Properties

Label 44.44.877...613.1
Degree $44$
Signature $[44, 0]$
Discriminant $8.775\times 10^{108}$
Root discriminant \(299.21\)
Ramified primes $23,37$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 312*x^42 + 303*x^41 + 45350*x^40 - 42623*x^39 - 4077513*x^38 + 3693906*x^37 + 253947067*x^36 - 220701913*x^35 - 11624898400*x^34 + 9638581183*x^33 + 405056137130*x^32 - 318308906483*x^31 - 10976588371921*x^30 + 8111808213574*x^29 + 234369143305301*x^28 - 161362869383135*x^27 - 3970563499772876*x^26 + 2518297675324661*x^25 + 53492986112604139*x^24 - 30828307034682190*x^23 - 572009925875972279*x^22 + 294555163094127792*x^21 + 4826078584950284457*x^20 - 2175082202480665232*x^19 - 31800584527656001615*x^18 + 12224850681757177737*x^17 + 161193984162806092803*x^16 - 51170566486435305249*x^15 - 615319732524023675934*x^14 + 154790608182698273409*x^13 + 1717658346766087000620*x^12 - 324640449052810836967*x^11 - 3366040964991975949283*x^10 + 445321559954531949468*x^9 + 4367921735840229125340*x^8 - 367209571759684618733*x^7 - 3436106010074360679257*x^6 + 161383741380219872801*x^5 + 1415445998956830381216*x^4 - 33374706535233304336*x^3 - 228324835255811811652*x^2 + 3748885669892449290*x + 9622931728201923763)
 
gp: K = bnfinit(y^44 - y^43 - 312*y^42 + 303*y^41 + 45350*y^40 - 42623*y^39 - 4077513*y^38 + 3693906*y^37 + 253947067*y^36 - 220701913*y^35 - 11624898400*y^34 + 9638581183*y^33 + 405056137130*y^32 - 318308906483*y^31 - 10976588371921*y^30 + 8111808213574*y^29 + 234369143305301*y^28 - 161362869383135*y^27 - 3970563499772876*y^26 + 2518297675324661*y^25 + 53492986112604139*y^24 - 30828307034682190*y^23 - 572009925875972279*y^22 + 294555163094127792*y^21 + 4826078584950284457*y^20 - 2175082202480665232*y^19 - 31800584527656001615*y^18 + 12224850681757177737*y^17 + 161193984162806092803*y^16 - 51170566486435305249*y^15 - 615319732524023675934*y^14 + 154790608182698273409*y^13 + 1717658346766087000620*y^12 - 324640449052810836967*y^11 - 3366040964991975949283*y^10 + 445321559954531949468*y^9 + 4367921735840229125340*y^8 - 367209571759684618733*y^7 - 3436106010074360679257*y^6 + 161383741380219872801*y^5 + 1415445998956830381216*y^4 - 33374706535233304336*y^3 - 228324835255811811652*y^2 + 3748885669892449290*y + 9622931728201923763, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 312*x^42 + 303*x^41 + 45350*x^40 - 42623*x^39 - 4077513*x^38 + 3693906*x^37 + 253947067*x^36 - 220701913*x^35 - 11624898400*x^34 + 9638581183*x^33 + 405056137130*x^32 - 318308906483*x^31 - 10976588371921*x^30 + 8111808213574*x^29 + 234369143305301*x^28 - 161362869383135*x^27 - 3970563499772876*x^26 + 2518297675324661*x^25 + 53492986112604139*x^24 - 30828307034682190*x^23 - 572009925875972279*x^22 + 294555163094127792*x^21 + 4826078584950284457*x^20 - 2175082202480665232*x^19 - 31800584527656001615*x^18 + 12224850681757177737*x^17 + 161193984162806092803*x^16 - 51170566486435305249*x^15 - 615319732524023675934*x^14 + 154790608182698273409*x^13 + 1717658346766087000620*x^12 - 324640449052810836967*x^11 - 3366040964991975949283*x^10 + 445321559954531949468*x^9 + 4367921735840229125340*x^8 - 367209571759684618733*x^7 - 3436106010074360679257*x^6 + 161383741380219872801*x^5 + 1415445998956830381216*x^4 - 33374706535233304336*x^3 - 228324835255811811652*x^2 + 3748885669892449290*x + 9622931728201923763);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 312*x^42 + 303*x^41 + 45350*x^40 - 42623*x^39 - 4077513*x^38 + 3693906*x^37 + 253947067*x^36 - 220701913*x^35 - 11624898400*x^34 + 9638581183*x^33 + 405056137130*x^32 - 318308906483*x^31 - 10976588371921*x^30 + 8111808213574*x^29 + 234369143305301*x^28 - 161362869383135*x^27 - 3970563499772876*x^26 + 2518297675324661*x^25 + 53492986112604139*x^24 - 30828307034682190*x^23 - 572009925875972279*x^22 + 294555163094127792*x^21 + 4826078584950284457*x^20 - 2175082202480665232*x^19 - 31800584527656001615*x^18 + 12224850681757177737*x^17 + 161193984162806092803*x^16 - 51170566486435305249*x^15 - 615319732524023675934*x^14 + 154790608182698273409*x^13 + 1717658346766087000620*x^12 - 324640449052810836967*x^11 - 3366040964991975949283*x^10 + 445321559954531949468*x^9 + 4367921735840229125340*x^8 - 367209571759684618733*x^7 - 3436106010074360679257*x^6 + 161383741380219872801*x^5 + 1415445998956830381216*x^4 - 33374706535233304336*x^3 - 228324835255811811652*x^2 + 3748885669892449290*x + 9622931728201923763)
 

