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Properties

Label 44.0.627...673.1
Degree $44$
Signature $[0, 22]$
Discriminant $6.272\times 10^{97}$
Root discriminant \(166.98\)
Ramified primes $17,23$
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 + 51*x^42 - 55*x^41 + 1248*x^40 - 1468*x^39 + 19662*x^38 - 25534*x^37 + 226772*x^36 - 328908*x^35 + 2076784*x^34 - 3392416*x^33 + 16113689*x^32 - 29683353*x^31 + 112279727*x^30 - 231013139*x^29 + 739331011*x^28 - 1663383567*x^27 + 4775371571*x^26 - 11428905839*x^25 + 30855186439*x^24 - 76570809795*x^23 + 200539909591*x^22 - 461675157780*x^21 + 1264576686904*x^20 - 2072873525231*x^19 + 7131979232127*x^18 - 5039435405121*x^17 + 33568030022304*x^16 + 5463244546413*x^15 + 128809321505673*x^14 + 104878030409751*x^13 + 410359478594376*x^12 + 503432848414096*x^11 + 1138005148281098*x^10 + 1631684206528590*x^9 + 2920336408155197*x^8 + 4348859129806158*x^7 + 7332486506596980*x^6 + 10508425239735849*x^5 + 18821520787059401*x^4 + 23360671914253394*x^3 + 51925411234007486*x^2 + 41540121306447536*x + 166161523629582937)
 
gp: K = bnfinit(y^44 - y^43 + 51*y^42 - 55*y^41 + 1248*y^40 - 1468*y^39 + 19662*y^38 - 25534*y^37 + 226772*y^36 - 328908*y^35 + 2076784*y^34 - 3392416*y^33 + 16113689*y^32 - 29683353*y^31 + 112279727*y^30 - 231013139*y^29 + 739331011*y^28 - 1663383567*y^27 + 4775371571*y^26 - 11428905839*y^25 + 30855186439*y^24 - 76570809795*y^23 + 200539909591*y^22 - 461675157780*y^21 + 1264576686904*y^20 - 2072873525231*y^19 + 7131979232127*y^18 - 5039435405121*y^17 + 33568030022304*y^16 + 5463244546413*y^15 + 128809321505673*y^14 + 104878030409751*y^13 + 410359478594376*y^12 + 503432848414096*y^11 + 1138005148281098*y^10 + 1631684206528590*y^9 + 2920336408155197*y^8 + 4348859129806158*y^7 + 7332486506596980*y^6 + 10508425239735849*y^5 + 18821520787059401*y^4 + 23360671914253394*y^3 + 51925411234007486*y^2 + 41540121306447536*y + 166161523629582937, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 + 51*x^42 - 55*x^41 + 1248*x^40 - 1468*x^39 + 19662*x^38 - 25534*x^37 + 226772*x^36 - 328908*x^35 + 2076784*x^34 - 3392416*x^33 + 16113689*x^32 - 29683353*x^31 + 112279727*x^30 - 231013139*x^29 + 739331011*x^28 - 1663383567*x^27 + 4775371571*x^26 - 11428905839*x^25 + 30855186439*x^24 - 76570809795*x^23 + 200539909591*x^22 - 461675157780*x^21 + 1264576686904*x^20 - 2072873525231*x^19 + 7131979232127*x^18 - 5039435405121*x^17 + 33568030022304*x^16 + 5463244546413*x^15 + 128809321505673*x^14 + 104878030409751*x^13 + 410359478594376*x^12 + 503432848414096*x^11 + 1138005148281098*x^10 + 1631684206528590*x^9 + 2920336408155197*x^8 + 4348859129806158*x^7 + 7332486506596980*x^6 + 10508425239735849*x^5 + 18821520787059401*x^4 + 23360671914253394*x^3 + 51925411234007486*x^2 + 41540121306447536*x + 166161523629582937);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 + 51*x^42 - 55*x^41 + 1248*x^40 - 1468*x^39 + 19662*x^38 - 25534*x^37 + 226772*x^36 - 328908*x^35 + 2076784*x^34 - 3392416*x^33 + 16113689*x^32 - 29683353*x^31 + 112279727*x^30 - 231013139*x^29 + 739331011*x^28 - 1663383567*x^27 + 4775371571*x^26 - 11428905839*x^25 + 30855186439*x^24 - 76570809795*x^23 + 200539909591*x^22 - 461675157780*x^21 + 1264576686904*x^20 - 2072873525231*x^19 + 7131979232127*x^18 - 5039435405121*x^17 + 33568030022304*x^16 + 5463244546413*x^15 + 128809321505673*x^14 + 104878030409751*x^13 + 410359478594376*x^12 + 503432848414096*x^11 + 1138005148281098*x^10 + 1631684206528590*x^9 + 2920336408155197*x^8 + 4348859129806158*x^7 + 7332486506596980*x^6 + 10508425239735849*x^5 + 18821520787059401*x^4 + 23360671914253394*x^3 + 51925411234007486*x^2 + 41540121306447536*x + 166161523629582937)
 

