Properties

Label 26.26.191...128.1
Degree $26$
Signature $[26, 0]$
Discriminant $1.920\times 10^{57}$
Root discriminant \(159.66\)
Ramified primes $2,79$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{26}$ (as 26T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447)
 
gp: K = bnfinit(y^26 - 2*y^25 - 97*y^24 + 274*y^23 + 3600*y^22 - 12928*y^21 - 64015*y^20 + 293118*y^19 + 533589*y^18 - 3564090*y^17 - 1089717*y^16 + 23968180*y^15 - 13363361*y^14 - 87417128*y^13 + 103695959*y^12 + 151824110*y^11 - 299732504*y^10 - 49085562*y^9 + 375988637*y^8 - 168890324*y^7 - 147518415*y^6 + 153068810*y^5 - 33898995*y^4 - 8099910*y^3 + 3461310*y^2 - 50580*y - 35447, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 2*x^25 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 2*x^25 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447)
 

\( x^{26} - 2 x^{25} - 97 x^{24} + 274 x^{23} + 3600 x^{22} - 12928 x^{21} - 64015 x^{20} + 293118 x^{19} + 533589 x^{18} - 3564090 x^{17} - 1089717 x^{16} + 23968180 x^{15} + \cdots - 35447 \) 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:  $[26, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1919641021107487828653877081110544587155912070949008048128\) \(\medspace = 2^{39}\cdot 79^{24}\) 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:  \(159.66\)
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:  $2^{3/2}79^{12/13}\approx 159.66170770416778$
Ramified primes:   \(2\), \(79\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $26$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(632=2^{3}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{632}(1,·)$, $\chi_{632}(97,·)$, $\chi_{632}(65,·)$, $\chi_{632}(457,·)$, $\chi_{632}(141,·)$, $\chi_{632}(337,·)$, $\chi_{632}(405,·)$, $\chi_{632}(89,·)$, $\chi_{632}(413,·)$, $\chi_{632}(289,·)$, $\chi_{632}(101,·)$, $\chi_{632}(433,·)$, $\chi_{632}(617,·)$, $\chi_{632}(225,·)$, $\chi_{632}(125,·)$, $\chi_{632}(301,·)$, $\chi_{632}(541,·)$, $\chi_{632}(561,·)$, $\chi_{632}(117,·)$, $\chi_{632}(417,·)$, $\chi_{632}(381,·)$, $\chi_{632}(441,·)$, $\chi_{632}(21,·)$, $\chi_{632}(317,·)$, $\chi_{632}(605,·)$, $\chi_{632}(245,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23}a^{17}-\frac{6}{23}a^{15}-\frac{11}{23}a^{14}+\frac{11}{23}a^{13}-\frac{2}{23}a^{11}+\frac{3}{23}a^{10}-\frac{7}{23}a^{9}+\frac{1}{23}a^{8}-\frac{10}{23}a^{7}+\frac{7}{23}a^{6}-\frac{11}{23}a^{5}+\frac{1}{23}a^{4}-\frac{8}{23}a^{3}-\frac{5}{23}a^{2}+\frac{9}{23}a+\frac{4}{23}$, $\frac{1}{23}a^{18}-\frac{6}{23}a^{16}-\frac{11}{23}a^{15}+\frac{11}{23}a^{14}-\frac{2}{23}a^{12}+\frac{3}{23}a^{11}-\frac{7}{23}a^{10}+\frac{1}{23}a^{9}-\frac{10}{23}a^{8}+\frac{7}{23}a^{7}-\frac{11}{23}a^{6}+\frac{1}{23}a^{5}-\frac{8}{23}a^{4}-\frac{5}{23}a^{3}+\frac{9}{23}a^{2}+\frac{4}{23}a$, $\frac{1}{23}a^{19}-\frac{11}{23}a^{16}-\frac{2}{23}a^{15}+\frac{3}{23}a^{14}-\frac{5}{23}a^{13}+\frac{3}{23}a^{12}+\frac{4}{23}a^{11}-\frac{4}{23}a^{10}-\frac{6}{23}a^{9}-\frac{10}{23}a^{8}-\frac{2}{23}a^{7}-\frac{3}{23}a^{6}-\frac{5}{23}a^{5}+\frac{1}{23}a^{4}+\frac{7}{23}a^{3}-\frac{3}{23}a^{2}+\frac{8}{23}a+\frac{1}{23}$, $\frac{1}{23}a^{20}-\frac{2}{23}a^{16}+\frac{6}{23}a^{15}-\frac{11}{23}a^{14}+\frac{9}{23}a^{13}+\frac{4}{23}a^{12}-\frac{3}{23}a^{11}+\frac{4}{23}a^{10}+\frac{5}{23}a^{9}+\frac{9}{23}a^{8}+\frac{2}{23}a^{7}+\frac{3}{23}a^{6}-\frac{5}{23}a^{5}-\frac{5}{23}a^{4}+\frac{1}{23}a^{3}-\frac{1}{23}a^{2}+\frac{8}{23}a-\frac{2}{23}$, $\frac{1}{23}a^{21}+\frac{6}{23}a^{16}+\frac{10}{23}a^{14}+\frac{3}{23}a^{13}-\frac{3}{23}a^{12}+\frac{11}{23}a^{10}-\frac{5}{23}a^{9}+\frac{4}{23}a^{8}+\frac{6}{23}a^{7}+\frac{9}{23}a^{6}-\frac{4}{23}a^{5}+\frac{3}{23}a^{4}+\frac{6}{23}a^{3}-\frac{2}{23}a^{2}-\frac{7}{23}a+\frac{8}{23}$, $\frac{1}{2369}a^{22}-\frac{17}{2369}a^{21}+\frac{4}{2369}a^{20}+\frac{9}{2369}a^{19}+\frac{1}{103}a^{18}+\frac{33}{2369}a^{17}-\frac{1014}{2369}a^{16}-\frac{882}{2369}a^{15}+\frac{830}{2369}a^{14}-\frac{893}{2369}a^{13}+\frac{623}{2369}a^{12}+\frac{326}{2369}a^{11}-\frac{453}{2369}a^{10}-\frac{1031}{2369}a^{9}-\frac{664}{2369}a^{8}-\frac{1017}{2369}a^{7}+\frac{1144}{2369}a^{6}+\frac{951}{2369}a^{5}+\frac{293}{2369}a^{4}-\frac{48}{103}a^{3}+\frac{436}{2369}a^{2}-\frac{745}{2369}a+\frac{1054}{2369}$, $\frac{1}{2369}a^{23}+\frac{24}{2369}a^{21}-\frac{26}{2369}a^{20}-\frac{30}{2369}a^{19}+\frac{12}{2369}a^{18}-\frac{41}{2369}a^{17}+\frac{523}{2369}a^{16}-\frac{465}{2369}a^{15}+\frac{651}{2369}a^{14}+\frac{480}{2369}a^{13}+\frac{308}{2369}a^{12}+\frac{145}{2369}a^{11}-\frac{801}{2369}a^{10}-\frac{990}{2369}a^{9}-\frac{666}{2369}a^{8}+\frac{232}{2369}a^{7}+\frac{108