Properties

Label 26.0.199...875.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.999\times 10^{45}$
Root discriminant $55.25$
Ramified primes $5, 691$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 6*x^25 + 34*x^24 - 149*x^23 + 724*x^22 - 3027*x^21 + 10234*x^20 - 26698*x^19 + 56427*x^18 - 101122*x^17 + 164719*x^16 - 254230*x^15 + 377196*x^14 - 535703*x^13 + 716143*x^12 - 954928*x^11 + 1568217*x^10 - 3148667*x^9 + 5745440*x^8 - 8171380*x^7 + 9122775*x^6 - 8433000*x^5 + 7096875*x^4 - 5099875*x^3 + 3012500*x^2 - 717500*x + 153125)
 
gp: K = bnfinit(x^26 - 6*x^25 + 34*x^24 - 149*x^23 + 724*x^22 - 3027*x^21 + 10234*x^20 - 26698*x^19 + 56427*x^18 - 101122*x^17 + 164719*x^16 - 254230*x^15 + 377196*x^14 - 535703*x^13 + 716143*x^12 - 954928*x^11 + 1568217*x^10 - 3148667*x^9 + 5745440*x^8 - 8171380*x^7 + 9122775*x^6 - 8433000*x^5 + 7096875*x^4 - 5099875*x^3 + 3012500*x^2 - 717500*x + 153125, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153125, -717500, 3012500, -5099875, 7096875, -8433000, 9122775, -8171380, 5745440, -3148667, 1568217, -954928, 716143, -535703, 377196, -254230, 164719, -101122, 56427, -26698, 10234, -3027, 724, -149, 34, -6, 1]);
 

