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Properties

Label 26.26.3830224792...4173.1
Degree $26$
Signature $[26, 0]$
Discriminant $13^{49}$
Root discriminant $125.71$
Ramified prime $13$
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![-7751, -343057, -4697355, -20621640, -27198223, 45373016, 179735062, 174681793, -49300303, -219905283, -124959328, 52742326, 85339592, 18065815, -18421481, -10051106, 906503, 1781858, 215267, -151905, -37323, 6331, 2470, -104, -78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 78*x^24 - 104*x^23 + 2470*x^22 + 6331*x^21 - 37323*x^20 - 151905*x^19 + 215267*x^18 + 1781858*x^17 + 906503*x^16 - 10051106*x^15 - 18421481*x^14 + 18065815*x^13 + 85339592*x^12 + 52742326*x^11 - 124959328*x^10 - 219905283*x^9 - 49300303*x^8 + 174681793*x^7 + 179735062*x^6 + 45373016*x^5 - 27198223*x^4 - 20621640*x^3 - 4697355*x^2 - 343057*x - 7751)
 
gp: K = bnfinit(x^26 - 78*x^24 - 104*x^23 + 2470*x^22 + 6331*x^21 - 37323*x^20 - 151905*x^19 + 215267*x^18 + 1781858*x^17 + 906503*x^16 - 10051106*x^15 - 18421481*x^14 + 18065815*x^13 + 85339592*x^12 + 52742326*x^11 - 124959328*x^10 - 219905283*x^9 - 49300303*x^8 + 174681793*x^7 + 179735062*x^6 + 45373016*x^5 - 27198223*x^4 - 20621640*x^3 - 4697355*x^2 - 343057*x - 7751, 1)
 

Normalized defining polynomial

\( x^{26} - 78 x^{24} - 104 x^{23} + 2470 x^{22} + 6331 x^{21} - 37323 x^{20} - 151905 x^{19} + 215267 x^{18} + 1781858 x^{17} + 906503 x^{16} - 10051106 x^{15} - 18421481 x^{14} + 18065815 x^{13} + 85339592 x^{12} + 52742326 x^{11} - 124959328 x^{10} - 219905283 x^{9} - 49300303 x^{8} + 174681793 x^{7} + 179735062 x^{6} + 45373016 x^{5} - 27198223 x^{4} - 20621640 x^{3} - 4697355 x^{2} - 343057 x - 7751 \)

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:  \(3830224792147131369362629348887201408953937846517364173=13^{49}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.71$
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:  $13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(169=13^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{169}(64,·)$, $\chi_{169}(1,·)$, $\chi_{169}(66,·)$, $\chi_{169}(131,·)$, $\chi_{169}(129,·)$, $\chi_{169}(12,·)$, $\chi_{169}(77,·)$, $\chi_{169}(142,·)$, $\chi_{169}(79,·)$, $\chi_{169}(144,·)$, $\chi_{169}(14,·)$, $\chi_{169}(25,·)$, $\chi_{169}(90,·)$, $\chi_{169}(155,·)$, $\chi_{169}(92,·)$, $\chi_{169}(157,·)$, $\chi_{169}(27,·)$, $\chi_{169}(38,·)$, $\chi_{169}(103,·)$, $\chi_{169}(40,·)$, $\chi_{169}(105,·)$, $\chi_{169}(168,·)$, $\chi_{169}(51,·)$, $\chi_{169}(116,·)$, $\chi_{169}(53,·)$, $\chi_{169}(118,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{5}{23} a^{17} - \frac{5}{23} a^{16} + \frac{7}{23} a^{15} - \frac{4}{23} a^{14} + \frac{5}{23} a^{13} + \frac{2}{23} a^{12} - \frac{1}{23} a^{7} + \frac{5}{23} a^{6} + \frac{5}{23} a^{5} - \frac{7}{23} a^{4} + \frac{4}{23} a^{3} - \frac{5}{23} a^{2} - \frac{2}{23} a$, $\frac{1}{23} a^{19} - \frac{7}{23} a^{17} + \frac{5}{23} a^{16} + \frac{8}{23} a^{15} + \frac{8}{23} a^{14} + \frac{4}{23} a^{13} + \frac{10}{23} a^{12} - \frac{1}{23} a^{8} + \frac{7}{23} a^{6} - \frac{5}{23} a^{5} - \frac{8}{23} a^{4} - \frac{8}{23} a^{3} - \frac{4}{23} a^{2} - \frac{10}{23} a$, $\frac{1}{23} a^{20} - \frac{7}{23} a^{17} - \frac{4}{23} a^{16} + \frac{11}{23} a^{15} - \frac{1}{23} a^{14} - \frac{1}{23} a^{13} - \frac{9}{23} a^{12} - \frac{1}{23} a^{9} + \frac{7}{23} a^{6} + \frac{4}{23} a^{5} - \frac{11}{23} a^{4} + \frac{1}{23} a^{3} + \frac{1}{23} a^{2} + \frac{9}{23} a$, $\frac{1}{23} a^{21} + \frac{7}{23} a^{17} - \frac{1}{23} a^{16} + \frac{2}{23} a^{15} - \frac{6}{23} a^{14} + \frac{3}{23} a^{13} - \frac{9}{23} a^{12} - \frac{1}{23} a^{10} - \frac{7}{23} a^{6} + \frac{1}{23} a^{5} - \frac{2}{23} a^{4} + \frac{6}{23} a^{3} - \frac{3}{23} a^{2} + \frac{9}{23} a$, $\frac{1}{23} a^{22} + \frac{11}{23} a^{17} - \frac{9}{23} a^{16} - \frac{9}{23} a^{15} + \frac{8}{23} a^{14} + \frac{2}{23} a^{13} + \frac{9}{23} a^{12} - \frac{1}{23} a^{11} - \frac{11}{23} a^{6} + \frac{9}{23} a^{5} + \frac{9}{23} a^{4} - \frac{8}{23} a^{3} - \frac{2}{23} a^{2} - \frac{9}{23} a$, $\frac{1}{23} a^{23} - \frac{1}{23} a$, $\frac{1}{34477} a^{24} - \frac{394}{34477} a^{23} + \frac{476}{34477} a^{22} - \frac{408}{34477} a^{21} + \frac{580}{34477} a^{20} + \frac{520}{34477} a^{19} + \frac{225}{34477} a^{18} - \frac{8009}{34477} a^{17} + \frac{2892}{34477} a^{16} - \frac{177}{1499} a^{15} - \frac{3116}{34477} a^{14} - \frac{13770}{34477} a^{13} - \frac{5115}{34477} a^{12} + \frac{6585}{34477} a^{11} - \frac{6446}{34477} a^{10} - \frac{7043}{34477} a^{9} + \frac{8933}{34477} a^{8} - \frac{340}{34477} a^{7} + \frac{1753}{34477} a^{6} + \frac{7642}{34477} a^{5} + \frac{481}{1499} a^{4} + \frac{15122}{34477} a^{3} - \frac{468}{34477} a^{2} + \frac{5969}{34477} a - \frac{38}{1499}$, $\frac{1}{26814668934265320178043805162355914796789} a^{25} + \frac{92860423872918691427623686627362799}{26814668934265320178043805162355914796789} a^{24} + \frac{47073347095592103697687911315693541000}{26814668934265320178043805162355914796789} a^{23} - \frac{358497429750042282663259898751818376230}{26814668934265320178043805162355914796789} a^{22} + \frac{71201565313461043088342785765497731547}{26814668934265320178043805162355914796789} a^{21} + \frac{216250412877673016886224033725896578837}{26814668934265320178043805162355914796789} a^{20} - \frac{206483657881543610221031287358112173057}{26814668934265320178043805162355914796789} a^{19} + \frac{235274641180612451871220770016890032}{50689355263261474816717968170805131941} a^{18} - \frac{7866916335456500367568430925335910930320}{26814668934265320178043805162355914796789} a^{17} - \frac{1605899162730146923695013538787801451569}{26814668934265320178043805162355914796789} a^{16} - \frac{10488080293804409889761156260910067567297}{26814668934265320178043805162355914796789} a^{15} + \frac{13310557366483590172165710164436776029971}{26814668934265320178043805162355914796789} a^{14} - \frac{1720003759770559059740193866143586561202}{26814668934265320178043805162355914796789} a^{13} + \frac{292939284247695891351414757758229241856}{1165855171055013920784513267928518034643} a^{12} - \frac{3266733463127694606336593112270264004933}{26814668934265320178043805162355914796789} a^{11} + \frac{10929570987560980612817310388255034929452}{26814668934265320178043805162355914796789} a^{10} - \frac{8105948517098252612833523757660403239735}{26814668934265320178043805162355914796789} a^{9} + \frac{9881228136213874800748648542830773246701}{26814668934265320178043805162355914796789} a^{8} + \frac{270967254008618395810808192307902765792}{1165855171055013920784513267928518034643} a^{7} + \frac{8924477730678686012206604206926016796208}{26814668934265320178043805162355914796789} a^{6} - \frac{13101469999469892826427854686306456616903}{26814668934265320178043805162355914796789} a^{5} + \frac{3571478243931909431593972718054658233837}{26814668934265320178043805162355914796789} a^{4} - \frac{3332373747833511396445935049520655962760}{26814668934265320178043805162355914796789} a^{3} + \frac{9542028672504614281965817320695909954159}{26814668934265320178043805162355914796789} a^{2} + \frac{368446791173088086688002451045384512885}{26814668934265320178043805162355914796789} a + \frac{170119615416993302629994530294780871821}{1165855171055013920784513267928518034643}$

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:  \( 23413899400308634000 \) (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

\(\Q(\sqrt{13}) \), 13.13.542800770374370512771595361.1

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 ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ $26$

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