}{2369}a^{6}-\frac{20}{2369}a^{5}-\frac{655}{2369}a^{4}-\frac{307}{2369}a^{3}+\frac{1002}{2369}a^{2}+\frac{28}{2369}a+\frac{717}{2369}$, $\frac{1}{1253201}a^{24}+\frac{185}{1253201}a^{23}-\frac{31}{1253201}a^{22}+\frac{7512}{1253201}a^{21}-\frac{26072}{1253201}a^{20}-\frac{12522}{1253201}a^{19}-\frac{16081}{1253201}a^{18}-\frac{26902}{1253201}a^{17}-\frac{188149}{1253201}a^{16}+\frac{457845}{1253201}a^{15}-\frac{618234}{1253201}a^{14}+\frac{211456}{1253201}a^{13}-\frac{60261}{1253201}a^{12}-\frac{370122}{1253201}a^{11}+\frac{304323}{1253201}a^{10}+\frac{35423}{1253201}a^{9}-\frac{296475}{1253201}a^{8}-\frac{180373}{1253201}a^{7}+\frac{147693}{1253201}a^{6}+\frac{615930}{1253201}a^{5}+\frac{300977}{1253201}a^{4}+\frac{213811}{1253201}a^{3}-\frac{478830}{1253201}a^{2}+\frac{623878}{1253201}a-\frac{5925}{12167}$, $\frac{1}{44\!\cdots\!29}a^{25}+\frac{74\!\cdots\!81}{19\!\cdots\!23}a^{24}-\frac{72\!\cdots\!32}{44\!\cdots\!29}a^{23}+\frac{86\!\cdots\!95}{44\!\cdots\!29}a^{22}+\frac{89\!\cdots\!20}{44\!\cdots\!29}a^{21}-\frac{43\!\cdots\!89}{44\!\cdots\!29}a^{20}-\frac{53\!\cdots\!03}{44\!\cdots\!29}a^{19}+\frac{13\!\cdots\!79}{44\!\cdots\!29}a^{18}+\frac{34\!\cdots\!56}{44\!\cdots\!29}a^{17}-\frac{20\!\cdots\!95}{44\!\cdots\!29}a^{16}-\frac{11\!\cdots\!73}{44\!\cdots\!29}a^{15}-\frac{84\!\cdots\!09}{44\!\cdots\!29}a^{14}-\frac{10\!\cdots\!13}{44\!\cdots\!29}a^{13}-\frac{11\!\cdots\!96}{44\!\cdots\!29}a^{12}-\frac{21\!\cdots\!15}{44\!\cdots\!29}a^{11}+\frac{15\!\cdots\!23}{44\!\cdots\!29}a^{10}-\frac{15\!\cdots\!19}{44\!\cdots\!29}a^{9}+\frac{15\!\cdots\!53}{44\!\cdots\!29}a^{8}-\frac{99\!\cdots\!71}{44\!\cdots\!29}a^{7}+\frac{17\!\cdots\!29}{44\!\cdots\!29}a^{6}+\frac{20\!\cdots\!28}{44\!\cdots\!29}a^{5}-\frac{28\!\cdots\!52}{44\!\cdots\!29}a^{4}+\frac{19\!\cdots\!98}{44\!\cdots\!29}a^{3}+\frac{31\!\cdots\!21}{44\!\cdots\!29}a^{2}-\frac{30\!\cdots\!68}{44\!\cdots\!29}a+\frac{22\!\cdots\!12}{44\!\cdots\!29}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $23$

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:  $25$
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:   $\frac{53\!\cdots\!12}{84\!\cdots\!01}a^{25}-\frac{73\!\cdots\!90}{84\!\cdots\!01}a^{24}-\frac{52\!\cdots\!50}{84\!\cdots\!01}a^{23}+\frac{11\!\cdots\!83}{84\!\cdots\!01}a^{22}+\frac{20\!\cdots\!26}{84\!\cdots\!01}a^{21}-\frac{56\!\cdots\!25}{84\!\cdots\!01}a^{20}-\frac{38\!\cdots\!96}{84\!\cdots\!01}a^{19}+\frac{13\!\cdots\!63}{84\!\cdots\!01}a^{18}+\frac{37\!\cdots\!20}{84\!\cdots\!01}a^{17}-\frac{16\!\cdots\!50}{84\!\cdots\!01}a^{16}-\frac{16\!\cdots\!06}{84\!\cdots\!01}a^{15}+\frac{11\!\cdots\!37}{84\!\cdots\!01}a^{14}+\frac{36\!\cdots\!82}{84\!\cdots\!01}a^{13}-\frac{46\!\cdots\!80}{84\!\cdots\!01}a^{12}+\frac{25\!\cdots\!84}{84\!\cdots\!01}a^{11}+\frac{98\!\cdots\!82}{84\!\cdots\!01}a^{10}-\frac{98\!\cdots\!20}{84\!\cdots\!01}a^{9}-\frac{89\!\cdots\!21}{84\!\cdots\!01}a^{8}+\frac{63\!\cdots\!54}{36\!\cdots\!87}a^{7}+\frac{18\!\cdots\!58}{84\!\cdots\!01}a^{6}-\frac{78\!\cdots\!62}{84\!\cdots\!01}a^{5}+\frac{32\!\cdots\!93}{84\!\cdots\!01}a^{4}+\frac{24\!\cdots\!42}{84\!\cdots\!01}a^{3}-\frac{27\!\cdots\!32}{84\!\cdots\!01}a^{2}+\frac{89\!\cdots\!53}{84\!\cdots\!01}a+\frac{29\!\cdots\!07}{84\!\cdots\!01}$, $\frac{10\!\cdots\!44}{44\!\cdots\!29}a^{25}-\frac{14\!\cdots\!95}{44\!\cdots\!29}a^{24}-\frac{10\!\cdots\!22}{44\!\cdots\!29}a^{23}+\frac{23\!\cdots\!32}{44\!\cdots\!29}a^{22}+\frac{40\!\cdots\!86}{44\!\cdots\!29}a^{21}-\frac{11\!\cdots\!64}{44\!\cdots\!29}a^{20}-\frac{77\!\cdots\!20}{44\!\cdots\!29}a^{19}+\frac{27\!\cdots\!76}{44\!\cdots\!29}a^{18}+\frac{32\!\cdots\!46}{19\!\cdots\!23}a^{17}-\frac{34\!\cdots\!26}{44\!\cdots\!29}a^{16}-\frac{33\!\cdots\!06}{44\!\cdots\!29}a^{15}+\frac{24\!\cdots\!30}{44\!\cdots\!29}a^{14}+\frac{84\!\cdots\!64}{44\!\cdots\!29}a^{13}-\frac{94\!\cdots\!07}{44\!\cdots\!29}a^{12}+\frac{52\!\cdots\!36}{44\!\cdots\!29}a^{11}+\frac{19\!\cdots\!98}{44\!\cdots\!29}a^{10}-\frac{19\!\cdots\!76}{44\!\cdots\!29}a^{9}-\frac{18\!\cdots\!34}{44\!\cdots\!29}a^{8}+\frac{29\!\cdots\!14}{44\!\cdots\!29}a^{7}+\frac{45\!\cdots\!27}{44\!\cdots\!29}a^{6}-\frac{15\!\cdots\!50}{44\!\cdots\!29}a^{5}+\frac{65\!\cdots\!04}{44\!\cdots\!29}a^{4}+\frac{50\!\cdots\!24}{44\!\cdots\!29}a^{3}-\frac{55\!\cdots\!84}{44\!\cdots\!29}a^{2}+\frac{17\!\cdots\!10}{44\!\cdots\!29}a+\frac{60\!\cdots\!71}{44\!\cdots\!29}$, $\frac{19\!\cdots\!06}{44\!\cdots\!29}a^{25}-\frac{25\!\cdots\!56}{44\!\cdots\!29}a^{24}-\frac{18\!\cdots\!60}{44\!\cdots\!29}a^{23}+\frac{40\!\cdots\!51}{44\!\cdots\!29}a^{22}+\frac{72\!\cdots\!22}{44\!\cdots\!29}a^{21}-\frac{20\!