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Properties

Label 44.44.158...625.1
Degree $44$
Signature $[44, 0]$
Discriminant $1.589\times 10^{99}$
Root discriminant \(179.71\)
Ramified primes $5,89$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581)
 
gp: K = bnfinit(y^44 - y^43 - 132*y^42 + 131*y^41 + 7981*y^40 - 7851*y^39 - 293502*y^38 + 285780*y^37 + 7352544*y^36 - 7074358*y^35 - 133171784*y^34 + 126368145*y^33 + 1806620424*y^32 - 1686827784*y^31 - 18764844858*y^30 + 17194056330*y^29 + 151268636295*y^28 - 135618356054*y^27 - 953714610433*y^26 + 833885620680*y^25 + 4717270351997*y^24 - 4008528403014*y^23 - 18288247142006*y^22 + 15048715494929*y^21 + 55312614944324*y^20 - 43911599382957*y^19 - 129393380339543*y^18 + 98733171381559*y^17 + 231114721012025*y^16 - 168838600437028*y^15 - 309586495847682*y^14 + 215575931040434*y^13 + 303617014545478*y^12 - 200381975387743*y^11 - 211144037309930*y^10 + 130979137701417*y^9 + 99728212374110*y^8 - 57361851429233*y^7 - 30095367088880*y^6 + 15665394240053*y^5 + 5275330691163*y^4 - 2357998728050*y^3 - 452559496938*y^2 + 146388138349*y + 12360161581, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581)
 

