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Properties

Label 26.26.1561373472...5625.1
Degree $26$
Signature $[26, 0]$
Discriminant $5^{13}\cdot 53^{25}$
Root discriminant $101.73$
Ramified primes $5, 53$
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![798709, 7895450, 14295304, -37579021, -117142919, 15619489, 250434704, 59188590, -274320987, -85848429, 183881042, 52642857, -80012336, -18225371, 23134676, 3900033, -4483638, -533706, 580404, 46749, -49206, -2532, 2608, 77, -78, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 78*x^24 + 77*x^23 + 2608*x^22 - 2532*x^21 - 49206*x^20 + 46749*x^19 + 580404*x^18 - 533706*x^17 - 4483638*x^16 + 3900033*x^15 + 23134676*x^14 - 18225371*x^13 - 80012336*x^12 + 52642857*x^11 + 183881042*x^10 - 85848429*x^9 - 274320987*x^8 + 59188590*x^7 + 250434704*x^6 + 15619489*x^5 - 117142919*x^4 - 37579021*x^3 + 14295304*x^2 + 7895450*x + 798709)
 
gp: K = bnfinit(x^26 - x^25 - 78*x^24 + 77*x^23 + 2608*x^22 - 2532*x^21 - 49206*x^20 + 46749*x^19 + 580404*x^18 - 533706*x^17 - 4483638*x^16 + 3900033*x^15 + 23134676*x^14 - 18225371*x^13 - 80012336*x^12 + 52642857*x^11 + 183881042*x^10 - 85848429*x^9 - 274320987*x^8 + 59188590*x^7 + 250434704*x^6 + 15619489*x^5 - 117142919*x^4 - 37579021*x^3 + 14295304*x^2 + 7895450*x + 798709, 1)
 

Normalized defining polynomial

\( x^{26} - x^{25} - 78 x^{24} + 77 x^{23} + 2608 x^{22} - 2532 x^{21} - 49206 x^{20} + 46749 x^{19} + 580404 x^{18} - 533706 x^{17} - 4483638 x^{16} + 3900033 x^{15} + 23134676 x^{14} - 18225371 x^{13} - 80012336 x^{12} + 52642857 x^{11} + 183881042 x^{10} - 85848429 x^{9} - 274320987 x^{8} + 59188590 x^{7} + 250434704 x^{6} + 15619489 x^{5} - 117142919 x^{4} - 37579021 x^{3} + 14295304 x^{2} + 7895450 x + 798709 \)