\cdots\!15}{44\!\cdots\!29}a^{20}-\frac{13\!\cdots\!26}{44\!\cdots\!29}a^{19}+\frac{47\!\cdots\!02}{44\!\cdots\!29}a^{18}+\frac{58\!\cdots\!24}{19\!\cdots\!23}a^{17}-\frac{59\!\cdots\!56}{44\!\cdots\!29}a^{16}-\frac{60\!\cdots\!78}{44\!\cdots\!29}a^{15}+\frac{42\!\cdots\!28}{44\!\cdots\!29}a^{14}+\frac{22\!\cdots\!90}{44\!\cdots\!29}a^{13}-\frac{16\!\cdots\!30}{44\!\cdots\!29}a^{12}+\frac{89\!\cdots\!14}{44\!\cdots\!29}a^{11}+\frac{35\!\cdots\!08}{44\!\cdots\!29}a^{10}-\frac{34\!\cdots\!48}{44\!\cdots\!29}a^{9}-\frac{32\!\cdots\!57}{44\!\cdots\!29}a^{8}+\frac{51\!\cdots\!60}{44\!\cdots\!29}a^{7}+\frac{13\!\cdots\!33}{44\!\cdots\!29}a^{6}-\frac{27\!\cdots\!20}{44\!\cdots\!29}a^{5}+\frac{11\!\cdots\!27}{44\!\cdots\!29}a^{4}+\frac{90\!\cdots\!06}{44\!\cdots\!29}a^{3}-\frac{95\!\cdots\!57}{44\!\cdots\!29}a^{2}+\frac{30\!\cdots\!98}{44\!\cdots\!29}a+\frac{10\!\cdots\!39}{44\!\cdots\!29}$, $\frac{26\!\cdots\!58}{44\!\cdots\!29}a^{25}-\frac{35\!\cdots\!05}{44\!\cdots\!29}a^{24}-\frac{26\!\cdots\!90}{44\!\cdots\!29}a^{23}+\frac{55\!\cdots\!32}{44\!\cdots\!29}a^{22}+\frac{10\!\cdots\!14}{44\!\cdots\!29}a^{21}-\frac{27\!\cdots\!05}{44\!\cdots\!29}a^{20}-\frac{18\!\cdots\!74}{44\!\cdots\!29}a^{19}+\frac{65\!\cdots\!37}{44\!\cdots\!29}a^{18}+\frac{81\!\cdots\!30}{19\!\cdots\!23}a^{17}-\frac{82\!\cdots\!95}{44\!\cdots\!29}a^{16}-\frac{84\!\cdots\!04}{44\!\cdots\!29}a^{15}+\frac{58\!\cdots\!38}{44\!\cdots\!29}a^{14}+\frac{32\!\cdots\!78}{44\!\cdots\!29}a^{13}-\frac{23\!\cdots\!95}{44\!\cdots\!29}a^{12}+\frac{12\!\cdots\!86}{44\!\cdots\!29}a^{11}+\frac{48\!\cdots\!53}{44\!\cdots\!29}a^{10}-\frac{47\!\cdots\!30}{44\!\cdots\!29}a^{9}-\frac{44\!\cdots\!44}{44\!\cdots\!29}a^{8}+\frac{70\!\cdots\!58}{44\!\cdots\!29}a^{7}+\frac{19\!\cdots\!02}{44\!\cdots\!29}a^{6}-\frac{38\!\cdots\!24}{44\!\cdots\!29}a^{5}+\frac{15\!\cdots\!57}{44\!\cdots\!29}a^{4}+\frac{12\!\cdots\!48}{44\!\cdots\!29}a^{3}-\frac{13\!\cdots\!08}{44\!\cdots\!29}a^{2}+\frac{41\!\cdots\!02}{44\!\cdots\!29}a+\frac{14\!\cdots\!59}{44\!\cdots\!29}$, $\frac{47\!\cdots\!06}{44\!\cdots\!29}a^{25}-\frac{64\!\cdots\!93}{44\!\cdots\!29}a^{24}-\frac{46\!\cdots\!66}{44\!\cdots\!29}a^{23}+\frac{10\!\cdots\!51}{44\!\cdots\!29}a^{22}+\frac{17\!\cdots\!10}{44\!\cdots\!29}a^{21}-\frac{49\!\cdots\!91}{44\!\cdots\!29}a^{20}-\frac{33\!\cdots\!78}{44\!\cdots\!29}a^{19}+\frac{11\!\cdots\!80}{44\!\cdots\!29}a^{18}+\frac{14\!\cdots\!32}{19\!\cdots\!23}a^{17}-\frac{14\!\cdots\!41}{44\!\cdots\!29}a^{16}-\frac{14\!\cdots\!20}{44\!\cdots\!29}a^{15}+\frac{10\!\cdots\!19}{44\!\cdots\!29}a^{14}+\frac{43\!\cdots\!00}{44\!\cdots\!29}a^{13}-\frac{41\!\cdots\!22}{44\!\cdots\!29}a^{12}+\frac{22\!\cdots\!82}{44\!\cdots\!29}a^{11}+\frac{86\!\cdots\!87}{44\!\cdots\!29}a^{10}-\frac{86\!\cdots\!42}{44\!\cdots\!29}a^{9}-\frac{79\!\cdots\!60}{44\!\cdots\!29}a^{8}+\frac{12\!\cdots\!06}{44\!\cdots\!29}a^{7}+\frac{24\!\cdots\!89}{44\!\cdots\!29}a^{6}-\frac{68\!\cdots\!32}{44\!\cdots\!29}a^{5}+\frac{28\!\cdots\!17}{44\!\cdots\!29}a^{4}+\frac{22\!\cdots\!10}{44\!\cdots\!29}a^{3}-\frac{24\!\cdots\!48}{44\!\cdots\!29}a^{2}+\frac{76\!\cdots\!00}{44\!\cdots\!29}a+\frac{25\!\cdots\!85}{44\!\cdots\!29}$, $\frac{61\!\cdots\!56}{44\!\cdots\!29}a^{25}-\frac{82\!\cdots\!58}{44\!\cdots\!29}a^{24}-\frac{60\!\cdots\!96}{44\!\cdots\!29}a^{23}+\frac{12\!\cdots\!36}{44\!\cdots\!29}a^{22}+\frac{22\!\cdots\!78}{44\!\cdots\!29}a^{21}-\frac{64\!\cdots\!56}{44\!\cdots\!29}a^{20}-\frac{43\!\cdots\!02}{44\!\cdots\!29}a^{19}+\frac{15\!\cdots\!00}{44\!\cdots\!29}a^{18}+\frac{18\!\cdots\!82}{19\!\cdots\!23}a^{17}-\frac{19\!\cdots\!68}{44\!\cdots\!29}a^{16}-\frac{19\!\cdots\!18}{44\!\cdots\!29}a^{15}+\frac{13\!\cdots\!59}{44\!\cdots\!29}a^{14}+\frac{53\!\cdots\!12}{44\!\cdots\!29}a^{13}-\frac{53\!\cdots\!88}{44\!\cdots\!29}a^{12}+\frac{29\!\cdots\!30}{44\!\cdots\!29}a^{11}+\frac{11\!\cdots\!02}{44\!\cdots\!29}a^{10}-\frac{11\!\cdots\!40}{44\!\cdots\!29}a^{9}-\frac{10\!\cdots\!48}{44\!\cdots\!29}a^{8}+\frac{16\!\cdots\!76}{44\!\cdots\!29}a^{7}+\frac{29\!\cdots\!60}{44\!\cdots\!29}a^{6}-\frac{88\!\cdots\!90}{44\!\cdots\!29}a^{5}+\frac{36\!\cdots\!88}{44\!\cdots\!29}a^{4}+\frac{28\!\cdots\!28}{44\!\cdots\!29}a^{3}-\frac{31\!\cdots\!22}{44\!\cdots\!29}a^{2}+\frac{99\!\cdots\!28}{44\!\cdots\!29}a+\frac{33\!\cdots\!48}{44\!\cdots\!29}$, $\frac{24\!\cdots\!68}{44\!\cdots\!29}a^{25}-\frac{32\!\cdots\!73}{44\!\cdots\!29}a^{24}-\frac{23\!\cdots\!74}{44\!\cdots\!29}a^{23}+\frac{50\!\cdots\!84}{44\!\cdots\!29}a^{22}+\frac{90\!\cdots\!04}{44\!\cdots\!29}a^{21}-\frac{25\!\cdots\!13}{44\!\cdots\!29}a^{20}-\frac{17\!\cdots\!44}{44\!\cdots\!29}a^{19}+\frac{59\!\cdots\!46}{44\!