\( x^{26} - 6 x^{25} + 34 x^{24} - 149 x^{23} + 724 x^{22} - 3027 x^{21} + 10234 x^{20} - 26698 x^{19} + 56427 x^{18} - 101122 x^{17} + 164719 x^{16} - 254230 x^{15} + 377196 x^{14} - 535703 x^{13} + 716143 x^{12} - 954928 x^{11} + 1568217 x^{10} - 3148667 x^{9} + 5745440 x^{8} - 8171380 x^{7} + 9122775 x^{6} - 8433000 x^{5} + 7096875 x^{4} - 5099875 x^{3} + 3012500 x^{2} - 717500 x + 153125 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1999192748751319434913358682729045110107421875\)\(\medspace = -\,5^{12}\cdot 691^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $55.25$
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:  $5, 691$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} + \frac{2}{25} a^{12} - \frac{1}{25} a^{11} - \frac{2}{25} a^{9} - \frac{11}{25} a^{8} + \frac{11}{25} a^{7} - \frac{1}{25} a^{6} - \frac{2}{25} a^{5} + \frac{4}{25} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{15} + \frac{1}{25} a^{13} + \frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{2}{25} a^{10} + \frac{2}{25} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{25} a^{6} + \frac{12}{25} a^{5} - \frac{1}{25} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{16} + \frac{2}{25} a^{13} + \frac{2}{25} a^{12} - \frac{1}{25} a^{11} + \frac{2}{25} a^{10} + \frac{2}{25} a^{9} + \frac{11}{25} a^{8} + \frac{11}{25} a^{7} - \frac{2}{25} a^{6} - \frac{4}{25} a^{5} + \frac{1}{25} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{17} - \frac{1}{25} a^{13} - \frac{1}{25} a^{11} + \frac{2}{25} a^{10} - \frac{2}{25} a^{8} + \frac{11}{25} a^{7} - \frac{2}{25} a^{6} - \frac{8}{25} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{125} a^{18} - \frac{1}{125} a^{17} + \frac{2}{125} a^{15} + \frac{12}{125} a^{13} - \frac{12}{125} a^{12} - \frac{2}{25} a^{11} - \frac{11}{125} a^{10} + \frac{2}{25} a^{9} + \frac{12}{125} a^{8} + \frac{3}{125} a^{7} - \frac{1}{5} a^{6} - \frac{31}{125} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{875} a^{19} + \frac{1}{875} a^{18} + \frac{13}{875} a^{17} + \frac{12}{875} a^{16} - \frac{1}{875} a^{15} - \frac{13}{875} a^{14} - \frac{38}{875} a^{13} + \frac{6}{875} a^{12} + \frac{24}{875} a^{11} - \frac{52}{875} a^{10} - \frac{33}{875} a^{9} - \frac{193}{875} a^{8} + \frac{306}{875} a^{7} - \frac{341}{875} a^{6} + \frac{113}{875} a^{5} + \frac{64}{175} a^{4} + \frac{11}{35} a^{3} - \frac{13}{35} a^{2} + \frac{2}{35} a$, $\frac{1}{875} a^{20} - \frac{2}{875} a^{18} + \frac{13}{875} a^{17} - \frac{13}{875} a^{16} - \frac{1}{175} a^{15} + \frac{2}{175} a^{14} + \frac{51}{875} a^{13} - \frac{59}{875} a^{12} - \frac{6}{875} a^{11} - \frac{72}{875} a^{10} + \frac{2}{35} a^{9} - \frac{54}{875} a^{8} - \frac{304}{875} a^{7} - \frac{36}{875} a^{6} - \frac{409}{875} a^{5} - \frac{23}{175} a^{4} + \frac{4}{35} a^{3} + \frac{3}{7} a^{2} - \frac{16}{35} a$, $\frac{1}{875} a^{21} + \frac{1}{875} a^{18} - \frac{8}{875} a^{17} - \frac{16}{875} a^{16} + \frac{3}{175} a^{15} - \frac{2}{175} a^{14} + \frac{82}{875} a^{13} + \frac{69}{875} a^{12} + \frac{11}{875} a^{11} + \frac{13}{175} a^{10} - \frac{3}{175} a^{9} - \frac{88}{875} a^{8} + \frac{429}{875} a^{7} + \frac{29}{875} a^{6} - \frac{9}{35} a^{5} - \frac{83}{175} a^{4} + \frac{2}{35} a^{3} - \frac{2}{5} a^{2} - \frac{2}{7} a$, $\frac{1}{4375} a^{22} + \frac{1}{4375} a^{21} - \frac{1}{4375} a^{18} - \frac{9}{4375} a^{17} + \frac{57}{4375} a^{16} + \frac{4}{875} a^{15} + \frac{3}{875} a^{14} - \frac{11}{625} a^{13} - \frac{2}{875} a^{12} + \frac{122}{4375} a^{11} - \frac{58}{875} a^{10} - \frac{12}{125} a^{9} - \frac{1412}{4375} a^{8} + \frac{208}{4375} a^{7} + \frac{351}{875} a^{6} + \frac{373}{875} a^{5} + \frac{6}{35} a^{4} - \frac{58}{175} a^{3} + \frac{9}{35} a^{2} - \frac{8}{35} a$, $\frac{1}{135625} a^{23} + \frac{3}{135625} a^{22} - \frac{58}{135625} a^{21} + \frac{1}{27125} a^{20} + \frac{44}{135625} a^{19} + \frac{69}{135625} a^{18} - \frac{98}{19375} a^{17} - \frac{1056}{135625} a^{16} - \frac{141}{27125} a^{15} - \frac{1032}{135625} a^{14} - \frac{12104}{135625} a^{13} - \frac{9173}{135625} a^{12} + \frac{1219}{135625} a^{11} - \frac{2101}{27125} a^{10} - \frac{7312}{135625} a^{9} - \frac{33731}{135625} a^{8} + \frac{1903}{19375} a^{7} - \frac{9168}{27125} a^{6} + \frac{11848}{27125} a^{5} + \frac{1698}{5425} a^{4} - \frac{81}{175} a^{3} - \frac{428}{1085} a^{2} + \frac{302}{1085} a + \frac{12}{31}$, $\frac{1}{90190625} a^{24} + \frac{6}{2576875} a^{23} + \frac{1214}{90190625} a^{22} - \frac{596}{18038125} a^{21} - \frac{6826}{90190625} a^{20} - \frac{30038}{90190625} a^{19} - \frac{92144}{90190625} a^{18} - \frac{326082}{90190625} a^{17} - \frac{3258}{721525} a^{16} - \frac{28081}{12884375} a^{15} - \frac{231599}{12884375} a^{14} - \frac{46282}{4746875} a^{13} - \frac{4246307}{90190625} a^{12} + \frac{177389}{2576875} a^{11} - \frac{35683}{1840625} a^{10} + \frac{984012}{18038125} a^{9} - \frac{26627088}{90190625} a^{8} + \frac{1089946}{2576875} a^{7} - \frac{562507}{18038125} a^{6} - \frac{610068}{3607625} a^{5} - \frac{120301}{515375} a^{4} + \frac{187897}{721525} a^{3} + \frac{261417}{721525} a^{2} - \frac{4773}{20615} a + \frac{156}{589}$, $\frac{1}{180224921982360164612618368005290822305676045089494034375} a^{25} + \frac{623084045465936049154241138379524569735389497136}{180224921982360164612618368005290822305676045089494034375} a^{24} + \frac{425291398389101165110863187609437592813341137437984}{180224921982360164612618368005290822305676045089494034375} a^{23} - \frac{2604196884626215333364024724456250892382002949886433}{25746417426051452087516909715041546043668006441356290625} a^{22} - \frac{12113125694374703416610582702335290532674931088921558}{25746417426051452087516909715041546043668006441356290625} a^{21} - \frac{1233641791980439166925229798835188580156665939568281}{9485522209597903400664124631857411700298739215236528125} a^{20} - \frac{5538206668811518987274762631075299611144236842387606}{25746417426051452087516909715041546043668006441356290625} a^{19} + \frac{566369398857862683710570896409954641071310655478409659}{180224921982360164612618368005290822305676045089494034375} a^{18} + \frac{3330725692149309743425556803609756148627967271408280688}{180224921982360164612618368005290822305676045089494034375} a^{17} - \frac{3350107057513105146079534564533028965401102887397389017}{180224921982360164612618368005290822305676045089494034375} a^{16} - \frac{21474746635735191811587105647242377160418823056304186}{5149283485210290417503381943008309208733601288271258125} a^{15} + \frac{2376294106789813704499471642738138718557353961142237604}{180224921982360164612618368005290822305676045089494034375} a^{14} - \frac{1994442809417682001615548329573476684585012664735682083}{36044984396472032922523673601058164461135209017898806875} a^{13} + \frac{8778192749856699302927254308352133322892930434039611728}{180224921982360164612618368005290822305676045089494034375} a^{12} - \frac{740595839566329704863042360544194708055452417949295586}{25746417426051452087516909715041546043668006441356290625} a^{11} + \frac{7383245918331555359150719111671509890230479461768692168}{180224921982360164612618368005290822305676045089494034375} a^{10} + \frac{138232510057532867505038425218052780229693018493454792}{5813707160721295632665108645331962009860517583532065625} a^{9} - \frac{72898450320037420757961828705299553876065010385268763023}{180224921982360164612618368005290822305676045089494034375} a^{8} - \frac{633238133747198441370372587496128898472432591269915801}{1441799375858881316900946944042326578445408360715952275} a^{7} - \frac{7859027894933160452263066064936320444752902583710173497}{36044984396472032922523673601058164461135209017898806875} a^{6} + \frac{2624457709953943806727471355494719088262260117279081214}{7208996879294406584504734720211632892227041803579761375} a^{5} + \frac{1886967312112291817217091782979113735264272967241413948}{7208996879294406584504734720211632892227041803579761375} a^{4} + \frac{260441629652795282025145886496125961136370980147046}{29424477058344516671447896817190338335620578790121475} a^{3} - \frac{646836555142502551913724001756394381906856160991338738}{1441799375858881316900946944042326578445408360715952275} a^{2} + \frac{119789827940960410047472705074304547170492352751412}{1176979082333780666857915872687613533424823151604859} a + \frac{300299492551604206194785897462978433228999338736659}{1176979082333780666857915872687613533424823151604859}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 4189632832285.5117 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 4189632832285.5117 \cdot 5}{2\sqrt{1999192748751319434913358682729045110107421875}}\approx 5.57220297316000$ (assuming GRH)

