↑

Properties

Label 26.26.3735995888...5632.1
Degree $26$
Signature $[26, 0]$
Discriminant $2^{26}\cdot 3^{13}\cdot 79^{24}$
Root discriminant $195.54$
Ramified primes $2, 3, 79$
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![336697, 419040, -11681058, 4496184, 102776883, -88275712, -332262756, 374426398, 436159945, -612670414, -231878250, 464489034, 34084476, -185004256, 13707903, 41264200, -6513691, -5295248, 1130070, 387876, -100019, -15458, 4698, 298, -110, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 110*x^24 + 298*x^23 + 4698*x^22 - 15458*x^21 - 100019*x^20 + 387876*x^19 + 1130070*x^18 - 5295248*x^17 - 6513691*x^16 + 41264200*x^15 + 13707903*x^14 - 185004256*x^13 + 34084476*x^12 + 464489034*x^11 - 231878250*x^10 - 612670414*x^9 + 436159945*x^8 + 374426398*x^7 - 332262756*x^6 - 88275712*x^5 + 102776883*x^4 + 4496184*x^3 - 11681058*x^2 + 419040*x + 336697)
 
gp: K = bnfinit(x^26 - 2*x^25 - 110*x^24 + 298*x^23 + 4698*x^22 - 15458*x^21 - 100019*x^20 + 387876*x^19 + 1130070*x^18 - 5295248*x^17 - 6513691*x^16 + 41264200*x^15 + 13707903*x^14 - 185004256*x^13 + 34084476*x^12 + 464489034*x^11 - 231878250*x^10 - 612670414*x^9 + 436159945*x^8 + 374426398*x^7 - 332262756*x^6 - 88275712*x^5 + 102776883*x^4 + 4496184*x^3 - 11681058*x^2 + 419040*x + 336697, 1)
 

Normalized defining polynomial

\( x^{26} - 2 x^{25} - 110 x^{24} + 298 x^{23} + 4698 x^{22} - 15458 x^{21} - 100019 x^{20} + 387876 x^{19} + 1130070 x^{18} - 5295248 x^{17} - 6513691 x^{16} + 41264200 x^{15} + 13707903 x^{14} - 185004256 x^{13} + 34084476 x^{12} + 464489034 x^{11} - 231878250 x^{10} - 612670414 x^{9} + 436159945 x^{8} + 374426398 x^{7} - 332262756 x^{6} - 88275712 x^{5} + 102776883 x^{4} + 4496184 x^{3} - 11681058 x^{2} + 419040 x + 336697 \)

