↑

Properties

Label 26.0.85837414394...2752.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,2^{26}\cdot 53^{25}$
Root discriminant $90.99$
Ramified primes $2, 53$
Class number $195474$ (GRH)
Class group $[195474]$ (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![53, 0, 2968, 0, 53318, 0, 365117, 0, 1209089, 0, 2133568, 0, 2105054, 0, 1229759, 0, 442444, 0, 99640, 0, 13939, 0, 1166, 0, 53, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 + 53*x^24 + 1166*x^22 + 13939*x^20 + 99640*x^18 + 442444*x^16 + 1229759*x^14 + 2105054*x^12 + 2133568*x^10 + 1209089*x^8 + 365117*x^6 + 53318*x^4 + 2968*x^2 + 53)
 
gp: K = bnfinit(x^26 + 53*x^24 + 1166*x^22 + 13939*x^20 + 99640*x^18 + 442444*x^16 + 1229759*x^14 + 2105054*x^12 + 2133568*x^10 + 1209089*x^8 + 365117*x^6 + 53318*x^4 + 2968*x^2 + 53, 1)
 

Normalized defining polynomial

\( x^{26} + 53 x^{24} + 1166 x^{22} + 13939 x^{20} + 99640 x^{18} + 442444 x^{16} + 1229759 x^{14} + 2105054 x^{12} + 2133568 x^{10} + 1209089 x^{8} + 365117 x^{6} + 53318 x^{4} + 2968 x^{2} + 53 \)

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:  \(-858374143948696578292647084480714777491390439882752=-\,2^{26}\cdot 53^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.99$
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:  $2, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(212=2^{2}\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{212}(123,·)$, $\chi_{212}(1,·)$, $\chi_{212}(131,·)$, $\chi_{212}(69,·)$, $\chi_{212}(7,·)$, $\chi_{212}(201,·)$, $\chi_{212}(11,·)$, $\chi_{212}(13,·)$, $\chi_{212}(77,·)$, $\chi_{212}(143,·)$, $\chi_{212}(81,·)$, $\chi_{212}(205,·)$, $\chi_{212}(211,·)$, $\chi_{212}(89,·)$, $\chi_{212}(153,·)$, $\chi_{212}(91,·)$, $\chi_{212}(135,·)$, $\chi_{212}(199,·)$, $\chi_{212}(97,·)$, $\chi_{212}(163,·)$, $\chi_{212}(169,·)$, $\chi_{212}(43,·)$, $\chi_{212}(49,·)$, $\chi_{212}(115,·)$, $\chi_{212}(121,·)$, $\chi_{212}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} - \frac{11}{23} a^{12} + \frac{5}{23} a^{10} - \frac{7}{23} a^{8} + \frac{9}{23} a^{6} - \frac{11}{23} a^{4} + \frac{9}{23} a^{2} - \frac{1}{23}$, $\frac{1}{23} a^{15} - \frac{11}{23} a^{13} + \frac{5}{23} a^{11} - \frac{7}{23} a^{9} + \frac{9}{23} a^{7} - \frac{11}{23} a^{5} + \frac{9}{23} a^{3} - \frac{1}{23} a$, $\frac{1}{23} a^{16} - \frac{1}{23} a^{12} + \frac{2}{23} a^{10} + \frac{1}{23} a^{8} - \frac{4}{23} a^{6} + \frac{3}{23} a^{4} + \frac{6}{23} a^{2} - \frac{11}{23}$, $\frac{1}{23} a^{17} - \frac{1}{23} a^{13} + \frac{2}{23} a^{11} + \frac{1}{23} a^{9} - \frac{4}{23} a^{7} + \frac{3}{23} a^{5} + \frac{6}{23} a^{3} - \frac{11}{23} a$, $\frac{1}{23} a^{18} - \frac{9}{23} a^{12} + \frac{6}{23} a^{10} - \frac{11}{23} a^{8} - \frac{11}{23} a^{6} - \frac{5}{23} a^{4} - \frac{2}{23} a^{2} - \frac{1}{23}$, $\frac{1}{23} a^{19} - \frac{9}{23} a^{13} + \frac{6}{23} a^{11} - \frac{11}{23} a^{9} - \frac{11}{23} a^{7} - \frac{5}{23} a^{5} - \frac{2}{23} a^{3} - \frac{1}{23} a$, $\frac{1}{23} a^{20} - \frac{1}{23} a^{12} + \frac{11}{23} a^{10} - \frac{5}{23} a^{8} + \frac{7}{23} a^{6} - \frac{9}{23} a^{4} + \frac{11}{23} a^{2} - \frac{9}{23}$, $\frac{1}{23} a^{21} - \frac{1}{23} a^{13} + \frac{11}{23} a^{11} - \frac{5}{23} a^{9} + \frac{7}{23} a^{7} - \frac{9}{23} a^{5} + \frac{11}{23} a^{3} - \frac{9}{23} a$, $\frac{1}{23} a^{22} - \frac{1}{23}$, $\frac{1}{23} a^{23} - \frac{1}{23} a$, $\frac{1}{953877272718409223} a^{24} - \frac{5951976017765939}{953877272718409223} a^{22} + \frac{3400273862401490}{953877272718409223} a^{20} + \frac{9619312240139629}{953877272718409223} a^{18} - \frac{3891034132342681}{953877272718409223} a^{16} + \frac{6025801841851293}{953877272718409223} a^{14} - \frac{316074545567462860}{953877272718409223} a^{12} - \frac{348293703142197190}{953877272718409223} a^{10} - \frac{124477184844099589}{953877272718409223} a^{8} - \frac{139786812110216495}{953877272718409223} a^{6} - \frac{48544943583040178}{953877272718409223} a^{4} - \frac{228842787930625108}{953877272718409223} a^{2} + \frac{119300292076422685}{953877272718409223}$, $\frac{1}{953877272718409223} a^{25} - \frac{5951976017765939}{953877272718409223} a^{23} + \frac{3400273862401490}{953877272718409223} a^{21} + \frac{9619312240139629}{953877272718409223} a^{19} - \frac{3891034132342681}{953877272718409223} a^{17} + \frac{6025801841851293}{953877272718409223} a^{15} - \frac{316074545567462860}{953877272718409223} a^{13} - \frac{348293703142197190}{953877272718409223} a^{11} - \frac{124477184844099589}{953877272718409223} a^{9} - \frac{139786812110216495}{953877272718409223} a^{7} - \frac{48544943583040178}{953877272718409223} a^{5} - \frac{228842787930625108}{953877272718409223} a^{3} + \frac{119300292076422685}{953877272718409223} a$

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

Class group and class number

$C_{195474}$, which has order $195474$ (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:  \( 5382739421.971964 \) (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

\(\Q(\sqrt{-53}) \), 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 ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ $26$ $26$ R $26$

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
53Data not computed