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Properties

Label 26.2.287...125.1
Degree $26$
Signature $[2, 12]$
Discriminant $2.877\times 10^{36}$
Root discriminant \(25.25\)
Ramified primes $5,191$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1)
 
gp: K = bnfinit(y^26 - 2*y^25 - 4*y^24 + 16*y^23 - 11*y^22 - 29*y^21 + 88*y^20 - 189*y^19 + 510*y^18 - 655*y^17 + 132*y^16 - 179*y^15 + 449*y^14 + 1090*y^13 + 114*y^12 - 781*y^11 - 543*y^10 - 58*y^9 + 245*y^8 + 173*y^7 + 196*y^6 + 159*y^5 + 34*y^4 - 4*y^3 + 6*y^2 - 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1)
 

\( x^{26} - 2 x^{25} - 4 x^{24} + 16 x^{23} - 11 x^{22} - 29 x^{21} + 88 x^{20} - 189 x^{19} + 510 x^{18} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2877467739962384875567767188720703125\) \(\medspace = 5^{13}\cdot 191^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(25.25\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}191^{1/2}\approx 30.903074280724887$
Ramified primes:   \(5\), \(191\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{7}a^{19}-\frac{2}{7}a^{18}-\frac{2}{7}a^{17}+\frac{1}{7}a^{16}+\frac{2}{7}a^{15}+\frac{1}{7}a^{14}+\frac{3}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}-\frac{2}{7}a^{8}+\frac{2}{7}a^{7}-\frac{3}{7}a^{6}-\frac{1}{7}a^{5}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{20}+\frac{1}{7}a^{18}-\frac{3}{7}a^{17}-\frac{3}{7}a^{16}-\frac{2}{7}a^{15}-\frac{2}{7}a^{14}-\frac{3}{7}a^{12}-\frac{2}{7}a^{11}+\frac{1}{7}a^{10}+\frac{3}{7}a^{9}-\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{5}+\frac{3}{7}a^{4}+\frac{3}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{21}-\frac{1}{7}a^{18}-\frac{1}{7}a^{17}-\frac{3}{7}a^{16}+\frac{3}{7}a^{15}-\frac{1}{7}a^{14}+\frac{1}{7}a^{13}-\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{2}{7}a^{7}+\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}-\frac{2}{7}a^{2}-\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{22}-\frac{3}{7}a^{18}+\frac{2}{7}a^{17}-\frac{3}{7}a^{16}+\frac{1}{7}a^{15}+\frac{2}{7}a^{14}-\frac{3}{7}a^{11}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}+\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{287}a^{23}-\frac{13}{287}a^{22}-\frac{19}{287}a^{21}-\frac{1}{287}a^{20}+\frac{6}{287}a^{19}-\frac{36}{287}a^{18}+\frac{94}{287}a^{17}-\frac{31}{287}a^{16}-\frac{104}{287}a^{15}-\frac{38}{287}a^{14}+\frac{127}{287}a^{13}+\frac{17}{287}a^{12}-\frac{76}{287}a^{11}+\frac{6}{41}a^{10}+\frac{124}{287}a^{9}-\frac{116}{287}a^{8}+\frac{17}{287}a^{7}-\frac{64}{287}a^{6}+\frac{138}{287}a^{5}-\frac{22}{287}a^{4}-\frac{4}{287}a^{3}-\frac{17}{