\cdots\!29}a^{18}+\frac{72\!\cdots\!42}{19\!\cdots\!23}a^{17}-\frac{75\!\cdots\!99}{44\!\cdots\!29}a^{16}-\frac{74\!\cdots\!28}{44\!\cdots\!29}a^{15}+\frac{52\!\cdots\!65}{44\!\cdots\!29}a^{14}+\frac{17\!\cdots\!88}{44\!\cdots\!29}a^{13}-\frac{20\!\cdots\!37}{44\!\cdots\!29}a^{12}+\frac{11\!\cdots\!30}{44\!\cdots\!29}a^{11}+\frac{43\!\cdots\!48}{44\!\cdots\!29}a^{10}-\frac{44\!\cdots\!34}{44\!\cdots\!29}a^{9}-\frac{40\!\cdots\!03}{44\!\cdots\!29}a^{8}+\frac{64\!\cdots\!84}{44\!\cdots\!29}a^{7}+\frac{90\!\cdots\!32}{44\!\cdots\!29}a^{6}-\frac{34\!\cdots\!44}{44\!\cdots\!29}a^{5}+\frac{14\!\cdots\!92}{44\!\cdots\!29}a^{4}+\frac{11\!\cdots\!36}{44\!\cdots\!29}a^{3}-\frac{12\!\cdots\!59}{44\!\cdots\!29}a^{2}+\frac{39\!\cdots\!76}{44\!\cdots\!29}a+\frac{13\!\cdots\!05}{44\!\cdots\!29}$, $\frac{53\!\cdots\!12}{84\!\cdots\!01}a^{25}-\frac{73\!\cdots\!90}{84\!\cdots\!01}a^{24}-\frac{52\!\cdots\!50}{84\!\cdots\!01}a^{23}+\frac{11\!\cdots\!83}{84\!\cdots\!01}a^{22}+\frac{20\!\cdots\!26}{84\!\cdots\!01}a^{21}-\frac{56\!\cdots\!25}{84\!\cdots\!01}a^{20}-\frac{38\!\cdots\!96}{84\!\cdots\!01}a^{19}+\frac{13\!\cdots\!63}{84\!\cdots\!01}a^{18}+\frac{37\!\cdots\!20}{84\!\cdots\!01}a^{17}-\frac{16\!\cdots\!50}{84\!\cdots\!01}a^{16}-\frac{16\!\cdots\!06}{84\!\cdots\!01}a^{15}+\frac{11\!\cdots\!37}{84\!\cdots\!01}a^{14}+\frac{36\!\cdots\!82}{84\!\cdots\!01}a^{13}-\frac{46\!\cdots\!80}{84\!\cdots\!01}a^{12}+\frac{25\!\cdots\!84}{84\!\cdots\!01}a^{11}+\frac{98\!\cdots\!82}{84\!\cdots\!01}a^{10}-\frac{98\!\cdots\!20}{84\!\cdots\!01}a^{9}-\frac{89\!\cdots\!21}{84\!\cdots\!01}a^{8}+\frac{63\!\cdots\!54}{36\!\cdots\!87}a^{7}+\frac{18\!\cdots\!58}{84\!\cdots\!01}a^{6}-\frac{78\!\cdots\!62}{84\!\cdots\!01}a^{5}+\frac{32\!\cdots\!93}{84\!\cdots\!01}a^{4}+\frac{24\!\cdots\!42}{84\!\cdots\!01}a^{3}-\frac{27\!\cdots\!32}{84\!\cdots\!01}a^{2}+\frac{89\!\cdots\!52}{84\!\cdots\!01}a+\frac{29\!\cdots\!10}{84\!\cdots\!01}$, $\frac{15\!\cdots\!50}{44\!\cdots\!29}a^{25}-\frac{20\!\cdots\!11}{44\!\cdots\!29}a^{24}-\frac{15\!\cdots\!12}{44\!\cdots\!29}a^{23}+\frac{32\!\cdots\!47}{44\!\cdots\!29}a^{22}+\frac{57\!\cdots\!50}{44\!\cdots\!29}a^{21}-\frac{16\!\cdots\!32}{44\!\cdots\!29}a^{20}-\frac{10\!\cdots\!42}{44\!\cdots\!29}a^{19}+\frac{38\!\cdots\!26}{44\!\cdots\!29}a^{18}+\frac{46\!\cdots\!94}{19\!\cdots\!23}a^{17}-\frac{48\!\cdots\!30}{44\!\cdots\!29}a^{16}-\frac{47\!\cdots\!20}{44\!\cdots\!29}a^{15}+\frac{33\!\cdots\!88}{44\!\cdots\!29}a^{14}+\frac{10\!\cdots\!84}{44\!\cdots\!29}a^{13}-\frac{13\!\cdots\!23}{44\!\cdots\!29}a^{12}+\frac{74\!\cdots\!24}{44\!\cdots\!29}a^{11}+\frac{28\!\cdots\!29}{44\!\cdots\!29}a^{10}-\frac{28\!\cdots\!20}{44\!\cdots\!29}a^{9}-\frac{25\!\cdots\!17}{44\!\cdots\!29}a^{8}+\frac{41\!\cdots\!06}{44\!\cdots\!29}a^{7}+\frac{53\!\cdots\!75}{44\!\cdots\!29}a^{6}-\frac{22\!\cdots\!16}{44\!\cdots\!29}a^{5}+\frac{92\!\cdots\!77}{44\!\cdots\!29}a^{4}+\frac{70\!\cdots\!14}{44\!\cdots\!29}a^{3}-\frac{79\!\cdots\!69}{44\!\cdots\!29}a^{2}+\frac{25\!\cdots\!84}{44\!\cdots\!29}a+\frac{85\!\cdots\!52}{44\!\cdots\!29}$, $\frac{77\!\cdots\!12}{44\!\cdots\!29}a^{25}-\frac{10\!\cdots\!08}{43\!\cdots\!43}a^{24}-\frac{75\!\cdots\!08}{44\!\cdots\!29}a^{23}+\frac{16\!\cdots\!14}{44\!\cdots\!29}a^{22}+\frac{28\!\cdots\!88}{44\!\cdots\!29}a^{21}-\frac{81\!\cdots\!34}{44\!\cdots\!29}a^{20}-\frac{54\!\cdots\!04}{44\!\cdots\!29}a^{19}+\frac{19\!\cdots\!52}{44\!\cdots\!29}a^{18}+\frac{23\!\cdots\!36}{19\!\cdots\!23}a^{17}-\frac{24\!\cdots\!87}{44\!\cdots\!29}a^{16}-\frac{24\!\cdots\!60}{44\!\cdots\!29}a^{15}+\frac{16\!\cdots\!36}{44\!\cdots\!29}a^{14}+\frac{73\!\cdots\!96}{44\!\cdots\!29}a^{13}-\frac{67\!\cdots\!56}{44\!\cdots\!29}a^{12}+\frac{36\!\cdots\!48}{44\!\cdots\!29}a^{11}+\frac{14\!\cdots\!24}{44\!\cdots\!29}a^{10}-\frac{13\!\cdots\!44}{44\!\cdots\!29}a^{9}-\frac{12\!\cdots\!56}{44\!\cdots\!29}a^{8}+\frac{20\!\cdots\!12}{44\!\cdots\!29}a^{7}+\frac{41\!\cdots\!46}{44\!\cdots\!29}a^{6}-\frac{11\!\cdots\!52}{44\!\cdots\!29}a^{5}+\frac{45\!\cdots\!22}{44\!\cdots\!29}a^{4}+\frac{36\!\cdots\!04}{44\!\cdots\!29}a^{3}-\frac{39\!\cdots\!02}{44\!\cdots\!29}a^{2}+\frac{12\!\cdots\!92}{44\!\cdots\!29}a+\frac{42\!\cdots\!86}{44\!\cdots\!29}$, $\frac{34\!\cdots\!86}{44\!\cdots\!29}a^{25}-\frac{46\!\cdots\!23}{44\!\cdots\!29}a^{24}-\frac{33\!\cdots\!12}{44\!\cdots\!29}a^{23}+\frac{72\!\cdots\!35}{44\!\cdots\!29}a^{22}+\frac{12\!\cdots\!34}{44\!\cdots\!29}a^{21}-\frac{36\!\cdots\!67}{44\!\cdots\!29}a^{20}-\frac{24\!\cdots\!40}{44\!\cdots\!29}a^{19}+\frac{85\!\cdots\!22}{44\!\cdots\!29}a^{18}+\frac{10\!\cdots\!88}{19\!\cdots\!23}a^{17}-\frac{10\!\cdots\!41}{44\!