\( x^{44} - x^{43} - 312 x^{42} + 303 x^{41} + 45350 x^{40} - 42623 x^{39} - 4077513 x^{38} + 3693906 x^{37} + 253947067 x^{36} - 220701913 x^{35} + \cdots + 96\!\cdots\!63 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(877\!\cdots\!613\) \(\medspace = 23^{42}\cdot 37^{33}\) 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:  \(299.21\)
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:  $23^{21/22}37^{3/4}\approx 299.21434397733475$
Ramified primes:   \(23\), \(37\) 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:  \(\Q(\sqrt{37}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(851=23\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{851}(1,·)$, $\chi_{851}(519,·)$, $\chi_{851}(401,·)$, $\chi_{851}(147,·)$, $\chi_{851}(413,·)$, $\chi_{851}(672,·)$, $\chi_{851}(290,·)$, $\chi_{851}(36,·)$, $\chi_{851}(549,·)$, $\chi_{851}(554,·)$, $\chi_{851}(43,·)$, $\chi_{851}(556,·)$, $\chi_{851}(813,·)$, $\chi_{851}(820,·)$, $\chi_{851}(697,·)$, $\chi_{851}(186,·)$, $\chi_{851}(443,·)$, $\chi_{851}(445,·)$, $\chi_{851}(702,·)$, $\chi_{851}(191,·)$, $\chi_{851}(68,·)$, $\chi_{851}(709,·)$, $\chi_{851}(327,·)$, $\chi_{851}(73,·)$, $\chi_{851}(586,·)$, $\chi_{851}(75,·)$, $\chi_{851}(845,·)$, $\chi_{851}(334,·)$, $\chi_{851}(591,·)$, $\chi_{851}(80,·)$, $\chi_{851}(593,·)$, $\chi_{851}(339,·)$, $\chi_{851}(475,·)$, $\chi_{851}(734,·)$, $\chi_{851}(223,·)$, $\chi_{851}(739,·)$, $\chi_{851}(228,·)$, $\chi_{851}(746,·)$, $\chi_{851}(364,·)$, $\chi_{851}(110,·)$, $\chi_{851}(369,·)$, $\chi_{851}(371,·)$, $\chi_{851}(630,·)$, $\chi_{851}(635,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{47}a^{12}+\frac{17}{47}a^{11}+\frac{10}{47}a^{10}+\frac{7}{47}a^{9}+\frac{14}{47}a^{8}-\frac{8}{47}a^{7}-\frac{17}{47}a^{6}+\frac{4}{47}a^{5}-\frac{3}{47}a^{4}-\frac{20}{47}a^{3}-\frac{21}{47}a^{2}-\frac{19}{47}a+\frac{16}{47}$, $\frac{1}{47}a^{13}+\frac{3}{47}a^{11}-\frac{22}{47}a^{10}-\frac{11}{47}a^{9}-\frac{11}{47}a^{8}-\frac{22}{47}a^{7}+\frac{11}{47}a^{6}+\frac{23}{47}a^{5}-\frac{16}{47}a^{4}-\frac{10}{47}a^{3}+\frac{9}{47}a^{2}+\frac{10}{47}a+\frac{10}{47}$, $\frac{1}{47}a^{14}+\frac{21}{47}a^{11}+\frac{6}{47}a^{10}+\frac{15}{47}a^{9}-\frac{17}{47}a^{8}-\frac{12}{47}a^{7}-\frac{20}{47}a^{6}+\frac{19}{47}a^{5}-\frac{1}{47}a^{4}+\frac{22}{47}a^{3}-\frac{21}{47}a^{2}+\frac{20}{47}a-\frac{1}{47}$, $\frac{1}{47}a^{15}-\frac{22}{47}a^{11}-\frac{7}{47}a^{10}-\frac{23}{47}a^{9}+\frac{23}{47}a^{8}+\frac{7}{47}a^{7}+\frac{9}{47}a^{5}-\frac{9}{47}a^{4}+\frac{23}{47}a^{3}-\frac{9}{47}a^{2}+\frac{22}{47}a-\frac{7}{47}$, $\frac{1}{47}a^{16}-\frac{9}{47}a^{11}+\frac{9}{47}a^{10}-\frac{11}{47}a^{9}-\frac{14}{47}a^{8}+\frac{12}{47}a^{7}+\frac{11}{47}a^{6}-\frac{15}{47}a^{5}+\frac{4}{47}a^{4}+\frac{21}{47}a^{3}-\frac{17}{47}a^{2}-\frac{2}{47}a+\frac{23}{47}$, $\frac{1}{47}a^{17}+\frac{21}{47}a^{11}-\frac{15}{47}a^{10}+\frac{2}{47}a^{9}-\frac{3}{47}a^{8}-\frac{14}{47}a^{7}+\frac{20}{47}a^{6}-\frac{7}{47}a^{5}-\frac{6}{47}a^{4}-\frac{9}{47}a^{3}-\frac{3}{47}a^{2}-\frac{7}{47}a+\frac{3}{47}$, $\frac{1}{47}a^{18}+\frac{4}{47}a^{11}-\frac{20}{47}a^{10}-\frac{9}{47}a^{9}+\frac{21}{47}a^{8}+\frac{21}{47}a^{6}+\frac{4}{47}a^{5}+\frac{7}{47}a^{4}-\frac{6}{47}a^{3}+\frac{11}{47}a^{2}-\frac{21}{47}a-\frac{7}{47}$, $\frac{1}{47}a^{19}+\frac{6}{47}a^{11}-\frac{2}{47}a^{10}-\frac{7}{47}a^{9}-\frac{9}{47}a^{8}+\frac{6}{47}a^{7}-\frac{22}{47}a^{6}-\frac{9}{47}a^{5}+\frac{6}{47}a^{4}-\frac{3}{47}a^{3}+\frac{16}{47}a^{2}+\frac{22}{47}a-\frac{17}{47}$, $\frac{1}{47}a^{20}-\frac{10}{47}a^{11}-\frac{20}{47}a^{10}-\frac{4}{47}a^{9}+\frac{16}{47}a^{8}-\frac{21}{47}a^{7}-\frac{1}{47}a^{6}-\frac{18}{47}a^{5}+\frac{15}{47}a^{4}-\frac{5}{47}a^{3}+\frac{7}{47}a^{2}+\frac{3}{47}a-\frac{2}{47}$, $\frac{1}{47}a^{21}+\frac{9}{47}a^{11}+\frac{2}{47}a^{10}-\frac{8}{47}a^{9}-\frac{22}{47}a^{8}+\frac{13}{47}a^{7}+\frac{8}{47}a^{5}+\frac{12}{47}a^{4}-\frac{5}{47}a^{3}-\frac{19}{47}a^{2}-\frac{4}{47}a+\frac{19}{47}$, $\frac{1}{47}a^{22}-\frac{10}{47}a^{11}-\frac{4}{47}a^{10}+\frac{9}{47}a^{9}-\frac{19}{47}a^{8}-\frac{22}{47}a^{7}+\frac{20}{47}a^{6}+\frac{23}{47}a^{5}+\frac{22}{47}a^{4}+\frac{20}{47}a^{3}-\frac{3}{47}a^{2}+\frac{2}{47}a-\frac{3}{47}$, $\frac{1}{3061119541}a^{23}-\frac{161}{3061119541}a^{21}+\frac{11270}{3061119541}a^{19}-\frac{449673}{3061119541}a^{17}+\frac{11265492}{3061119541}a^{15}+\frac{11387573}{3061119541}a^{13}-\frac{505032858}{3061119541}a^{11}-\frac{19}{47}a^{10}-\frac{386863629}{3061119541}a^{9}+\frac{14}{47}a^{8}+\frac{543665872}{3061119541}a^{7}-\frac{14}{47}a^{6}+\frac{337991306}{3061119541}a^{5}-\frac{1}{47}a^{4}+\frac{167071018}{3061119541}a^{3}-\frac{20}{47}a^{2}-\frac{668935425}{3061119541}a+\frac{688493931}{3061119541}$, $\frac{1}{3061119541}a^{24}-\frac{161}{3061119541}a^{22}+\frac{11270}{3061119541}a^{20}-\frac{449673}{3061119541}a^{18}+\frac{11265492}{3061119541}a^{16}+\frac{11387573}{3061119541}a^{14}+\frac{16008766}{3061119541}a^{12}+\frac{23}{47}a^{11}-\frac{1298686471}{3061119541}a^{10}+\frac{23}{47}a^{9}-\frac{1345110015}{3061119541}a^{8}+\frac{16}{47}a^{7}+\frac{663642321}{3061119541}a^{6}-\frac{16}{47}a^{5}-\frac{1396053854}{3061119541}a^{4}+\frac{8}{47}a^{3}+\frac{633668635}{3061119541}a^{2}-\frac{27938302}{3061119541}a-\frac{13}{47}$, $\frac{1}{3061119541}a^{25}-\frac{14651}{3061119541}a^{21}+\frac{1364797}{3061119541}a^{19}+\frac{3998342}{3061119541}a^{17}+\frac{1486101}{3061119541}a^{15}+\frac{25762335}{3061119541}a^{13}+\frac{1408985261}{3061119541}a^{11}+\frac{22}{47}a^{10}+\frac{262574859}{3061119541}a^{9}+\frac{11}{47}a^{8}+\frac{1114766302}{3061119541}a^{7}-\frac{5}{47}a^{6}+\frac{460472591}{3061119541}a^{5}-\frac{1}{47}a^{4}-\frac{473884757}{3061119541}a^{3}-\frac{939761144}{3061119541}a^{2}-\frac{1145591317}{3061119541}a-\frac{1046165863}{3061119541}$, $\frac{1}{3061119541}a^{26}-\frac{14651}{3061119541}a^{22}+\frac{1364797}{3061119541}a^{20}+\frac{3998342}{3061119541}a^{18}+\frac{1486101}{3061119541}a^{16}+\frac{25762335}{3061119541}a^{14}-\frac{23879205}{3061119541}a^{12}-\frac{23}{47}a^{11}+\frac{1239527904}{3061119541}a^{10}-\frac{2}{47}a^{9}-\frac{578618976}{3061119541}a^{8}-\frac{17}{47}a^{7}+\frac{330212185}{3061119541}a^{6}+\frac{5}{47}a^{5}+\frac{763589100}{3061119541}a^{4}+\frac{167452307}{3061119541}a^{3}+\frac{1394486600}{3061119541}a^{2}-\frac{1371816878}{3061119541}a-\frac{23}{47}$, $\frac{1}{3061119541}a^{27}-\frac{994014}{3061119541}a^{21}-\frac{26275497}{3061119541}a^{19}-\frac{8522519}{3061119541}a^{17}-\frac{28578978}{3061119541}a^{15}+\frac{17002935}{3061119541}a^{13}+\frac{1315227770}{3061119541}a^{11}+\frac{15}{47}a^{10}+\frac{936263289}{3061119541}a^{9}-\frac{19}{47}a^{8}+\frac{24815551}{3061119541}a^{7}-\frac{16}{47}a^{6}-\frac{803375859}{3061119541}a^{5}-\frac{483849723}{3061119541}a^{4}-\frac{1502176963}{3061119541}a^{3}+\frac{1103130836}{3061119541}a^{2}-\frac{1082568295}{3061119541}a+\frac{800825689}{3061119541}$, $\frac{1}{3061119541}a^{28}-\frac{994014}{3061119541}a^{22}-\frac{26275497}{3061119541}a^{20}-\frac{8522519}{3061119541}a^{18}-\frac{28578978}{3061119541}a^{16}+\frac{17002935}{3061119541}a^{14}+\frac{12623710}{3061119541}a^{12}+\frac{4}{47}a^{11}+\frac{154700853}{3061119541}a^{10}-\frac{18}{47}a^{9}+\frac{155075957}{3061119541}a^{8}+\frac{3}{47}a^{7}-\frac{86943626}{3061119541}a^{6}+\frac{427973119}{3061119541}a^{5}-\frac{655484324}{3061119541}a^{4}-\frac{394863833}{3061119541}a^{3}-\frac{1277958904}{3061119541}a^{2}+\frac{1061346501}{3061119541}a+\frac{9}{47}$, $\frac{1}{3061119541}a^{29}+\frac{9078858}{3061119541}a^{21}-\frac{8379655}{3061119541}a^{19}-\frac{21253211}{3061119541}a^{17}-\frac{22554779}{3061119541}a^{15}-\frac{14279059}{3061119541}a^{13}-\frac{1472875473}{3061119541}a^{11}+\frac{20}{47}a^{10}+\frac{419071645}{3061119541}a^{9}-\frac{19}{47}a^{8}+\frac{901467021}{3061119541}a^{7}+\frac{1079275149}{3061119541}a^{6}-\frac{139467037}{3061119541}a^{5}+\frac{126177791}{3061119541}a^{4}-\frac{1162327736}{3061119541}a^{3}-\frac{371517965}{3061119541}a^{2}-\frac{717698403}{3061119541}a-\frac{1173020103}{3061119541}$, $\frac{1}{3061119541}a^{30}+\frac{9078858}{3061119541}a^{22}-\frac{8379655}{3061119541}a^{20}-\frac{21253211}{3061119541}a^{18}-\frac{22554779}{3061119541}a^{16}-\frac{14279059}{3061119541}a^{14}+\frac{25119196}{3061119541}a^{12}-\frac{12}{47}a^{11}+\frac{93420630}{3061119541}a^{10}+\frac{1}{47}a^{9}+\frac{445555600}{3061119541}a^{8}+\frac{1339795961}{3061119541}a^{7}-\frac{1116420082}{3061119541}a^{6}-\frac{4082615}{3061119541}a^{5}+\frac{465927339}{3061119541}a^{4}+\frac{279784065}{3061119541}a^{3}+\frac{1496728499}{3061119541}a^{2}+\frac{976276596}{3061119541}a-\frac{8}{47}$, $\frac{1}{3061119541}a^{31}+\frac{20452017}{3061119541}a^{21}-\frac{20433958}{3061119541}a^{19}+\frac{3374209}{3061119541}a^{17}-\frac{15994318}{3061119541}a^{15}-\frac{11995110}{3061119541}a^{13}-\frac{566645978}{3061119541}a^{11}+\frac{1271091959}{3061119541}a^{9}-\frac{614110129}{3061119541}a^{8}-\frac{1353214721}{3061119541}a^{7}+\frac{972870430}{3061119541