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Properties

Label 26.26.1368525640...6896.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{26}\cdot 3^{13}\cdot 53^{25}$
Root discriminant $157.60$
Ramified primes $2, 3, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![-84499119, 0, 1577316888, 0, -9445123746, 0, 21559793733, 0, -23798498787, 0, 13998339648, 0, -4603753098, 0, 896494311, 0, -107513892, 0, 8070840, 0, -376353, 0, 10494, 0, -159, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 159*x^24 + 10494*x^22 - 376353*x^20 + 8070840*x^18 - 107513892*x^16 + 896494311*x^14 - 4603753098*x^12 + 13998339648*x^10 - 23798498787*x^8 + 21559793733*x^6 - 9445123746*x^4 + 1577316888*x^2 - 84499119)
 
gp: K = bnfinit(x^26 - 159*x^24 + 10494*x^22 - 376353*x^20 + 8070840*x^18 - 107513892*x^16 + 896494311*x^14 - 4603753098*x^12 + 13998339648*x^10 - 23798498787*x^8 + 21559793733*x^6 - 9445123746*x^4 + 1577316888*x^2 - 84499119, 1)
 

Normalized defining polynomial

\( x^{26} - 159 x^{24} + 10494 x^{22} - 376353 x^{20} + 8070840 x^{18} - 107513892 x^{16} + 896494311 x^{14} - 4603753098 x^{12} + 13998339648 x^{10} - 23798498787 x^{8} + 21559793733 x^{6} - 9445123746 x^{4} + 1577316888 x^{2} - 84499119 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $26$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[26, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1368525640302717774793267977670546626194406080285188816896=2^{26}\cdot 3^{13}\cdot 53^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $157.60$
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:  $2, 3, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(636=2^{2}\cdot 3\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{636}(1,·)$, $\chi_{636}(131,·)$, $\chi_{636}(577,·)$, $\chi_{636}(335,·)$, $\chi_{636}(11,·)$, $\chi_{636}(13,·)$, $\chi_{636}(493,·)$, $\chi_{636}(143,·)$, $\chi_{636}(635,·)$, $\chi_{636}(205,·)$, $\chi_{636}(515,·)$, $\chi_{636}(625,·)$, $\chi_{636}(539,·)$, $\chi_{636}(289,·)$, $\chi_{636}(587,·)$, $\chi_{636}(347,·)$, $\chi_{636}(623,·)$, $\chi_{636}(97,·)$, $\chi_{636}(169,·)$, $\chi_{636}(301,·)$, $\chi_{636}(431,·)$, $\chi_{636}(49,·)$, $\chi_{636}(467,·)$, $\chi_{636}(121,·)$, $\chi_{636}(505,·)$, $\chi_{636}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{50301} a^{14} + \frac{11}{16767} a^{12} + \frac{5}{5589} a^{10} + \frac{7}{1863} a^{8} + \frac{1}{69} a^{6} + \frac{11}{207} a^{4} + \frac{3}{23} a^{2} + \frac{1}{23}$, $\frac{1}{50301} a^{15} + \frac{11}{16767} a^{13} + \frac{5}{5589} a^{11} + \frac{7}{1863} a^{9} + \frac{1}{69} a^{7} + \frac{11}{207} a^{5} + \frac{3}{23} a^{3} + \frac{1}{23} a$, $\frac{1}{150903} a^{16} - \frac{1}{16767} a^{12} - \frac{2}{5589} a^{10} + \frac{1}{1863} a^{8} + \frac{4}{621} a^{6} + \frac{1}{69} a^{4} - \frac{2}{23} a^{2} - \frac{11}{23}$, $\frac{1}{150903} a^{17} - \frac{1}{16767} a^{13} - \frac{2}{5589} a^{11} + \frac{1}{1863} a^{9} + \frac{4}{621} a^{7} + \frac{1}{69} a^{5} - \frac{2}{23} a^{3} - \frac{11}{23} a$, $\frac{1}{452709} a^{18} + \frac{1}{1863} a^{12} + \frac{2}{1863} a^{10} + \frac{11}{1863} a^{8} - \frac{11}{621} a^{6} + \frac{5}{207} a^{4} - \frac{2}{69} a^{2} + \frac{1}{23}$, $\frac{1}{452709} a^{19} + \frac{1}{1863} a^{13} + \frac{2}{1863} a^{11} + \frac{11}{1863} a^{9} - \frac{11}{621} a^{7} + \frac{5}{207} a^{5} - \frac{2}{69} a^{3} + \frac{1}{23} a$, $\frac{1}{1358127} a^{20} - \frac{1}{16767} a^{12} - \frac{11}{5589} a^{10} - \frac{5}{1863} a^{8} - \frac{7}{621} a^{6} - \frac{1}{23} a^{4} - \frac{11}{69} a^{2} - \frac{9}{23}$, $\frac{1}{1358127} a^{21} - \frac{1}{16767} a^{13} - \frac{11}{5589} a^{11} - \frac{5}{1863} a^{9} - \frac{7}{621} a^{7} - \frac{1}{23} a^{5} - \frac{11}{69} a^{3} - \frac{9}{23} a$, $\frac{1}{4074381} a^{22} + \frac{1}{23}$, $\frac{1}{4074381} a^{23} + \frac{1}{23} a$, $\frac{1}{506929491690744115880343} a^{24} + \frac{5951976017765939}{168976497230248038626781} a^{22} + \frac{3400273862401490}{56325499076749346208927} a^{20} - \frac{9619312240139629}{18775166358916448736309} a^{18} - \frac{3891034132342681}{6258388786305482912103} a^{16} - \frac{669533537983477}{231792177270573441189} a^{14} - \frac{316074545567462860}{695376531811720323567} a^{12} + \frac{348293703142197190}{231792177270573441189} a^{10} - \frac{124477184844099589}{77264059090191147063} a^{8} + \frac{139786812110216495}{25754686363397049021} a^{6} - \frac{48544943583040178}{8584895454465683007} a^{4} + \frac{228842787930625108}{2861631818155227669} a^{2} + \frac{119300292076422685}{953877272718409223}$, $\frac{1}{506929491690744115880343} a^{25} + \frac{5951976017765939}{168976497230248038626781} a^{23} + \frac{3400273862401490}{56325499076749346208927} a^{21} - \frac{9619312240139629}{18775166358916448736309} a^{19} - \frac{3891034132342681}{6258388786305482912103} a^{17} - \frac{669533537983477}{231792177270573441189} a^{15} - \frac{316074545567462860}{695376531811720323567} a^{13} + \frac{348293703142197190}{231792177270573441189} a^{11} - \frac{124477184844099589}{77264059090191147063} a^{9} + \frac{139786812110216495}{25754686363397049021} a^{7} - \frac{48544943583040178}{8584895454465683007} a^{5} + \frac{228842787930625108}{2861631818155227669} a^{3} + \frac{119300292076422685}{953877272718409223} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $25$
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:  \( 65471383026909620000 \) (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

\(\Q(\sqrt{159}) \), 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 R R ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ R ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
3Data not computed
53Data not computed