\( x^{44} - x^{43} + 51 x^{42} - 55 x^{41} + 1248 x^{40} - 1468 x^{39} + 19662 x^{38} + \cdots + 16\!\cdots\!37 \) 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:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(627\!\cdots\!673\) \(\medspace = 17^{33}\cdot 23^{42}\) 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:  \(166.98\)
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:  $17^{3/4}23^{21/22}\approx 166.98128822979427$
Ramified primes:   \(17\), \(23\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(391=17\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{391}(256,·)$, $\chi_{391}(1,·)$, $\chi_{391}(387,·)$, $\chi_{391}(132,·)$, $\chi_{391}(268,·)$, $\chi_{391}(271,·)$, $\chi_{391}(16,·)$, $\chi_{391}(18,·)$, $\chi_{391}(149,·)$, $\chi_{391}(89,·)$, $\chi_{391}(154,·)$, $\chi_{391}(157,·)$, $\chi_{391}(30,·)$, $\chi_{391}(288,·)$, $\chi_{391}(35,·)$, $\chi_{391}(293,·)$, $\chi_{391}(38,·)$, $\chi_{391}(169,·)$, $\chi_{391}(305,·)$, $\chi_{391}(50,·)$, $\chi_{391}(307,·)$, $\chi_{391}(52,·)$, $\chi_{391}(310,·)$, $\chi_{391}(183,·)$, $\chi_{391}(186,·)$, $\chi_{391}(188,·)$, $\chi_{391}(191,·)$, $\chi_{391}(324,·)$, $\chi_{391}(327,·)$, $\chi_{391}(336,·)$, $\chi_{391}(344,·)$, $\chi_{391}(217,·)$, $\chi_{391}(220,·)$, $\chi_{391}(251,·)$, $\chi_{391}(101,·)$, $\chi_{391}(358,·)$, $\chi_{391}(106,·)$, $\chi_{391}(239,·)$, $\chi_{391}(118,·)$, $\chi_{391}(378,·)$, $\chi_{391}(319,·)$, $\chi_{391}(166,·)$, $\chi_{391}(254,·)$, $\chi_{391}(21,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{10957134863}a^{23}+\frac{23}{10957134863}a^{21}+\frac{230}{10957134863}a^{19}+\frac{1311}{10957134863}a^{17}+\frac{4692}{10957134863}a^{15}+\frac{10948}{10957134863}a^{13}+\frac{16744}{10957134863}a^{11}+\frac{16445}{10957134863}a^{9}+\frac{9867}{10957134863}a^{7}+\frac{3289}{10957134863}a^{5}+\frac{506}{10957134863}a^{3}+\frac{23}{10957134863}a-\frac{102528546}{233130529}$, $\frac{1}{10957134863}a^{24}+\frac{23}{10957134863}a^{22}+\frac{230}{10957134863}a^{20}+\frac{1311}{10957134863}a^{18}+\frac{4692}{10957134863}a^{16}+\frac{10948}{10957134863}a^{14}+\frac{16744}{10957134863}a^{12}+\frac{16445}{10957134863}a^{10}+\frac{9867}{10957134863}a^{8}+\frac{3289}{10957134863}a^{6}+\frac{506}{10957134863}a^{4}+\frac{23}{10957134863}a^{2}-\frac{102528546}{233130529}a$, $\frac{1}{10957134863}a^{25}-\frac{299}{10957134863}a^{21}-\frac{3979}{10957134863}a^{19}-\frac{25461}{10957134863}a^{17}-\frac{96968}{10957134863}a^{15}-\frac{235060}{10957134863}a^{13}-\frac{368667}{10957134863}a^{11}-\frac{368368}{10957134863}a^{9}-\frac{223652}{10957134863}a^{7}-\frac{75141}{10957134863}a^{5}-\frac{11615}{10957134863}a^{3}-\frac{102528546}{233130529}a^{2}-\frac{529}{10957134863}a+\frac{26851268}{233130529}$, $\frac{1}{10957134863}a^{26}-\frac{299}{10957134863}a^{22}-\frac{3979}{10957134863}a^{20}-\frac{25461}{10957134863}a^{18}-\frac{96968}{10957134863}a^{16}-\frac{235060}{10957134863}a^{14}-\frac{368667}{10957134863}a^{12}-\frac{368368}{10957134863}a^{10}-\frac{223652}{10957134863}a^{8}-\frac{75141}{10957134863}a^{6}-\frac{11615}{10957134863}a^{4}-\frac{102528546}{233130529}a^{3}-\frac{529}{10957134863}a^{2}+\frac{26851268}{233130529}a$, $\frac{1}{10957134863}a^{27}+\frac{2898}{10957134863}a^{21}+\frac{43309}{10957134863}a^{19}+\frac{295021}{10957134863}a^{17}+\frac{1167848}{10957134863}a^{15}+\frac{2904785}{10957134863}a^{13}+\frac{4638088}{10957134863}a^{11}+\frac{4693403}{10957134863}a^{9}+\frac{2875092}{10957134863}a^{7}+\frac{971796}{10957134863}a^{5}-\frac{102528546}{233130529}a^{4}+\frac{150765}{10957134863}a^{3}+\frac{26851268}{233130529}a^{2}+\frac{6877}{10957134863}a-\frac{115935955}{233130529}$, $\frac{1}{10957134863}a^{28}+\frac{2898}{10957134863}a^{22}+\frac{43309}{10957134863}a^{20}+\frac{295021}{10957134863}a^{18}+\frac{1167848}{10957134863}a^{16}+\frac{2904785}{10957134863}a^{14}+\frac{4638088}{10957134863}a^{12}+\frac{4693403}{10957134863}a^{10}+\frac{2875092}{10957134863}a^{8}+\frac{971796}{10957134863}a^{6}-\frac{102528546}{233130529}a^{5}+\frac{150765}{10957134863}a^{4}+\frac{26851268}{233130529}a^{3}+\frac{6877}{10957134863}a^{2}-\frac{115935955}{233130529}a$, $\frac{1}{10957134863}a^{29}-\frac{23345}{10957134863}a^{21}-\frac{371519}{10957134863}a^{19}-\frac{2631430}{10957134863}a^{17}-\frac{10692631}{10957134863}a^{15}-\frac{27089216}{10957134863}a^{13}-\frac{43830709}{10957134863}a^{11}-\frac{44782518}{10957134863}a^{9}-\frac{27622770}{10957134863}a^{7}-\frac{102528546}{233130529}a^{6}-\frac{9380757}{10957134863}a^{5}+\frac{26851268}{233130529}a^{4}-\frac{1459511}{10957134863}a^{3}-\frac{115935955}{233130529}a^{2}-\frac{66654}{10957134863}a-\frac{113698167}{233130529}$, $\frac{1}{10957134863}a^{30}-\frac{23345}{10957134863}a^{22}-\frac{371519}{10957134863}a^{20}-\frac{2631430}{10957134863}a^{18}-\frac{10692631}{10957134863}a^{16}-\frac{27089216}{10957134863}a^{14}-\frac{43830709}{10957134863}a^{12}-\frac{44782518}{10957134863}a^{10}-\frac{27622770}{10957134863}a^{8}-\frac{102528546}{233130529}a^{7}-\frac{9380757}{10957134863}a^{6}+\frac{26851268}{233130529}a^{5}-\frac{1459511}{10957134863}a^{4}-\frac{115935955}{233130529}a^{3}-\frac{66654}{10957134863}a^{2}-\frac{113698167}{233130529}a$, $\frac{1}{10957134863}a^{31}+\frac{165416}{10957134863}a^{21}+\frac{2737920}{10957134863}a^{19}+\frac{19912664}{10957134863}a^{17}+\frac{82445524}{10957134863}a^{15}+\frac{211750351}{10957134863}a^{13}+\frac{346106162}{10957134863}a^{11}+\frac{356285755}{10957134863}a^{9}-\frac{102528546}{233130529}a^{8}+\frac{220964358}{10957134863}a^{7}+\frac{26851268}{233130529}a^{6}+\frac{75322194}{10957134863}a^{5}-\frac{115935955}{233130529}a^{4}+\frac{11745916}{10957134863}a^{3}-\frac{113698167}{233130529}a^{2}+\frac{536935}{10957134863}a+\frac{22234873}{233130529}$, $\frac{1}{10957134863}a^{32}+\frac{165416}{10957134863}a^{22}+\frac{2737920}{10957134863}a^{20}+\frac{19912664}{10957134863}a^{18}+\frac{82445524}{10957134863}a^{16}+\frac{211750351}{10957134863}a^{14}+\frac{346106162}{10957134863}a^{12}+\frac{356285755}{10957134863}a^{10}-\frac{102528546}{233130529}a^{9}+\frac{220964358}{10957134863}a^{8}+\frac{26851268}{233130529}a^{7}+\frac{75322194}{10957134863}a^{6}-\frac{115935955}{233130529}a^{5}+\frac{11745916}{10957134863}a^{4}-\frac{113698167}{233130529}a^{3}+\frac{536935}{10957134863}a^{2}+\frac{22234873}{233130529}a$, $\frac{1}{21914269726}a^{33}-\frac{1}{21914269726}a^{31}-\frac{1}{21914269726}a^{30}-\frac{1}{21914269726}a^{29}-\frac{1}{21914269726}a^{27}-\frac{1}{21914269726}a^{26}-\frac{1}{21914269726}a^{24}-\frac{5478555621}{10957134863}a^{22}+\frac{5477961623}{10957134863}a^{21}+\frac{187634}{10957134863}a^{20}-\frac{10271363}{10957134863}a^{19}+\frac{1327790}{10957134863}a^{18}-\frac{151991107}{21914269726}a^{17}-\frac{5473174978}{10957134863}a^{16}+\frac{10319832601}{21914269726}a^{15}-\frac{10929821535}{21914269726}a^{14}+\frac{9304700737}{21914269726}a^{13}-\frac{232190473}{466261058}a^{12}+\frac{8236781573}{21914269726}a^{11}-\frac{4773707221}{21914269726}a^{10}+\frac{8141636461}{21914269726}a^{9}+\frac{6108687813}{21914269726}a^{8}-\frac{7891347359}{21914269726}a^{7}+\frac{9074429653}{21914269726}a^{6}-\frac{1861230137}{21914269726}a^{5}-\frac{7293656141}{21914269726}a^{4}-\frac{782904047}{21914269726}a^{3}+\frac{10575900329}{21914269726}a^{2}-\frac{1030385337}{10957134863}a-\frac{176620365}{466261058}$, $\frac{1}{17\!\cdots\!86}a^{34}+\frac{17\!\cdots\!68}{88\!\cdots\!43}a^{33}+\frac{64\!\cdots\!79}{17\!\cdots\!86}a^{32}+\frac{40\!\cdots\!93}{17\!\cdots\!86}a^{31}+\frac{638700053746789}{37\!\cdots\!38}a^{30}-\frac{22\!\cdots\!77}{88\!\cdots\!43}a^{29}+\frac{22\!\cdots\!35}{17\!\cdots\!86}a^{28}-\frac{26\!\cdots\!95}{17\!\cdots\!86}a^{27}+\frac{37\!\cdots\!17}{88\!\cdots\!43}a^{26}-\frac{42\!