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Properties

Label 26.26.1057581319...1693.1
Degree $26$
Signature $[26, 0]$
Discriminant $13^{13}\cdot 79^{24}$
Root discriminant $203.53$
Ramified primes $13, 79$
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![221651, -44114, -9474756, 6250371, 105773239, -47073727, -429440544, 219989261, 763726138, -497778327, -575076162, 432764524, 215561192, -183515795, -44883247, 43853516, 5301193, -6318949, -311575, 560201, 822, -29884, 1038, 880, -57, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 - 57*x^24 + 880*x^23 + 1038*x^22 - 29884*x^21 + 822*x^20 + 560201*x^19 - 311575*x^18 - 6318949*x^17 + 5301193*x^16 + 43853516*x^15 - 44883247*x^14 - 183515795*x^13 + 215561192*x^12 + 432764524*x^11 - 575076162*x^10 - 497778327*x^9 + 763726138*x^8 + 219989261*x^7 - 429440544*x^6 - 47073727*x^5 + 105773239*x^4 + 6250371*x^3 - 9474756*x^2 - 44114*x + 221651)
 
gp: K = bnfinit(x^26 - 11*x^25 - 57*x^24 + 880*x^23 + 1038*x^22 - 29884*x^21 + 822*x^20 + 560201*x^19 - 311575*x^18 - 6318949*x^17 + 5301193*x^16 + 43853516*x^15 - 44883247*x^14 - 183515795*x^13 + 215561192*x^12 + 432764524*x^11 - 575076162*x^10 - 497778327*x^9 + 763726138*x^8 + 219989261*x^7 - 429440544*x^6 - 47073727*x^5 + 105773239*x^4 + 6250371*x^3 - 9474756*x^2 - 44114*x + 221651, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} - 57 x^{24} + 880 x^{23} + 1038 x^{22} - 29884 x^{21} + 822 x^{20} + 560201 x^{19} - 311575 x^{18} - 6318949 x^{17} + 5301193 x^{16} + 43853516 x^{15} - 44883247 x^{14} - 183515795 x^{13} + 215561192 x^{12} + 432764524 x^{11} - 575076162 x^{10} - 497778327 x^{9} + 763726138 x^{8} + 219989261 x^{7} - 429440544 x^{6} - 47073727 x^{5} + 105773239 x^{4} + 6250371 x^{3} - 9474756 x^{2} - 44114 x + 221651 \)

