↑

Properties

Label 26.26.1851216316...9136.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{26}\cdot 79^{25}$
Root discriminant $133.56$
Ramified primes $2, 79$
Class number $3$ (GRH)
Class group $[3]$ (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![-79, 0, 10586, 0, -310233, 0, 2900801, 0, -11220923, 0, 20149108, 0, -17752643, 0, 8601283, 0, -2445287, 0, 417989, 0, -42739, 0, 2528, 0, -79, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 79*x^24 + 2528*x^22 - 42739*x^20 + 417989*x^18 - 2445287*x^16 + 8601283*x^14 - 17752643*x^12 + 20149108*x^10 - 11220923*x^8 + 2900801*x^6 - 310233*x^4 + 10586*x^2 - 79)
 
gp: K = bnfinit(x^26 - 79*x^24 + 2528*x^22 - 42739*x^20 + 417989*x^18 - 2445287*x^16 + 8601283*x^14 - 17752643*x^12 + 20149108*x^10 - 11220923*x^8 + 2900801*x^6 - 310233*x^4 + 10586*x^2 - 79, 1)
 

Normalized defining polynomial

\( x^{26} - 79 x^{24} + 2528 x^{22} - 42739 x^{20} + 417989 x^{18} - 2445287 x^{16} + 8601283 x^{14} - 17752643 x^{12} + 20149108 x^{10} - 11220923 x^{8} + 2900801 x^{6} - 310233 x^{4} + 10586 x^{2} - 79 \)

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:  \(18512163167418400691364293140592409959145148145138139136=2^{26}\cdot 79^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.56$
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:  $2, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(316=2^{2}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{316}(1,·)$, $\chi_{316}(245,·)$, $\chi_{316}(97,·)$, $\chi_{316}(71,·)$, $\chi_{316}(141,·)$, $\chi_{316}(15,·)$, $\chi_{316}(227,·)$, $\chi_{316}(21,·)$, $\chi_{316}(215,·)$, $\chi_{316}(225,·)$, $\chi_{316}(89,·)$, $\chi_{316}(27,·)$, $\chi_{316}(289,·)$, $\chi_{316}(219,·)$, $\chi_{316}(101,·)$, $\chi_{316}(65,·)$, $\chi_{316}(295,·)$, $\chi_{316}(199,·)$, $\chi_{316}(301,·)$, $\chi_{316}(175,·)$, $\chi_{316}(91,·)$, $\chi_{316}(117,·)$, $\chi_{316}(251,·)$, $\chi_{316}(315,·)$, $\chi_{316}(125,·)$, $\chi_{316}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} + \frac{4}{23} a^{16} - \frac{9}{23} a^{14} + \frac{2}{23} a^{12} + \frac{3}{23} a^{10} + \frac{8}{23} a^{8} + \frac{3}{23} a^{6} - \frac{4}{23} a^{4} + \frac{1}{23} a^{2} - \frac{11}{23}$, $\frac{1}{23} a^{19} + \frac{4}{23} a^{17} - \frac{9}{23} a^{15} + \frac{2}{23} a^{13} + \frac{3}{23} a^{11} + \frac{8}{23} a^{9} + \frac{3}{23} a^{7} - \frac{4}{23} a^{5} + \frac{1}{23} a^{3} - \frac{11}{23} a$, $\frac{1}{23} a^{20} - \frac{2}{23} a^{16} - \frac{8}{23} a^{14} - \frac{5}{23} a^{12} - \frac{4}{23} a^{10} - \frac{6}{23} a^{8} + \frac{7}{23} a^{6} - \frac{6}{23} a^{4} + \frac{8}{23} a^{2} - \frac{2}{23}$, $\frac{1}{23} a^{21} - \frac{2}{23} a^{17} - \frac{8}{23} a^{15} - \frac{5}{23} a^{13} - \frac{4}{23} a^{11} - \frac{6}{23} a^{9} + \frac{7}{23} a^{7} - \frac{6}{23} a^{5} + \frac{8}{23} a^{3} - \frac{2}{23} a$, $\frac{1}{2369} a^{22} - \frac{36}{2369} a^{20} - \frac{15}{2369} a^{18} + \frac{58}{2369} a^{16} - \frac{1003}{2369} a^{14} - \frac{172}{2369} a^{12} + \frac{306}{2369} a^{10} - \frac{1077}{2369} a^{8} - \frac{642}{2369} a^{6} + \frac{9}{103} a^{4} - \frac{579}{2369} a^{2} + \frac{376}{2369}$, $\frac{1}{2369} a^{23} - \frac{36}{2369} a^{21} - \frac{15}{2369} a^{19} + \frac{58}{2369} a^{17} - \frac{1003}{2369} a^{15} - \frac{172}{2369} a^{13} + \frac{306}{2369} a^{11} - \frac{1077}{2369} a^{9} - \frac{642}{2369} a^{7} + \frac{9}{103} a^{5} - \frac{579}{2369} a^{3} + \frac{376}{2369} a$, $\frac{1}{6130694585174677034818193668507} a^{24} + \frac{972631654876513918949647338}{6130694585174677034818193668507} a^{22} - \frac{87689094975678379996736073722}{6130694585174677034818193668507} a^{20} + \frac{16180434431672850839923533208}{6130694585174677034818193668507} a^{18} - \frac{1019935547245848613060855708348}{6130694585174677034818193668507} a^{16} - \frac{219859669085799716065385828768}{6130694585174677034818193668507} a^{14} + \frac{1910602514060879423963960022814}{6130694585174677034818193668507} a^{12} + \frac{350360693222680941252877495513}{6130694585174677034818193668507} a^{10} - \frac{2087866601719585700045759064802}{6130694585174677034818193668507} a^{8} + \frac{2694598909210971410206000920261}{6130694585174677034818193668507} a^{6} + \frac{2107344757015685327256300253190}{6130694585174677034818193668507} a^{4} - \frac{916026493504859916570579613102}{6130694585174677034818193668507} a^{2} + \frac{383023767409824919122278615787}{6130694585174677034818193668507}$, $\frac{1}{6130694585174677034818193668507} a^{25} + \frac{972631654876513918949647338}{6130694585174677034818193668507} a^{23} - \frac{87689094975678379996736073722}{6130694585174677034818193668507} a^{21} + \frac{16180434431672850839923533208}{6130694585174677034818193668507} a^{19} - \frac{1019935547245848613060855708348}{6130694585174677034818193668507} a^{17} - \frac{219859669085799716065385828768}{6130694585174677034818193668507} a^{15} + \frac{1910602514060879423963960022814}{6130694585174677034818193668507} a^{13} + \frac{350360693222680941252877495513}{6130694585174677034818193668507} a^{11} - \frac{2087866601719585700045759064802}{6130694585174677034818193668507} a^{9} + \frac{2694598909210971410206000920261}{6130694585174677034818193668507} a^{7} + \frac{2107344757015685327256300253190}{6130694585174677034818193668507} a^{5} - \frac{916026493504859916570579613102}{6130694585174677034818193668507} a^{3} + \frac{383023767409824919122278615787}{6130694585174677034818193668507} a$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 6734782845265950000 \) (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

\(\Q(\sqrt{79}) \), 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 R ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ $26$ $26$ $26$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
2Data not computed
79Data not computed