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Properties

Label 26.26.294...125.1
Degree $26$
Signature $[26, 0]$
Discriminant $2.946\times 10^{50}$
Root discriminant $87.32$
Ramified primes $5, 53$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{26}$ (as 26T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 - 7*x^24 + 468*x^23 - 1046*x^22 - 6960*x^21 + 26796*x^20 + 40357*x^19 - 278503*x^18 + 355*x^17 + 1499785*x^16 - 1077692*x^15 - 4479644*x^14 + 5284410*x^13 + 7470556*x^12 - 11963101*x^11 - 6678940*x^10 + 14762504*x^9 + 2744933*x^8 - 10366844*x^7 - 29320*x^6 + 4079010*x^5 - 378384*x^4 - 821546*x^3 + 120702*x^2 + 64121*x - 11789)
 
gp: K = bnfinit(x^26 - 11*x^25 - 7*x^24 + 468*x^23 - 1046*x^22 - 6960*x^21 + 26796*x^20 + 40357*x^19 - 278503*x^18 + 355*x^17 + 1499785*x^16 - 1077692*x^15 - 4479644*x^14 + 5284410*x^13 + 7470556*x^12 - 11963101*x^11 - 6678940*x^10 + 14762504*x^9 + 2744933*x^8 - 10366844*x^7 - 29320*x^6 + 4079010*x^5 - 378384*x^4 - 821546*x^3 + 120702*x^2 + 64121*x - 11789, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11789, 64121, 120702, -821546, -378384, 4079010, -29320, -10366844, 2744933, 14762504, -6678940, -11963101, 7470556, 5284410, -4479644, -1077692, 1499785, 355, -278503, 40357, 26796, -6960, -1046, 468, -7, -11, 1]);
 

