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Properties

Label 26.26.7031800987...4384.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{39}\cdot 53^{25}$
Root discriminant $128.68$
Ramified primes $2, 53$
Class number $2$ (GRH)
Class group $[2]$ (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![-434176, 0, 12156928, 0, -109195264, 0, 373879808, 0, -619053568, 0, 546193408, 0, -269446912, 0, 78704576, 0, -14158208, 0, 1594240, 0, -111512, 0, 4664, 0, -106, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 106*x^24 + 4664*x^22 - 111512*x^20 + 1594240*x^18 - 14158208*x^16 + 78704576*x^14 - 269446912*x^12 + 546193408*x^10 - 619053568*x^8 + 373879808*x^6 - 109195264*x^4 + 12156928*x^2 - 434176)
 
gp: K = bnfinit(x^26 - 106*x^24 + 4664*x^22 - 111512*x^20 + 1594240*x^18 - 14158208*x^16 + 78704576*x^14 - 269446912*x^12 + 546193408*x^10 - 619053568*x^8 + 373879808*x^6 - 109195264*x^4 + 12156928*x^2 - 434176, 1)
 

Normalized defining polynomial

\( x^{26} - 106 x^{24} + 4664 x^{22} - 111512 x^{20} + 1594240 x^{18} - 14158208 x^{16} + 78704576 x^{14} - 269446912 x^{12} + 546193408 x^{10} - 619053568 x^{8} + 373879808 x^{6} - 109195264 x^{4} + 12156928 x^{2} - 434176 \)

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:  \(7031800987227722369373364916066015457209470483519504384=2^{39}\cdot 53^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.68$
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:  $2, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(424=2^{3}\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{424}(1,·)$, $\chi_{424}(325,·)$, $\chi_{424}(225,·)$, $\chi_{424}(201,·)$, $\chi_{424}(269,·)$, $\chi_{424}(417,·)$, $\chi_{424}(81,·)$, $\chi_{424}(37,·)$, $\chi_{424}(149,·)$, $\chi_{424}(281,·)$, $\chi_{424}(153,·)$, $\chi_{424}(29,·)$, $\chi_{424}(197,·)$, $\chi_{424}(289,·)$, $\chi_{424}(229,·)$, $\chi_{424}(165,·)$, $\chi_{424}(97,·)$, $\chi_{424}(169,·)$, $\chi_{424}(237,·)$, $\chi_{424}(93,·)$, $\chi_{424}(221,·)$, $\chi_{424}(49,·)$, $\chi_{424}(117,·)$, $\chi_{424}(89,·)$, $\chi_{424}(121,·)$, $\chi_{424}(317,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{2944} a^{14} + \frac{11}{1472} a^{12} + \frac{5}{736} a^{10} + \frac{7}{368} a^{8} + \frac{9}{184} a^{6} + \frac{11}{92} a^{4} + \frac{9}{46} a^{2} + \frac{1}{23}$, $\frac{1}{2944} a^{15} + \frac{11}{1472} a^{13} + \frac{5}{736} a^{11} + \frac{7}{368} a^{9} + \frac{9}{184} a^{7} + \frac{11}{92} a^{5} + \frac{9}{46} a^{3} + \frac{1}{23} a$, $\frac{1}{5888} a^{16} - \frac{1}{1472} a^{12} - \frac{1}{368} a^{10} + \frac{1}{368} a^{8} + \frac{1}{46} a^{6} + \frac{3}{92} a^{4} - \frac{3}{23} a^{2} - \frac{11}{23}$, $\frac{1}{5888} a^{17} - \frac{1}{1472} a^{13} - \frac{1}{368} a^{11} + \frac{1}{368} a^{9} + \frac{1}{46} a^{7} + \frac{3}{92} a^{5} - \frac{3}{23} a^{3} - \frac{11}{23} a$, $\frac{1}{11776} a^{18} + \frac{9}{1472} a^{12} + \frac{3}{368} a^{10} + \frac{11}{368} a^{8} - \frac{11}{184} a^{6} + \frac{5}{92} a^{4} - \frac{1}{23} a^{2} + \frac{1}{23}$, $\frac{1}{11776} a^{19} + \frac{9}{1472} a^{13} + \frac{3}{368} a^{11} + \frac{11}{368} a^{9} - \frac{11}{184} a^{7} + \frac{5}{92} a^{5} - \frac{1}{23} a^{3} + \frac{1}{23} a$, $\frac{1}{23552} a^{20} - \frac{1}{1472} a^{12} - \frac{11}{736} a^{10} - \frac{5}{368} a^{8} - \frac{7}{184} a^{6} - \frac{9}{92} a^{4} - \frac{11}{46} a^{2} - \frac{9}{23}$, $\frac{1}{23552} a^{21} - \frac{1}{1472} a^{13} - \frac{11}{736} a^{11} - \frac{5}{368} a^{9} - \frac{7}{184} a^{7} - \frac{9}{92} a^{5} - \frac{11}{46} a^{3} - \frac{9}{23} a$, $\frac{1}{47104} a^{22} + \frac{1}{23}$, $\frac{1}{47104} a^{23} + \frac{1}{23} a$, $\frac{1}{3907081309054604177408} a^{24} + \frac{5951976017765939}{1953540654527302088704} a^{22} + \frac{1700136931200745}{488385163631825522176} a^{20} - \frac{9619312240139629}{488385163631825522176} a^{18} - \frac{3891034132342681}{244192581815912761088} a^{16} - \frac{6025801841851293}{122096290907956380544} a^{14} - \frac{79018636391865715}{15262036363494547568} a^{12} + \frac{174146851571098595}{15262036363494547568} a^{10} - \frac{124477184844099589}{15262036363494547568} a^{8} + \frac{139786812110216495}{7631018181747273784} a^{6} - \frac{24272471791520089}{1907754545436818446} a^{4} + \frac{114421393965312554}{953877272718409223} a^{2} + \frac{119300292076422685}{953877272718409223}$, $\frac{1}{3907081309054604177408} a^{25} + \frac{5951976017765939}{1953540654527302088704} a^{23} + \frac{1700136931200745}{488385163631825522176} a^{21} - \frac{9619312240139629}{488385163631825522176} a^{19} - \frac{3891034132342681}{244192581815912761088} a^{17} - \frac{6025801841851293}{122096290907956380544} a^{15} - \frac{79018636391865715}{15262036363494547568} a^{13} + \frac{174146851571098595}{15262036363494547568} a^{11} - \frac{124477184844099589}{15262036363494547568} a^{9} + \frac{139786812110216495}{7631018181747273784} a^{7} - \frac{24272471791520089}{1907754545436818446} a^{5} + \frac{114421393965312554}{953877272718409223} a^{3} + \frac{119300292076422685}{953877272718409223} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 5192842746653208000 \) (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

\(\Q(\sqrt{106}) \), 13.13.491258904256726154641.1

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 ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ $26$ $26$ $26$ $26$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ R $26$

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
53Data not computed