}a^{6}-\frac{848617656}{3061119541}a^{5}+\frac{149523659}{3061119541}a^{4}-\frac{659748794}{3061119541}a^{3}+\frac{129583957}{3061119541}a^{2}+\frac{594377489}{3061119541}a+\frac{1036504142}{3061119541}$, $\frac{1}{3061119541}a^{32}+\frac{20452017}{3061119541}a^{22}-\frac{20433958}{3061119541}a^{20}+\frac{3374209}{3061119541}a^{18}-\frac{15994318}{3061119541}a^{16}-\frac{11995110}{3061119541}a^{14}+\frac{19525849}{3061119541}a^{12}+\frac{12}{47}a^{11}+\frac{1010571147}{3061119541}a^{10}+\frac{427973119}{3061119541}a^{9}+\frac{730951775}{3061119541}a^{8}-\frac{655384645}{3061119541}a^{7}+\frac{1430939449}{3061119541}a^{6}-\frac{566908574}{3061119541}a^{5}+\frac{642855266}{3061119541}a^{4}+\frac{650625581}{3061119541}a^{3}+\frac{529247286}{3061119541}a^{2}-\frac{917401948}{3061119541}a+\frac{3}{47}$, $\frac{1}{9183358623}a^{33}+\frac{1}{9183358623}a^{32}+\frac{1}{9183358623}a^{30}+\frac{1}{9183358623}a^{29}-\frac{1}{9183358623}a^{27}-\frac{1}{9183358623}a^{26}-\frac{11861559}{3061119541}a^{22}+\frac{30344568}{3061119541}a^{21}-\frac{10059470}{3061119541}a^{20}+\frac{29319027}{3061119541}a^{19}-\frac{87007547}{9183358623}a^{18}-\frac{84329387}{9183358623}a^{17}-\frac{13345066}{3061119541}a^{16}+\frac{26186608}{9183358623}a^{15}+\frac{13093699}{9183358623}a^{14}-\frac{27028067}{3061119541}a^{13}-\frac{61736156}{9183358623}a^{12}+\frac{562291483}{9183358623}a^{11}-\frac{1335818083}{9183358623}a^{10}-\frac{3403607576}{9183358623}a^{9}+\frac{1126432937}{3061119541}a^{8}+\frac{1289740494}{3061119541}a^{7}+\frac{360948528}{3061119541}a^{6}-\frac{312408898}{9183358623}a^{5}-\frac{1455272941}{9183358623}a^{4}+\frac{3281906642}{9183358623}a^{3}+\frac{1951860007}{9183358623}a^{2}+\frac{964314135}{3061119541}a-\frac{1781631988}{9183358623}$, $\frac{1}{50\!\cdots\!01}a^{34}-\frac{128468056793242}{50\!\cdots\!01}a^{33}-\frac{753951573597764}{50\!\cdots\!01}a^{32}-\frac{398563161084014}{50\!\cdots\!01}a^{31}+\frac{545804795404235}{50\!\cdots\!01}a^{30}+\frac{798770140862929}{50\!\cdots\!01}a^{29}+\frac{669658208442740}{50\!\cdots\!01}a^{28}-\frac{539969055879227}{50\!\cdots\!01}a^{27}+\frac{823238592678440}{50\!\cdots\!01}a^{26}-\frac{124529930475052}{16\!\cdots\!67}a^{25}-\frac{41084176508536}{16\!\cdots\!67}a^{24}-\frac{145263012485774}{16\!\cdots\!67}a^{23}+\frac{34\!\cdots\!62}{16\!\cdots\!67}a^{22}-\frac{64\!\cdots\!78}{16\!\cdots\!67}a^{21}+\frac{12\!\cdots\!14}{16\!\cdots\!67}a^{20}+\frac{40\!\cdots\!34}{50\!\cdots\!01}a^{19}+\frac{35\!\cdots\!55}{50\!\cdots\!01}a^{18}+\frac{48\!\cdots\!68}{50\!\cdots\!01}a^{17}-\frac{43\!\cdots\!79}{50\!\cdots\!01}a^{16}+\frac{70\!\cdots\!81}{50\!\cdots\!01}a^{15}+\frac{51\!\cdots\!82}{50\!\cdots\!01}a^{14}+\frac{32\!\cdots\!37}{50\!\cdots\!01}a^{13}+\frac{21\!\cdots\!38}{50\!\cdots\!01}a^{12}+\frac{10\!\cdots\!23}{16\!\cdots\!67}a^{11}+\frac{72\!\cdots\!53}{16\!\cdots\!67}a^{10}-\frac{15\!\cdots\!84}{50\!\cdots\!01}a^{9}-\frac{74\!\cdots\!36}{16\!\cdots\!67}a^{8}+\frac{61\!\cdots\!87}{16\!\cdots\!67}a^{7}+\frac{13\!\cdots\!50}{50\!\cdots\!01}a^{6}+\frac{34\!\cdots\!06}{50\!\cdots\!01}a^{5}+\frac{17\!\cdots\!91}{50\!\cdots\!01}a^{4}-\frac{13\!\cdots\!00}{16\!\cdots\!67}a^{3}-\frac{59\!\cdots\!06}{50\!\cdots\!01}a^{2}+\frac{24\!\cdots\!22}{50\!\cdots\!01}a+\frac{24\!\cdots\!54}{50\!\cdots\!01}$, $\frac{1}{23\!\cdots\!47}a^{35}+\frac{17}{23\!\cdots\!47}a^{34}-\frac{18\!\cdots\!81}{23\!\cdots\!47}a^{33}-\frac{24\!\cdots\!70}{23\!\cdots\!47}a^{32}-\frac{15\!\cdots\!46}{23\!\cdots\!47}a^{31}+\frac{57\!\cdots\!49}{23\!\cdots\!47}a^{30}-\frac{13\!\cdots\!28}{79\!\cdots\!49}a^{29}+\frac{28\!\cdots\!01}{23\!\cdots\!47}a^{28}-\frac{24\!\cdots\!59}{23\!\cdots\!47}a^{27}-\frac{37\!\cdots\!24}{23\!\cdots\!47}a^{26}-\frac{16\!\cdots\!17}{79\!\cdots\!49}a^{25}+\frac{11\!\cdots\!64}{79\!\cdots\!49}a^{24}+\frac{34\!\cdots\!90}{79\!\cdots\!49}a^{23}+\frac{87\!\cdots\!99}{79\!\cdots\!49}a^{22}+\frac{35\!\cdots\!85}{79\!\cdots\!49}a^{21}+\frac{59\!\cdots\!60}{23\!\cdots\!47}a^{20}+\frac{72\!\cdots\!91}{23\!\cdots\!47}a^{19}+\frac{12\!\cdots\!80}{23\!\cdots\!47}a^{18}+\frac{83\!\cdots\!35}{23\!\cdots\!47}a^{17}+\frac{20\!\cdots\!03}{23\!\cdots\!47}a^{16}-\frac{25\!\cdots\!16}{23\!\cdots\!47}a^{15}+\frac{15\!\cdots\!90}{23\!\cdots\!47}a^{14}+\frac{19\!\cdots\!19}{23\!\cdots\!47}a^{13}-\frac{13\!\cdots\!35}{23\!\cdots\!47}a^{12}-\frac{12\!\cdots\!69}{23\!\cdots\!47}a^{11}+\frac{20\!\cdots\!86}{79\!\cdots\!49}a^{10}-\frac{10\!\cdots\!49}{23\!\cdots\!47}a^{9}-\frac{34\!