\(x^{26} - 11 x^{25} - 7 x^{24} + 468 x^{23} - 1046 x^{22} - 6960 x^{21} + 26796 x^{20} + 40357 x^{19} - 278503 x^{18} + 355 x^{17} + 1499785 x^{16} - 1077692 x^{15} - 4479644 x^{14} + 5284410 x^{13} + 7470556 x^{12} - 11963101 x^{11} - 6678940 x^{10} + 14762504 x^{9} + 2744933 x^{8} - 10366844 x^{7} - 29320 x^{6} + 4079010 x^{5} - 378384 x^{4} - 821546 x^{3} + 120702 x^{2} + 64121 x - 11789\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[26, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(294598768324608440893618976324163386033790283203125\)\(\medspace = 5^{13}\cdot 53^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $87.32$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 53$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $26$
This field is Galois and abelian over $\Q$.
Conductor:  \(265=5\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{265}(256,·)$, $\chi_{265}(1,·)$, $\chi_{265}(66,·)$, $\chi_{265}(259,·)$, $\chi_{265}(261,·)$, $\chi_{265}(134,·)$, $\chi_{265}(201,·)$, $\chi_{265}(206,·)$, $\chi_{265}(16,·)$, $\chi_{265}(81,·)$, $\chi_{265}(46,·)$, $\chi_{265}(24,·)$, $\chi_{265}(89,·)$, $\chi_{265}(69,·)$, $\chi_{265}(99,·)$, $\chi_{265}(36,·)$, $\chi_{265}(169,·)$, $\chi_{265}(44,·)$, $\chi_{265}(174,·)$, $\chi_{265}(49,·)$, $\chi_{265}(116,·)$, $\chi_{265}(236,·)$, $\chi_{265}(54,·)$, $\chi_{265}(119,·)$, $\chi_{265}(121,·)$, $\chi_{265}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22} - \frac{5}{23} a^{21} + \frac{1}{23} a^{20} + \frac{5}{23} a^{19} - \frac{7}{23} a^{18} - \frac{3}{23} a^{17} - \frac{9}{23} a^{16} + \frac{2}{23} a^{15} + \frac{6}{23} a^{14} - \frac{5}{23} a^{13} + \frac{7}{23} a^{12} - \frac{6}{23} a^{11} - \frac{2}{23} a^{10} - \frac{5}{23} a^{9} - \frac{2}{23} a^{8} + \frac{6}{23} a^{7} + \frac{2}{23} a^{6} - \frac{3}{23} a^{5} - \frac{11}{23} a^{4} - \frac{1}{23} a^{3} + \frac{7}{23} a^{2} + \frac{4}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{23} - \frac{1}{23} a^{21} + \frac{10}{23} a^{20} - \frac{5}{23} a^{19} + \frac{8}{23} a^{18} - \frac{1}{23} a^{17} + \frac{3}{23} a^{16} - \frac{7}{23} a^{15} + \frac{2}{23} a^{14} + \frac{5}{23} a^{13} + \frac{6}{23} a^{12} - \frac{9}{23} a^{11} + \frac{8}{23} a^{10} - \frac{4}{23} a^{9} - \frac{4}{23} a^{8} + \frac{9}{23} a^{7} + \frac{7}{23} a^{6} - \frac{3}{23} a^{5} - \frac{10}{23} a^{4} + \frac{2}{23} a^{3} - \frac{7}{23} a^{2} - \frac{2}{23} a + \frac{5}{23}$, $\frac{1}{174477107807} a^{24} - \frac{365329292}{174477107807} a^{23} + \frac{2231143202}{174477107807} a^{22} - \frac{78088812650}{174477107807} a^{21} + \frac{65286242610}{174477107807} a^{20} + \frac{21598286683}{174477107807} a^{19} + \frac{15723475173}{174477107807} a^{18} + \frac{53262712179}{174477107807} a^{17} - \frac{82793813622}{174477107807} a^{16} - \frac{81208199961}{174477107807} a^{15} + \frac{1330987541}{7585961209} a^{14} + \frac{41169875659}{174477107807} a^{13} + \frac{44103404856}{174477107807} a^{12} + \frac{61518323431}{174477107807} a^{11} + \frac{73469624576}{174477107807} a^{10} + \frac{60231931393}{174477107807} a^{9} - \frac{1367575550}{7585961209} a^{8} + \frac{2547796509}{174477107807} a^{7} + \frac{14774394850}{174477107807} a^{6} + \frac{19947426538}{174477107807} a^{5} + \frac{32663636974}{174477107807} a^{4} - \frac{32054666520}{174477107807} a^{3} + \frac{63699582248}{174477107807} a^{2} - \frac{52202505252}{174477107807} a - \frac{76184508139}{174477107807}$, $\frac{1}{8568869712761194354633618302459979000860051958828963} a^{25} - \frac{18595760542811582806276595658706872423820}{8568869712761194354633618302459979000860051958828963} a^{24} + \frac{124920192736096390946550140704013230017528068461636}{8568869712761194354633618302459979000860051958828963} a^{23} + \frac{159669101380909158997754666792144726671387485817901}{8568869712761194354633618302459979000860051958828963} a^{22} + \frac{86069562696069203528594086971956181624835869409162}{8568869712761194354633618302459979000860051958828963} a^{21} - \frac{1343538972415379497392480824389479684033550404441837}{8568869712761194354633618302459979000860051958828963} a^{20} - \frac{172496180944969158360236980048859898209163716709220}{8568869712761194354633618302459979000860051958828963} a^{19} - \frac{4234420612251925076322096828082893032620997712486536}{8568869712761194354633618302459979000860051958828963} a^{18} + \frac{3597403192813367163285053433076057964321042117007131}{8568869712761194354633618302459979000860051958828963} a^{17} - \frac{985484172624135209883307255995493444878570419954431}{8568869712761194354633618302459979000860051958828963} a^{16} - \frac{391851720026793997647253402380124192105448738944359}{8568869712761194354633618302459979000860051958828963} a^{15} + \frac{3894038707666536437728648383371256693480670711477814}{8568869712761194354633618302459979000860051958828963} a^{14} - \frac{292114783538278108135094655179919524437480679666971}{8568869712761194354633618302459979000860051958828963} a^{13} + \frac{1351063902135824626083777525657563085257156392859602}{8568869712761194354633618302459979000860051958828963} a^{12} + \frac{2166240524088170270250281271653454347096262126800800}{8568869712761194354633618302459979000860051958828963} a^{11} + \frac{5119948020312990642017922298782421829310221093359}{11917760379361883664302668014547954104116901194477} a^{10} - \frac{2258481498751637316920494317964059643476661768088127}{8568869712761194354633618302459979000860051958828963} a^{9} + \frac{1365268416842128439469658710943855649734217886976503}{8568869712761194354633618302459979000860051958828963} a^{8} + \frac{2024044766054667709858561848968007657979649362417799}{8568869712761194354633618302459979000860051958828963} a^{7} - \frac{1147137683725970102491939169965150701005981285264678}{8568869712761194354633618302459979000860051958828963} a^{6} - \frac{2254522796441229858017826591545693814748291616024445}{8568869712761194354633618302459979000860051958828963} a^{5} + \frac{3970062655264977251770524493757845046887989751619556}{8568869712761194354633618302459979000860051958828963} a^{4} + \frac{3901241697906332467259738598953755670619397961248670}{8568869712761194354633618302459979000860051958828963} a^{3} - \frac{2915990909422131241126483370726812704415806739454657}{8568869712761194354633618302459979000860051958828963} a^{2} - \frac{7069295869210807452712444113023980420280939276829}{103239394129652944031730340993493722901928336853361} a - \frac{2411825627503713245228752071202950677499858745303795}{8568869712761194354633618302459979000860051958828963}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $25$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 53967208334298856 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{26}\cdot(2\pi)^{0}\cdot 53967208334298856 \cdot 1}{2\sqrt{294598768324608440893618976324163386033790283203125}}\approx 0.105502896075004$ (assuming GRH)

Galois group

$C_{26}$ (as 26T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 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 $26$ $26$ R $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ $26$ R ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
53Data not computed