\cdots\!67}{17\!\cdots\!86}a^{25}+\frac{38\!\cdots\!27}{88\!\cdots\!43}a^{24}-\frac{26\!\cdots\!77}{88\!\cdots\!43}a^{23}+\frac{26\!\cdots\!89}{88\!\cdots\!43}a^{22}-\frac{39\!\cdots\!09}{88\!\cdots\!43}a^{21}+\frac{41\!\cdots\!97}{88\!\cdots\!43}a^{20}+\frac{25\!\cdots\!88}{88\!\cdots\!43}a^{19}-\frac{97\!\cdots\!37}{17\!\cdots\!86}a^{18}-\frac{22\!\cdots\!53}{88\!\cdots\!43}a^{17}-\frac{63\!\cdots\!21}{17\!\cdots\!86}a^{16}-\frac{32\!\cdots\!15}{17\!\cdots\!86}a^{15}+\frac{22\!\cdots\!53}{17\!\cdots\!86}a^{14}-\frac{39\!\cdots\!31}{17\!\cdots\!86}a^{13}+\frac{45\!\cdots\!03}{17\!\cdots\!86}a^{12}+\frac{45\!\cdots\!41}{17\!\cdots\!86}a^{11}+\frac{40\!\cdots\!33}{17\!\cdots\!86}a^{10}+\frac{34\!\cdots\!61}{17\!\cdots\!86}a^{9}+\frac{47\!\cdots\!57}{17\!\cdots\!86}a^{8}+\frac{82\!\cdots\!49}{17\!\cdots\!86}a^{7}+\frac{19\!\cdots\!21}{17\!\cdots\!86}a^{6}-\frac{35\!\cdots\!61}{17\!\cdots\!86}a^{5}+\frac{70\!\cdots\!25}{17\!\cdots\!86}a^{4}-\frac{73\!\cdots\!31}{17\!\cdots\!86}a^{3}-\frac{30\!\cdots\!05}{88\!\cdots\!43}a^{2}-\frac{61\!\cdots\!95}{17\!\cdots\!86}a+\frac{59\!\cdots\!53}{18\!\cdots\!69}$, $\frac{1}{17\!\cdots\!86}a^{35}+\frac{21\!\cdots\!73}{17\!\cdots\!86}a^{33}-\frac{65\!\cdots\!91}{17\!\cdots\!86}a^{32}-\frac{17\!\cdots\!91}{17\!\cdots\!86}a^{31}+\frac{77\!\cdots\!59}{88\!\cdots\!43}a^{30}-\frac{16\!\cdots\!79}{17\!\cdots\!86}a^{29}-\frac{75\!\cdots\!65}{17\!\cdots\!86}a^{28}-\frac{16\!\cdots\!02}{88\!\cdots\!43}a^{27}-\frac{24\!\cdots\!07}{17\!\cdots\!86}a^{26}-\frac{10\!\cdots\!04}{88\!\cdots\!43}a^{25}+\frac{34\!\cdots\!86}{88\!\cdots\!43}a^{24}+\frac{24\!\cdots\!08}{88\!\cdots\!43}a^{23}+\frac{40\!\cdots\!13}{88\!\cdots\!43}a^{22}+\frac{39\!\cdots\!05}{88\!\cdots\!43}a^{21}+\frac{25\!\cdots\!63}{88\!\cdots\!43}a^{20}+\frac{73\!\cdots\!63}{17\!\cdots\!86}a^{19}-\frac{11\!\cdots\!41}{88\!\cdots\!43}a^{18}+\frac{22\!\cdots\!51}{17\!\cdots\!86}a^{17}+\frac{83\!\cdots\!51}{17\!\cdots\!86}a^{16}+\frac{15\!\cdots\!17}{37\!\cdots\!38}a^{15}-\frac{51\!\cdots\!19}{17\!\cdots\!86}a^{14}-\frac{48\!\cdots\!85}{17\!\cdots\!86}a^{13}-\frac{28\!\cdots\!37}{17\!\cdots\!86}a^{12}+\frac{73\!\cdots\!65}{17\!\cdots\!86}a^{11}+\frac{35\!\cdots\!65}{17\!\cdots\!86}a^{10}+\frac{30\!\cdots\!83}{17\!\cdots\!86}a^{9}+\frac{30\!\cdots\!11}{17\!\cdots\!86}a^{8}+\frac{84\!\cdots\!01}{17\!\cdots\!86}a^{7}-\frac{18\!\cdots\!21}{17\!\cdots\!86}a^{6}-\frac{33\!\cdots\!31}{17\!\cdots\!86}a^{5}-\frac{45\!\cdots\!47}{17\!\cdots\!86}a^{4}-\frac{16\!\cdots\!08}{88\!\cdots\!43}a^{3}-\frac{70\!\cdots\!53}{17\!\cdots\!86}a^{2}-\frac{28\!\cdots\!25}{88\!\cdots\!43}a-\frac{37\!\cdots\!68}{18\!\cdots\!69}$, $\frac{1}{17\!\cdots\!86}a^{36}+\frac{133390832906764}{88\!\cdots\!43}a^{33}-\frac{25\!\cdots\!71}{88\!\cdots\!43}a^{32}-\frac{17\!\cdots\!45}{88\!\cdots\!43}a^{31}-\frac{32\!\cdots\!43}{17\!\cdots\!86}a^{30}+\frac{29\!\cdots\!45}{88\!\cdots\!43}a^{29}-\frac{32\!\cdots\!95}{17\!\cdots\!86}a^{28}-\frac{80\!\cdots\!95}{17\!\cdots\!86}a^{27}-\frac{70\!\cdots\!95}{17\!\cdots\!86}a^{26}-\frac{15\!\cdots\!59}{17\!\cdots\!86}a^{25}+\frac{60\!\cdots\!21}{17\!\cdots\!86}a^{24}-\frac{22\!\cdots\!85}{88\!\cdots\!43}a^{23}-\frac{17\!\cdots\!22}{88\!\cdots\!43}a^{22}-\frac{35\!\cdots\!64}{88\!\cdots\!43}a^{21}+\frac{72\!\cdots\!19}{17\!\cdots\!86}a^{20}+\frac{41\!\cdots\!83}{88\!\cdots\!43}a^{19}+\frac{10\!\cdots\!92}{88\!\cdots\!43}a^{18}+\frac{76\!\cdots\!94}{88\!\cdots\!43}a^{17}-\frac{40\!\cdots\!85}{88\!\cdots\!43}a^{16}-\frac{20\!\cdots\!37}{17\!\cdots\!86}a^{15}+\frac{44\!\cdots\!05}{17\!\cdots\!86}a^{14}+\frac{30\!\cdots\!77}{17\!\cdots\!86}a^{13}-\frac{66\!\cdots\!35}{17\!\cdots\!86}a^{12}-\frac{27\!\cdots\!53}{17\!\cdots\!86}a^{11}+\frac{87\!\cdots\!71}{17\!\cdots\!86}a^{10}+\frac{64\!\cdots\!33}{17\!\cdots\!86}a^{9}-\frac{22\!\cdots\!35}{17\!