287}a^{2}+\frac{132}{287}a-\frac{64}{287}$, $\frac{1}{3157}a^{24}-\frac{5}{3157}a^{23}+\frac{1}{77}a^{22}+\frac{216}{3157}a^{21}-\frac{166}{3157}a^{20}+\frac{31}{451}a^{19}-\frac{1055}{3157}a^{18}+\frac{1049}{3157}a^{17}-\frac{114}{287}a^{16}+\frac{565}{3157}a^{15}+\frac{168}{451}a^{14}+\frac{582}{3157}a^{13}+\frac{32}{451}a^{12}-\frac{976}{3157}a^{11}+\frac{95}{451}a^{10}+\frac{999}{3157}a^{9}-\frac{255}{3157}a^{8}+\frac{933}{3157}a^{7}-\frac{1030}{3157}a^{6}-\frac{1296}{3157}a^{5}+\frac{27}{451}a^{4}-\frac{172}{3157}a^{3}-\frac{127}{3157}a^{2}+\frac{90}{3157}a-\frac{120}{451}$, $\frac{1}{72\!\cdots\!69}a^{25}-\frac{49\!\cdots\!19}{72\!\cdots\!69}a^{24}+\frac{78\!\cdots\!89}{66\!\cdots\!79}a^{23}-\frac{16\!\cdots\!05}{66\!\cdots\!79}a^{22}+\frac{34\!\cdots\!96}{10\!\cdots\!67}a^{21}+\frac{23\!\cdots\!14}{72\!\cdots\!69}a^{20}+\frac{21\!\cdots\!53}{72\!\cdots\!69}a^{19}+\frac{38\!\cdots\!23}{72\!\cdots\!69}a^{18}+\frac{20\!\cdots\!86}{72\!\cdots\!69}a^{17}+\frac{35\!\cdots\!96}{72\!\cdots\!69}a^{16}-\frac{10\!\cdots\!61}{10\!\cdots\!67}a^{15}-\frac{12\!\cdots\!51}{72\!\cdots\!69}a^{14}+\frac{19\!\cdots\!82}{72\!\cdots\!69}a^{13}-\frac{20\!\cdots\!80}{72\!\cdots\!69}a^{12}+\frac{12\!\cdots\!95}{72\!\cdots\!69}a^{11}-\frac{62\!\cdots\!12}{72\!\cdots\!69}a^{10}-\frac{35\!\cdots\!98}{10\!\cdots\!67}a^{9}-\frac{64\!\cdots\!63}{72\!\cdots\!69}a^{8}-\frac{48\!\cdots\!51}{72\!\cdots\!69}a^{7}+\frac{19\!\cdots\!80}{72\!\cdots\!69}a^{6}+\frac{13\!\cdots\!31}{72\!\cdots\!69}a^{5}+\frac{17\!\cdots\!90}{72\!\cdots\!69}a^{4}-\frac{34\!\cdots\!81}{72\!\cdots\!69}a^{3}-\frac{24\!\cdots\!85}{72\!\cdots\!69}a^{2}-\frac{12\!\cdots\!48}{72\!\cdots\!69}a+\frac{46\!\cdots\!89}{10\!\cdots\!67}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{68\!\cdots\!00}{10\!\cdots\!67}a^{25}-\frac{19\!\cdots\!55}{72\!\cdots\!69}a^{24}-\frac{60\!\cdots\!69}{72\!\cdots\!69}a^{23}+\frac{11\!\cdots\!51}{72\!\cdots\!69}a^{22}-\frac{20\!\cdots\!84}{72\!\cdots\!69}a^{21}-\frac{43\!\cdots\!53}{72\!\cdots\!69}a^{20}+\frac{72\!\cdots\!28}{72\!\cdots\!69}a^{19}-\frac{25\!\cdots\!46}{10\!\cdots\!67}a^{18}+\frac{41\!\cdots\!16}{72\!\cdots\!69}a^{17}-\frac{78\!\cdots\!87}{72\!\cdots\!69}a^{16}+\frac{65\!\cdots\!78}{72\!\cdots\!69}a^{15}-\frac{11\!\cdots\!02}{72\!\cdots\!69}a^{14}+\frac{27\!\cdots\!88}{72\!\cdots\!69}a^{13}+\frac{13\!\cdots\!70}{72\!\cdots\!69}a^{12}-\frac{10\!\cdots\!91}{72\!\cdots\!69}a^{11}-\frac{38\!\cdots\!94}{72\!\cdots\!69}a^{10}+\frac{63\!\cdots\!76}{72\!\cdots\!69}a^{9}+\frac{55\!\cdots\!15}{72\!\cdots\!69}a^{8}+\frac{32\!\cdots\!06}{72\!\cdots\!69}a^{7}-\frac{27\!\cdots\!51}{72\!