↑

Properties

Label 26.26.4397983479...1677.1
Degree $26$
Signature $[26, 0]$
Discriminant $3^{13}\cdot 79^{25}$
Root discriminant $115.67$
Ramified primes $3, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{26}$ (as 26T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89293, 586590, -1571682, -22484313, -68803449, -69761804, 55690317, 182685173, 97930228, -102611855, -138127674, -12621780, 59401398, 27660187, -9071733, -9437473, -437386, 1465307, 311697, -113646, -40490, 4025, 2487, -31, -77, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 77*x^24 - 31*x^23 + 2487*x^22 + 4025*x^21 - 40490*x^20 - 113646*x^19 + 311697*x^18 + 1465307*x^17 - 437386*x^16 - 9437473*x^15 - 9071733*x^14 + 27660187*x^13 + 59401398*x^12 - 12621780*x^11 - 138127674*x^10 - 102611855*x^9 + 97930228*x^8 + 182685173*x^7 + 55690317*x^6 - 69761804*x^5 - 68803449*x^4 - 22484313*x^3 - 1571682*x^2 + 586590*x + 89293)
 
gp: K = bnfinit(x^26 - x^25 - 77*x^24 - 31*x^23 + 2487*x^22 + 4025*x^21 - 40490*x^20 - 113646*x^19 + 311697*x^18 + 1465307*x^17 - 437386*x^16 - 9437473*x^15 - 9071733*x^14 + 27660187*x^13 + 59401398*x^12 - 12621780*x^11 - 138127674*x^10 - 102611855*x^9 + 97930228*x^8 + 182685173*x^7 + 55690317*x^6 - 69761804*x^5 - 68803449*x^4 - 22484313*x^3 - 1571682*x^2 + 586590*x + 89293, 1)
 

Normalized defining polynomial

\( x^{26} - x^{25} - 77 x^{24} - 31 x^{23} + 2487 x^{22} + 4025 x^{21} - 40490 x^{20} - 113646 x^{19} + 311697 x^{18} + 1465307 x^{17} - 437386 x^{16} - 9437473 x^{15} - 9071733 x^{14} + 27660187 x^{13} + 59401398 x^{12} - 12621780 x^{11} - 138127674 x^{10} - 102611855 x^{9} + 97930228 x^{8} + 182685173 x^{7} + 55690317 x^{6} - 69761804 x^{5} - 68803449 x^{4} - 22484313 x^{3} - 1571682 x^{2} + 586590 x + 89293 \)

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:  \(439798347913742167434960513305495870460482985171691677=3^{13}\cdot 79^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.67$
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:  $3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(237=3\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{237}(64,·)$, $\chi_{237}(1,·)$, $\chi_{237}(67,·)$, $\chi_{237}(196,·)$, $\chi_{237}(71,·)$, $\chi_{237}(137,·)$, $\chi_{237}(10,·)$, $\chi_{237}(140,·)$, $\chi_{237}(14,·)$, $\chi_{237}(17,·)$, $\chi_{237}(22,·)$, $\chi_{237}(215,·)$, $\chi_{237}(220,·)$, $\chi_{237}(223,·)$, $\chi_{237}(97,·)$, $\chi_{237}(227,·)$, $\chi_{237}(100,·)$, $\chi_{237}(166,·)$, $\chi_{237}(41,·)$, $\chi_{237}(170,·)$, $\chi_{237}(236,·)$, $\chi_{237}(173,·)$, $\chi_{237}(46,·)$, $\chi_{237}(52,·)$, $\chi_{237}(185,·)$, $\chi_{237}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{71478321117840575353002064443565153832319315874105957} a^{25} + \frac{7476761018254716809477131567519648534588753713004717}{71478321117840575353002064443565153832319315874105957} a^{24} - \frac{8344387611319691510352349274710586036837504462346588}{71478321117840575353002064443565153832319315874105957} a^{23} - \frac{8962128680400321866296844642733074958230743144054849}{71478321117840575353002064443565153832319315874105957} a^{22} + \frac{26407251837399988664967572687805747612835021882147901}{71478321117840575353002064443565153832319315874105957} a^{21} + \frac{34245118702357946289340549163771845783600907212704698}{71478321117840575353002064443565153832319315874105957} a^{20} + \frac{24236406082006017417886089784777556846336787876517016}{71478321117840575353002064443565153832319315874105957} a^{19} + \frac{14421616362708225355493388912717127775813232722791348}{71478321117840575353002064443565153832319315874105957} a^{18} + \frac{22219788406764866544216505445381704305370491251464926}{71478321117840575353002064443565153832319315874105957} a^{17} - \frac{21895055141656985480592621345372930153733740233971222}{71478321117840575353002064443565153832319315874105957} a^{16} - \frac{14812588260971991619794938727517944453940054215187788}{71478321117840575353002064443565153832319315874105957} a^{15} + \frac{24697235714269640143671471770492859496958500313156087}{71478321117840575353002064443565153832319315874105957} a^{14} + \frac{26226163822408209136171684149146321588879610131277883}{71478321117840575353002064443565153832319315874105957} a^{13} + \frac{31088694037304016655945674525541789582033119332065988}{71478321117840575353002064443565153832319315874105957} a^{12} + \frac{21884774725600228877593264316090514594664312086573972}{71478321117840575353002064443565153832319315874105957} a^{11} - \frac{34857595447277816415526877978635161735837216471383496}{71478321117840575353002064443565153832319315874105957} a^{10} - \frac{19259566504395234290957710150024885689907335734353154}{71478321117840575353002064443565153832319315874105957} a^{9} - \frac{12724903981680728928788211909464457227258314171221151}{71478321117840575353002064443565153832319315874105957} a^{8} + \frac{2668561916673541468163597774276253617679412745276359}{71478321117840575353002064443565153832319315874105957} a^{7} - \frac{1607572692402525331261996027269696228386853052843452}{71478321117840575353002064443565153832319315874105957} a^{6} - \frac{16342478368359524520550526296044262427741445426395486}{71478321117840575353002064443565153832319315874105957} a^{5} + \frac{28204925227119770134409078667952400012951015299839210}{71478321117840575353002064443565153832319315874105957} a^{4} - \frac{18180774065732386622057773291355140077755721147510181}{71478321117840575353002064443565153832319315874105957} a^{3} - \frac{33027685846000803998387156454415643355501851746986002}{71478321117840575353002064443565153832319315874105957} a^{2} - \frac{4321993611788332956907558538513589179307714567329082}{71478321117840575353002064443565153832319315874105957} a + \frac{8477010171817637250911986237792619175807422402571479}{71478321117840575353002064443565153832319315874105957}$

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:  \( 2262965364986346500 \) (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

\(\Q(\sqrt{237}) \), 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 $26$ R $26$ $26$ $26$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
3Data not computed
79Data not computed