\cdots\!86}a^{8}-\frac{71\!\cdots\!23}{17\!\cdots\!86}a^{7}+\frac{24\!\cdots\!37}{17\!\cdots\!86}a^{6}+\frac{63\!\cdots\!47}{17\!\cdots\!86}a^{5}-\frac{33\!\cdots\!52}{88\!\cdots\!43}a^{4}+\frac{30\!\cdots\!03}{17\!\cdots\!86}a^{3}-\frac{82\!\cdots\!33}{17\!\cdots\!86}a^{2}-\frac{30\!\cdots\!87}{17\!\cdots\!86}a-\frac{12\!\cdots\!27}{37\!\cdots\!38}$, $\frac{1}{17\!\cdots\!86}a^{37}+\frac{26\!\cdots\!19}{88\!\cdots\!43}a^{33}-\frac{26\!\cdots\!86}{88\!\cdots\!43}a^{32}+\frac{611040529728891}{17\!\cdots\!86}a^{31}+\frac{30\!\cdots\!05}{88\!\cdots\!43}a^{30}+\frac{60\!\cdots\!03}{17\!\cdots\!86}a^{29}+\frac{37459392274107}{37\!\cdots\!38}a^{28}-\frac{66\!\cdots\!87}{17\!\cdots\!86}a^{27}+\frac{82\!\cdots\!57}{17\!\cdots\!86}a^{26}-\frac{935280949653927}{17\!\cdots\!86}a^{25}+\frac{19\!\cdots\!65}{88\!\cdots\!43}a^{24}-\frac{11\!\cdots\!25}{88\!\cdots\!43}a^{23}+\frac{80\!\cdots\!84}{88\!\cdots\!43}a^{22}+\frac{62\!\cdots\!47}{17\!\cdots\!86}a^{21}+\frac{92\!\cdots\!77}{88\!\cdots\!43}a^{20}-\frac{36\!\cdots\!60}{88\!\cdots\!43}a^{19}-\frac{18\!\cdots\!06}{88\!\cdots\!43}a^{18}+\frac{25\!\cdots\!91}{88\!\cdots\!43}a^{17}+\frac{57\!\cdots\!05}{17\!\cdots\!86}a^{16}+\frac{63\!\cdots\!67}{17\!\cdots\!86}a^{15}+\frac{53\!\cdots\!39}{17\!\cdots\!86}a^{14}+\frac{86\!\cdots\!63}{17\!\cdots\!86}a^{13}-\frac{15\!\cdots\!59}{17\!\cdots\!86}a^{12}+\frac{75\!\cdots\!83}{17\!\cdots\!86}a^{11}+\frac{18\!\cdots\!59}{37\!\cdots\!38}a^{10}-\frac{51\!\cdots\!77}{17\!\cdots\!86}a^{9}+\frac{51\!\cdots\!85}{17\!\cdots\!86}a^{8}+\frac{33\!\cdots\!69}{17\!\cdots\!86}a^{7}+\frac{26\!\cdots\!41}{17\!\cdots\!86}a^{6}-\frac{37\!\cdots\!32}{88\!\cdots\!43}a^{5}-\frac{53\!\cdots\!87}{17\!\cdots\!86}a^{4}+\frac{65\!\cdots\!27}{17\!\cdots\!86}a^{3}-\frac{82\!\cdots\!29}{17\!\cdots\!86}a^{2}-\frac{85\!\cdots\!81}{17\!\cdots\!86}a+\frac{90\!\cdots\!18}{18\!\cdots\!69}$, $\frac{1}{17\!\cdots\!86}a^{38}-\frac{19\!\cdots\!29}{88\!\cdots\!43}a^{33}-\frac{52\!\cdots\!89}{17\!\cdots\!86}a^{32}+\frac{31\!\cdots\!91}{88\!\cdots\!43}a^{31}-\frac{13\!\cdots\!77}{17\!\cdots\!86}a^{30}+\frac{48\!\cdots\!19}{17\!\cdots\!86}a^{29}+\frac{61\!\cdots\!55}{17\!\cdots\!86}a^{28}+\frac{59\!\cdots\!99}{17\!\cdots\!86}a^{27}-\frac{41\!\cdots\!85}{17\!\cdots\!86}a^{26}+\frac{30\!\cdots\!14}{88\!\cdots\!43}a^{25}+\frac{33\!\cdots\!12}{88\!\cdots\!43}a^{24}+\frac{10\!\cdots\!39}{88\!\cdots\!43}a^{23}-\frac{14\!\cdots\!39}{17\!\cdots\!86}a^{22}-\frac{27\!\cdots\!49}{88\!\cdots\!43}a^{21}+\frac{14\!\cdots\!44}{88\!\cdots\!43}a^{20}-\frac{23\!\cdots\!73}{88\!\cdots\!43}a^{19}-\frac{27\!\cdots\!02}{88\!\cdots\!43}a^{18}-\frac{16\!\cdots\!61}{37\!\cdots\!38}a^{17}+\frac{49\!\cdots\!91}{17\!\cdots\!86}a^{16}-\frac{33\!\cdots\!79}{17\!\cdots\!86}a^{15}-\frac{20\!\cdots\!99}{17\!\cdots\!86}a^{14}+\frac{49\!\cdots\!07}{17\!\cdots\!86}a^{13}+\frac{19\!\cdots\!67}{17\!\cdots\!86}a^{12}+\frac{61\!\cdots\!37}{17\!\cdots\!86}a^{11}-\frac{20\!\cdots\!97}{17\!\cdots\!86}a^{10}+\frac{66\!\cdots\!65}{17\!\cdots\!86}a^{9}+\frac{81\!\cdots\!75}{17\!\cdots\!86}a^{8}+\frac{75\!\cdots\!23}{17\!\cdots\!86}a^{7}+\frac{20\!\cdots\!05}{88\!\cdots\!43}a^{6}+\frac{20\!\cdots\!17}{17\!\cdots\!86}a^{5}+\frac{33\!\cdots\!53}{17\!\cdots\!86}a^{4}-\frac{54\!\cdots\!05}{17\!\cdots\!86}a^{3}-\frac{44\!\cdots\!93}{17\!\cdots\!86}a^{2}+\frac{43\!\cdots\!42}{18\!\cdots\!69}a-\frac{87\!\cdots\!84}{18\!\cdots\!69}$, $\frac{1}{17\!\cdots\!86}a^{39}-\frac{14\!\cdots\!21}{88\!\cdots\!43}a^{33}+\frac{19\!\cdots\!39}{88\!\cdots\!43}a^{32}+\frac{25\!\cdots\!83}{88\!\cdots\!43}a^{31}+\frac{35\!\cdots\!65}{88\!\cdots\!43}a^{30}-\frac{45\!\cdots\!34}{88\!\cdots\!43}a^{29}-\frac{37\!\cdots\!89}{17\!\cdots\!86}a^{28}-\frac{36\!\cdots\!18}{88\!\cdots\!43}a^{27}+\frac{20\!\cdots\!63}{17\!\cdots\!86}a^{26}-\frac{15\!\cdots\!42}{88\!\cdots\!43}a^{25}+\frac{78\!