\cdots\!50}{79\!\cdots\!49}a^{8}-\frac{52\!\cdots\!30}{23\!\cdots\!47}a^{7}+\frac{36\!\cdots\!45}{23\!\cdots\!47}a^{6}+\frac{19\!\cdots\!86}{79\!\cdots\!49}a^{5}-\frac{53\!\cdots\!47}{23\!\cdots\!47}a^{4}+\frac{13\!\cdots\!59}{79\!\cdots\!49}a^{3}-\frac{12\!\cdots\!40}{79\!\cdots\!49}a^{2}-\frac{11\!\cdots\!98}{23\!\cdots\!47}a+\frac{81\!\cdots\!11}{23\!\cdots\!47}$, $\frac{1}{23\!\cdots\!47}a^{36}-\frac{17}{23\!\cdots\!47}a^{34}+\frac{12\!\cdots\!43}{23\!\cdots\!47}a^{33}+\frac{38\!\cdots\!89}{23\!\cdots\!47}a^{32}-\frac{44\!\cdots\!84}{23\!\cdots\!47}a^{31}-\frac{356872387364752}{79\!\cdots\!49}a^{30}+\frac{218104517879780}{23\!\cdots\!47}a^{29}+\frac{21\!\cdots\!21}{23\!\cdots\!47}a^{28}+\frac{28\!\cdots\!35}{23\!\cdots\!47}a^{27}+\frac{12\!\cdots\!65}{23\!\cdots\!47}a^{26}-\frac{55\!\cdots\!48}{79\!\cdots\!49}a^{25}-\frac{96\!\cdots\!70}{79\!\cdots\!49}a^{24}+\frac{57\!\cdots\!31}{79\!\cdots\!49}a^{23}-\frac{54\!\cdots\!96}{79\!\cdots\!49}a^{22}+\frac{17\!\cdots\!56}{23\!\cdots\!47}a^{21}+\frac{69\!\cdots\!71}{79\!\cdots\!49}a^{20}+\frac{62\!\cdots\!65}{23\!\cdots\!47}a^{19}+\frac{58\!\cdots\!38}{23\!\cdots\!47}a^{18}+\frac{39\!\cdots\!49}{23\!\cdots\!47}a^{17}-\frac{18\!\cdots\!06}{23\!\cdots\!47}a^{16}+\frac{23\!\cdots\!17}{23\!\cdots\!47}a^{15}+\frac{11\!\cdots\!58}{23\!\cdots\!47}a^{14}-\frac{32\!\cdots\!60}{79\!\cdots\!49}a^{13}-\frac{24\!\cdots\!63}{79\!\cdots\!49}a^{12}-\frac{93\!\cdots\!96}{23\!\cdots\!47}a^{11}-\frac{23\!\cdots\!97}{79\!\cdots\!49}a^{10}-\frac{34\!\cdots\!63}{23\!\cdots\!47}a^{9}-\frac{10\!\cdots\!82}{23\!\cdots\!47}a^{8}-\frac{17\!\cdots\!89}{79\!\cdots\!49}a^{7}-\frac{87\!\cdots\!17}{50\!\cdots\!01}a^{6}-\frac{43\!\cdots\!46}{23\!\cdots\!47}a^{5}+\frac{97\!\cdots\!08}{23\!\cdots\!47}a^{4}-\frac{24\!\cdots\!62}{23\!\cdots\!47}a^{3}+\frac{12\!\cdots\!45}{79\!\cdots\!49}a^{2}+\frac{90\!\cdots\!34}{23\!\cdots\!47}a+\frac{17\!\cdots\!99}{23\!\cdots\!47}$, $\frac{1}{23\!\cdots\!47}a^{37}-\frac{5}{79\!\cdots\!49}a^{34}+\frac{22\!\cdots\!77}{79\!\cdots\!49}a^{33}+\frac{61\!\cdots\!03}{23\!\cdots\!47}a^{32}-\frac{88\!\cdots\!02}{23\!\cdots\!47}a^{31}+\frac{11\!\cdots\!59}{79\!\cdots\!49}a^{30}+\frac{18\!\cdots\!55}{23\!\cdots\!47}a^{29}+\frac{42\!\cdots\!83}{23\!\cdots\!47}a^{28}-\frac{99\!\cdots\!41}{79\!\cdots\!49}a^{27}+\frac{27\!\cdots\!95}{23\!\cdots\!47}a^{26}+\frac{75\!\cdots\!98}{79\!\cdots\!49}a^{25}+\frac{98\!\cdots\!48}{79\!\cdots\!49}a^{24}+\frac{14\!\cdots\!71}{79\!\cdots\!49}a^{23}-\frac{19\!\cdots\!73}{23\!\cdots\!47}a^{22}+\frac{36\!\cdots\!17}{79\!\cdots\!49}a^{21}-\frac{42\!\cdots\!00}{79\!\cdots\!49}a^{20}+\frac{80\!\cdots\!58}{79\!\cdots\!49}a^{19}+\frac{24\!\cdots\!25}{79\!\cdots\!49}a^{18}-\frac{16\!\cdots\!73}{23\!\cdots\!47}a^{17}+\frac{12\!\cdots\!30}{79\!\cdots\!49}a^{16}+\frac{33\!\cdots\!27}{79\!\cdots\!49}a^{15}+\frac{98\!\cdots\!44}{23\!\cdots\!47}a^{14}-\frac{19\!\cdots\!40}{23\!\cdots\!47}a^{13}-\frac{19\!\cdots\!63}{23\!\cdots\!47}a^{12}-\frac{38\!\cdots\!28}{23\!\cdots\!47}a^{11}+\frac{43\!\cdots\!25}{23\!\cdots\!47}a^{10}+\frac{42\!\cdots\!90}{23\!\cdots\!47}a^{9}-\frac{16\!\cdots\!02}{79\!\cdots\!49}a^{8}-\frac{51\!\cdots\!65}{23\!\cdots\!47}a^{7}+\frac{11\!\cdots\!48}{79\!\cdots\!49}a^{6}+\frac{61\!\cdots\!19}{23\!\cdots\!47}a^{5}+\frac{39\!\cdots\!28}{79\!\cdots\!49}a^{4}-\frac{55\!\cdots\!29}{79\!\cdots\!49}a^{3}+\frac{27\!\cdots\!67}{23\!\cdots\!47}a^{2}-\frac{21\!\cdots\!03}{23\!\cdots\!47}a-\frac{15\!\cdots\!96}{23\!\cdots\!47}$, $\frac{1}{23\!\cdots\!47}a^{38}+\frac{4}{23\!\cdots\!47}a^{34}-\frac{89\!\cdots\!80}{23\!\cdots\!47}a^{33}-\frac{39\!\cdots\!70}{23\!\cdots\!47}a^{32}+\frac{24\!\cdots\!45}{23\!\cdots\!47}a^{31}-\frac{17\!\cdots\!62}{23\!\cdots\!47}a^{30}+\frac{26\!\cdots\!56}{23\!\cdots\!47}a^{29}+\frac{22\!\cdots\!99}{23\!\cdots\!47}a^{28}-\frac{12\!\cdots\!32}{23\!\cdots\!47}a^{27}-\frac{14\!\cdots\!38}{23\!\cdots\!47}a^{26}+\frac{90\!\cdots\!03}{79\!\cdots\!49}a^{25}+\frac{72\!\cdots\!49}{79\!\cdots\!49}a^{24}-\frac{11\!\cdots\!47}{23\!\cdots\!47}a^{23}+\frac{41\!\cdots\!38}{79\!\cdots\!49}a^{22}-\frac{12\!\cdots\!62}{79\!\cdots\!49}a^{21}+\frac{75\!\cdots\!96}{79\!\cdots\!49}a^{20}-\frac{67\!\cdots\!96}{23\!\cdots\!47}a^{19}+\frac{24\!\cdots\!57}{23\!\cdots\!47}a^{18}-\frac{31\!\cdots\!17}{23\!\cdots\!47}a^{17}-\frac{22\!\cdots\!45}{23\!\cdots\!47}a^{16}+\frac{55\!\cdots\!40}{79\!\cdots\!49}a^{15}-\frac{46\!\cdots\!41}{23\!\cdots\!47}a^{14}-\frac{10\!\cdots\!34}{23\!\cdots\!47}a^{13}-\frac{76\!\cdots\!79}{23\!