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:  \(1057581319194999682024401130977042071156641159890952121581693=13^{13}\cdot 79^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $203.53$
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:  $13, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1027=13\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{1027}(64,·)$, $\chi_{1027}(1,·)$, $\chi_{1027}(259,·)$, $\chi_{1027}(196,·)$, $\chi_{1027}(326,·)$, $\chi_{1027}(324,·)$, $\chi_{1027}(857,·)$, $\chi_{1027}(144,·)$, $\chi_{1027}(337,·)$, $\chi_{1027}(131,·)$, $\chi_{1027}(599,·)$, $\chi_{1027}(729,·)$, $\chi_{1027}(220,·)$, $\chi_{1027}(222,·)$, $\chi_{1027}(417,·)$, $\chi_{1027}(482,·)$, $\chi_{1027}(870,·)$, $\chi_{1027}(38,·)$, $\chi_{1027}(168,·)$, $\chi_{1027}(495,·)$, $\chi_{1027}(1000,·)$, $\chi_{1027}(1013,·)$, $\chi_{1027}(1015,·)$, $\chi_{1027}(441,·)$, $\chi_{1027}(378,·)$, $\chi_{1027}(571,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23} a^{15} - \frac{11}{23} a^{14} - \frac{10}{23} a^{13} + \frac{10}{23} a^{12} - \frac{5}{23} a^{11} - \frac{7}{23} a^{10} + \frac{6}{23} a^{9} + \frac{2}{23} a^{8} + \frac{4}{23} a^{7} + \frac{10}{23} a^{5} - \frac{9}{23} a^{4} + \frac{4}{23} a^{3} - \frac{8}{23} a^{2} - \frac{10}{23} a$, $\frac{1}{23} a^{16} + \frac{7}{23} a^{14} - \frac{8}{23} a^{13} - \frac{10}{23} a^{12} + \frac{7}{23} a^{11} - \frac{2}{23} a^{10} - \frac{1}{23} a^{9} + \frac{3}{23} a^{8} - \frac{2}{23} a^{7} + \frac{10}{23} a^{6} + \frac{9}{23} a^{5} - \frac{3}{23} a^{4} - \frac{10}{23} a^{3} - \frac{6}{23} a^{2} + \frac{5}{23} a$, $\frac{1}{23} a^{17} - \frac{9}{23} a^{13} + \frac{6}{23} a^{12} + \frac{10}{23} a^{11} + \frac{2}{23} a^{10} + \frac{7}{23} a^{9} + \frac{7}{23} a^{8} + \frac{5}{23} a^{7} + \frac{9}{23} a^{6} - \frac{4}{23} a^{5} + \frac{7}{23} a^{4} - \frac{11}{23} a^{3} - \frac{8}{23} a^{2} + \frac{1}{23} a$, $\frac{1}{23} a^{18} - \frac{9}{23} a^{14} + \frac{6}{23} a^{13} + \frac{10}{23} a^{12} + \frac{2}{23} a^{11} + \frac{7}{23} a^{10} + \frac{7}{23} a^{9} + \frac{5}{23} a^{8} + \frac{9}{23} a^{7} - \frac{4}{23} a^{6} + \frac{7}{23} a^{5} - \frac{11}{23} a^{4} - \frac{8}{23} a^{3} + \frac{1}{23} a^{2}$, $\frac{1}{23} a^{19} - \frac{1}{23} a^{14} - \frac{11}{23} a^{13} + \frac{8}{23} a^{11} - \frac{10}{23} a^{10} - \frac{10}{23} a^{9} + \frac{4}{23} a^{8} + \frac{9}{23} a^{7} + \frac{7}{23} a^{6} + \frac{10}{23} a^{5} + \frac{3}{23} a^{4} - \frac{9}{23} a^{3} - \frac{3}{23} a^{2} + \frac{2}{23} a$, $\frac{1}{23} a^{20} + \frac{1}{23} a^{14} - \frac{10}{23} a^{13} - \frac{5}{23} a^{12} + \frac{8}{23} a^{11} + \frac{6}{23} a^{10} + \frac{10}{23} a^{9} + \frac{11}{23} a^{8} + \frac{11}{23} a^{7} + \frac{10}{23} a^{6} - \frac{10}{23} a^{5} + \frac{5}{23} a^{4} + \frac{1}{23} a^{3} - \frac{6}{23} a^{2} - \frac{10}{23} a$, $\frac{1}{23} a^{21} + \frac{1}{23} a^{14} + \frac{5}{23} a^{13} - \frac{2}{23} a^{12} + \frac{11}{23} a^{11} - \frac{6}{23} a^{10} + \frac{5}{23} a^{9} + \frac{9}{23} a^{8} + \frac{6}{23} a^{7} - \frac{10}{23} a^{6} - \frac{5}{23} a^{5} + \frac{10}{23} a^{4} - \frac{10}{23} a^{3} - \frac{2}{23} a^{2} + \frac{10}{23} a$, $\frac{1}{54487} a^{22} + \frac{836}{54487} a^{21} - \frac{1158}{54487} a^{20} - \frac{501}{54487} a^{19} + \frac{10}{54487} a^{18} - \frac{387}{54487} a^{17} + \frac{632}{54487} a^{16} - \frac{719}{54487} a^{15} - \frac{948}{54487} a^{14} + \frac{650}{2369} a^{13} - \frac{26425}{54487} a^{12} - \frac{1078}{2369} a^{11} + \frac{8083}{54487} a^{10} - \frac{2175}{54487} a^{9} - \frac{24066}{54487} a^{8} + \frac{10221}{54487} a^{7} + \frac{5565}{54487} a^{6} - \frac{15158}{54487} a^{5} + \frac{11999}{54487} a^{4} + \frac{20832}{54487} a^{3} - \frac{11183}{54487} a^{2} + \frac{1110}{2369} a + \frac{20}{103}$, $\frac{1}{1253201} a^{23} - \frac{11}{1253201} a^{22} - \frac{5657}{1253201} a^{21} - \frac{14655}{1253201} a^{20} + \frac{12151}{1253201} a^{19} - \frac{15964}{1253201} a^{18} + \frac{3868}{1253201} a^{17} + \frac{25430}{1253201} a^{16} - \frac{24478}{1253201} a^{15} - \frac{314476}{1253201} a^{14} - \frac{164172}{1253201} a^{13} + \frac{88581}{1253201} a^{12} + \frac{2832}{12167} a^{11} + \frac{40576}{1253201} a^{10} - \frac{141004}{1253201} a^{9} - \frac{852}{2369} a^{8} - \frac{68735}{1253201} a^{7} + \frac{352792}{1253201} a^{6} + \frac{479907}{1253201} a^{5} + \frac{179412}{1253201} a^{4} - \frac{340866}{1253201} a^{3} + \frac{507176}{1253201} a^{2} + \frac{3770}{54487} a + \frac{364}{2369}$, $\frac{1}{819080894791} a^{24} - \frac{29108}{819080894791} a^{23} + \frac{65107}{35612212817} a^{22} + \frac{13509961622}{819080894791} a^{21} + \frac{5076342849}{819080894791} a^{20} + \frac{10469888565}{819080894791} a^{19} - \frac{1854560487}{819080894791} a^{18} + \frac{14987419581}{819080894791} a^{17} + \frac{16125744462}{819080894791} a^{16} - \frac{9546199251}{819080894791} a^{15} + \frac{328396152474}{819080894791} a^{14} - \frac{175451042021}{819080894791} a^{13} + \frac{120624943198}{819080894791} a^{12} - \frac{309177469518}{819080894791} a^{11} + \frac{264510168945}{819080894791} a^{10} - \frac{22300028851}{819080894791} a^{9} + \frac{327241331345}{819080894791} a^{8} - \frac{365366175016}{819080894791} a^{7} - \frac{165339436681}{819080894791} a^{6} - \frac{71347238079}{819080894791} a^{5} + \frac{17921720467}{819080894791} a^{4} - \frac{394544768748}{819080894791} a^{3} - \frac{58527999013}{819080894791} a^{2} + \frac{7462750273}{35612212817} a + \frac{358839476}{1548357079}$, $\frac{1}{105121059071329924886851820337791824960325695897902321937393} a^{25} + \frac{53506322404394880746189277781252696837320972417}{105121059071329924886851820337791824960325695897902321937393} a^{24} - \frac{38616552446200118535440012386194058188033928665426020}{105121059071329924886851820337791824960325695897902321937393} a^{23} - \frac{670505536933482957625163974832693011793112760571529372}{105121059071329924886851820337791824960325695897902321937393} a^{22} - \frac{1551137926085009338118499565504665828552174235722311923890}{105121059071329924886851820337791824960325695897902321937393} a^{21} - \frac{1938730042985676737369945051033023004661529517611630079526}{105121059071329924886851820337791824960325695897902321937393} a^{20} - \frac{1427662892715206536507739843475229207048045107681313540596}{105121059071329924886851820337791824960325695897902321937393} a^{19} - \frac{2066782024962243939222913529258626589393866060803179173630}{105121059071329924886851820337791824960325695897902321937393} a^{18} + \frac{711238097771130488077139406169564893182464785101176957602}{105121059071329924886851820337791824960325695897902321937393} a^{17} - \frac{109666576470970348876773696905118325875457500474419650766}{105121059071329924886851820337791824960325695897902321937393} a^{16} - \frac{1401763609654939023744613387233704000313640636897980430418}{105121059071329924886851820337791824960325695897902321937393} a^{15} - \frac{24045831715151573602571100520536517432385486517357161847481}{105121059071329924886851820337791824960325695897902321937393} a^{14} - \frac{20646782587884308620391476460853716444395256892946315064990}{105121059071329924886851820337791824960325695897902321937393} a^{13} + \frac{19220639426329922481095989967839692572460020837656684015337}{105121059071329924886851820337791824960325695897902321937393} a^{12} + \frac{19950760672722178412690698543908890342175157068736952300813}{105121059071329924886851820337791824960325695897902321937393} a^{11} - \frac{35812880488375923561297847258262816062645944238949435734499}{105121059071329924886851820337791824960325695897902321937393} a^{10} + \frac{46557473587640493096859082915374094082633436274488906543496}{105121059071329924886851820337791824960325695897902321937393} a^{9} - \frac{19962534494988576374854291045901404567613361196109099208997}{105121059071329924886851820337791824960325695897902321937393} a^{8} + \frac{19414166322005493321138635901041207165241769506749570291673}{105121059071329924886851820337791824960325695897902321937393} a^{7} - \frac{21718951819644805656257040377012339938726852339312147470300}{105121059071329924886851820337791824960325695897902321937393} a^{6} - \frac{11447632441589053345756509841769952364335113367020664477391}{105121059071329924886851820337791824960325695897902321937393} a^{5} - \frac{207714829557229339539376052518495886642313147376727984110}{580779331885800689982606742197744889283567380651394043853} a^{4} + \frac{13786980595008810480068826249261374297281165208594398751662}{105121059071329924886851820337791824960325695897902321937393} a^{3} + \frac{26217846884455273135419031798523819492660319189671706151125}{105121059071329924886851820337791824960325695897902321937393} a^{2} + \frac{2035481108394692015409457706153022747879107045907653344565}{4570480829188257603776166101643122824361986778169666171191} a - \frac{142767136606342230430040354165151362396064931524363817}{474263861075880212075974483931007868046278590657846443}$

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:  \( 15248476620472010000000 \) (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

\(\Q(\sqrt{13}) \), 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 $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ $26$ $26$ R ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ $26$

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
13Data 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}$