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:  \(15613734721204247367361805745180659459790885009765625=5^{13}\cdot 53^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.73$
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:  $5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(265=5\cdot 53\)
Dirichlet character group:    $\lbrace$$\chi_{265}(256,·)$, $\chi_{265}(1,·)$, $\chi_{265}(66,·)$, $\chi_{265}(4,·)$, $\chi_{265}(261,·)$, $\chi_{265}(199,·)$, $\chi_{265}(264,·)$, $\chi_{265}(9,·)$, $\chi_{265}(206,·)$, $\chi_{265}(144,·)$, $\chi_{265}(81,·)$, $\chi_{265}(149,·)$, $\chi_{265}(121,·)$, $\chi_{265}(219,·)$, $\chi_{265}(29,·)$, $\chi_{265}(16,·)$, $\chi_{265}(36,·)$, $\chi_{265}(229,·)$, $\chi_{265}(64,·)$, $\chi_{265}(236,·)$, $\chi_{265}(46,·)$, $\chi_{265}(116,·)$, $\chi_{265}(201,·)$, $\chi_{265}(184,·)$, $\chi_{265}(249,·)$, $\chi_{265}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} + \frac{4}{23} a^{15} + \frac{8}{23} a^{14} + \frac{11}{23} a^{13} + \frac{1}{23} a^{12} - \frac{2}{23} a^{11} + \frac{1}{23} a^{5} + \frac{4}{23} a^{4} + \frac{8}{23} a^{3} + \frac{11}{23} a^{2} + \frac{1}{23} a - \frac{2}{23}$, $\frac{1}{23} a^{17} - \frac{8}{23} a^{15} + \frac{2}{23} a^{14} + \frac{3}{23} a^{13} - \frac{6}{23} a^{12} + \frac{8}{23} a^{11} + \frac{1}{23} a^{6} - \frac{8}{23} a^{4} + \frac{2}{23} a^{3} + \frac{3}{23} a^{2} - \frac{6}{23} a + \frac{8}{23}$, $\frac{1}{23} a^{18} + \frac{11}{23} a^{15} - \frac{2}{23} a^{14} - \frac{10}{23} a^{13} - \frac{7}{23} a^{12} + \frac{7}{23} a^{11} + \frac{1}{23} a^{7} + \frac{11}{23} a^{4} - \frac{2}{23} a^{3} - \frac{10}{23} a^{2} - \frac{7}{23} a + \frac{7}{23}$, $\frac{1}{23} a^{19} - \frac{6}{23} a^{14} + \frac{10}{23} a^{13} - \frac{4}{23} a^{12} - \frac{1}{23} a^{11} + \frac{1}{23} a^{8} - \frac{6}{23} a^{3} + \frac{10}{23} a^{2} - \frac{4}{23} a - \frac{1}{23}$, $\frac{1}{23} a^{20} - \frac{6}{23} a^{15} + \frac{10}{23} a^{14} - \frac{4}{23} a^{13} - \frac{1}{23} a^{12} + \frac{1}{23} a^{9} - \frac{6}{23} a^{4} + \frac{10}{23} a^{3} - \frac{4}{23} a^{2} - \frac{1}{23} a$, $\frac{1}{23} a^{21} + \frac{11}{23} a^{15} - \frac{2}{23} a^{14} - \frac{4}{23} a^{13} + \frac{6}{23} a^{12} + \frac{11}{23} a^{11} + \frac{1}{23} a^{10} + \frac{11}{23} a^{4} - \frac{2}{23} a^{3} - \frac{4}{23} a^{2} + \frac{6}{23} a + \frac{11}{23}$, $\frac{1}{529} a^{22} - \frac{2}{529} a^{21} + \frac{9}{529} a^{20} + \frac{3}{529} a^{19} - \frac{7}{529} a^{18} - \frac{5}{529} a^{17} + \frac{9}{529} a^{16} + \frac{130}{529} a^{15} - \frac{147}{529} a^{14} - \frac{120}{529} a^{13} + \frac{32}{529} a^{12} + \frac{98}{529} a^{11} - \frac{255}{529} a^{10} + \frac{262}{529} a^{9} + \frac{187}{529} a^{8} - \frac{53}{529} a^{7} - \frac{258}{529} a^{6} - \frac{198}{529} a^{5} - \frac{146}{529} a^{4} - \frac{170}{529} a^{3} + \frac{248}{529} a^{2} + \frac{147}{529} a + \frac{120}{529}$, $\frac{1}{111619} a^{23} + \frac{30}{111619} a^{22} + \frac{313}{111619} a^{21} - \frac{813}{111619} a^{20} + \frac{273}{111619} a^{19} + \frac{1036}{111619} a^{18} - \frac{634}{111619} a^{17} + \frac{1200}{111619} a^{16} - \frac{23127}{111619} a^{15} - \frac{31872}{111619} a^{14} - \frac{52660}{111619} a^{13} - \frac{24753}{111619} a^{12} - \frac{30032}{111619} a^{11} + \frac{3579}{111619} a^{10} - \frac{36969}{111619} a^{9} + \frac{7173}{111619} a^{8} + \frac{14123}{111619} a^{7} + \frac{12223}{111619} a^{6} + \frac{7525}{111619} a^{5} + \frac{47368}{111619} a^{4} - \frac{46523}{111619} a^{3} + \frac{19537}{111619} a^{2} + \frac{45074}{111619} a - \frac{53407}{111619}$, $\frac{1}{357515657} a^{24} + \frac{440}{357515657} a^{23} + \frac{146176}{357515657} a^{22} + \frac{5053101}{357515657} a^{21} + \frac{5925836}{357515657} a^{20} - \frac{762684}{357515657} a^{19} + \frac{386990}{357515657} a^{18} - \frac{53367}{15544159} a^{17} - \frac{1881456}{357515657} a^{16} - \frac{43422697}{357515657} a^{15} - \frac{96541773}{357515657} a^{14} - \frac{66134876}{357515657} a^{13} + \frac{3762430}{357515657} a^{12} - \frac{166920635}{357515657} a^{11} + \frac{16430833}{357515657} a^{10} - \frac{117637248}{357515657} a^{9} + \frac{149324276}{357515657} a^{8} - \frac{123940614}{357515657} a^{7} + \frac{939285}{15544159} a^{6} + \frac{22247108}{357515657} a^{5} + \frac{24168277}{357515657} a^{4} - \frac{101418532}{357515657} a^{3} - \frac{102392284}{357515657} a^{2} + \frac{171736579}{357515657} a + \frac{85163264}{357515657}$, $\frac{1}{196902342334003258375054771035819659279807324213} a^{25} - \frac{260247938482579431532797654638084070757}{196902342334003258375054771035819659279807324213} a^{24} + \frac{446526684568846818844635594899864994088391}{196902342334003258375054771035819659279807324213} a^{23} - \frac{4794671997983058383911474015003737663532533}{196902342334003258375054771035819659279807324213} a^{22} + \frac{1095824999822659137501721306120062876203583703}{196902342334003258375054771035819659279807324213} a^{21} - \frac{3976271441945252588101668986149122311375744001}{196902342334003258375054771035819659279807324213} a^{20} - \frac{4112487547788198492231855983240904109697936083}{196902342334003258375054771035819659279807324213} a^{19} + \frac{444022165195251933880938425567260477080725889}{196902342334003258375054771035819659279807324213} a^{18} + \frac{1769618169425360234246575751297152028312009003}{196902342334003258375054771035819659279807324213} a^{17} - \frac{2461244367210872715470525576822556640328505738}{196902342334003258375054771035819659279807324213} a^{16} + \frac{97555717124352452731244838397937448468844625341}{196902342334003258375054771035819659279807324213} a^{15} - \frac{9428210170398290715459675031908412892222312682}{196902342334003258375054771035819659279807324213} a^{14} + \frac{10627444980998923934174414769418562347243546407}{196902342334003258375054771035819659279807324213} a^{13} - \frac{41238840443539570310727352387894832729144929621}{196902342334003258375054771035819659279807324213} a^{12} - \frac{48874831226910915490864940267530398788199958310}{196902342334003258375054771035819659279807324213} a^{11} + \frac{5611353632479806558168467810921412053695163294}{196902342334003258375054771035819659279807324213} a^{10} + \frac{85590302210188729211796985071807828084745678252}{196902342334003258375054771035819659279807324213} a^{9} - \frac{77006539031310737512535105759580522125488075158}{196902342334003258375054771035819659279807324213} a^{8} - \frac{39946058125356454727994012317780937701332429404}{196902342334003258375054771035819659279807324213} a^{7} + \frac{24746585892409225414155192171283643554522830380}{196902342334003258375054771035819659279807324213} a^{6} - \frac{36951919748115780383038621541529268711811367513}{196902342334003258375054771035819659279807324213} a^{5} + \frac{20568224382613928471930883726947423416705640399}{196902342334003258375054771035819659279807324213} a^{4} - \frac{65406959628297385394031262585017663961159164322}{196902342334003258375054771035819659279807324213} a^{3} - \frac{30503244112320864112259685542665494060630714530}{196902342334003258375054771035819659279807324213} a^{2} + \frac{69979915134531222103799407259715571014533684478}{196902342334003258375054771035819659279807324213} a + \frac{457345432516753158009818519583192704389552979}{2372317377518111546687406879949634449154305111}$

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:  \( 796588150034711700 \) (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

\(\Q(\sqrt{265}) \), 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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ R $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ $26$ $26$ $26$ $26$ 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.

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
5Data not computed
53Data not computed