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Properties

Label 26.26.5735381071...6864.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{26}\cdot 131^{25}$
Root discriminant $217.20$
Ramified primes $2, 131$
Class number $1$ (GRH)
Class group Trivial (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![-487451, 0, 71742543, 0, -484104605, 0, 1210493317, 0, -1544541090, 0, 1135945802, 0, -509684058, 0, 143325004, 0, -25376141, 0, 2784143, 0, -182745, 0, 6812, 0, -131, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 131*x^24 + 6812*x^22 - 182745*x^20 + 2784143*x^18 - 25376141*x^16 + 143325004*x^14 - 509684058*x^12 + 1135945802*x^10 - 1544541090*x^8 + 1210493317*x^6 - 484104605*x^4 + 71742543*x^2 - 487451)
 
gp: K = bnfinit(x^26 - 131*x^24 + 6812*x^22 - 182745*x^20 + 2784143*x^18 - 25376141*x^16 + 143325004*x^14 - 509684058*x^12 + 1135945802*x^10 - 1544541090*x^8 + 1210493317*x^6 - 484104605*x^4 + 71742543*x^2 - 487451, 1)
 

Normalized defining polynomial

\( x^{26} - 131 x^{24} + 6812 x^{22} - 182745 x^{20} + 2784143 x^{18} - 25376141 x^{16} + 143325004 x^{14} - 509684058 x^{12} + 1135945802 x^{10} - 1544541090 x^{8} + 1210493317 x^{6} - 484104605 x^{4} + 71742543 x^{2} - 487451 \)

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:  \(5735381071083349848637397344247707950917141114970532021796864=2^{26}\cdot 131^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $217.20$
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:  $2, 131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(524=2^{2}\cdot 131\)
Dirichlet character group:    $\lbrace$$\chi_{524}(1,·)$, $\chi_{524}(331,·)$, $\chi_{524}(453,·)$, $\chi_{524}(199,·)$, $\chi_{524}(79,·)$, $\chi_{524}(523,·)$, $\chi_{524}(45,·)$, $\chi_{524}(19,·)$, $\chi_{524}(473,·)$, $\chi_{524}(155,·)$, $\chi_{524}(477,·)$, $\chi_{524}(479,·)$, $\chi_{524}(163,·)$, $\chi_{524}(325,·)$, $\chi_{524}(113,·)$, $\chi_{524}(361,·)$, $\chi_{524}(193,·)$, $\chi_{524}(71,·)$, $\chi_{524}(301,·)$, $\chi_{524}(47,·)$, $\chi_{524}(369,·)$, $\chi_{524}(51,·)$, $\chi_{524}(505,·)$, $\chi_{524}(223,·)$, $\chi_{524}(411,·)$, $\chi_{524}(445,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{61} a^{19} + \frac{29}{61} a^{17} - \frac{18}{61} a^{15} - \frac{8}{61} a^{13} + \frac{1}{61} a^{11} + \frac{14}{61} a^{9} + \frac{6}{61} a^{7} - \frac{14}{61} a^{5} + \frac{21}{61} a^{3} - \frac{5}{61} a$, $\frac{1}{3233} a^{20} - \frac{1435}{3233} a^{18} + \frac{165}{3233} a^{16} + \frac{785}{3233} a^{14} + \frac{611}{3233} a^{12} + \frac{746}{3233} a^{10} + \frac{921}{3233} a^{8} - \frac{1539}{3233} a^{6} - \frac{223}{3233} a^{4} + \frac{1337}{3233} a^{2} + \frac{21}{53}$, $\frac{1}{3233} a^{21} - \frac{4}{3233} a^{19} - \frac{365}{3233} a^{17} + \frac{891}{3233} a^{15} - \frac{1138}{3233} a^{13} - \frac{1056}{3233} a^{11} + \frac{1557}{3233} a^{9} + \frac{581}{3233} a^{7} - \frac{859}{3233} a^{5} - \frac{942}{3233} a^{3} + \frac{592}{3233} a$, $\frac{1}{3233} a^{22} + \frac{361}{3233} a^{18} + \frac{1551}{3233} a^{16} - \frac{1231}{3233} a^{14} + \frac{1388}{3233} a^{12} + \frac{1308}{3233} a^{10} + \frac{1032}{3233} a^{8} - \frac{9}{53} a^{6} + \frac{1399}{3233} a^{4} - \frac{526}{3233} a^{2} - \frac{22}{53}$, $\frac{1}{3233} a^{23} - \frac{10}{3233} a^{19} + \frac{491}{3233} a^{17} - \frac{1019}{3233} a^{15} + \frac{1123}{3233} a^{13} + \frac{937}{3233} a^{11} - \frac{929}{3233} a^{9} + \frac{458}{3233} a^{7} + \frac{127}{3233} a^{5} + \frac{1382}{3233} a^{3} + \frac{513}{3233} a$, $\frac{1}{3582728933281103016202342024330513304136659} a^{24} + \frac{540815178205352027042403993688378088292}{3582728933281103016202342024330513304136659} a^{22} + \frac{299670385028762930815262407062024749154}{3582728933281103016202342024330513304136659} a^{20} - \frac{182304914667665090123007295824569101454807}{3582728933281103016202342024330513304136659} a^{18} - \frac{414689724408535729744848398832034820241042}{3582728933281103016202342024330513304136659} a^{16} + \frac{642900840919561253161734621906487691769202}{3582728933281103016202342024330513304136659} a^{14} + \frac{393552372555690350130396627550851475060921}{3582728933281103016202342024330513304136659} a^{12} - \frac{670537112026538006576573261367650857353797}{3582728933281103016202342024330513304136659} a^{10} + \frac{1508201529137378139898123423469340881797384}{3582728933281103016202342024330513304136659} a^{8} - \frac{225233072294797018015392585668194491724694}{3582728933281103016202342024330513304136659} a^{6} + \frac{147998683895603772609521990752885902714238}{3582728933281103016202342024330513304136659} a^{4} - \frac{1549245086026672407887863359775691111191102}{3582728933281103016202342024330513304136659} a^{2} - \frac{18451572757930587129254299734202897558209}{58733261201329557642661344661155955805519}$, $\frac{1}{3582728933281103016202342024330513304136659} a^{25} + \frac{540815178205352027042403993688378088292}{3582728933281103016202342024330513304136659} a^{23} + \frac{299670385028762930815262407062024749154}{3582728933281103016202342024330513304136659} a^{21} - \frac{6105131063676417195023261841101234038250}{3582728933281103016202342024330513304136659} a^{19} + \frac{1112375066826032768964346562358020030702452}{3582728933281103016202342024330513304136659} a^{17} + \frac{1054033669328868156660364034534579382407835}{3582728933281103016202342024330513304136659} a^{15} - \frac{1016045896276219033293475644316891464271535}{3582728933281103016202342024330513304136659} a^{13} - \frac{494337328422549333648589227384182989937240}{3582728933281103016202342024330513304136659} a^{11} + \frac{392269566312116544687557874907377721492523}{3582728933281103016202342024330513304136659} a^{9} + \frac{831965629329135019552511618232612712774648}{3582728933281103016202342024330513304136659} a^{7} + \frac{1263930646720865367820087539314849063019099}{3582728933281103016202342024330513304136659} a^{5} - \frac{1431778563624013292602540670453379199580064}{3582728933281103016202342024330513304136659} a^{3} + \frac{1576184077027393836677909570626797216003125}{3582728933281103016202342024330513304136659} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 24334596273964407000000 \) (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 R $26$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{26}$ $26$

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
131Data not computed