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:  \(373599588829974770195670809275806491434103418053178144325632=2^{26}\cdot 3^{13}\cdot 79^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.54$
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:  $2, 3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(948=2^{2}\cdot 3\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{948}(1,·)$, $\chi_{948}(131,·)$, $\chi_{948}(877,·)$, $\chi_{948}(179,·)$, $\chi_{948}(97,·)$, $\chi_{948}(457,·)$, $\chi_{948}(599,·)$, $\chi_{948}(143,·)$, $\chi_{948}(337,·)$, $\chi_{948}(275,·)$, $\chi_{948}(791,·)$, $\chi_{948}(539,·)$, $\chi_{948}(541,·)$, $\chi_{948}(719,·)$, $\chi_{948}(289,·)$, $\chi_{948}(721,·)$, $\chi_{948}(299,·)$, $\chi_{948}(301,·)$, $\chi_{948}(733,·)$, $\chi_{948}(433,·)$, $\chi_{948}(563,·)$, $\chi_{948}(757,·)$, $\chi_{948}(887,·)$, $\chi_{948}(697,·)$, $\chi_{948}(383,·)$, $\chi_{948}(575,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{3}{23} a^{15} + \frac{9}{23} a^{14} + \frac{8}{23} a^{13} - \frac{9}{23} a^{12} - \frac{11}{23} a^{11} + \frac{6}{23} a^{10} - \frac{6}{23} a^{8} + \frac{2}{23} a^{7} - \frac{4}{23} a^{6} + \frac{2}{23} a^{5} + \frac{1}{23} a^{4} + \frac{8}{23} a^{3} + \frac{2}{23} a^{2} - \frac{6}{23} a$, $\frac{1}{23} a^{17} - \frac{11}{23} a^{14} - \frac{8}{23} a^{13} + \frac{8}{23} a^{12} - \frac{4}{23} a^{11} - \frac{5}{23} a^{10} - \frac{6}{23} a^{9} + \frac{7}{23} a^{8} + \frac{2}{23} a^{7} - \frac{10}{23} a^{6} + \frac{7}{23} a^{5} + \frac{11}{23} a^{4} + \frac{3}{23} a^{3} + \frac{5}{23} a$, $\frac{1}{23} a^{18} - \frac{11}{23} a^{15} - \frac{8}{23} a^{14} + \frac{8}{23} a^{13} - \frac{4}{23} a^{12} - \frac{5}{23} a^{11} - \frac{6}{23} a^{10} + \frac{7}{23} a^{9} + \frac{2}{23} a^{8} - \frac{10}{23} a^{7} + \frac{7}{23} a^{6} + \frac{11}{23} a^{5} + \frac{3}{23} a^{4} + \frac{5}{23} a^{2}$, $\frac{1}{23} a^{19} + \frac{5}{23} a^{15} - \frac{8}{23} a^{14} - \frac{8}{23} a^{13} + \frac{11}{23} a^{12} + \frac{11}{23} a^{11} + \frac{4}{23} a^{10} + \frac{2}{23} a^{9} - \frac{7}{23} a^{8} + \frac{6}{23} a^{7} - \frac{10}{23} a^{6} + \frac{2}{23} a^{5} + \frac{11}{23} a^{4} + \frac{1}{23} a^{3} - \frac{1}{23} a^{2} + \frac{3}{23} a$, $\frac{1}{23} a^{20} + \frac{7}{23} a^{15} - \frac{7}{23} a^{14} - \frac{6}{23} a^{13} + \frac{10}{23} a^{12} - \frac{10}{23} a^{11} - \frac{5}{23} a^{10} - \frac{7}{23} a^{9} - \frac{10}{23} a^{8} + \frac{3}{23} a^{7} - \frac{1}{23} a^{6} + \frac{1}{23} a^{5} - \frac{4}{23} a^{4} + \frac{5}{23} a^{3} - \frac{7}{23} a^{2} + \frac{7}{23} a$, $\frac{1}{23} a^{21} - \frac{9}{23} a^{15} + \frac{7}{23} a^{12} + \frac{3}{23} a^{11} - \frac{3}{23} a^{10} - \frac{10}{23} a^{9} - \frac{1}{23} a^{8} + \frac{8}{23} a^{7} + \frac{6}{23} a^{6} + \frac{5}{23} a^{5} - \frac{2}{23} a^{4} + \frac{6}{23} a^{3} - \frac{7}{23} a^{2} - \frac{4}{23} a$, $\frac{1}{529} a^{22} - \frac{2}{529} a^{21} + \frac{4}{529} a^{20} + \frac{1}{529} a^{19} + \frac{1}{529} a^{17} + \frac{162}{529} a^{15} - \frac{242}{529} a^{14} - \frac{30}{529} a^{13} + \frac{197}{529} a^{12} + \frac{227}{529} a^{11} + \frac{259}{529} a^{10} + \frac{194}{529} a^{9} - \frac{84}{529} a^{8} - \frac{179}{529} a^{7} + \frac{48}{529} a^{6} - \frac{96}{529} a^{5} - \frac{251}{529} a^{4} + \frac{238}{529} a^{3} - \frac{254}{529} a^{2} + \frac{197}{529} a - \frac{6}{23}$, $\frac{1}{529} a^{23} + \frac{9}{529} a^{20} + \frac{2}{529} a^{19} + \frac{1}{529} a^{18} + \frac{2}{529} a^{17} + \frac{1}{529} a^{16} + \frac{36}{529} a^{15} + \frac{153}{529} a^{14} - \frac{93}{529} a^{13} - \frac{2}{23} a^{12} - \frac{7}{23} a^{11} - \frac{254}{529} a^{10} - \frac{225}{529} a^{9} + \frac{90}{529} a^{8} - \frac{103}{529} a^{7} + \frac{5}{23} a^{6} - \frac{236}{529} a^{5} + \frac{104}{529} a^{4} - \frac{8}{529} a^{3} - \frac{104}{529} a^{2} + \frac{164}{529} a + \frac{11}{23}$, $\frac{1}{196752557} a^{24} + \frac{165118}{196752557} a^{23} - \frac{49401}{196752557} a^{22} - \frac{2869845}{196752557} a^{21} + \frac{27932}{8554459} a^{20} + \frac{1348059}{196752557} a^{19} - \frac{2623354}{196752557} a^{18} + \frac{470333}{196752557} a^{17} + \frac{3283586}{196752557} a^{16} + \frac{6390949}{196752557} a^{15} - \frac{94470199}{196752557} a^{14} + \frac{4259601}{196752557} a^{13} + \frac{3172130}{196752557} a^{12} - \frac{6365007}{196752557} a^{11} + \frac{51375076}{196752557} a^{10} + \frac{16947790}{196752557} a^{9} - \frac{33282875}{196752557} a^{8} - \frac{32980777}{196752557} a^{7} + \frac{4161313}{8554459} a^{6} - \frac{88484479}{196752557} a^{5} - \frac{86590666}{196752557} a^{4} + \frac{60132308}{196752557} a^{3} + \frac{12539519}{196752557} a^{2} + \frac{35104252}{196752557} a + \frac{3030818}{8554459}$, $\frac{1}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{25} + \frac{4187839274092115963706209345652293357008054533268558123109450}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{24} + \frac{889284587820575735163540882882251464589198424706162130551593210423}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{23} + \frac{908248598583869916259025083180576569913928187174047569139335580266}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{22} - \frac{3553014189794018978037749204352738722442032196302166405035237847648}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{21} - \frac{32701205746976214226457328307822392555464006675281137466022702269361}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{20} + \frac{35906561168729180693763859435958587188921188309433728646569792828871}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{19} + \frac{28602559832412124970583469631060315662373631050926954166110623739073}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{18} - \frac{27163711349188305187028388031968422073822952628516231286149548547835}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{17} + \frac{23934188809553452919819274394983937763987861925022355053370123988601}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{16} + \frac{712939900076239835019899655866068928232978563476431297160360767056671}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{15} + \frac{283786703235408576573837653410723568316255224138310929783772339712834}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{14} + \frac{632557039350399367407096584990335137950325933581707640722545916013865}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{13} + \frac{596408476387063073504527834282782098931067503945662678406588588028181}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{12} - \frac{281939255355152259948654218772193955836804670281729308082879117289509}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{11} + \frac{611956883828544716918488973254795283635486574615107523173069485817321}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{10} - \frac{436258483194827337177176041548157806486603716975314999459599393493424}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{9} - \frac{26213540373143575461718126511877395577093558587311032594287223470679}{81602117989477039723000832947661443259663244100680533704195196668493} a^{8} + \frac{286885038331098936153155009871982497803947174135334379567909130405606}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{7} + \frac{201146291288106628784729384015904790985770024776805153363745480467395}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{6} + \frac{129804928252582407341481871278724977981544377989069046675033500483517}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{5} + \frac{917231394177413959097172709891607011188160813147151971509696068546964}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{4} + \frac{645870957851296492636674434303108984931867622445874021288533100562232}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{3} + \frac{183256307430220805856028463677614364287970256860472618872089178441414}{1876848713757971913629019157796213194972254614315652275196489523375339} a^{2} - \frac{499142704951695278811027445637902032090726122616870562276045193987243}{1876848713757971913629019157796213194972254614315652275196489523375339} a - \frac{15866137260200828658323069967567507060158484009656167809115445188128}{81602117989477039723000832947661443259663244100680533704195196668493}$

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:  \( 7436031871254272000000 \) (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

\(\Q(\sqrt{3}) \), 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 R R $26$ $26$ ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ $26$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
2Data not computed
3Data not computed
79Data not computed