 8746*x^2 + 10048*x + 4289, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4289, 10048, -8746, -10788, 25775, -25107, 43270, 8338, 14468, 39165, -6997, 35758, 15676, -10353, 23537, 5397, -3755, 6414, 990, -597, 827, 75, -41, 48, 2, -1, 1]);
 

\( x^{26} - x^{25} + 2 x^{24} + 48 x^{23} - 41 x^{22} + 75 x^{21} + 827 x^{20} - 597 x^{19} + 990 x^{18} + 6414 x^{17} - 3755 x^{16} + 5397 x^{15} + 23537 x^{14} - 10353 x^{13} + 15676 x^{12} + 35758 x^{11} - 6997 x^{10} + 39165 x^{9} + 14468 x^{8} + 8338 x^{7} + 43270 x^{6} - 25107 x^{5} + 25775 x^{4} - 10788 x^{3} - 8746 x^{2} + 10048 x + 4289 \)

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:  \(-275852727404510985186163978883510976421015681999\)\(\medspace = -\,79^{25}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $66.78$
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:  $79$
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:  \(79\)
Dirichlet character group:    $\lbrace$$\chi_{79}(64,·)$, $\chi_{79}(1,·)$, $\chi_{79}(67,·)$, $\chi_{79}(69,·)$, $\chi_{79}(65,·)$, $\chi_{79}(8,·)$, $\chi_{79}(10,·)$, $\chi_{79}(12,·)$, $\chi_{79}(14,·)$, $\chi_{79}(15,·)$, $\chi_{79}(17,·)$, $\chi_{79}(18,·)$, $\chi_{79}(21,·)$, $\chi_{79}(22,·)$, $\chi_{79}(27,·)$, $\chi_{79}(33,·)$, $\chi_{79}(38,·)$, $\chi_{79}(41,·)$, $\chi_{79}(71,·)$, $\chi_{79}(78,·)$, $\chi_{79}(46,·)$, $\chi_{79}(52,·)$, $\chi_{79}(57,·)$, $\chi_{79}(58,·)$, $\chi_{79}(61,·)$, $\chi_{79}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23} a^{17} - \frac{7}{23} a^{15} - \frac{8}{23} a^{14} - \frac{2}{23} a^{13} - \frac{10}{23} a^{12} - \frac{2}{23} a^{11} - \frac{9}{23} a^{10} - \frac{1}{23} a^{9} + \frac{5}{23} a^{8} - \frac{2}{23} a^{7} + \frac{9}{23} a^{6} - \frac{4}{23} a^{5} + \frac{4}{23} a^{4} - \frac{4}{23} a^{2} - \frac{6}{23} a - \frac{10}{23}$, $\frac{1}{23} a^{18} - \frac{7}{23} a^{16} - \frac{8}{23} a^{15} - \frac{2}{23} a^{14} - \frac{10}{23} a^{13} - \frac{2}{23} a^{12} - \frac{9}{23} a^{11} - \frac{1}{23} a^{10} + \frac{5}{23} a^{9} - \frac{2}{23} a^{8} + \frac{9}{23} a^{7} - \frac{4}{23} a^{6} + \frac{4}{23} a^{5} - \frac{4}{23} a^{3} - \frac{6}{23} a^{2} - \frac{10}{23} a$, $\frac{1}{23} a^{19} - \frac{8}{23} a^{16} - \frac{5}{23} a^{15} + \frac{3}{23} a^{14} + \frac{7}{23} a^{13} - \frac{10}{23} a^{12} + \frac{8}{23} a^{11} + \frac{11}{23} a^{10} - \frac{9}{23} a^{9} - \frac{2}{23} a^{8} + \frac{5}{23} a^{7} - \frac{2}{23} a^{6} - \frac{5}{23} a^{5} + \frac{1}{23} a^{4} - \frac{6}{23} a^{3} + \frac{8}{23} a^{2} + \frac{4}{23} a - \frac{1}{23}$, $\frac{1}{23} a^{20} - \frac{5}{23} a^{16} - \frac{7}{23} a^{15} - \frac{11}{23} a^{14} - \frac{3}{23} a^{13} - \frac{3}{23} a^{12} - \frac{5}{23} a^{11} + \frac{11}{23} a^{10} - \frac{10}{23} a^{9} - \frac{1}{23} a^{8} + \frac{5}{23} a^{7} - \frac{2}{23} a^{6} - \frac{8}{23} a^{5} + \frac{3}{23} a^{4} + \frac{8}{23} a^{3} - \frac{5}{23} a^{2} - \frac{3}{23} a - \frac{11}{23}$, $\frac{1}{23} a^{21} - \frac{7}{23} a^{16} + \frac{3}{23} a^{14} + \frac{10}{23} a^{13} - \frac{9}{23} a^{12} + \frac{1}{23} a^{11} - \frac{9}{23} a^{10} - \frac{6}{23} a^{9} + \frac{7}{23} a^{8} + \frac{11}{23} a^{7} - \frac{9}{23} a^{6} + \frac{6}{23} a^{5} + \frac{5}{23} a^{4} - \frac{5}{23} a^{3} + \frac{5}{23} a - \frac{4}{23}$, $\frac{1}{23} a^{22} - \frac{1}{23}$, $\frac{1}{23} a^{23} - \frac{1}{23} a$, $\frac{1}{529} a^{24} - \frac{3}{529} a^{23} - \frac{8}{529} a^{22} - \frac{3}{529} a^{21} + \frac{1}{529} a^{20} + \frac{6}{529} a^{19} - \frac{6}{529} a^{18} + \frac{9}{529} a^{17} + \frac{217}{529} a^{16} + \frac{201}{529} a^{15} - \frac{246}{529} a^{14} - \frac{202}{529} a^{13} + \frac{70}{529} a^{12} + \frac{99}{529} a^{11} + \frac{190}{529} a^{10} + \frac{260}{529} a^{9} + \frac{2}{23} a^{8} + \frac{206}{529} a^{7} - \frac{250}{529} a^{6} - \frac{93}{529} a^{5} - \frac{177}{529} a^{4} - \frac{242}{529} a^{3} + \frac{19}{529} a^{2} + \frac{199}{529} a + \frac{235}{529}$, $\frac{1}{36432810425294767351714415418299842796503405351772669} a^{25} - \frac{17839857856621827873061919016957308631878955352041}{36432810425294767351714415418299842796503405351772669} a^{24} + \frac{440145998647082491741624978150882103033417228436444}{36432810425294767351714415418299842796503405351772669} a^{23} - \frac{7129346072081157614126464182607660028638853592258}{36432810425294767351714415418299842796503405351772669} a^{22} - \frac{633417308567108127494434948438699468463256059499958}{36432810425294767351714415418299842796503405351772669} a^{21} - \frac{427112901546216272995042014816696727630945672693889}{36432810425294767351714415418299842796503405351772669} a^{20} + \frac{447798965901514624635926441593782881627315954212004}{36432810425294767351714415418299842796503405351772669} a^{19} - \frac{439394605514417214247918863342591219200972610033569}{36432810425294767351714415418299842796503405351772669} a^{18} + \frac{491109329063796105314881888138274513552389182356292}{36432810425294767351714415418299842796503405351772669} a^{17} + \frac{8724560591415570329195561717309076957811567098896631}{36432810425294767351714415418299842796503405351772669} a^{16} + \frac{7071518812524006663579908388735472057691166638022678}{36432810425294767351714415418299842796503405351772669} a^{15} - \frac{7339095751398517139537744989791783804973113754123829}{36432810425294767351714415418299842796503405351772669} a^{14} - \frac{9292828366718200822278380328065407988290083669738782}{36432810425294767351714415418299842796503405351772669} a^{13} + \frac{16907556580108194542622990709052254475173712866041237}{36432810425294767351714415418299842796503405351772669} a^{12} + \frac{15450502886024479287848744756723832895361767226614390}{36432810425294767351714415418299842796503405351772669} a^{11} + \frac{9215466889122503292219423966592992253708970192713592}{36432810425294767351714415418299842796503405351772669} a^{10} - \frac{60603329868202371124991734801813227732741566021967}{201286245443617499180742626620441120422670747799849} a^{9} + \frac{11795635643733095308586680141355747676319555391573418}{36432810425294767351714415418299842796503405351772669} a^{8} - \frac{10219699546671010514304307820280883967793345974041303}{36432810425294767351714415418299842796503405351772669} a^{7} + \frac{9119792296009641541272570166263836823925601506926547}{36432810425294767351714415418299842796503405351772669} a^{6} - \frac{4449323217715615151074512330648758920635445866560267}{36432810425294767351714415418299842796503405351772669} a^{5} - \frac{1105232197434425023488823310473678647973298608928299}{36432810425294767351714415418299842796503405351772669} a^{4} + \frac{16138364221509949153371248887137094214361821414353125}{36432810425294767351714415418299842796503405351772669} a^{3} - \frac{2111627529911607880355394610566290018377151170188012}{36432810425294767351714415418299842796503405351772669} a^{2} + \frac{9406461793776214260150072643657474835663067760117159}{36432810425294767351714415418299842796503405351772669} a + \frac{2014414019775677796445492827843631830157997776164}{8494476667123983994337704690673780087783493903421}$

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

Class group and class number

$C_{265}$, which has order $265$ (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:  \( -1 \) (order $2$)
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:  \( 57529828940.82975 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 57529828940.82975 \cdot 265}{2\sqrt{275852727404510985186163978883510976421015681999}}\approx 0.345230063779195$ (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

\(\Q(\sqrt{-79}) \), 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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ $26$ $26$ $26$ $26$ $26$

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
79Data not computed