\cdots\!91}{17\!\cdots\!86}a^{24}-\frac{18\!\cdots\!05}{17\!\cdots\!86}a^{23}+\frac{11\!\cdots\!77}{88\!\cdots\!43}a^{22}+\frac{38\!\cdots\!12}{88\!\cdots\!43}a^{21}+\frac{17\!\cdots\!27}{88\!\cdots\!43}a^{20}-\frac{32\!\cdots\!40}{88\!\cdots\!43}a^{19}+\frac{16\!\cdots\!85}{17\!\cdots\!86}a^{18}+\frac{25\!\cdots\!03}{88\!\cdots\!43}a^{17}-\frac{75\!\cdots\!81}{17\!\cdots\!86}a^{16}-\frac{28\!\cdots\!36}{88\!\cdots\!43}a^{15}-\frac{12\!\cdots\!35}{88\!\cdots\!43}a^{14}+\frac{30\!\cdots\!07}{88\!\cdots\!43}a^{13}-\frac{28\!\cdots\!68}{88\!\cdots\!43}a^{12}+\frac{16\!\cdots\!30}{88\!\cdots\!43}a^{11}+\frac{39\!\cdots\!26}{88\!\cdots\!43}a^{10}+\frac{14\!\cdots\!78}{88\!\cdots\!43}a^{9}+\frac{18\!\cdots\!95}{88\!\cdots\!43}a^{8}+\frac{15\!\cdots\!79}{17\!\cdots\!86}a^{7}+\frac{20\!\cdots\!19}{88\!\cdots\!43}a^{6}+\frac{30\!\cdots\!00}{88\!\cdots\!43}a^{5}+\frac{22\!\cdots\!20}{88\!\cdots\!43}a^{4}-\frac{18\!\cdots\!96}{88\!\cdots\!43}a^{3}-\frac{53\!\cdots\!75}{17\!\cdots\!86}a^{2}+\frac{18\!\cdots\!69}{88\!\cdots\!43}a+\frac{10\!\cdots\!53}{37\!\cdots\!38}$, $\frac{1}{17\!\cdots\!86}a^{40}-\frac{13\!\cdots\!00}{88\!\cdots\!43}a^{33}+\frac{35\!\cdots\!96}{88\!\cdots\!43}a^{32}-\frac{15\!\cdots\!36}{88\!\cdots\!43}a^{31}+\frac{10\!\cdots\!64}{88\!\cdots\!43}a^{30}+\frac{72\!\cdots\!95}{17\!\cdots\!86}a^{29}+\frac{13\!\cdots\!08}{88\!\cdots\!43}a^{28}+\frac{312617511327795}{17\!\cdots\!86}a^{27}-\frac{22\!\cdots\!32}{88\!\cdots\!43}a^{26}-\frac{58\!\cdots\!49}{17\!\cdots\!86}a^{25}+\frac{20\!\cdots\!55}{17\!\cdots\!86}a^{24}+\frac{98\!\cdots\!04}{88\!\cdots\!43}a^{23}-\frac{16\!\cdots\!04}{88\!\cdots\!43}a^{22}+\frac{24\!\cdots\!38}{88\!\cdots\!43}a^{21}-\frac{19\!\cdots\!08}{88\!\cdots\!43}a^{20}-\frac{78\!\cdots\!93}{17\!\cdots\!86}a^{19}+\frac{39\!\cdots\!78}{88\!\cdots\!43}a^{18}+\frac{79\!\cdots\!65}{17\!\cdots\!86}a^{17}-\frac{16\!\cdots\!51}{88\!\cdots\!43}a^{16}-\frac{14\!\cdots\!38}{88\!\cdots\!43}a^{15}+\frac{75\!\cdots\!66}{88\!\cdots\!43}a^{14}-\frac{37\!\cdots\!14}{88\!\cdots\!43}a^{13}-\frac{59\!\cdots\!51}{88\!\cdots\!43}a^{12}+\frac{36\!\cdots\!65}{88\!\cdots\!43}a^{11}+\frac{11\!\cdots\!28}{88\!\cdots\!43}a^{10}+\frac{12\!\cdots\!32}{88\!\cdots\!43}a^{9}+\frac{74\!\cdots\!85}{17\!\cdots\!86}a^{8}+\frac{39\!\cdots\!17}{88\!\cdots\!43}a^{7}-\frac{43\!\cdots\!85}{88\!\cdots\!43}a^{6}+\frac{13\!\cdots\!14}{88\!\cdots\!43}a^{5}+\frac{32\!\cdots\!00}{88\!\cdots\!43}a^{4}+\frac{31\!\cdots\!23}{17\!\cdots\!86}a^{3}+\frac{11\!\cdots\!65}{88\!\cdots\!43}a^{2}+\frac{53\!\cdots\!35}{37\!\cdots\!38}a-\frac{13\!\cdots\!86}{18\!\cdots\!69}$, $\frac{1}{17\!\cdots\!86}a^{41}-\frac{11\!\cdots\!84}{88\!\cdots\!43}a^{33}-\frac{17\!\cdots\!72}{88\!\cdots\!43}a^{32}-\frac{38\!\cdots\!20}{88\!\cdots\!43}a^{31}+\frac{71\!\cdots\!87}{17\!\cdots\!86}a^{30}-\frac{24\!\cdots\!79}{88\!\cdots\!43}a^{29}-\frac{54\!\cdots\!57}{17\!\cdots\!86}a^{28}+\frac{56\!\cdots\!24}{88\!\cdots\!43}a^{27}-\frac{32\!\cdots\!47}{17\!\cdots\!86}a^{26}-\frac{21\!\cdots\!43}{17\!\cdots\!86}a^{25}-\frac{39\!\cdots\!16}{88\!\cdots\!43}a^{24}-\frac{21\!\cdots\!11}{88\!\cdots\!43}a^{23}+\frac{36\!\cdots\!62}{88\!\cdots\!43}a^{22}+\frac{18\!\cdots\!75}{88\!\cdots\!43}a^{21}+\frac{44\!\cdots\!61}{17\!\cdots\!86}a^{20}+\frac{79\!\cdots\!72}{88\!\cdots\!43}a^{19}+\frac{37\!\cdots\!47}{17\!\cdots\!86}a^{18}-\frac{34\!\cdots\!13}{88\!\cdots\!43}a^{17}+\frac{17\!\cdots\!41}{88\!\cdots\!43}a^{16}-\frac{29\!\cdots\!91}{88\!\cdots\!43}a^{15}-\frac{12\!\cdots\!74}{88\!\cdots\!43}a^{14}-\frac{31\!\cdots\!72}{88\!\cdots\!43}a^{13}+\frac{33\!\cdots\!21}{88\!\cdots\!43}a^{12}-\frac{38\!\cdots\!14}{88\!\cdots\!43}a^{11}+\frac{54\!\cdots\!43}{88\!\cdots\!43}a^{10}-\frac{17\!\cdots\!55}{17\!\cdots\!86}a^{9}+\frac{35\!\cdots\!21}{88\!\cdots\!43}a^{8}+\frac{15\!\cdots\!42}{88\!\cdots\!43}a^{7}-\frac{32\!