\( x^{44} - x^{43} - 132 x^{42} + 131 x^{41} + 7981 x^{40} - 7851 x^{39} - 293502 x^{38} + \cdots + 12360161581 \) 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:   \(158\!\cdots\!625\) \(\medspace = 5^{22}\cdot 89^{43}\) 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:  \(179.71\)
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:  $5^{1/2}89^{43/44}\approx 179.70935708086185$
Ramified primes:   \(5\), \(89\) 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:  \(\Q(\sqrt{89}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(445=5\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{445}(256,·)$, $\chi_{445}(1,·)$, $\chi_{445}(129,·)$, $\chi_{445}(9,·)$, $\chi_{445}(266,·)$, $\chi_{445}(11,·)$, $\chi_{445}(271,·)$, $\chi_{445}(16,·)$, $\chi_{445}(401,·)$, $\chi_{445}(146,·)$, $\chi_{445}(406,·)$, $\chi_{445}(409,·)$, $\chi_{445}(284,·)$, $\chi_{445}(34,·)$, $\chi_{445}(424,·)$, $\chi_{445}(156,·)$, $\chi_{445}(176,·)$, $\chi_{445}(49,·)$, $\chi_{445}(306,·)$, $\chi_{445}(309,·)$, $\chi_{445}(311,·)$, $\chi_{445}(441,·)$, $\chi_{445}(314,·)$, $\chi_{445}(69,·)$, $\chi_{445}(199,·)$, $\chi_{445}(331,·)$, $\chi_{445}(79,·)$, $\chi_{445}(81,·)$, $\chi_{445}(339,·)$, $\chi_{445}(84,·)$, $\chi_{445}(214,·)$, $\chi_{445}(121,·)$, $\chi_{445}(91,·)$, $\chi_{445}(186,·)$, $\chi_{445}(94,·)$, $\chi_{445}(144,·)$, $\chi_{445}(99,·)$, $\chi_{445}(109,·)$, $\chi_{445}(111,·)$, $\chi_{445}(374,·)$, $\chi_{445}(169,·)$, $\chi_{445}(249,·)$, $\chi_{445}(251,·)$, $\chi_{445}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $\frac{1}{179}a^{38}+\frac{26}{179}a^{37}-\frac{8}{179}a^{36}-\frac{39}{179}a^{35}+\frac{46}{179}a^{34}-\frac{36}{179}a^{33}-\frac{49}{179}a^{32}+\frac{39}{179}a^{31}-\frac{44}{179}a^{30}-\frac{69}{179}a^{29}-\frac{24}{179}a^{28}-\frac{59}{179}a^{27}+\frac{82}{179}a^{26}+\frac{25}{179}a^{25}+\frac{12}{179}a^{24}+\frac{66}{179}a^{23}+\frac{81}{179}a^{22}-\frac{77}{179}a^{21}+\frac{79}{179}a^{20}-\frac{5}{179}a^{19}-\frac{76}{179}a^{18}+\frac{18}{179}a^{17}-\frac{34}{179}a^{16}+\frac{22}{179}a^{15}+\frac{26}{179}a^{14}-\frac{49}{179}a^{13}-\frac{44}{179}a^{12}-\frac{68}{179}a^{11}+\frac{62}{179}a^{10}-\frac{11}{179}a^{9}+\frac{82}{179}a^{8}-\frac{67}{179}a^{7}+\frac{1}{179}a^{6}-\frac{61}{179}a^{5}-\frac{16}{179}a^{4}-\frac{11}{179}a^{3}+\frac{42}{179}a^{2}-\frac{86}{179}a-\frac{40}{179}$, $\frac{1}{179}a^{39}+\frac{32}{179}a^{37}-\frac{10}{179}a^{36}-\frac{14}{179}a^{35}+\frac{21}{179}a^{34}-\frac{8}{179}a^{33}+\frac{60}{179}a^{32}+\frac{16}{179}a^{31}+\frac{1}{179}a^{30}-\frac{20}{179}a^{29}+\frac{28}{179}a^{28}+\frac{5}{179}a^{27}+\frac{41}{179}a^{26}+\frac{78}{179}a^{25}-\frac{67}{179}a^{24}-\frac{24}{179}a^{23}-\frac{35}{179}a^{22}-\frac{67}{179}a^{21}+\frac{89}{179}a^{20}+\frac{54}{179}a^{19}+\frac{25}{179}a^{18}+\frac{35}{179}a^{17}+\frac{11}{179}a^{16}-\frac{9}{179}a^{15}-\frac{9}{179}a^{14}-\frac{23}{179}a^{13}+\frac{2}{179}a^{12}+\frac{40}{179}a^{11}-\frac{12}{179}a^{10}+\frac{10}{179}a^{9}-\frac{51}{179}a^{8}-\frac{47}{179}a^{7}-\frac{87}{179}a^{6}-\frac{41}{179}a^{5}+\frac{47}{179}a^{4}-\frac{30}{179}a^{3}+\frac{75}{179}a^{2}+\frac{48}{179}a-\frac{34}{179}$, $\frac{1}{179}a^{40}+\frac{53}{179}a^{37}+\frac{63}{179}a^{36}+\frac{16}{179}a^{35}-\frac{48}{179}a^{34}-\frac{41}{179}a^{33}-\frac{27}{179}a^{32}+\frac{6}{179}a^{31}-\frac{44}{179}a^{30}+\frac{88}{179}a^{29}+\frac{57}{179}a^{28}-\frac{40}{179}a^{27}-\frac{40}{179}a^{26}+\frac{28}{179}a^{25}-\frac{50}{179}a^{24}+\frac{1}{179}a^{23}+\frac{26}{179}a^{22}+\frac{47}{179}a^{21}+\frac{32}{179}a^{20}+\frac{6}{179}a^{19}-\frac{39}{179}a^{18}-\frac{28}{179}a^{17}+\frac{5}{179}a^{16}+\frac{3}{179}a^{15}+\frac{40}{179}a^{14}-\frac{41}{179}a^{13}+\frac{16}{179}a^{12}+\frac{16}{179}a^{11}-\frac{5}{179}a^{10}-\frac{57}{179}a^{9}+\frac{14}{179}a^{8}+\frac{88}{179}a^{7}-\frac{73}{179}a^{6}+\frac{30}{179}a^{5}-\frac{55}{179}a^{4}+\frac{69}{179}a^{3}-\frac{43}{179}a^{2}+\frac{33}{179}a+\frac{27}{179}$, $\frac{1}{44584783}a^{41}-\frac{79331}{44584783}a^{40}+\frac{120426}{44584783}a^{39}-\frac{56178}{44584783}a^{38}-\frac{12998743}{44584783}a^{37}-\frac{4993110}{44584783}a^{36}-\frac{13595346}{44584783}a^{35}-\frac{3268423}{44584783}a^{34}-\frac{7765468}{44584783}a^{33}-\frac{6751833}{44584783}a^{32}-\frac{1606256}{44584783}a^{31}+\frac{8423877}{44584783}a^{30}+\frac{12158722}{44584783}a^{29}+\frac{10313602}{44584783}a^{28}-\frac{10726491}{44584783}a^{27}-\frac{19603559}{44584783}a^{26}-\frac{16165303}{44584783}a^{25}+\frac{10500887}{44584783}a^{24}-\frac{16156856}{44584783}a^{23}-\frac{12664245}{44584783}a^{22}-\frac{17213293}{44584