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Properties

Label 26.0.15165164066...2112.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,2^{39}\cdot 79^{25}$
Root discriminant $188.88$
Ramified primes $2, 79$
Class number $217593176$ (GRH)
Class group $[217593176]$ (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![647168, 0, 43360256, 0, 635357184, 0, 2970420224, 0, 5745112576, 0, 5158171648, 0, 2272338304, 0, 550482112, 0, 78249184, 0, 6687824, 0, 341912, 0, 10112, 0, 158, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 + 158*x^24 + 10112*x^22 + 341912*x^20 + 6687824*x^18 + 78249184*x^16 + 550482112*x^14 + 2272338304*x^12 + 5158171648*x^10 + 5745112576*x^8 + 2970420224*x^6 + 635357184*x^4 + 43360256*x^2 + 647168)
 
gp: K = bnfinit(x^26 + 158*x^24 + 10112*x^22 + 341912*x^20 + 6687824*x^18 + 78249184*x^16 + 550482112*x^14 + 2272338304*x^12 + 5158171648*x^10 + 5745112576*x^8 + 2970420224*x^6 + 635357184*x^4 + 43360256*x^2 + 647168, 1)
 

Normalized defining polynomial

\( x^{26} + 158 x^{24} + 10112 x^{22} + 341912 x^{20} + 6687824 x^{18} + 78249184 x^{16} + 550482112 x^{14} + 2272338304 x^{12} + 5158171648 x^{10} + 5745112576 x^{8} + 2970420224 x^{6} + 635357184 x^{4} + 43360256 x^{2} + 647168 \)

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:  \(-151651640667491538463656289407733022385317053604971635802112=-\,2^{39}\cdot 79^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.88$
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:  \(632=2^{3}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{632}(1,·)$, $\chi_{632}(97,·)$, $\chi_{632}(453,·)$, $\chi_{632}(65,·)$, $\chi_{632}(457,·)$, $\chi_{632}(333,·)$, $\chi_{632}(417,·)$, $\chi_{632}(337,·)$, $\chi_{632}(89,·)$, $\chi_{632}(157,·)$, $\chi_{632}(69,·)$, $\chi_{632}(289,·)$, $\chi_{632}(229,·)$, $\chi_{632}(373,·)$, $\chi_{632}(357,·)$, $\chi_{632}(561,·)$, $\chi_{632}(617,·)$, $\chi_{632}(225,·)$, $\chi_{632}(173,·)$, $\chi_{632}(93,·)$, $\chi_{632}(433,·)$, $\chi_{632}(565,·)$, $\chi_{632}(349,·)$, $\chi_{632}(441,·)$, $\chi_{632}(61,·)$, $\chi_{632}(501,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{11776} a^{18} - \frac{1}{1472} a^{16} - \frac{9}{2944} a^{14} - \frac{1}{736} a^{12} + \frac{3}{736} a^{10} - \frac{1}{46} a^{8} + \frac{3}{184} a^{6} + \frac{1}{23} a^{4} + \frac{1}{46} a^{2} + \frac{11}{23}$, $\frac{1}{11776} a^{19} - \frac{1}{1472} a^{17} - \frac{9}{2944} a^{15} - \frac{1}{736} a^{13} + \frac{3}{736} a^{11} - \frac{1}{46} a^{9} + \frac{3}{184} a^{7} + \frac{1}{23} a^{5} + \frac{1}{46} a^{3} + \frac{11}{23} a$, $\frac{1}{23552} a^{20} - \frac{1}{2944} a^{16} + \frac{1}{368} a^{14} - \frac{5}{1472} a^{12} + \frac{1}{184} a^{10} - \frac{3}{184} a^{8} - \frac{7}{184} a^{6} - \frac{3}{46} a^{4} - \frac{4}{23} a^{2} - \frac{2}{23}$, $\frac{1}{23552} a^{21} - \frac{1}{2944} a^{17} + \frac{1}{368} a^{15} - \frac{5}{1472} a^{13} + \frac{1}{184} a^{11} - \frac{3}{184} a^{9} - \frac{7}{184} a^{7} - \frac{3}{46} a^{5} - \frac{4}{23} a^{3} - \frac{2}{23} a$, $\frac{1}{4851712} a^{22} + \frac{9}{606464} a^{20} - \frac{15}{1212928} a^{18} - \frac{29}{303232} a^{16} - \frac{1003}{303232} a^{14} + \frac{43}{37904} a^{12} + \frac{153}{37904} a^{10} + \frac{1077}{37904} a^{8} - \frac{321}{9476} a^{6} - \frac{9}{412} a^{4} - \frac{579}{4738} a^{2} - \frac{376}{2369}$, $\frac{1}{4851712} a^{23} + \frac{9}{606464} a^{21} - \frac{15}{1212928} a^{19} - \frac{29}{303232} a^{17} - \frac{1003}{303232} a^{15} + \frac{43}{37904} a^{13} + \frac{153}{37904} a^{11} + \frac{1077}{37904} a^{9} - \frac{321}{9476} a^{7} - \frac{9}{412} a^{5} - \frac{579}{4738} a^{3} - \frac{376}{2369} a$, $\frac{1}{25111325020875477134615321266204672} a^{24} - \frac{486315827438256959474823669}{6277831255218869283653830316551168} a^{22} - \frac{43844547487839189998368036861}{3138915627609434641826915158275584} a^{20} - \frac{2022554303959106354990441651}{392364453451179330228364394784448} a^{18} - \frac{254983886811462153265213927087}{392364453451179330228364394784448} a^{16} + \frac{6870614658931241127043307149}{24522778340698708139272774674028} a^{14} + \frac{955301257030439711981980011407}{196182226725589665114182197392224} a^{12} - \frac{350360693222680941252877495513}{196182226725589665114182197392224} a^{10} - \frac{1043933300859792850022879532401}{49045556681397416278545549348056} a^{8} - \frac{2694598909210971410206000920261}{49045556681397416278545549348056} a^{6} + \frac{1053672378507842663628150126595}{12261389170349354069636387337014} a^{4} + \frac{458013246752429958285289806551}{6130694585174677034818193668507} a^{2} + \frac{383023767409824919122278615787}{6130694585174677034818193668507}$, $\frac{1}{25111325020875477134615321266204672} a^{25} - \frac{486315827438256959474823669}{6277831255218869283653830316551168} a^{23} - \frac{43844547487839189998368036861}{3138915627609434641826915158275584} a^{21} - \frac{2022554303959106354990441651}{392364453451179330228364394784448} a^{19} - \frac{254983886811462153265213927087}{392364453451179330228364394784448} a^{17} + \frac{6870614658931241127043307149}{24522778340698708139272774674028} a^{15} + \frac{955301257030439711981980011407}{196182226725589665114182197392224} a^{13} - \frac{350360693222680941252877495513}{196182226725589665114182197392224} a^{11} - \frac{1043933300859792850022879532401}{49045556681397416278545549348056} a^{9} - \frac{2694598909210971410206000920261}{49045556681397416278545549348056} a^{7} + \frac{1053672378507842663628150126595}{12261389170349354069636387337014} a^{5} + \frac{458013246752429958285289806551}{6130694585174677034818193668507} a^{3} + \frac{383023767409824919122278615787}{6130694585174677034818193668507} a$

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

Class group and class number

$C_{217593176}$, which has order $217593176$ (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:  \( 57529828940.82975 \) (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

\(\Q(\sqrt{-158}) \), 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 ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ $26$ $26$ $26$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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