\cdots\!69}a^{6}-\frac{10\!\cdots\!88}{10\!\cdots\!67}a^{5}-\frac{35\!\cdots\!76}{72\!\cdots\!69}a^{4}-\frac{10\!\cdots\!96}{72\!\cdots\!69}a^{3}-\frac{15\!\cdots\!18}{72\!\cdots\!69}a^{2}+\frac{26\!\cdots\!59}{10\!\cdots\!67}a-\frac{67\!\cdots\!65}{72\!\cdots\!69}$, $\frac{28\!\cdots\!97}{72\!\cdots\!69}a^{25}-\frac{94\!\cdots\!32}{10\!\cdots\!67}a^{24}-\frac{64\!\cdots\!54}{66\!\cdots\!79}a^{23}+\frac{40\!\cdots\!23}{66\!\cdots\!79}a^{22}-\frac{56\!\cdots\!16}{72\!\cdots\!69}a^{21}-\frac{30\!\cdots\!19}{72\!\cdots\!69}a^{20}+\frac{25\!\cdots\!35}{72\!\cdots\!69}a^{19}-\frac{70\!\cdots\!32}{72\!\cdots\!69}a^{18}+\frac{18\!\cdots\!30}{72\!\cdots\!69}a^{17}-\frac{27\!\cdots\!70}{72\!\cdots\!69}a^{16}+\frac{21\!\cdots\!43}{72\!\cdots\!69}a^{15}-\frac{20\!\cdots\!06}{72\!\cdots\!69}a^{14}+\frac{12\!\cdots\!30}{72\!\cdots\!69}a^{13}+\frac{32\!\cdots\!61}{72\!\cdots\!69}a^{12}-\frac{36\!\cdots\!05}{72\!\cdots\!69}a^{11}+\frac{11\!\cdots\!69}{72\!\cdots\!69}a^{10}-\frac{24\!\cdots\!64}{10\!\cdots\!67}a^{9}-\frac{19\!\cdots\!45}{72\!\cdots\!69}a^{8}-\frac{61\!\cdots\!40}{72\!\cdots\!69}a^{7}+\frac{84\!\cdots\!70}{72\!\cdots\!69}a^{6}+\frac{13\!\cdots\!40}{72\!\cdots\!69}a^{5}+\frac{73\!\cdots\!23}{72\!\cdots\!69}a^{4}+\frac{17\!\cdots\!26}{72\!\cdots\!69}a^{3}+\frac{16\!\cdots\!12}{72\!\cdots\!69}a^{2}-\frac{10\!\cdots\!48}{72\!\cdots\!69}a-\frac{84\!\cdots\!21}{72\!\cdots\!69}$, $\frac{52\!\cdots\!65}{72\!\cdots\!69}a^{25}-\frac{15\!\cdots\!98}{72\!\cdots\!69}a^{24}-\frac{10\!\cdots\!63}{72\!\cdots\!69}a^{23}+\frac{14\!\cdots\!94}{10\!\cdots\!67}a^{22}-\frac{13\!\cdots\!72}{72\!\cdots\!69}a^{21}-\frac{87\!\cdots\!78}{72\!\cdots\!69}a^{20}+\frac{82\!\cdots\!07}{10\!\cdots\!67}a^{19}-\frac{14\!\cdots\!85}{72\!\cdots\!69}a^{18}+\frac{36\!\cdots\!72}{72\!\cdots\!69}a^{17}-\frac{61\!\cdots\!75}{72\!\cdots\!69}a^{16}+\frac{63\!\cdots\!67}{10\!\cdots\!67}a^{15}-\frac{26\!\cdots\!54}{72\!\cdots\!69}a^{14}+\frac{43\!\cdots\!48}{72\!\cdots\!69}a^{13}+\frac{25\!\cdots\!11}{72\!\cdots\!69}a^{12}-\frac{37\!\cdots\!96}{72\!\cdots\!69}a^{11}-\frac{48\!\cdots\!53}{72\!\cdots\!69}a^{10}+\frac{80\!\cdots\!60}{72\!\cdots\!69}a^{9}+\frac{23\!\cdots\!52}{72\!\cdots\!69}a^{8}+\frac{24\!\cdots\!28}{10\!\cdots\!67}a^{7}-\frac{40\!\cdots\!47}{72\!\cdots\!69}a^{6}-\frac{78\!\cdots\!94}{72\!\cdots\!69}a^{5}+\frac{28\!\cdots\!73}{66\!\cdots\!79}a^{4}-\frac{36\!\cdots\!27}{72\!\cdots\!69}a^{3}-\frac{95\!\cdots\!92}{72\!\cdots\!69}a^{2}+\frac{58\!\cdots\!39}{72\!\cdots\!69}a-\frac{47\!\cdots\!94}{72\!\cdots\!69}$, $\frac{43\!\cdots\!65}{72\!\cdots\!69}a^{25}-\frac{12\!\cdots\!99}{72\!