\cdots\!47}a^{12}+\frac{65\!\cdots\!55}{23\!\cdots\!47}a^{11}-\frac{49\!\cdots\!73}{23\!\cdots\!47}a^{10}-\frac{81\!\cdots\!02}{23\!\cdots\!47}a^{9}+\frac{64\!\cdots\!46}{23\!\cdots\!47}a^{8}+\frac{35\!\cdots\!85}{79\!\cdots\!49}a^{7}-\frac{35\!\cdots\!35}{79\!\cdots\!49}a^{6}-\frac{10\!\cdots\!40}{23\!\cdots\!47}a^{5}-\frac{75\!\cdots\!08}{23\!\cdots\!47}a^{4}-\frac{55\!\cdots\!99}{23\!\cdots\!47}a^{3}-\frac{18\!\cdots\!39}{79\!\cdots\!49}a^{2}-\frac{10\!\cdots\!49}{79\!\cdots\!49}a+\frac{10\!\cdots\!51}{23\!\cdots\!47}$, $\frac{1}{23\!\cdots\!47}a^{39}+\frac{2}{23\!\cdots\!47}a^{34}-\frac{10\!\cdots\!53}{23\!\cdots\!47}a^{33}+\frac{62\!\cdots\!49}{79\!\cdots\!49}a^{32}-\frac{83\!\cdots\!40}{23\!\cdots\!47}a^{31}-\frac{14\!\cdots\!21}{23\!\cdots\!47}a^{30}+\frac{92\!\cdots\!78}{79\!\cdots\!49}a^{29}+\frac{10\!\cdots\!68}{23\!\cdots\!47}a^{28}+\frac{54\!\cdots\!59}{79\!\cdots\!49}a^{27}+\frac{44\!\cdots\!27}{79\!\cdots\!49}a^{26}+\frac{12\!\cdots\!36}{79\!\cdots\!49}a^{25}-\frac{25\!\cdots\!91}{23\!\cdots\!47}a^{24}+\frac{358619484460536}{79\!\cdots\!49}a^{23}+\frac{46\!\cdots\!25}{79\!\cdots\!49}a^{22}-\frac{35\!\cdots\!03}{79\!\cdots\!49}a^{21}-\frac{65\!\cdots\!89}{79\!\cdots\!49}a^{20}-\frac{19\!\cdots\!68}{23\!\cdots\!47}a^{19}+\frac{55\!\cdots\!50}{79\!\cdots\!49}a^{18}-\frac{65\!\cdots\!26}{79\!\cdots\!49}a^{17}-\frac{13\!\cdots\!84}{23\!\cdots\!47}a^{16}+\frac{18\!\cdots\!79}{23\!\cdots\!47}a^{15}+\frac{23\!\cdots\!31}{23\!\cdots\!47}a^{14}+\frac{24\!\cdots\!68}{23\!\cdots\!47}a^{13}+\frac{17\!\cdots\!24}{23\!\cdots\!47}a^{12}-\frac{47\!\cdots\!92}{23\!\cdots\!47}a^{11}+\frac{61\!\cdots\!91}{79\!\cdots\!49}a^{10}+\frac{87\!\cdots\!44}{23\!\cdots\!47}a^{9}+\frac{19\!\cdots\!73}{79\!\cdots\!49}a^{8}-\frac{11\!\cdots\!83}{23\!\cdots\!47}a^{7}+\frac{18\!\cdots\!74}{79\!\cdots\!49}a^{6}-\frac{22\!\cdots\!43}{79\!\cdots\!49}a^{5}+\frac{11\!\cdots\!54}{23\!\cdots\!47}a^{4}-\frac{74\!\cdots\!93}{23\!\cdots\!47}a^{3}-\frac{14\!\cdots\!34}{23\!\cdots\!47}a^{2}+\frac{26\!\cdots\!55}{79\!\cdots\!49}a+\frac{89\!\cdots\!04}{79\!\cdots\!49}$, $\frac{1}{23\!\cdots\!47}a^{40}+\frac{1}{23\!\cdots\!47}a^{34}-\frac{25\!\cdots\!85}{79\!\cdots\!49}a^{33}+\frac{37\!\cdots\!59}{23\!\cdots\!47}a^{32}-\frac{93\!\cdots\!57}{23\!\cdots\!47}a^{31}+\frac{12\!\cdots\!95}{79\!\cdots\!49}a^{30}+\frac{16\!\cdots\!83}{23\!\cdots\!47}a^{29}+\frac{20\!\cdots\!64}{79\!\cdots\!49}a^{28}-\frac{30\!\cdots\!80}{79\!\cdots\!49}a^{27}+\frac{22\!\cdots\!12}{23\!\cdots\!47}a^{26}+\frac{19\!\cdots\!66}{23\!\cdots\!47}a^{25}-\frac{10\!\cdots\!53}{79\!\cdots\!49}a^{24}-\frac{10\!\cdots\!97}{79\!\cdots\!49}a^{23}-\frac{16\!\cdots\!56}{79\!\cdots\!49}a^{22}+\frac{24\!\cdots\!55}{79\!\cdots\!49}a^{21}+\frac{17\!\cdots\!89}{79\!\cdots\!49}a^{20}+\frac{14\!\cdots\!34}{79\!\cdots\!49}a^{19}-\frac{22\!\cdots\!49}{79\!\cdots\!49}a^{18}+\frac{63\!\cdots\!37}{79\!\cdots\!49}a^{17}-\frac{94\!\cdots\!86}{23\!\cdots\!47}a^{16}-\frac{39\!\cdots\!73}{23\!\cdots\!47}a^{15}+\frac{24\!\cdots\!09}{23\!\cdots\!47}a^{14}-\frac{78\!\cdots\!45}{23\!\cdots\!47}a^{13}-\frac{21\!\cdots\!32}{79\!\cdots\!49}a^{12}-\frac{17\!\cdots\!65}{79\!\cdots\!49}a^{11}-\frac{36\!\cdots\!91}{79\!\cdots\!49}a^{10}-\frac{87\!\cdots\!69}{23\!\cdots\!47}a^{9}+\frac{10\!\cdots\!52}{23\!\cdots\!47}a^{8}-\frac{61\!\cdots\!14}{23\!\cdots\!47}a^{7}+\frac{32\!\cdots\!88}{79\!\cdots\!49}a^{6}+\frac{28\!\cdots\!08}{79\!\cdots\!49}a^{5}-\frac{35\!\cdots\!40}{79\!\cdots\!49}a^{4}+\frac{56\!\cdots\!22}{23\!\cdots\!47}a^{3}+\frac{25\!\cdots\!66}{79\!\cdots\!49}a^{2}+\frac{15\!\cdots\!71}{23\!\cdots\!47}a+\frac{48\!\cdots\!24}{23\!\cdots\!47}$, $\frac{1}{23\!\cdots\!47}a^{41}+\frac{20}{23\!\cdots\!47}a^{34}-\frac{72\!\cdots\!60}{23\!\cdots\!47}a^{33}-\frac{11\!\cdots\!90}{23\!\cdots\!47}a^{32}-\frac{21\!\cdots\!75}{23\!\cdots\!47}a^{31}-\frac{30\!\cdots\!20}{23\!\cdots\!47}a^{30}+\frac{27\!\cdots\!75}{23\!\cdots\!47}a^{29}+\frac{20\!\cdots\!33}{23\!\cdots\!47}a^{28}+\frac{17\!\cdots\!70}{23\!\cdots\!47}a^{27}-\frac{19\!\cdots\!98}{79\!\cdots\!49}a^{26}+\frac{11\!\cdots\!47}{79\!\cdots\!49}a^{25}-\frac{12\!\cdots\!97}{79\!\cdots\!49}a^{24}-\frac{10\!\cdots\!96}{79\!\cdots\!49}a^{23}+\frac{63\!\cdots\!18}{79\!\cdots\!49}a^{22}+\frac{52\!\cdots\!35}{79\!\cdots\!49}a^{21}+\frac{13\!\cdots\!14}{23\!\cdots\!47}a^{20}-\frac{61\!\cdots\!34}{23\!\cdots\!47}a^{19}-\frac{10\!\cdots\!45}{23\!\cdots\!47}a^{18}+\frac{13\!\cdots\!45}{23\!\cdots\!47}a^{17}-\frac{11\!\cdots\!82}{79\!