783}a^{21}+\frac{12188922}{44584783}a^{20}+\frac{1708192}{44584783}a^{19}+\frac{2449640}{44584783}a^{18}-\frac{9591032}{44584783}a^{17}+\frac{6497}{44584783}a^{16}+\frac{495408}{44584783}a^{15}+\frac{6164152}{44584783}a^{14}+\frac{13299161}{44584783}a^{13}-\frac{7724255}{44584783}a^{12}+\frac{21747653}{44584783}a^{11}+\frac{14048049}{44584783}a^{10}+\frac{6302975}{44584783}a^{9}+\frac{10187438}{44584783}a^{8}+\frac{81386}{249077}a^{7}-\frac{16053468}{44584783}a^{6}+\frac{22133691}{44584783}a^{5}+\frac{14006446}{44584783}a^{4}+\frac{20290377}{44584783}a^{3}+\frac{18114497}{44584783}a^{2}-\frac{21985514}{44584783}a+\frac{16673536}{44584783}$, $\frac{1}{713137215452423}a^{42}-\frac{7661074}{713137215452423}a^{41}+\frac{846438090937}{713137215452423}a^{40}+\frac{736764066663}{713137215452423}a^{39}-\frac{1336688695915}{713137215452423}a^{38}-\frac{271666641505040}{713137215452423}a^{37}-\frac{176918309180665}{713137215452423}a^{36}+\frac{210202469578433}{713137215452423}a^{35}-\frac{272273272043377}{713137215452423}a^{34}+\frac{264357471954782}{713137215452423}a^{33}+\frac{21811035011433}{713137215452423}a^{32}-\frac{83739465977495}{713137215452423}a^{31}+\frac{298513681399637}{713137215452423}a^{30}+\frac{218039355148153}{713137215452423}a^{29}-\frac{72574156634696}{713137215452423}a^{28}+\frac{17537367116411}{713137215452423}a^{27}+\frac{229284790379740}{713137215452423}a^{26}+\frac{133333095091348}{713137215452423}a^{25}-\frac{189448319036215}{713137215452423}a^{24}-\frac{218635469112457}{713137215452423}a^{23}-\frac{117571537394431}{713137215452423}a^{22}+\frac{65939842132560}{713137215452423}a^{21}-\frac{173626213310051}{713137215452423}a^{20}+\frac{270088104960685}{713137215452423}a^{19}-\frac{10502856229368}{713137215452423}a^{18}-\frac{234848510943493}{713137215452423}a^{17}-\frac{177922375241404}{713137215452423}a^{16}-\frac{166618651811567}{713137215452423}a^{15}+\frac{351305906238201}{713137215452423}a^{14}+\frac{77946527290229}{713137215452423}a^{13}+\frac{37833220484599}{713137215452423}a^{12}+\frac{57877312867455}{713137215452423}a^{11}+\frac{84543277163859}{713137215452423}a^{10}+\frac{62507080680728}{713137215452423}a^{9}-\frac{69177779221382}{713137215452423}a^{8}-\frac{302937258844845}{713137215452423}a^{7}-\frac{263353923826093}{713137215452423}a^{6}-\frac{167895378112273}{713137215452423}a^{5}+\frac{220148712778010}{713137215452423}a^{4}-\frac{228599237937500}{713137215452423}a^{3}-\frac{333721693413459}{713137215452423}a^{2}+\frac{109405514909733}{713137215452423}a+\frac{109626420583699}{713137215452423}$, $\frac{1}{30\!\cdots\!51}a^{43}+\frac{83\!\cdots\!25}{30\!\cdots\!51}a^{42}-\frac{17\!\cdots\!73}{30\!\cdots\!51}a^{41}+\frac{70\!\cdots\!76}{30\!\cdots\!51}a^{40}-\frac{36\!\cdots\!91}{30\!\cdots\!51}a^{39}-\frac{97\!\cdots\!54}{30\!\cdots\!51}a^{38}+\frac{25\!\cdots\!73}{30\!\cdots\!51}a^{37}+\frac{63\!\cdots\!61}{30\!\cdots\!51}a^{36}+\frac{81\!\cdots\!86}{30\!\cdots\!51}a^{35}+\frac{67\!\cdots\!08}{30\!\cdots\!51}a^{34}+\frac{12\!\cdots\!11}{30\!\cdots\!51}a^{33}+\frac{11\!\cdots\!62}{30\!\cdots\!51}a^{32}-\frac{80\!\cdots\!34}{30\!\cdots\!51}a^{31}-\frac{11\!\cdots\!88}{30\!\cdots\!51}a^{30}+\frac{98\!\cdots\!59}{30\!\cdots\!51}a^{29}+\frac{72\!\cdots\!98}{30\!\cdots\!51}a^{28}-\frac{35\!\cdots\!70}{30\!\cdots\!51}a^{27}+\frac{10\!\cdots\!01}{30\!\cdots\!51}a^{26}+\frac{14\!\cdots\!49}{30\!\cdots\!51}a^{25}-\frac{78\!\cdots\!90}{30\!\cdots\!51}a^{24}+\frac{10\!\cdots\!05}{30\!\cdots\!51}a^{23}+\frac{11\!\cdots\!62}{30\!\cdots\!51}a^{22}-\frac{65\!\cdots\!49}{30\!\cdots\!51}a^{21}+\frac{31\!\cdots\!35}{30\!\cdots\!51}a^{20}-\frac{14\!\cdots\!03}{30\!\cdots\!51}a^{19}+\frac{10\!\cdots\!45}{30\!\cdots\!51}a^{18}-\frac{82\!\cdots\!43}{30\!\cdots\!51}a^{17}-\frac{13\!\cdots\!30}{30\!\cdots\!51}a^{16}-\frac{28\!\cdots\!78}{30\!\cdots\!51}a^{15}-\frac{11\!\cdots\!36}{30\!\cdots\!51}a^{14}+\frac{15\!\cdots\!94}{30\!\cdots\!51}a^{13}-\frac{61\!\cdots\!81}{30\!\cdots\!51}a^{12}-\frac{91\!\cdots\!50}{30\!\cdots\!51}a^{11}+\frac{12\!\cdots\!70}{30\!\cdots\!51}a^{10}+\frac{10\!\cdots\!73}{30\!\cdots\!51}a^{9}-\frac{10\!\cdots\!52}{30\!\cdots\!51}a^{8}-\frac{94\!\cdots\!52}{30\!\cdots\!51}a^{7}+\frac{57\!\cdots\!74}{30\!\cdots\!51}a^{6}-\frac{83\!\cdots\!75}{30\!\cdots\!51}a^{5}-\frac{12\!\cdots\!37}{30\!\cdots\!51}a^{4}+\frac{29\!\cdots\!37}{30\!\cdots\!51}a^{3}+\frac{20\!\cdots\!59}{30\!\cdots\!51}a^{2}+\frac{12\!\cdots\!31}{30\!\cdots\!51}a+\frac{12\!\cdots\!96}{30\!\cdots\!51}$ 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 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581)
 