\cdots\!69}a^{24}-\frac{15\!\cdots\!37}{10\!\cdots\!67}a^{23}+\frac{86\!\cdots\!37}{72\!\cdots\!69}a^{22}-\frac{15\!\cdots\!87}{10\!\cdots\!67}a^{21}-\frac{98\!\cdots\!58}{72\!\cdots\!69}a^{20}+\frac{51\!\cdots\!06}{72\!\cdots\!69}a^{19}-\frac{11\!\cdots\!09}{72\!\cdots\!69}a^{18}+\frac{28\!\cdots\!85}{72\!\cdots\!69}a^{17}-\frac{45\!\cdots\!42}{72\!\cdots\!69}a^{16}+\frac{22\!\cdots\!24}{66\!\cdots\!79}a^{15}-\frac{27\!\cdots\!12}{72\!\cdots\!69}a^{14}+\frac{23\!\cdots\!22}{10\!\cdots\!67}a^{13}+\frac{35\!\cdots\!90}{72\!\cdots\!69}a^{12}-\frac{39\!\cdots\!86}{72\!\cdots\!69}a^{11}-\frac{39\!\cdots\!55}{72\!\cdots\!69}a^{10}+\frac{22\!\cdots\!54}{10\!\cdots\!67}a^{9}+\frac{18\!\cdots\!83}{72\!\cdots\!69}a^{8}+\frac{13\!\cdots\!97}{10\!\cdots\!67}a^{7}-\frac{61\!\cdots\!67}{72\!\cdots\!69}a^{6}+\frac{37\!\cdots\!05}{72\!\cdots\!69}a^{5}+\frac{70\!\cdots\!01}{72\!\cdots\!69}a^{4}-\frac{54\!\cdots\!09}{10\!\cdots\!67}a^{3}-\frac{22\!\cdots\!84}{10\!\cdots\!67}a^{2}+\frac{70\!\cdots\!94}{72\!\cdots\!69}a-\frac{42\!\cdots\!83}{72\!\cdots\!69}$, $\frac{18\!\cdots\!01}{72\!\cdots\!69}a^{25}-\frac{38\!\cdots\!95}{72\!\cdots\!69}a^{24}-\frac{69\!\cdots\!94}{72\!\cdots\!69}a^{23}+\frac{30\!\cdots\!36}{72\!\cdots\!69}a^{22}-\frac{32\!\cdots\!65}{10\!\cdots\!67}a^{21}-\frac{53\!\cdots\!04}{72\!\cdots\!69}a^{20}+\frac{16\!\cdots\!42}{72\!\cdots\!69}a^{19}-\frac{35\!\cdots\!91}{72\!\cdots\!69}a^{18}+\frac{12\!\cdots\!44}{94\!\cdots\!97}a^{17}-\frac{12\!\cdots\!26}{72\!\cdots\!69}a^{16}+\frac{28\!\cdots\!37}{72\!\cdots\!69}a^{15}-\frac{21\!\cdots\!79}{72\!\cdots\!69}a^{14}+\frac{87\!\cdots\!26}{72\!\cdots\!69}a^{13}+\frac{18\!\cdots\!66}{72\!\cdots\!69}a^{12}-\frac{20\!\cdots\!41}{72\!\cdots\!69}a^{11}-\frac{17\!\cdots\!67}{72\!\cdots\!69}a^{10}-\frac{85\!\cdots\!69}{72\!\cdots\!69}a^{9}+\frac{42\!\cdots\!57}{10\!\cdots\!67}a^{8}+\frac{95\!\cdots\!03}{10\!\cdots\!67}a^{7}+\frac{39\!\cdots\!34}{10\!\cdots\!67}a^{6}+\frac{19\!\cdots\!21}{72\!\cdots\!69}a^{5}+\frac{17\!\cdots\!04}{72\!\cdots\!69}a^{4}-\frac{20\!\cdots\!34}{72\!\cdots\!69}a^{3}-\frac{52\!\cdots\!47}{72\!\cdots\!69}a^{2}+\frac{25\!\cdots\!39}{72\!\cdots\!69}a-\frac{10\!\cdots\!30}{66\!\cdots\!79}$, $\frac{19\!\cdots\!03}{72\!\cdots\!69}a^{25}-\frac{12\!\cdots\!31}{72\!\cdots\!69}a^{24}+\frac{10\!\cdots\!75}{10\!\cdots\!67}a^{23}+\frac{10\!\cdots\!14}{10\!\cdots\!67}a^{22}-\frac{15\!\cdots\!68}{72\!\cdots\!69}a^{21}-\frac{12\!\cdots\!34}{72\!\cdots\!69}a^{20}+\frac{47\!\cdots\!93}{72\!\cdots\!69}a^{19}-\frac{10\!\cdots\!90}{72\!\cdots\!69}a^{18}+\frac{23\!\cdots\!86}{72\!\cdots\!69}a^{17}-\frac{72\!