Galois group

$D_{26}$ (as 26T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{-691}) \), 13.1.1700937320873056890625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 26 sibling: 26.2.14465938847694062481283347921338966064453125.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ $26$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
691Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.3455.2t1.a.a$1$ $ 5 \cdot 691 $ \(\Q(\sqrt{-3455}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.691.2t1.a.a$1$ $ 691 $ \(\Q(\sqrt{-691}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3455.26t3.a.a$2$ $ 5 \cdot 691 $ 26.0.1999192748751319434913358682729045110107421875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.13t2.a.a$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.13t2.a.f$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.26t3.a.f$2$ $ 5 \cdot 691 $ 26.0.1999192748751319434913358682729045110107421875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.13t2.a.c$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.26t3.a.c$2$ $ 5 \cdot 691 $ 26.0.1999192748751319434913358682729045110107421875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.13t2.a.b$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.13t2.a.e$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.13t2.a.d$2$ $ 5 \cdot 691 $ 13.1.1700937320873056890625.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3455.26t3.a.b$2$ $ 5 \cdot 691 $ 26.0.1999192748751319434913358682729045110107421875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.26t3.a.d$2$ $ 5 \cdot 691 $ 26.0.1999192748751319434913358682729045110107421875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3455.26t3.a.e$2$ $ 5 \cdot 691 $ 26.0.1999192748751319434913358682729045110107421875.1 $D_{26}$ (as 26T3) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.