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 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581, 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 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581);
 
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 - 132*x^42 + 131*x^41 + 7981*x^40 - 7851*x^39 - 293502*x^38 + 285780*x^37 + 7352544*x^36 - 7074358*x^35 - 133171784*x^34 + 126368145*x^33 + 1806620424*x^32 - 1686827784*x^31 - 18764844858*x^30 + 17194056330*x^29 + 151268636295*x^28 - 135618356054*x^27 - 953714610433*x^26 + 833885620680*x^25 + 4717270351997*x^24 - 4008528403014*x^23 - 18288247142006*x^22 + 15048715494929*x^21 + 55312614944324*x^20 - 43911599382957*x^19 - 129393380339543*x^18 + 98733171381559*x^17 + 231114721012025*x^16 - 168838600437028*x^15 - 309586495847682*x^14 + 215575931040434*x^13 + 303617014545478*x^12 - 200381975387743*x^11 - 211144037309930*x^10 + 130979137701417*x^9 + 99728212374110*x^8 - 57361851429233*x^7 - 30095367088880*x^6 + 15665394240053*x^5 + 5275330691163*x^4 - 2357998728050*x^3 - 452559496938*x^2 + 146388138349*x + 12360161581);
 
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

\(\Q(\sqrt{89}) \), 4.4.17624225.2, 11.11.31181719929966183601.1, 22.22.86534669543385676516186776267386878120889.1

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 $22^{2}$ $44$ R $44$ ${\href{/padicField/11.11.0.1}{11} }^{4}$ $44$ ${\href{/padicField/17.11.0.1}{11} }^{4}$ $44$ $44$ $44$ $44$ ${\href{/padicField/37.4.0.1}{4} }^{11}$ $44$ $44$ ${\href{/padicField/47.11.0.1}{11} }^{4}$ ${\href{/padicField/53.11.0.1}{11} }^{4}$ $44$

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
\(5\) Copy content Toggle raw display 5.22.11.2$x^{22} + 29296875 x^{2} - 146484375$$2$$11$$11$22T1$[\ ]_{2}^{11}$
5.22.11.2$x^{22} + 29296875 x^{2} - 146484375$$2$$11$$11$22T1$[\ ]_{2}^{11}$
\(89\) Copy content Toggle raw display Deg $44$$44$$1$$43$