\cdots\!31}{10\!\cdots\!67}a^{16}+\frac{43\!\cdots\!35}{72\!\cdots\!69}a^{15}+\frac{87\!\cdots\!64}{66\!\cdots\!79}a^{14}+\frac{14\!\cdots\!58}{72\!\cdots\!69}a^{13}-\frac{27\!\cdots\!13}{10\!\cdots\!67}a^{12}-\frac{10\!\cdots\!86}{72\!\cdots\!69}a^{11}-\frac{50\!\cdots\!63}{72\!\cdots\!69}a^{10}+\frac{73\!\cdots\!84}{66\!\cdots\!79}a^{9}+\frac{68\!\cdots\!60}{66\!\cdots\!79}a^{8}+\frac{93\!\cdots\!46}{72\!\cdots\!69}a^{7}-\frac{35\!\cdots\!21}{72\!\cdots\!69}a^{6}-\frac{19\!\cdots\!96}{66\!\cdots\!79}a^{5}-\frac{10\!\cdots\!99}{72\!\cdots\!69}a^{4}-\frac{12\!\cdots\!22}{72\!\cdots\!69}a^{3}-\frac{32\!\cdots\!30}{72\!\cdots\!69}a^{2}+\frac{28\!\cdots\!61}{72\!\cdots\!69}a+\frac{16\!\cdots\!56}{72\!\cdots\!69}$, $\frac{50\!\cdots\!15}{72\!\cdots\!69}a^{25}-\frac{81\!\cdots\!99}{72\!\cdots\!69}a^{24}-\frac{23\!\cdots\!21}{72\!\cdots\!69}a^{23}+\frac{71\!\cdots\!62}{72\!\cdots\!69}a^{22}-\frac{40\!\cdots\!36}{10\!\cdots\!67}a^{21}-\frac{15\!\cdots\!34}{72\!\cdots\!69}a^{20}+\frac{38\!\cdots\!47}{72\!\cdots\!69}a^{19}-\frac{80\!\cdots\!60}{72\!\cdots\!69}a^{18}+\frac{22\!\cdots\!16}{72\!\cdots\!69}a^{17}-\frac{24\!\cdots\!19}{72\!\cdots\!69}a^{16}-\frac{22\!\cdots\!94}{66\!\cdots\!79}a^{15}-\frac{14\!\cdots\!71}{10\!\cdots\!67}a^{14}+\frac{18\!\cdots\!75}{72\!\cdots\!69}a^{13}+\frac{88\!\cdots\!86}{10\!\cdots\!67}a^{12}+\frac{41\!\cdots\!96}{10\!\cdots\!67}a^{11}-\frac{28\!\cdots\!50}{72\!\cdots\!69}a^{10}-\frac{37\!\cdots\!42}{72\!\cdots\!69}a^{9}-\frac{17\!\cdots\!54}{72\!\cdots\!69}a^{8}+\frac{58\!\cdots\!68}{72\!\cdots\!69}a^{7}+\frac{10\!\cdots\!01}{72\!\cdots\!69}a^{6}+\frac{13\!\cdots\!91}{72\!\cdots\!69}a^{5}+\frac{13\!\cdots\!63}{72\!\cdots\!69}a^{4}+\frac{66\!\cdots\!47}{72\!\cdots\!69}a^{3}+\frac{22\!\cdots\!02}{72\!\cdots\!69}a^{2}+\frac{11\!\cdots\!10}{72\!\cdots\!69}a-\frac{31\!\cdots\!87}{72\!\cdots\!69}$, $\frac{15\!\cdots\!21}{66\!\cdots\!79}a^{25}-\frac{17\!\cdots\!43}{72\!\cdots\!69}a^{24}-\frac{96\!\cdots\!13}{72\!\cdots\!69}a^{23}+\frac{19\!\cdots\!86}{72\!\cdots\!69}a^{22}+\frac{69\!\cdots\!57}{72\!\cdots\!69}a^{21}-\frac{61\!\cdots\!02}{72\!\cdots\!69}a^{20}+\frac{13\!\cdots\!32}{10\!\cdots\!67}a^{19}-\frac{25\!\cdots\!42}{10\!\cdots\!67}a^{18}+\frac{58\!\cdots\!51}{72\!\cdots\!69}a^{17}-\frac{33\!\cdots\!09}{66\!\cdots\!79}a^{16}-\frac{62\!\cdots\!21}{72\!\cdots\!69}a^{15}-\frac{45\!\cdots\!81}{72\!\cdots\!69}a^{14}+\frac{11\!\cdots\!05}{10\!\cdots\!67}a^{13}+\frac{23\!\cdots\!18}{72\!\cdots\!69}a^{12}+\frac{20\!\cdots\!45}{72\!\cdots\!69}a^{11}-\frac{11\!\cdots\!40}{72\!\cdots\!69}a^{10}-\frac{25\!\cdots\!75}{72\!\cdots\!69}a^{9}-\frac{99\!