\cdots\!49}a^{16}-\frac{20\!\cdots\!84}{23\!\cdots\!47}a^{15}-\frac{20\!\cdots\!15}{23\!\cdots\!47}a^{14}+\frac{17\!\cdots\!80}{23\!\cdots\!47}a^{13}+\frac{98\!\cdots\!81}{23\!\cdots\!47}a^{12}+\frac{31\!\cdots\!74}{79\!\cdots\!49}a^{11}+\frac{18\!\cdots\!88}{79\!\cdots\!49}a^{10}+\frac{26\!\cdots\!12}{23\!\cdots\!47}a^{9}-\frac{23\!\cdots\!89}{23\!\cdots\!47}a^{8}-\frac{49\!\cdots\!90}{10\!\cdots\!83}a^{7}-\frac{44\!\cdots\!05}{23\!\cdots\!47}a^{6}-\frac{35\!\cdots\!46}{79\!\cdots\!49}a^{5}-\frac{19\!\cdots\!40}{79\!\cdots\!49}a^{4}+\frac{21\!\cdots\!78}{50\!\cdots\!01}a^{3}+\frac{19\!\cdots\!55}{79\!\cdots\!49}a^{2}+\frac{11\!\cdots\!47}{23\!\cdots\!47}a-\frac{31\!\cdots\!89}{23\!\cdots\!47}$, $\frac{1}{23\!\cdots\!47}a^{42}+\frac{2}{23\!\cdots\!47}a^{34}+\frac{54\!\cdots\!51}{23\!\cdots\!47}a^{33}+\frac{12\!\cdots\!67}{23\!\cdots\!47}a^{32}+\frac{11\!\cdots\!74}{23\!\cdots\!47}a^{31}-\frac{46\!\cdots\!32}{79\!\cdots\!49}a^{30}+\frac{26\!\cdots\!61}{23\!\cdots\!47}a^{29}-\frac{25\!\cdots\!64}{23\!\cdots\!47}a^{28}+\frac{38\!\cdots\!91}{23\!\cdots\!47}a^{27}-\frac{33\!\cdots\!80}{23\!\cdots\!47}a^{26}+\frac{10\!\cdots\!58}{79\!\cdots\!49}a^{25}+\frac{47\!\cdots\!08}{79\!\cdots\!49}a^{24}-\frac{60\!\cdots\!47}{79\!\cdots\!49}a^{23}-\frac{58\!\cdots\!30}{79\!\cdots\!49}a^{22}-\frac{13\!\cdots\!95}{23\!\cdots\!47}a^{21}-\frac{79\!\cdots\!35}{79\!\cdots\!49}a^{20}-\frac{11\!\cdots\!50}{23\!\cdots\!47}a^{19}-\frac{13\!\cdots\!26}{23\!\cdots\!47}a^{18}-\frac{39\!\cdots\!86}{79\!\cdots\!49}a^{17}+\frac{24\!\cdots\!20}{23\!\cdots\!47}a^{16}+\frac{76\!\cdots\!47}{79\!\cdots\!49}a^{15}-\frac{20\!\cdots\!66}{23\!\cdots\!47}a^{14}+\frac{10\!\cdots\!36}{23\!\cdots\!47}a^{13}-\frac{78\!\cdots\!01}{79\!\cdots\!49}a^{12}-\frac{30\!\cdots\!80}{23\!\cdots\!47}a^{11}-\frac{50\!\cdots\!77}{23\!\cdots\!47}a^{10}+\frac{43\!\cdots\!33}{23\!\cdots\!47}a^{9}-\frac{25\!\cdots\!56}{23\!\cdots\!47}a^{8}+\frac{17\!\cdots\!00}{79\!\cdots\!49}a^{7}-\frac{15\!\cdots\!51}{79\!\cdots\!49}a^{6}+\frac{31\!\cdots\!43}{23\!\cdots\!47}a^{5}-\frac{46\!\cdots\!55}{23\!\cdots\!47}a^{4}-\frac{13\!\cdots\!48}{79\!\cdots\!49}a^{3}-\frac{27\!\cdots\!70}{79\!\cdots\!49}a^{2}-\frac{24\!\cdots\!11}{79\!\cdots\!49}a+\frac{25\!\cdots\!89}{23\!\cdots\!47}$, $\frac{1}{23\!\cdots\!47}a^{43}-\frac{17}{23\!\cdots\!47}a^{34}-\frac{10\!\cdots\!39}{23\!\cdots\!47}a^{33}+\frac{34\!\cdots\!65}{23\!\cdots\!47}a^{32}-\frac{36\!\cdots\!74}{23\!\cdots\!47}a^{31}-\frac{25\!\cdots\!07}{23\!\cdots\!47}a^{30}+\frac{19\!\cdots\!04}{23\!\cdots\!47}a^{29}+\frac{65\!\cdots\!31}{79\!\cdots\!49}a^{28}-\frac{11\!\cdots\!40}{23\!\cdots\!47}a^{27}+\frac{28\!\cdots\!00}{23\!\cdots\!47}a^{26}+\frac{528659615786432}{79\!\cdots\!49}a^{25}+\frac{28\!\cdots\!95}{79\!\cdots\!49}a^{24}-\frac{37\!\cdots\!36}{79\!\cdots\!49}a^{23}+\frac{18\!\cdots\!44}{23\!\cdots\!47}a^{22}-\frac{23\!\cdots\!61}{79\!\cdots\!49}a^{21}+\frac{82\!\cdots\!64}{23\!\cdots\!47}a^{20}-\frac{94\!\cdots\!18}{23\!\cdots\!47}a^{19}-\frac{54\!\cdots\!92}{79\!\cdots\!49}a^{18}+\frac{18\!\cdots\!66}{23\!\cdots\!47}a^{17}+\frac{25\!\cdots\!58}{23\!\cdots\!47}a^{16}-\frac{38\!\cdots\!86}{23\!\cdots\!47}a^{15}+\frac{20\!\cdots\!93}{23\!\cdots\!47}a^{14}+\frac{10\!\cdots\!87}{23\!\cdots\!47}a^{13}+\frac{72\!\cdots\!46}{79\!\cdots\!49}a^{12}+\frac{36\!\cdots\!74}{23\!\cdots\!47}a^{11}+\frac{79\!\cdots\!25}{79\!\cdots\!49}a^{10}-\frac{26\!\cdots\!79}{79\!\cdots\!49}a^{9}-\frac{24\!\cdots\!19}{79\!\cdots\!49}a^{8}+\frac{25\!\cdots\!67}{23\!\cdots\!47}a^{7}-\frac{10\!\cdots\!19}{23\!\cdots\!47}a^{6}+\frac{65\!\cdots\!41}{23\!\cdots\!47}a^{5}-\frac{82\!\cdots\!23}{23\!\cdots\!47}a^{4}+\frac{10\!\cdots\!66}{23\!\cdots\!47}a^{3}+\frac{58\!\cdots\!95}{23\!\cdots\!47}a^{2}+\frac{29\!\cdots\!19}{79\!\cdots\!49}a-\frac{14\!\cdots\!58}{23\!\cdots\!47}$ 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

not computed

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:  $43$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 312*x^42 + 303*x^41 + 45350*x^40 - 42623*x^39 - 4077513*x^38 + 3693906*x^37 + 253947067*x^36 - 220701913*x^35 - 11624898400*x^34 + 9638581183*x^33 + 405056137130*x^32 - 318308906483*x^31 - 10976588371921*x^30 + 8111808213574*x^29 + 234369143305301*x^28 - 161362869383135*x^27 - 3970563499772876*x^26 + 2518297675324661*x^25 + 53492986112604139*x^24 - 30828307034682190*x^23 - 572009925875972279*x^22 + 294555163094127792*x^21 + 4826078584950284457*x^20 - 2175082202480665232*x^19 - 31800584527656001615*x^18 + 12224850681757177737*x^17 + 161193984162806092803*x^16 - 51170566486435305249*x^15 - 615319732524023675934*x^14 + 154790608182698273409*x^13 + 1717658346766087000620*x^12 - 324640449052810836967*x^11 - 3366040964991975949283*x^10 + 445321559954531949468*x^9 + 4367921735840229125340*x^8 - 367209571759684618733*x^7 - 3436106010074360679257*x^6 + 161383741380219872801*x^5 + 1415445998956830381216*x^4 - 33374706535233304336*x^3 - 228324835255811811652*x^2 + 3748885669892449290*x + 9622931728201923763)
 