\cdots\!46}{88\!\cdots\!43}a^{6}+\frac{28\!\cdots\!65}{88\!\cdots\!43}a^{5}+\frac{72\!\cdots\!17}{17\!\cdots\!86}a^{4}-\frac{15\!\cdots\!45}{88\!\cdots\!43}a^{3}+\frac{28\!\cdots\!71}{17\!\cdots\!86}a^{2}-\frac{39\!\cdots\!98}{88\!\cdots\!43}a+\frac{23\!\cdots\!45}{18\!\cdots\!69}$, $\frac{1}{17\!\cdots\!86}a^{42}+\frac{24\!\cdots\!41}{88\!\cdots\!43}a^{33}+\frac{55\!\cdots\!90}{88\!\cdots\!43}a^{32}+\frac{66\!\cdots\!09}{17\!\cdots\!86}a^{31}-\frac{22\!\cdots\!49}{88\!\cdots\!43}a^{30}+\frac{34\!\cdots\!51}{17\!\cdots\!86}a^{29}+\frac{33\!\cdots\!26}{88\!\cdots\!43}a^{28}+\frac{31\!\cdots\!87}{17\!\cdots\!86}a^{27}+\frac{89\!\cdots\!75}{17\!\cdots\!86}a^{26}+\frac{19\!\cdots\!61}{88\!\cdots\!43}a^{25}-\frac{31\!\cdots\!35}{88\!\cdots\!43}a^{24}+\frac{12\!\cdots\!17}{88\!\cdots\!43}a^{23}-\frac{12\!\cdots\!43}{88\!\cdots\!43}a^{22}-\frac{22\!\cdots\!29}{17\!\cdots\!86}a^{21}+\frac{39\!\cdots\!14}{88\!\cdots\!43}a^{20}-\frac{76\!\cdots\!69}{17\!\cdots\!86}a^{19}+\frac{37\!\cdots\!47}{88\!\cdots\!43}a^{18}+\frac{26\!\cdots\!76}{88\!\cdots\!43}a^{17}-\frac{40\!\cdots\!64}{88\!\cdots\!43}a^{16}-\frac{23\!\cdots\!49}{88\!\cdots\!43}a^{15}+\frac{30\!\cdots\!34}{88\!\cdots\!43}a^{14}-\frac{26\!\cdots\!23}{88\!\cdots\!43}a^{13}+\frac{21\!\cdots\!77}{88\!\cdots\!43}a^{12}-\frac{24\!\cdots\!30}{88\!\cdots\!43}a^{11}-\frac{19\!\cdots\!99}{17\!\cdots\!86}a^{10}+\frac{13\!\cdots\!29}{88\!\cdots\!43}a^{9}-\frac{10\!\cdots\!11}{88\!\cdots\!43}a^{8}-\frac{22\!\cdots\!45}{88\!\cdots\!43}a^{7}+\frac{33\!\cdots\!55}{88\!\cdots\!43}a^{6}-\frac{11\!\cdots\!43}{17\!\cdots\!86}a^{5}-\frac{10\!\cdots\!85}{88\!\cdots\!43}a^{4}+\frac{85\!\cdots\!23}{17\!\cdots\!86}a^{3}+\frac{34\!\cdots\!16}{88\!\cdots\!43}a^{2}-\frac{85\!\cdots\!86}{88\!\cdots\!43}a+\frac{60\!\cdots\!12}{18\!\cdots\!69}$, $\frac{1}{17\!\cdots\!86}a^{43}-\frac{37\!\cdots\!79}{17\!\cdots\!86}a^{33}+\frac{19\!\cdots\!81}{17\!\cdots\!86}a^{32}+\frac{37\!\cdots\!19}{17\!\cdots\!86}a^{31}-\frac{21\!\cdots\!34}{88\!\cdots\!43}a^{30}+\frac{17\!\cdots\!47}{17\!\cdots\!86}a^{29}-\frac{29\!\cdots\!49}{17\!\cdots\!86}a^{28}-\frac{37\!\cdots\!25}{88\!\cdots\!43}a^{27}-\frac{25\!\cdots\!75}{17\!\cdots\!86}a^{26}+\frac{26\!\cdots\!18}{88\!\cdots\!43}a^{25}+\frac{32\!\cdots\!31}{17\!\cdots\!86}a^{24}-\frac{10\!\cdots\!92}{88\!\cdots\!43}a^{23}+\frac{80\!\cdots\!37}{17\!\cdots\!86}a^{22}-\frac{36\!\cdots\!62}{88\!\cdots\!43}a^{21}+\frac{35\!\cdots\!01}{17\!\cdots\!86}a^{20}-\frac{43\!\cdots\!03}{88\!\cdots\!43}a^{19}+\frac{25\!\cdots\!28}{88\!\cdots\!43}a^{18}-\frac{18\!\cdots\!83}{17\!\cdots\!86}a^{17}+\frac{13\!\cdots\!33}{88\!\cdots\!43}a^{16}+\frac{74\!\cdots\!35}{17\!\cdots\!86}a^{15}-\frac{38\!\cdots\!43}{17\!\cdots\!86}a^{14}+\frac{65\!\cdots\!61}{17\!\cdots\!86}a^{13}-\frac{57\!\cdots\!63}{17\!\cdots\!86}a^{12}+\frac{23\!\cdots\!77}{88\!\cdots\!43}a^{11}-\frac{72\!\cdots\!89}{17\!\cdots\!86}a^{10}-\frac{58\!\cdots\!01}{17\!\cdots\!86}a^{9}-\frac{62\!\cdots\!01}{17\!\cdots\!86}a^{8}+\frac{10\!\cdots\!49}{17\!\cdots\!86}a^{7}+\frac{84\!\cdots\!54}{88\!\cdots\!43}a^{6}+\frac{84\!\cdots\!79}{17\!\cdots\!86}a^{5}-\frac{37\!\cdots\!65}{88\!\cdots\!43}a^{4}-\frac{57\!\cdots\!55}{17\!\cdots\!86}a^{3}+\frac{35\!\cdots\!53}{17\!\cdots\!86}a^{2}+\frac{31\!\cdots\!61}{88\!\cdots\!43}a+\frac{86\!\cdots\!75}{37\!\cdots\!38}$ 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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:  $21$