\cdots\!68}{72\!\cdots\!69}a^{8}+\frac{55\!\cdots\!37}{72\!\cdots\!69}a^{7}+\frac{12\!\cdots\!85}{10\!\cdots\!67}a^{6}+\frac{58\!\cdots\!01}{72\!\cdots\!69}a^{5}+\frac{44\!\cdots\!23}{72\!\cdots\!69}a^{4}+\frac{29\!\cdots\!40}{72\!\cdots\!69}a^{3}+\frac{22\!\cdots\!76}{72\!\cdots\!69}a^{2}-\frac{25\!\cdots\!66}{72\!\cdots\!69}a+\frac{64\!\cdots\!59}{72\!\cdots\!69}$, $\frac{21\!\cdots\!98}{72\!\cdots\!69}a^{25}-\frac{42\!\cdots\!95}{72\!\cdots\!69}a^{24}-\frac{88\!\cdots\!85}{72\!\cdots\!69}a^{23}+\frac{34\!\cdots\!95}{72\!\cdots\!69}a^{22}-\frac{22\!\cdots\!15}{72\!\cdots\!69}a^{21}-\frac{63\!\cdots\!15}{72\!\cdots\!69}a^{20}+\frac{18\!\cdots\!49}{72\!\cdots\!69}a^{19}-\frac{40\!\cdots\!42}{72\!\cdots\!69}a^{18}+\frac{10\!\cdots\!54}{72\!\cdots\!69}a^{17}-\frac{13\!\cdots\!97}{72\!\cdots\!69}a^{16}+\frac{35\!\cdots\!73}{10\!\cdots\!67}a^{15}-\frac{42\!\cdots\!38}{72\!\cdots\!69}a^{14}+\frac{99\!\cdots\!40}{72\!\cdots\!69}a^{13}+\frac{23\!\cdots\!21}{72\!\cdots\!69}a^{12}+\frac{49\!\cdots\!82}{72\!\cdots\!69}a^{11}-\frac{24\!\cdots\!11}{10\!\cdots\!67}a^{10}-\frac{11\!\cdots\!18}{72\!\cdots\!69}a^{9}-\frac{30\!\cdots\!57}{72\!\cdots\!69}a^{8}+\frac{48\!\cdots\!23}{72\!\cdots\!69}a^{7}+\frac{39\!\cdots\!98}{72\!\cdots\!69}a^{6}+\frac{52\!\cdots\!92}{72\!\cdots\!69}a^{5}+\frac{48\!\cdots\!37}{94\!\cdots\!97}a^{4}+\frac{84\!\cdots\!83}{72\!\cdots\!69}a^{3}-\frac{86\!\cdots\!90}{72\!\cdots\!69}a^{2}+\frac{20\!\cdots\!31}{72\!\cdots\!69}a-\frac{16\!\cdots\!03}{72\!\cdots\!69}$, $\frac{20\!\cdots\!57}{72\!\cdots\!69}a^{25}-\frac{51\!\cdots\!69}{72\!\cdots\!69}a^{24}-\frac{74\!\cdots\!79}{72\!\cdots\!69}a^{23}+\frac{39\!\cdots\!20}{72\!\cdots\!69}a^{22}-\frac{34\!\cdots\!55}{72\!\cdots\!69}a^{21}-\frac{66\!\cdots\!11}{72\!\cdots\!69}a^{20}+\frac{20\!\cdots\!52}{66\!\cdots\!79}a^{19}-\frac{64\!\cdots\!70}{10\!\cdots\!67}a^{18}+\frac{11\!\cdots\!03}{72\!\cdots\!69}a^{17}-\frac{16\!\cdots\!84}{72\!\cdots\!69}a^{16}+\frac{42\!\cdots\!31}{72\!\cdots\!69}a^{15}+\frac{10\!\cdots\!75}{72\!\cdots\!69}a^{14}+\frac{97\!\cdots\!07}{66\!\cdots\!79}a^{13}+\frac{19\!\cdots\!85}{72\!\cdots\!69}a^{12}-\frac{12\!\cdots\!79}{72\!\cdots\!69}a^{11}-\frac{27\!\cdots\!55}{72\!\cdots\!69}a^{10}-\frac{33\!\cdots\!37}{72\!\cdots\!69}a^{9}+\frac{14\!\cdots\!10}{72\!\cdots\!69}a^{8}+\frac{16\!\cdots\!68}{10\!\cdots\!67}a^{7}-\frac{14\!\cdots\!48}{72\!\cdots\!69}a^{6}-\frac{26\!\cdots\!78}{72\!\cdots\!69}a^{5}-\frac{16\!\cdots\!44}{72\!\cdots\!69}a^{4}-\frac{86\!\cdots\!42}{72\!\cdots\!69}a^{3}-\frac{61\!\cdots\!61}{66\!\cdots\!79}a^{2}+\frac{23\!\cdots\!55}{72\!\cdots\!69}a+\frac{18\!