\cdots\!29}a^{16}-\frac{10\!\cdots\!88}{44\!\cdots\!29}a^{15}+\frac{75\!\cdots\!84}{44\!\cdots\!29}a^{14}+\frac{29\!\cdots\!28}{44\!\cdots\!29}a^{13}-\frac{30\!\cdots\!01}{44\!\cdots\!29}a^{12}+\frac{16\!\cdots\!88}{44\!\cdots\!29}a^{11}+\frac{63\!\cdots\!41}{44\!\cdots\!29}a^{10}-\frac{62\!\cdots\!06}{44\!\cdots\!29}a^{9}-\frac{57\!\cdots\!52}{44\!\cdots\!29}a^{8}+\frac{92\!\cdots\!84}{44\!\cdots\!29}a^{7}+\frac{15\!\cdots\!72}{44\!\cdots\!29}a^{6}-\frac{49\!\cdots\!68}{44\!\cdots\!29}a^{5}+\frac{20\!\cdots\!90}{44\!\cdots\!29}a^{4}+\frac{16\!\cdots\!86}{44\!\cdots\!29}a^{3}-\frac{17\!\cdots\!00}{44\!\cdots\!29}a^{2}+\frac{56\!\cdots\!62}{44\!\cdots\!29}a+\frac{18\!\cdots\!93}{44\!\cdots\!29}$, $\frac{26\!\cdots\!24}{44\!\cdots\!29}a^{25}-\frac{35\!\cdots\!94}{44\!\cdots\!29}a^{24}-\frac{25\!\cdots\!76}{44\!\cdots\!29}a^{23}+\frac{55\!\cdots\!00}{44\!\cdots\!29}a^{22}+\frac{98\!\cdots\!40}{44\!\cdots\!29}a^{21}-\frac{27\!\cdots\!04}{44\!\cdots\!29}a^{20}-\frac{18\!\cdots\!80}{44\!\cdots\!29}a^{19}+\frac{65\!\cdots\!08}{44\!\cdots\!29}a^{18}+\frac{79\!\cdots\!00}{19\!\cdots\!23}a^{17}-\frac{81\!\cdots\!12}{44\!\cdots\!29}a^{16}-\frac{81\!\cdots\!36}{44\!\cdots\!29}a^{15}+\frac{57\!\cdots\!68}{44\!\cdots\!29}a^{14}+\frac{23\!\cdots\!28}{44\!\cdots\!29}a^{13}-\frac{22\!\cdots\!18}{44\!\cdots\!29}a^{12}+\frac{12\!\cdots\!36}{44\!\cdots\!29}a^{11}+\frac{48\!\cdots\!04}{44\!\cdots\!29}a^{10}-\frac{47\!\cdots\!44}{44\!\cdots\!29}a^{9}-\frac{43\!\cdots\!49}{44\!\cdots\!29}a^{8}+\frac{70\!\cdots\!64}{44\!\cdots\!29}a^{7}+\frac{12\!\cdots\!66}{44\!\cdots\!29}a^{6}-\frac{37\!\cdots\!16}{44\!\cdots\!29}a^{5}+\frac{15\!\cdots\!38}{44\!\cdots\!29}a^{4}+\frac{12\!\cdots\!36}{44\!\cdots\!29}a^{3}-\frac{13\!\cdots\!32}{44\!\cdots\!29}a^{2}+\frac{42\!\cdots\!88}{44\!\cdots\!29}a+\frac{14\!\cdots\!34}{44\!\cdots\!29}$, $\frac{14\!\cdots\!84}{44\!\cdots\!29}a^{25}-\frac{18\!\cdots\!32}{44\!\cdots\!29}a^{24}-\frac{13\!\cdots\!30}{44\!\cdots\!29}a^{23}+\frac{29\!\cdots\!64}{44\!\cdots\!29}a^{22}+\frac{52\!\cdots\!86}{44\!\cdots\!29}a^{21}-\frac{14\!\cdots\!72}{44\!\cdots\!29}a^{20}-\frac{99\!\cdots\!50}{44\!\cdots\!29}a^{19}+\frac{34\!\cdots\!16}{44\!\cdots\!29}a^{18}+\frac{42\!\cdots\!00}{19\!\cdots\!23}a^{17}-\frac{43\!\cdots\!84}{44\!\cdots\!29}a^{16}-\frac{43\!\cdots\!50}{44\!\cdots\!29}a^{15}+\frac{30\!\cdots\!66}{44\!\cdots\!29}a^{14}+\frac{13\!\cdots\!60}{44\!\cdots\!29}a^{13}-\frac{12\!\cdots\!08}{44\!\cdots\!29}a^{12}+\frac{65\!\cdots\!40}{44\!\cdots\!29}a^{11}+\frac{25\!\cdots\!34}{44\!\cdots\!29}a^{10}-\frac{25\!\cdots\!34}{44\!\cdots\!29}a^{9}-\frac{23\!\cdots\!18}{44\!\cdots\!29}a^{8}+\frac{37\!\cdots\!20}{44\!\cdots\!29}a^{7}+\frac{79\!\cdots\!68}{44\!\cdots\!29}a^{6}-\frac{20\!\cdots\!62}{44\!\cdots\!29}a^{5}+\frac{82\!\cdots\!76}{44\!\cdots\!29}a^{4}+\frac{65\!\cdots\!64}{44\!\cdots\!29}a^{3}-\frac{70\!\cdots\!43}{44\!\cdots\!29}a^{2}+\frac{22\!\cdots\!96}{44\!\cdots\!29}a+\frac{75\!\cdots\!02}{44\!\cdots\!29}$, $\frac{33\!\cdots\!03}{44\!\cdots\!29}a^{25}-\frac{40\!\cdots\!63}{44\!\cdots\!29}a^{24}-\frac{14\!\cdots\!34}{19\!\cdots\!23}a^{23}+\frac{65\!\cdots\!01}{44\!\cdots\!29}a^{22}+\frac{12\!\cdots\!75}{44\!\cdots\!29}a^{21}-\frac{33\!\cdots\!16}{44\!\cdots\!29}a^{20}-\frac{24\!\cdots\!79}{44\!\cdots\!29}a^{19}+\frac{79\!\cdots\!80}{44\!\cdots\!29}a^{18}+\frac{24\!\cdots\!88}{44\!\cdots\!29}a^{17}-\frac{10\!\cdots\!45}{44\!\cdots\!29}a^{16}-\frac{11\!\cdots\!78}{44\!\cdots\!29}a^{15}+\frac{71\!\cdots\!79}{44\!\cdots\!29}a^{14}+\frac{12\!\cdots\!49}{44\!\cdots\!29}a^{13}-\frac{28\!\cdots\!45}{44\!\cdots\!29}a^{12}+\frac{11\!\cdots\!15}{44\!\cdots\!29}a^{11}+\frac{61\!\cdots\!24}{44\!\cdots\!29}a^{10}-\frac{51\!\cdots\!34}{44\!\cdots\!29}a^{9}-\frac{59\!\cdots\!50}{44\!\cdots\!29}a^{8}+\frac{34\!\cdots\!22}{19\!\cdots\!23}a^{7}+\frac{85\!\cdots\!66}{44\!\cdots\!29}a^{6}-\frac{43\!\cdots\!61}{44\!\cdots\!29}a^{5}+\frac{15\!\cdots\!02}{44\!\cdots\!29}a^{4}+\frac{17\!\cdots\!92}{44\!\cdots\!29}a^{3}-\frac{13\!\cdots\!74}{44\!\cdots\!29}a^{2}+\frac{36\!\cdots\!85}{44\!\cdots\!29}a+\frac{13\!\cdots\!85}{44\!\cdots\!29}$, $\frac{34\!\cdots\!42}{44\!\cdots\!29}a^{25}-\frac{46\!\cdots\!75}{44\!\cdots\!29}a^{24}-\frac{33\!\cdots\!60}{44\!\cdots\!29}a^{23}+\frac{72\!\cdots\!46}{44\!\cdots\!29}a^{22}+\frac{12\!\cdots\!52}{43\!\cdots\!43}a^{21}-\frac{36\!\cdots\!06}{44\!\cdots\!29}a^{20}-\frac{24\!\cdots\!32}{44\!\cdots\!29}a^{19}+\frac{85\!\cdots\!34}{44\!\cdots\!29}a^{18}+\frac{23\!\cdots\!78}{44\!\cdots\!29}a^{17}-\frac{10\!\cdots\!84}{44\!\cdots\!29}a^{16}-\frac{10\!\cdots\!29}{44\!\cdots\!29}a^{15}+\frac{75\!\cdots\!24}{44\!