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 + 51*x^42 - 55*x^41 + 1248*x^40 - 1468*x^39 + 19662*x^38 - 25534*x^37 + 226772*x^36 - 328908*x^35 + 2076784*x^34 - 3392416*x^33 + 16113689*x^32 - 29683353*x^31 + 112279727*x^30 - 231013139*x^29 + 739331011*x^28 - 1663383567*x^27 + 4775371571*x^26 - 11428905839*x^25 + 30855186439*x^24 - 76570809795*x^23 + 200539909591*x^22 - 461675157780*x^21 + 1264576686904*x^20 - 2072873525231*x^19 + 7131979232127*x^18 - 5039435405121*x^17 + 33568030022304*x^16 + 5463244546413*x^15 + 128809321505673*x^14 + 104878030409751*x^13 + 410359478594376*x^12 + 503432848414096*x^11 + 1138005148281098*x^10 + 1631684206528590*x^9 + 2920336408155197*x^8 + 4348859129806158*x^7 + 7332486506596980*x^6 + 10508425239735849*x^5 + 18821520787059401*x^4 + 23360671914253394*x^3 + 51925411234007486*x^2 + 41540121306447536*x + 166161523629582937)
 
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 + 51*x^42 - 55*x^41 + 1248*x^40 - 1468*x^39 + 19662*x^38 - 25534*x^37 + 226772*x^36 - 328908*x^35 + 2076784*x^34 - 3392416*x^33 + 16113689*x^32 - 29683353*x^31 + 112279727*x^30 - 231013139*x^29 + 739331011*x^28 - 1663383567*x^27 + 4775371571*x^26 - 11428905839*x^25 + 30855186439*x^24 - 76570809795*x^23 + 200539909591*x^22 - 461675157780*x^21 + 1264576686904*x^20 - 2072873525231*x^19 + 7131979232127*x^18 - 5039435405121*x^17 + 33568030022304*x^16 + 5463244546413*x^15 + 128809321505673*x^14 + 104878030409751*x^13 + 410359478594376*x^12 + 503432848414096*x^11 + 1138005148281098*x^10 + 1631684206528590*x^9 + 2920336408155197*x^8 + 4348859129806158*x^7 + 7332486506596980*x^6 + 10508425239735849*x^5 + 18821520787059401*x^4 + 23360671914253394*x^3 + 51925411234007486*x^2 + 41540121306447536*x + 166161523629582937, 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 + 51*x^42 - 55*x^41 + 1248*x^40 - 1468*x^39 + 19662*x^38 - 25534*x^37 + 226772*x^36 - 328908*x^35 + 2076784*x^34 - 3392416*x^33 + 16113689*x^32 - 29683353*x^31 + 112279727*x^30 - 231013139*x^29 + 739331011*x^28 - 1663383567*x^27 + 4775371571*x^26 - 11428905839*x^25 + 30855186439*x^24 - 76570809795*x^23 + 200539909591*x^22 - 461675157780*x^21 + 1264576686904*x^20 - 2072873525231*x^19 + 7131979232127*x^18 - 5039435405121*x^17 + 33568030022304*x^16 + 5463244546413*x^15 + 128809321505673*x^14 + 104878030409751*x^13 + 410359478594376*x^12 + 503432848414096*x^11 + 1138005148281098*x^10 + 1631684206528590*x^9 + 2920336408155197*x^8 + 4348859129806158*x^7 + 7332486506596980*x^6 + 10508425239735849*x^5 + 18821520787059401*x^4 + 23360671914253394*x^3 + 51925411234007486*x^2 + 41540121306447536*x + 166161523629582937);
 
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 + 51*x^42 - 55*x^41 + 1248*x^40 - 1468*x^39 + 19662*x^38 - 25534*x^37 + 226772*x^36 - 328908*x^35 + 2076784*x^34 - 3392416*x^33 + 16113689*x^32 - 29683353*x^31 + 112279727*x^30 - 231013139*x^29 + 739331011*x^28 - 1663383567*x^27 + 4775371571*x^26 - 11428905839*x^25 + 30855186439*x^24 - 76570809795*x^23 + 200539909591*x^22 - 461675157780*x^21 + 1264576686904*x^20 - 2072873525231*x^19 + 7131979232127*x^18 - 5039435405121*x^17 + 33568030022304*x^16 + 5463244546413*x^15 + 128809321505673*x^14 + 104878030409751*x^13 + 410359478594376*x^12 + 503432848414096*x^11 + 1138005148281098*x^10 + 1631684206528590*x^9 + 2920336408155197*x^8 + 4348859129806158*x^7 + 7332486506596980*x^6 + 10508425239735849*x^5 + 18821520787059401*x^4 + 23360671914253394*x^3 + 51925411234007486*x^2 + 41540121306447536*x + 166161523629582937);
 
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 $C_{44}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.2598977.1, \(\Q(\zeta_{23})^+\), 22.22.58815914699238651208660872676277748369233.1

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 $22^{2}$ $44$ $44$ $44$ $44$ ${\href{/padicField/13.11.0.1}{11} }^{4}$ R ${\href{/padicField/19.11.0.1}{11} }^{4}$ R $44$ $44$ $44$ $44$ ${\href{/padicField/43.11.0.1}{11} }^{4}$ ${\href{/padicField/47.1.0.1}{1} }^{44}$ ${\href{/padicField/53.11.0.1}{11} }^{4}$ $22^{2}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(17\) Copy content Toggle raw display Deg $44$$4$$11$$33$
\(23\) Copy content Toggle raw display Deg $44$$22$$2$$42$