\cdots\!31}{72\!\cdots\!69}$, $\frac{41\!\cdots\!19}{17\!\cdots\!09}a^{25}-\frac{85\!\cdots\!53}{17\!\cdots\!09}a^{24}-\frac{17\!\cdots\!18}{17\!\cdots\!09}a^{23}+\frac{69\!\cdots\!76}{17\!\cdots\!09}a^{22}-\frac{43\!\cdots\!78}{17\!\cdots\!09}a^{21}-\frac{13\!\cdots\!88}{17\!\cdots\!09}a^{20}+\frac{34\!\cdots\!62}{16\!\cdots\!19}a^{19}-\frac{75\!\cdots\!76}{17\!\cdots\!09}a^{18}+\frac{20\!\cdots\!52}{17\!\cdots\!09}a^{17}-\frac{26\!\cdots\!85}{17\!\cdots\!09}a^{16}+\frac{12\!\cdots\!58}{17\!\cdots\!09}a^{15}-\frac{14\!\cdots\!87}{17\!\cdots\!09}a^{14}+\frac{21\!\cdots\!00}{16\!\cdots\!19}a^{13}+\frac{40\!\cdots\!14}{17\!\cdots\!09}a^{12}-\frac{14\!\cdots\!66}{17\!\cdots\!09}a^{11}-\frac{55\!\cdots\!46}{17\!\cdots\!09}a^{10}-\frac{21\!\cdots\!61}{17\!\cdots\!09}a^{9}+\frac{12\!\cdots\!74}{17\!\cdots\!09}a^{8}+\frac{24\!\cdots\!38}{17\!\cdots\!09}a^{7}+\frac{59\!\cdots\!01}{17\!\cdots\!09}a^{6}+\frac{11\!\cdots\!90}{17\!\cdots\!09}a^{5}+\frac{19\!\cdots\!00}{17\!\cdots\!09}a^{4}-\frac{15\!\cdots\!98}{17\!\cdots\!09}a^{3}-\frac{22\!\cdots\!05}{16\!\cdots\!19}a^{2}-\frac{29\!\cdots\!47}{17\!\cdots\!09}a-\frac{56\!\cdots\!23}{17\!\cdots\!09}$, $\frac{83\!\cdots\!56}{94\!\cdots\!97}a^{25}-\frac{12\!\cdots\!62}{72\!\cdots\!69}a^{24}-\frac{32\!\cdots\!20}{72\!\cdots\!69}a^{23}+\frac{11\!\cdots\!07}{72\!\cdots\!69}a^{22}-\frac{36\!\cdots\!57}{72\!\cdots\!69}a^{21}-\frac{29\!\cdots\!50}{72\!\cdots\!69}a^{20}+\frac{61\!\cdots\!01}{72\!\cdots\!69}a^{19}-\frac{97\!\cdots\!72}{72\!\cdots\!69}a^{18}+\frac{26\!\cdots\!10}{72\!\cdots\!69}a^{17}-\frac{25\!\cdots\!73}{66\!\cdots\!79}a^{16}-\frac{24\!\cdots\!07}{72\!\cdots\!69}a^{15}+\frac{35\!\cdots\!82}{10\!\cdots\!67}a^{14}+\frac{27\!\cdots\!01}{72\!\cdots\!69}a^{13}+\frac{74\!\cdots\!56}{72\!\cdots\!69}a^{12}-\frac{70\!\cdots\!41}{72\!\cdots\!69}a^{11}-\frac{11\!\cdots\!36}{72\!\cdots\!69}a^{10}-\frac{49\!\cdots\!58}{72\!\cdots\!69}a^{9}+\frac{51\!\cdots\!08}{72\!\cdots\!69}a^{8}+\frac{45\!\cdots\!54}{72\!\cdots\!69}a^{7}+\frac{19\!\cdots\!35}{10\!\cdots\!67}a^{6}+\frac{15\!\cdots\!38}{72\!\cdots\!69}a^{5}-\frac{72\!\cdots\!33}{72\!\cdots\!69}a^{4}-\frac{10\!\cdots\!22}{72\!\cdots\!69}a^{3}-\frac{50\!\cdots\!81}{72\!\cdots\!69}a^{2}-\frac{58\!\cdots\!28}{72\!\cdots\!69}a-\frac{10\!\cdots\!49}{72\!\cdots\!69}$, $\frac{86\!\cdots\!14}{72\!\cdots\!69}a^{25}-\frac{12\!\cdots\!15}{72\!\cdots\!69}a^{24}-\frac{46\!\cdots\!42}{72\!\cdots\!69}a^{23}+\frac{12\!\cdots\!32}{72\!\cdots\!69}a^{22}-\frac{12\!\cdots\!54}{72\!\cdots\!69}a^{21}-\frac{49\!\cdots\!39}{10\!\cdots\!67}a^{20}+\frac{67\!\cdots\!65}{72\!\cdots\!69}a^{19}-\frac{11\!