\cdots\!29}a^{14}+\frac{30\!\cdots\!52}{44\!\cdots\!29}a^{13}-\frac{29\!\cdots\!57}{44\!\cdots\!29}a^{12}+\frac{16\!\cdots\!41}{44\!\cdots\!29}a^{11}+\frac{62\!\cdots\!78}{44\!\cdots\!29}a^{10}-\frac{62\!\cdots\!17}{44\!\cdots\!29}a^{9}-\frac{57\!\cdots\!35}{44\!\cdots\!29}a^{8}+\frac{92\!\cdots\!72}{44\!\cdots\!29}a^{7}+\frac{16\!\cdots\!57}{44\!\cdots\!29}a^{6}-\frac{49\!\cdots\!78}{44\!\cdots\!29}a^{5}+\frac{20\!\cdots\!76}{44\!\cdots\!29}a^{4}+\frac{15\!\cdots\!83}{44\!\cdots\!29}a^{3}-\frac{17\!\cdots\!24}{44\!\cdots\!29}a^{2}+\frac{55\!\cdots\!96}{44\!\cdots\!29}a+\frac{18\!\cdots\!33}{44\!\cdots\!29}$, $\frac{22\!\cdots\!01}{44\!\cdots\!29}a^{25}-\frac{30\!\cdots\!83}{44\!\cdots\!29}a^{24}-\frac{22\!\cdots\!14}{44\!\cdots\!29}a^{23}+\frac{47\!\cdots\!22}{44\!\cdots\!29}a^{22}+\frac{84\!\cdots\!91}{44\!\cdots\!29}a^{21}-\frac{23\!\cdots\!81}{44\!\cdots\!29}a^{20}-\frac{16\!\cdots\!57}{44\!\cdots\!29}a^{19}+\frac{55\!\cdots\!41}{44\!\cdots\!29}a^{18}+\frac{15\!\cdots\!23}{44\!\cdots\!29}a^{17}-\frac{70\!\cdots\!37}{44\!\cdots\!29}a^{16}-\frac{70\!\cdots\!06}{44\!\cdots\!29}a^{15}+\frac{49\!\cdots\!59}{44\!\cdots\!29}a^{14}+\frac{22\!\cdots\!59}{44\!\cdots\!29}a^{13}-\frac{19\!\cdots\!56}{44\!\cdots\!29}a^{12}+\frac{10\!\cdots\!99}{44\!\cdots\!29}a^{11}+\frac{41\!\cdots\!29}{44\!\cdots\!29}a^{10}-\frac{40\!\cdots\!50}{44\!\cdots\!29}a^{9}-\frac{37\!\cdots\!99}{44\!\cdots\!29}a^{8}+\frac{60\!\cdots\!90}{44\!\cdots\!29}a^{7}+\frac{12\!\cdots\!63}{44\!\cdots\!29}a^{6}-\frac{32\!\cdots\!73}{44\!\cdots\!29}a^{5}+\frac{13\!\cdots\!32}{44\!\cdots\!29}a^{4}+\frac{45\!\cdots\!13}{19\!\cdots\!23}a^{3}-\frac{11\!\cdots\!53}{44\!\cdots\!29}a^{2}+\frac{15\!\cdots\!25}{19\!\cdots\!23}a+\frac{12\!\cdots\!18}{44\!\cdots\!29}$, $\frac{98\!\cdots\!91}{44\!\cdots\!29}a^{25}-\frac{13\!\cdots\!62}{44\!\cdots\!29}a^{24}-\frac{96\!\cdots\!03}{44\!\cdots\!29}a^{23}+\frac{20\!\cdots\!85}{44\!\cdots\!29}a^{22}+\frac{36\!\cdots\!25}{44\!\cdots\!29}a^{21}-\frac{10\!\cdots\!75}{44\!\cdots\!29}a^{20}-\frac{69\!\cdots\!30}{44\!\cdots\!29}a^{19}+\frac{24\!\cdots\!27}{44\!\cdots\!29}a^{18}+\frac{68\!\cdots\!71}{44\!\cdots\!29}a^{17}-\frac{30\!\cdots\!53}{44\!\cdots\!29}a^{16}-\frac{30\!\cdots\!20}{44\!\cdots\!29}a^{15}+\frac{21\!\cdots\!96}{44\!\cdots\!29}a^{14}+\frac{94\!\cdots\!63}{44\!\cdots\!29}a^{13}-\frac{85\!\cdots\!32}{44\!\cdots\!29}a^{12}+\frac{46\!\cdots\!03}{44\!\cdots\!29}a^{11}+\frac{17\!\cdots\!38}{44\!\cdots\!29}a^{10}-\frac{17\!\cdots\!96}{44\!\cdots\!29}a^{9}-\frac{16\!\cdots\!82}{44\!\cdots\!29}a^{8}+\frac{26\!\cdots\!10}{44\!\cdots\!29}a^{7}+\frac{53\!\cdots\!28}{44\!\cdots\!29}a^{6}-\frac{14\!\cdots\!00}{44\!\cdots\!29}a^{5}+\frac{25\!\cdots\!40}{19\!\cdots\!23}a^{4}+\frac{45\!\cdots\!89}{44\!\cdots\!29}a^{3}-\frac{49\!\cdots\!59}{44\!\cdots\!29}a^{2}+\frac{15\!\cdots\!23}{44\!\cdots\!29}a+\frac{53\!\cdots\!40}{44\!\cdots\!29}$, $\frac{53\!\cdots\!60}{44\!\cdots\!29}a^{25}-\frac{72\!\cdots\!33}{44\!\cdots\!29}a^{24}-\frac{22\!\cdots\!84}{19\!\cdots\!23}a^{23}+\frac{11\!\cdots\!19}{44\!\cdots\!29}a^{22}+\frac{19\!\cdots\!54}{44\!\cdots\!29}a^{21}-\frac{56\!\cdots\!93}{44\!\cdots\!29}a^{20}-\frac{37\!\cdots\!27}{44\!\cdots\!29}a^{19}+\frac{13\!\cdots\!84}{44\!\cdots\!29}a^{18}+\frac{37\!\cdots\!25}{44\!\cdots\!29}a^{17}-\frac{16\!\cdots\!46}{44\!\cdots\!29}a^{16}-\frac{16\!\cdots\!05}{44\!\cdots\!29}a^{15}+\frac{11\!\cdots\!33}{44\!\cdots\!29}a^{14}+\frac{45\!\cdots\!21}{44\!\cdots\!29}a^{13}-\frac{46\!\cdots\!53}{44\!\cdots\!29}a^{12}+\frac{25\!\cdots\!67}{44\!\cdots\!29}a^{11}+\frac{97\!\cdots\!62}{44\!\cdots\!29}a^{10}-\frac{97\!\cdots\!28}{44\!\cdots\!29}a^{9}-\frac{89\!\cdots\!01}{44\!\cdots\!29}a^{8}+\frac{62\!\cdots\!42}{19\!\cdots\!23}a^{7}+\frac{24\!\cdots\!96}{44\!\cdots\!29}a^{6}-\frac{77\!\cdots\!62}{44\!\cdots\!29}a^{5}+\frac{31\!\cdots\!84}{44\!\cdots\!29}a^{4}+\frac{24\!\cdots\!64}{44\!\cdots\!29}a^{3}-\frac{27\!\cdots\!81}{44\!\cdots\!29}a^{2}+\frac{87\!\cdots\!59}{44\!\cdots\!29}a+\frac{29\!\cdots\!00}{44\!\cdots\!29}$, $\frac{95\!\cdots\!81}{44\!\cdots\!29}a^{25}-\frac{56\!\cdots\!15}{19\!\cdots\!23}a^{24}-\frac{93\!\cdots\!04}{44\!\cdots\!29}a^{23}+\frac{20\!\cdots\!94}{44\!\cdots\!29}a^{22}+\frac{35\!\cdots\!30}{44\!\cdots\!29}a^{21}-\frac{10\!\cdots\!53}{44\!\cdots\!29}a^{20}-\frac{67\!\cdots\!02}{44\!\cdots\!29}a^{19}+\frac{23\!\cdots\!29}{43\!\cdots\!43}a^{18}+\frac{66\!\cdots\!12}{44\!\cdots\!29}a^{17}-\frac{29\!\cdots\!53}{44\!\cdots\!29}a^{16}-\frac{29\!\cdots\!92}{44\!\cdots\!29}a^{15}+\frac{21\!\cdots\!51}{44\!\cdots\!29}a^{14}+\frac{74\!\cdots\!65}{44\!\cdots\!29}a^{13}-\frac{83\!\cdots\!75}{44\!