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^44 - x^43 - 312*x^42 + 303*x^41 + 45350*x^40 - 42623*x^39 - 4077513*x^38 + 3693906*x^37 + 253947067*x^36 - 220701913*x^35 - 11624898400*x^34 + 9638581183*x^33 + 405056137130*x^32 - 318308906483*x^31 - 10976588371921*x^30 + 8111808213574*x^29 + 234369143305301*x^28 - 161362869383135*x^27 - 3970563499772876*x^26 + 2518297675324661*x^25 + 53492986112604139*x^24 - 30828307034682190*x^23 - 572009925875972279*x^22 + 294555163094127792*x^21 + 4826078584950284457*x^20 - 2175082202480665232*x^19 - 31800584527656001615*x^18 + 12224850681757177737*x^17 + 161193984162806092803*x^16 - 51170566486435305249*x^15 - 615319732524023675934*x^14 + 154790608182698273409*x^13 + 1717658346766087000620*x^12 - 324640449052810836967*x^11 - 3366040964991975949283*x^10 + 445321559954531949468*x^9 + 4367921735840229125340*x^8 - 367209571759684618733*x^7 - 3436106010074360679257*x^6 + 161383741380219872801*x^5 + 1415445998956830381216*x^4 - 33374706535233304336*x^3 - 228324835255811811652*x^2 + 3748885669892449290*x + 9622931728201923763, 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^44 - x^43 - 312*x^42 + 303*x^41 + 45350*x^40 - 42623*x^39 - 4077513*x^38 + 3693906*x^37 + 253947067*x^36 - 220701913*x^35 - 11624898400*x^34 + 9638581183*x^33 + 405056137130*x^32 - 318308906483*x^31 - 10976588371921*x^30 + 8111808213574*x^29 + 234369143305301*x^28 - 161362869383135*x^27 - 3970563499772876*x^26 + 2518297675324661*x^25 + 53492986112604139*x^24 - 30828307034682190*x^23 - 572009925875972279*x^22 + 294555163094127792*x^21 + 4826078584950284457*x^20 - 2175082202480665232*x^19 - 31800584527656001615*x^18 + 12224850681757177737*x^17 + 161193984162806092803*x^16 - 51170566486435305249*x^15 - 615319732524023675934*x^14 + 154790608182698273409*x^13 + 1717658346766087000620*x^12 - 324640449052810836967*x^11 - 3366040964991975949283*x^10 + 445321559954531949468*x^9 + 4367921735840229125340*x^8 - 367209571759684618733*x^7 - 3436106010074360679257*x^6 + 161383741380219872801*x^5 + 1415445998956830381216*x^4 - 33374706535233304336*x^3 - 228324835255811811652*x^2 + 3748885669892449290*x + 9622931728201923763);
 
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^44 - x^43 - 312*x^42 + 303*x^41 + 45350*x^40 - 42623*x^39 - 4077513*x^38 + 3693906*x^37 + 253947067*x^36 - 220701913*x^35 - 11624898400*x^34 + 9638581183*x^33 + 405056137130*x^32 - 318308906483*x^31 - 10976588371921*x^30 + 8111808213574*x^29 + 234369143305301*x^28 - 161362869383135*x^27 - 3970563499772876*x^26 + 2518297675324661*x^25 + 53492986112604139*x^24 - 30828307034682190*x^23 - 572009925875972279*x^22 + 294555163094127792*x^21 + 4826078584950284457*x^20 - 2175082202480665232*x^19 - 31800584527656001615*x^18 + 12224850681757177737*x^17 + 161193984162806092803*x^16 - 51170566486435305249*x^15 - 615319732524023675934*x^14 + 154790608182698273409*x^13 + 1717658346766087000620*x^12 - 324640449052810836967*x^11 - 3366040964991975949283*x^10 + 445321559954531949468*x^9 + 4367921735840229125340*x^8 - 367209571759684618733*x^7 - 3436106010074360679257*x^6 + 161383741380219872801*x^5 + 1415445998956830381216*x^4 - 33374706535233304336*x^3 - 228324835255811811652*x^2 + 3748885669892449290*x + 9622931728201923763);
 
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

$C_{44}$ (as 44T1):

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 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 4.4.26795437.1, \(\Q(\zeta_{23})^+\), 22.22.305334364114002390216524630254855801940178013.1

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 $44$ $22^{2}$ $44$ $22^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{4}$ $44$ $44$ $44$ R $44$ $44$ R $22^{2}$ $44$ ${\href{/padicField/47.1.0.1}{1} }^{44}$ $22^{2}$ $44$

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
\(23\) Copy content Toggle raw display Deg $44$$22$$2$$42$
\(37\) Copy content Toggle raw display Deg $44$$4$$11$$33$