\cdots\!26}{72\!\cdots\!69}a^{18}+\frac{32\!\cdots\!76}{72\!\cdots\!69}a^{17}-\frac{26\!\cdots\!44}{72\!\cdots\!69}a^{16}-\frac{31\!\cdots\!06}{66\!\cdots\!79}a^{15}+\frac{12\!\cdots\!88}{72\!\cdots\!69}a^{14}+\frac{19\!\cdots\!03}{72\!\cdots\!69}a^{13}+\frac{12\!\cdots\!01}{72\!\cdots\!69}a^{12}+\frac{53\!\cdots\!83}{72\!\cdots\!69}a^{11}-\frac{94\!\cdots\!52}{72\!\cdots\!69}a^{10}-\frac{73\!\cdots\!55}{72\!\cdots\!69}a^{9}-\frac{15\!\cdots\!21}{72\!\cdots\!69}a^{8}+\frac{26\!\cdots\!35}{72\!\cdots\!69}a^{7}+\frac{35\!\cdots\!49}{72\!\cdots\!69}a^{6}+\frac{15\!\cdots\!84}{72\!\cdots\!69}a^{5}+\frac{13\!\cdots\!81}{72\!\cdots\!69}a^{4}+\frac{63\!\cdots\!51}{72\!\cdots\!69}a^{3}+\frac{21\!\cdots\!55}{72\!\cdots\!69}a^{2}+\frac{11\!\cdots\!06}{72\!\cdots\!69}a-\frac{10\!\cdots\!66}{72\!\cdots\!69}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 62790605.5500462 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 62790605.5500462 \cdot 1}{2\cdot\sqrt{2877467739962384875567767188720703125}}\cr\approx \mathstrut & 0.280270801626651 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{5}) \), 13.1.48551226272641.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 26 sibling: 26.0.549596338332815511233443533045654296875.1
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/7.2.0.1}{2} }^{13}$ ${\href{/padicField/11.2.0.1}{2} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $26$ $26$ ${\href{/padicField/19.2.0.1}{2} }^{12}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $26$ ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{12}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{13}$ ${\href{/padicField/41.2.0.1}{2} }^{12}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $26$ ${\href{/padicField/47.2.0.1}{2} }^{13}$ ${\href{/padicField/53.2.0.1}{2} }^{13}$ ${\href{/padicField/59.13.0.1}{13} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $26$$2$$13$$13$
\(191\) Copy content Toggle raw display $\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.955.2t1.a.a$1$ $ 5 \cdot 191 $ \(\Q(\sqrt{-955}) \) $C_2$ (as 2T1) $1$ $-1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.4775.26t3.b.c$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.b.f$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.4775.26t3.b.b$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.b.d$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.4775.26t3.b.a$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.4775.26t3.b.e$2$ $ 5^{2} \cdot 191 $ 26.2.2877467739962384875567767188720703125.1 $D_{26}$ (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 *.