\cdots\!29}a^{12}+\frac{45\!\cdots\!99}{44\!\cdots\!29}a^{11}+\frac{17\!\cdots\!78}{44\!\cdots\!29}a^{10}-\frac{17\!\cdots\!21}{44\!\cdots\!29}a^{9}-\frac{15\!\cdots\!81}{44\!\cdots\!29}a^{8}+\frac{25\!\cdots\!12}{44\!\cdots\!29}a^{7}+\frac{40\!\cdots\!18}{44\!\cdots\!29}a^{6}-\frac{13\!\cdots\!69}{44\!\cdots\!29}a^{5}+\frac{57\!\cdots\!61}{44\!\cdots\!29}a^{4}+\frac{44\!\cdots\!10}{44\!\cdots\!29}a^{3}-\frac{49\!\cdots\!70}{44\!\cdots\!29}a^{2}+\frac{15\!\cdots\!68}{44\!\cdots\!29}a+\frac{52\!\cdots\!80}{44\!\cdots\!29}$, $\frac{49\!\cdots\!68}{44\!\cdots\!29}a^{25}-\frac{66\!\cdots\!98}{44\!\cdots\!29}a^{24}-\frac{48\!\cdots\!71}{44\!\cdots\!29}a^{23}+\frac{45\!\cdots\!90}{19\!\cdots\!23}a^{22}+\frac{80\!\cdots\!78}{19\!\cdots\!23}a^{21}-\frac{51\!\cdots\!50}{44\!\cdots\!29}a^{20}-\frac{33\!\cdots\!70}{43\!\cdots\!43}a^{19}+\frac{12\!\cdots\!27}{44\!\cdots\!29}a^{18}+\frac{34\!\cdots\!29}{44\!\cdots\!29}a^{17}-\frac{15\!\cdots\!36}{44\!\cdots\!29}a^{16}-\frac{15\!\cdots\!16}{44\!\cdots\!29}a^{15}+\frac{10\!\cdots\!31}{44\!\cdots\!29}a^{14}+\frac{17\!\cdots\!55}{19\!\cdots\!23}a^{13}-\frac{42\!\cdots\!88}{44\!\cdots\!29}a^{12}+\frac{23\!\cdots\!56}{44\!\cdots\!29}a^{11}+\frac{89\!\cdots\!05}{44\!\cdots\!29}a^{10}-\frac{89\!\cdots\!48}{44\!\cdots\!29}a^{9}-\frac{82\!\cdots\!57}{44\!\cdots\!29}a^{8}+\frac{13\!\cdots\!91}{44\!\cdots\!29}a^{7}+\frac{21\!\cdots\!21}{44\!\cdots\!29}a^{6}-\frac{71\!\cdots\!13}{44\!\cdots\!29}a^{5}+\frac{29\!\cdots\!17}{44\!\cdots\!29}a^{4}+\frac{22\!\cdots\!19}{44\!\cdots\!29}a^{3}-\frac{25\!\cdots\!72}{44\!\cdots\!29}a^{2}+\frac{80\!\cdots\!16}{44\!\cdots\!29}a+\frac{27\!\cdots\!11}{44\!\cdots\!29}$, $\frac{78\!\cdots\!60}{44\!\cdots\!29}a^{25}-\frac{46\!\cdots\!06}{19\!\cdots\!23}a^{24}-\frac{77\!\cdots\!59}{44\!\cdots\!29}a^{23}+\frac{16\!\cdots\!97}{44\!\cdots\!29}a^{22}+\frac{29\!\cdots\!83}{44\!\cdots\!29}a^{21}-\frac{82\!\cdots\!49}{44\!\cdots\!29}a^{20}-\frac{55\!\cdots\!84}{44\!\cdots\!29}a^{19}+\frac{19\!\cdots\!00}{44\!\cdots\!29}a^{18}+\frac{54\!\cdots\!20}{44\!\cdots\!29}a^{17}-\frac{24\!\cdots\!93}{44\!\cdots\!29}a^{16}-\frac{24\!\cdots\!56}{44\!\cdots\!29}a^{15}+\frac{17\!\cdots\!53}{44\!\cdots\!29}a^{14}+\frac{68\!\cdots\!80}{44\!\cdots\!29}a^{13}-\frac{68\!\cdots\!43}{44\!\cdots\!29}a^{12}+\frac{37\!\cdots\!70}{44\!\cdots\!29}a^{11}+\frac{14\!\cdots\!75}{44\!\cdots\!29}a^{10}-\frac{14\!\cdots\!82}{44\!\cdots\!29}a^{9}-\frac{13\!\cdots\!39}{44\!\cdots\!29}a^{8}+\frac{21\!\cdots\!01}{44\!\cdots\!29}a^{7}+\frac{37\!\cdots\!63}{44\!\cdots\!29}a^{6}-\frac{11\!\cdots\!35}{44\!\cdots\!29}a^{5}+\frac{46\!\cdots\!48}{44\!\cdots\!29}a^{4}+\frac{36\!\cdots\!85}{44\!\cdots\!29}a^{3}-\frac{40\!\cdots\!32}{44\!\cdots\!29}a^{2}+\frac{12\!\cdots\!31}{44\!\cdots\!29}a+\frac{43\!\cdots\!98}{44\!\cdots\!29}$, $\frac{15\!\cdots\!18}{44\!\cdots\!29}a^{25}-\frac{20\!\cdots\!20}{44\!\cdots\!29}a^{24}-\frac{15\!\cdots\!67}{44\!\cdots\!29}a^{23}+\frac{32\!\cdots\!33}{44\!\cdots\!29}a^{22}+\frac{57\!\cdots\!89}{44\!\cdots\!29}a^{21}-\frac{16\!\cdots\!11}{44\!\cdots\!29}a^{20}-\frac{10\!\cdots\!00}{44\!\cdots\!29}a^{19}+\frac{38\!\cdots\!24}{44\!\cdots\!29}a^{18}+\frac{10\!\cdots\!19}{44\!\cdots\!29}a^{17}-\frac{47\!\cdots\!99}{44\!\cdots\!29}a^{16}-\frac{47\!\cdots\!32}{44\!\cdots\!29}a^{15}+\frac{33\!\cdots\!34}{44\!\cdots\!29}a^{14}+\frac{13\!\cdots\!10}{44\!\cdots\!29}a^{13}-\frac{13\!\cdots\!13}{44\!\cdots\!29}a^{12}+\frac{73\!\cdots\!58}{44\!\cdots\!29}a^{11}+\frac{28\!\cdots\!90}{44\!\cdots\!29}a^{10}-\frac{27\!\cdots\!36}{44\!\cdots\!29}a^{9}-\frac{24\!\cdots\!73}{43\!\cdots\!43}a^{8}+\frac{41\!\cdots\!26}{44\!\cdots\!29}a^{7}+\frac{73\!\cdots\!77}{44\!\cdots\!29}a^{6}-\frac{22\!\cdots\!21}{44\!\cdots\!29}a^{5}+\frac{91\!\cdots\!47}{44\!\cdots\!29}a^{4}+\frac{71\!\cdots\!00}{44\!\cdots\!29}a^{3}-\frac{78\!\cdots\!48}{44\!\cdots\!29}a^{2}+\frac{25\!\cdots\!87}{44\!\cdots\!29}a+\frac{84\!\cdots\!39}{44\!\cdots\!29}$, $\frac{22\!\cdots\!93}{44\!\cdots\!29}a^{25}-\frac{13\!\cdots\!80}{19\!\cdots\!23}a^{24}-\frac{22\!\cdots\!27}{44\!\cdots\!29}a^{23}+\frac{48\!\cdots\!20}{44\!\cdots\!29}a^{22}+\frac{85\!\cdots\!10}{44\!\cdots\!29}a^{21}-\frac{24\!\cdots\!02}{44\!\cdots\!29}a^{20}-\frac{16\!\cdots\!22}{44\!\cdots\!29}a^{19}+\frac{56\!\cdots\!75}{44\!\cdots\!29}a^{18}+\frac{15\!\cdots\!78}{44\!\cdots\!29}a^{17}-\frac{71\!\cdots\!61}{44\!\cdots\!29}a^{16}-\frac{71\!\cdots\!98}{44\!\cdots\!29}a^{15}+\frac{50\!\cdots\!58}{44\!\cdots\!29}a^{14}+\frac{20\!\cdots\!38}{44\!\cdots\!29}a^{13}-\frac{19\!\cdots\!24}{44\!\cdots\!29}a^{12}+\frac{10\!\cdots\!62}{44\!\cdots\!29}a^{11}+\frac{41\!\cdots\!80}{44\!\cdots\!29}a^{10}-\frac{41\!