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Properties

Label 26.26.4262459321...3125.1
Degree $26$
Signature $[26, 0]$
Discriminant $5^{13}\cdot 79^{24}$
Root discriminant $126.22$
Ramified primes $5, 79$
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![631, -17478, 104206, 177547, -2127423, -1051271, 15898634, 10936597, -46847964, -49003867, 40567618, 57442428, -14262346, -31294019, 1790481, 9584292, 140843, -1788553, -58361, 209025, 4190, -15056, 162, 616, -31, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 - 31*x^24 + 616*x^23 + 162*x^22 - 15056*x^21 + 4190*x^20 + 209025*x^19 - 58361*x^18 - 1788553*x^17 + 140843*x^16 + 9584292*x^15 + 1790481*x^14 - 31294019*x^13 - 14262346*x^12 + 57442428*x^11 + 40567618*x^10 - 49003867*x^9 - 46847964*x^8 + 10936597*x^7 + 15898634*x^6 - 1051271*x^5 - 2127423*x^4 + 177547*x^3 + 104206*x^2 - 17478*x + 631)
 
gp: K = bnfinit(x^26 - 11*x^25 - 31*x^24 + 616*x^23 + 162*x^22 - 15056*x^21 + 4190*x^20 + 209025*x^19 - 58361*x^18 - 1788553*x^17 + 140843*x^16 + 9584292*x^15 + 1790481*x^14 - 31294019*x^13 - 14262346*x^12 + 57442428*x^11 + 40567618*x^10 - 49003867*x^9 - 46847964*x^8 + 10936597*x^7 + 15898634*x^6 - 1051271*x^5 - 2127423*x^4 + 177547*x^3 + 104206*x^2 - 17478*x + 631, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} - 31 x^{24} + 616 x^{23} + 162 x^{22} - 15056 x^{21} + 4190 x^{20} + 209025 x^{19} - 58361 x^{18} - 1788553 x^{17} + 140843 x^{16} + 9584292 x^{15} + 1790481 x^{14} - 31294019 x^{13} - 14262346 x^{12} + 57442428 x^{11} + 40567618 x^{10} - 49003867 x^{9} - 46847964 x^{8} + 10936597 x^{7} + 15898634 x^{6} - 1051271 x^{5} - 2127423 x^{4} + 177547 x^{3} + 104206 x^{2} - 17478 x + 631 \)

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:  \(4262459321296958211564292098551086834037153907470703125=5^{13}\cdot 79^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.22$
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:  $5, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(395=5\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{395}(64,·)$, $\chi_{395}(1,·)$, $\chi_{395}(131,·)$, $\chi_{395}(324,·)$, $\chi_{395}(326,·)$, $\chi_{395}(204,·)$, $\chi_{395}(141,·)$, $\chi_{395}(334,·)$, $\chi_{395}(144,·)$, $\chi_{395}(146,·)$, $\chi_{395}(259,·)$, $\chi_{395}(21,·)$, $\chi_{395}(89,·)$, $\chi_{395}(176,·)$, $\chi_{395}(196,·)$, $\chi_{395}(159,·)$, $\chi_{395}(289,·)$, $\chi_{395}(354,·)$, $\chi_{395}(101,·)$, $\chi_{395}(166,·)$, $\chi_{395}(299,·)$, $\chi_{395}(301,·)$, $\chi_{395}(46,·)$, $\chi_{395}(304,·)$, $\chi_{395}(179,·)$, $\chi_{395}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23} a^{20} - \frac{11}{23} a^{19} - \frac{4}{23} a^{18} + \frac{1}{23} a^{17} - \frac{8}{23} a^{16} - \frac{9}{23} a^{15} + \frac{8}{23} a^{14} + \frac{3}{23} a^{13} + \frac{10}{23} a^{11} + \frac{3}{23} a^{10} - \frac{11}{23} a^{9} + \frac{5}{23} a^{8} + \frac{8}{23} a^{7} - \frac{2}{23} a^{6} + \frac{2}{23} a^{5} + \frac{6}{23} a^{4} - \frac{4}{23} a^{3} + \frac{11}{23} a^{2} + \frac{7}{23} a + \frac{8}{23}$, $\frac{1}{23} a^{21} - \frac{10}{23} a^{19} + \frac{3}{23} a^{18} + \frac{3}{23} a^{17} - \frac{5}{23} a^{16} + \frac{1}{23} a^{15} - \frac{1}{23} a^{14} + \frac{10}{23} a^{13} + \frac{10}{23} a^{12} - \frac{2}{23} a^{11} - \frac{1}{23} a^{10} - \frac{1}{23} a^{9} - \frac{6}{23} a^{8} - \frac{6}{23} a^{7} + \frac{3}{23} a^{6} + \frac{5}{23} a^{5} - \frac{7}{23} a^{4} - \frac{10}{23} a^{3} - \frac{10}{23} a^{2} - \frac{7}{23} a - \frac{4}{23}$, $\frac{1}{23} a^{22} + \frac{8}{23} a^{19} + \frac{9}{23} a^{18} + \frac{5}{23} a^{17} - \frac{10}{23} a^{16} + \frac{1}{23} a^{15} - \frac{2}{23} a^{14} - \frac{6}{23} a^{13} - \frac{2}{23} a^{12} + \frac{7}{23} a^{11} + \frac{6}{23} a^{10} - \frac{1}{23} a^{9} - \frac{2}{23} a^{8} - \frac{9}{23} a^{7} + \frac{8}{23} a^{6} - \frac{10}{23} a^{5} + \frac{4}{23} a^{4} - \frac{4}{23} a^{3} + \frac{11}{23} a^{2} - \frac{3}{23} a + \frac{11}{23}$, $\frac{1}{4163} a^{23} + \frac{28}{4163} a^{22} + \frac{62}{4163} a^{21} + \frac{86}{4163} a^{20} - \frac{1567}{4163} a^{19} + \frac{2040}{4163} a^{18} - \frac{296}{4163} a^{17} - \frac{477}{4163} a^{16} + \frac{536}{4163} a^{15} + \frac{1535}{4163} a^{14} - \frac{1938}{4163} a^{13} + \frac{1330}{4163} a^{12} - \frac{1028}{4163} a^{11} + \frac{1650}{4163} a^{10} - \frac{1824}{4163} a^{9} + \frac{597}{4163} a^{8} + \frac{675}{4163} a^{7} + \frac{1854}{4163} a^{6} - \frac{1788}{4163} a^{5} - \frac{249}{4163} a^{4} - \frac{1654}{4163} a^{3} - \frac{653}{4163} a^{2} + \frac{775}{4163} a + \frac{1029}{4163}$, $\frac{1}{226829381} a^{24} + \frac{22021}{226829381} a^{23} + \frac{3995860}{226829381} a^{22} + \frac{31197}{2202227} a^{21} + \frac{2168571}{226829381} a^{20} - \frac{24737309}{226829381} a^{19} + \frac{11754189}{226829381} a^{18} + \frac{103860861}{226829381} a^{17} + \frac{81723221}{226829381} a^{16} + \frac{91580375}{226829381} a^{15} - \frac{71747583}{226829381} a^{14} + \frac{96641382}{226829381} a^{13} - \frac{1663871}{226829381} a^{12} - \frac{59431604}{226829381} a^{11} - \frac{29796655}{226829381} a^{10} + \frac{108105903}{226829381} a^{9} - \frac{17791725}{226829381} a^{8} - \frac{76886205}{226829381} a^{7} + \frac{79357185}{226829381} a^{6} - \frac{77584599}{226829381} a^{5} - \frac{12603881}{226829381} a^{4} - \frac{68500955}{226829381} a^{3} - \frac{106131817}{226829381} a^{2} + \frac{26944494}{226829381} a + \frac{50320901}{226829381}$, $\frac{1}{34032705252010179604795025181570770367834525340105817231039} a^{25} + \frac{14379070825166015995285177526268790747884147955123}{34032705252010179604795025181570770367834525340105817231039} a^{24} + \frac{4051297043085877716109056127885615596445774211929721011}{34032705252010179604795025181570770367834525340105817231039} a^{23} - \frac{472060834176915934970715539179678402381356440407860065960}{34032705252010179604795025181570770367834525340105817231039} a^{22} - \frac{528448691529054017742268898484961022911925998185834028619}{34032705252010179604795025181570770367834525340105817231039} a^{21} + \frac{124855792687644380418765807644563977941333973235073473281}{34032705252010179604795025181570770367834525340105817231039} a^{20} - \frac{9208888255688710421737731440276599982214842641258021739992}{34032705252010179604795025181570770367834525340105817231039} a^{19} - \frac{11780548664328537056982800643887176943731203783217230027730}{34032705252010179604795025181570770367834525340105817231039} a^{18} + \frac{14394964218412428425557714188535600615622358031814706077756}{34032705252010179604795025181570770367834525340105817231039} a^{17} + \frac{5199382054220203847100382783174524483457145797309391162092}{34032705252010179604795025181570770367834525340105817231039} a^{16} - \frac{4918683188459619643364065579615361805178509756169208146699}{34032705252010179604795025181570770367834525340105817231039} a^{15} + \frac{669401229832693702329705869850510430308331179539980561909}{1479682837043920852382392399198729146427588058265470314393} a^{14} - \frac{7077898894037704141065615862799660901114767206418771035320}{34032705252010179604795025181570770367834525340105817231039} a^{13} - \frac{12058877407204250776023913630431869082073733786012041235507}{34032705252010179604795025181570770367834525340105817231039} a^{12} + \frac{978862757179115854147128287918677933755869448404782446791}{34032705252010179604795025181570770367834525340105817231039} a^{11} - \frac{998831706800167573564038011083948619172721459939407483661}{34032705252010179604795025181570770367834525340105817231039} a^{10} - \frac{7277400956983931601488912444338358823015982603293468373299}{34032705252010179604795025181570770367834525340105817231039} a^{9} + \frac{13313514440796137384869315779521242449520118701364101793247}{34032705252010179604795025181570770367834525340105817231039} a^{8} + \frac{4097975720616177426135594129206054335790570414185775260719}{34032705252010179604795025181570770367834525340105817231039} a^{7} - \frac{4472337164378412377150346725986877040948794737999576923462}{34032705252010179604795025181570770367834525340105817231039} a^{6} + \frac{6455029493232747488359377031503757470206657760385913186333}{34032705252010179604795025181570770367834525340105817231039} a^{5} - \frac{3571627769633389777472758513103087437117449862898890055168}{34032705252010179604795025181570770367834525340105817231039} a^{4} + \frac{4208021127195925083446705985679009420911238509316170786339}{34032705252010179604795025181570770367834525340105817231039} a^{3} + \frac{408647374785799375874097216827014198313797354038839441601}{34032705252010179604795025181570770367834525340105817231039} a^{2} + \frac{9155951918180833335857565904150114152167134743688407816188}{34032705252010179604795025181570770367834525340105817231039} a - \frac{2720154081732635316279588067812732558364663210167121598633}{34032705252010179604795025181570770367834525340105817231039}$

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:  \( 7085626941768992000 \) (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

\(\Q(\sqrt{5}) \), 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 $26$ $26$ R $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
5Data not computed
$79$79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$
79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$