\cdots\!96}{44\!\cdots\!29}a^{9}-\frac{38\!\cdots\!14}{44\!\cdots\!29}a^{8}+\frac{61\!\cdots\!66}{44\!\cdots\!29}a^{7}+\frac{11\!\cdots\!66}{44\!\cdots\!29}a^{6}-\frac{33\!\cdots\!71}{44\!\cdots\!29}a^{5}+\frac{13\!\cdots\!30}{44\!\cdots\!29}a^{4}+\frac{10\!\cdots\!15}{44\!\cdots\!29}a^{3}-\frac{11\!\cdots\!07}{44\!\cdots\!29}a^{2}+\frac{37\!\cdots\!13}{44\!\cdots\!29}a+\frac{12\!\cdots\!39}{44\!\cdots\!29}$, $\frac{60\!\cdots\!14}{44\!\cdots\!29}a^{25}-\frac{35\!\cdots\!19}{19\!\cdots\!23}a^{24}-\frac{58\!\cdots\!18}{44\!\cdots\!29}a^{23}+\frac{12\!\cdots\!69}{44\!\cdots\!29}a^{22}+\frac{22\!\cdots\!90}{44\!\cdots\!29}a^{21}-\frac{63\!\cdots\!10}{44\!\cdots\!29}a^{20}-\frac{42\!\cdots\!65}{44\!\cdots\!29}a^{19}+\frac{14\!\cdots\!53}{44\!\cdots\!29}a^{18}+\frac{41\!\cdots\!30}{44\!\cdots\!29}a^{17}-\frac{18\!\cdots\!98}{44\!\cdots\!29}a^{16}-\frac{18\!\cdots\!57}{44\!\cdots\!29}a^{15}+\frac{13\!\cdots\!34}{44\!\cdots\!29}a^{14}+\frac{57\!\cdots\!14}{44\!\cdots\!29}a^{13}-\frac{52\!\cdots\!43}{44\!\cdots\!29}a^{12}+\frac{28\!\cdots\!10}{44\!\cdots\!29}a^{11}+\frac{10\!\cdots\!47}{44\!\cdots\!29}a^{10}-\frac{10\!\cdots\!48}{44\!\cdots\!29}a^{9}-\frac{10\!\cdots\!11}{44\!\cdots\!29}a^{8}+\frac{16\!\cdots\!13}{44\!\cdots\!29}a^{7}+\frac{32\!\cdots\!82}{44\!\cdots\!29}a^{6}-\frac{86\!\cdots\!82}{44\!\cdots\!29}a^{5}+\frac{35\!\cdots\!73}{44\!\cdots\!29}a^{4}+\frac{28\!\cdots\!19}{44\!\cdots\!29}a^{3}-\frac{30\!\cdots\!77}{44\!\cdots\!29}a^{2}+\frac{96\!\cdots\!03}{44\!\cdots\!29}a+\frac{32\!\cdots\!68}{44\!\cdots\!29}$, $\frac{27\!\cdots\!45}{44\!\cdots\!29}a^{25}-\frac{37\!\cdots\!45}{44\!\cdots\!29}a^{24}-\frac{26\!\cdots\!04}{44\!\cdots\!29}a^{23}+\frac{58\!\cdots\!54}{44\!\cdots\!29}a^{22}+\frac{99\!\cdots\!56}{43\!\cdots\!43}a^{21}-\frac{28\!\cdots\!72}{44\!\cdots\!29}a^{20}-\frac{84\!\cdots\!20}{19\!\cdots\!23}a^{19}+\frac{68\!\cdots\!00}{44\!\cdots\!29}a^{18}+\frac{19\!\cdots\!44}{44\!\cdots\!29}a^{17}-\frac{85\!\cdots\!16}{44\!\cdots\!29}a^{16}-\frac{85\!\cdots\!58}{44\!\cdots\!29}a^{15}+\frac{60\!\cdots\!34}{44\!\cdots\!29}a^{14}+\frac{22\!\cdots\!35}{44\!\cdots\!29}a^{13}-\frac{23\!\cdots\!92}{44\!\cdots\!29}a^{12}+\frac{13\!\cdots\!02}{44\!\cdots\!29}a^{11}+\frac{50\!\cdots\!35}{44\!\cdots\!29}a^{10}-\frac{50\!\cdots\!62}{44\!\cdots\!29}a^{9}-\frac{45\!\cdots\!97}{44\!\cdots\!29}a^{8}+\frac{73\!\cdots\!33}{44\!\cdots\!29}a^{7}+\frac{12\!\cdots\!25}{44\!\cdots\!29}a^{6}-\frac{39\!\cdots\!67}{44\!\cdots\!29}a^{5}+\frac{16\!\cdots\!08}{44\!\cdots\!29}a^{4}+\frac{12\!\cdots\!90}{44\!\cdots\!29}a^{3}-\frac{14\!\cdots\!50}{44\!\cdots\!29}a^{2}+\frac{45\!\cdots\!13}{44\!\cdots\!29}a+\frac{15\!\cdots\!82}{44\!\cdots\!29}$ 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:  \( 514750610132899270000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{26}\cdot(2\pi)^{0}\cdot 514750610132899270000 \cdot 1}{2\cdot\sqrt{1919641021107487828653877081110544587155912070949008048128}}\cr\approx \mathstrut & 0.394218272712406 \end{aligned}\] (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 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447)
 
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 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447, 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 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447);
 
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 - 97*x^24 + 274*x^23 + 3600*x^22 - 12928*x^21 - 64015*x^20 + 293118*x^19 + 533589*x^18 - 3564090*x^17 - 1089717*x^16 + 23968180*x^15 - 13363361*x^14 - 87417128*x^13 + 103695959*x^12 + 151824110*x^11 - 299732504*x^10 - 49085562*x^9 + 375988637*x^8 - 168890324*x^7 - 147518415*x^6 + 153068810*x^5 - 33898995*x^4 - 8099910*x^3 + 3461310*x^2 - 50580*x - 35447);
 
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

$C_{26}$ (as 26T1):

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 cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 13.13.59091511031674153381441.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $26$ $26$ ${\href{/padicField/7.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/padicField/17.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/23.1.0.1}{1} }^{26}$ $26$ ${\href{/padicField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/47.13.0.1}{13} }^{2}$ $26$ $26$

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
\(2\) Copy content Toggle raw display Deg $26$$2$$13$$39$
\(79\) Copy content Toggle raw display 79.13.12.1$x^{13} + 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$
79.13.12.1$x^{13} + 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$