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Properties

Label 44.0.281...333.1
Degree $44$
Signature $[0, 22]$
Discriminant $2.814\times 10^{104}$
Root discriminant \(236.51\)
Ramified primes $3,13,23$
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 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809)
 
gp: K = bnfinit(y^44 - y^43 + 188*y^42 - 191*y^41 + 15682*y^40 - 16255*y^39 + 767101*y^38 - 815866*y^37 + 24539323*y^36 - 26986921*y^35 + 543368874*y^34 - 624329637*y^33 + 8632506022*y^32 - 10505494933*y^31 + 101311538399*y^30 - 132828023198*y^29 + 907079143955*y^28 - 1305563213549*y^27 + 6469389498694*y^26 - 10386079139341*y^25 + 38961713404851*y^24 - 70119950822874*y^23 + 212107487819539*y^22 - 267306271081404*y^21 + 954204965731533*y^20 - 348452808851934*y^19 + 3286452427237125*y^18 + 1220729003449623*y^17 + 8721817319211705*y^16 + 7424451290225763*y^15 + 18808628124207888*y^14 + 21088254372919965*y^13 + 35377638646744746*y^12 + 44100561928942881*y^11 + 62049209117584005*y^10 + 79953228494360262*y^9 + 106199279503507242*y^8 + 136817944804830789*y^7 + 181780810904989953*y^6 + 226679666465622069*y^5 + 318662867727691812*y^4 + 354165989590339128*y^3 + 601822619090434404*y^2 + 452107998974921166*y + 1353359858390092809, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809)
 

\( x^{44} - x^{43} + 188 x^{42} - 191 x^{41} + 15682 x^{40} - 16255 x^{39} + 767101 x^{38} + \cdots + 13\!\cdots\!09 \) 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:   \(281\!\cdots\!333\) \(\medspace = 3^{22}\cdot 13^{33}\cdot 23^{42}\) 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:  \(236.51\)
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:  $3^{1/2}13^{3/4}23^{21/22}\approx 236.50986369052094$
Ramified primes:   \(3\), \(13\), \(23\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(897=3\cdot 13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{897}(1,·)$, $\chi_{897}(259,·)$, $\chi_{897}(5,·)$, $\chi_{897}(64,·)$, $\chi_{897}(398,·)$, $\chi_{897}(532,·)$, $\chi_{897}(25,·)$, $\chi_{897}(281,·)$, $\chi_{897}(415,·)$, $\chi_{897}(547,·)$, $\chi_{897}(551,·)$, $\chi_{897}(44,·)$, $\chi_{897}(430,·)$, $\chi_{897}(434,·)$, $\chi_{897}(824,·)$, $\chi_{897}(827,·)$, $\chi_{897}(703,·)$, $\chi_{897}(320,·)$, $\chi_{897}(707,·)$, $\chi_{897}(196,·)$, $\chi_{897}(710,·)$, $\chi_{897}(203,·)$, $\chi_{897}(844,·)$, $\chi_{897}(610,·)$, $\chi_{897}(590,·)$, $\chi_{897}(632,·)$, $\chi_{897}(83,·)$, $\chi_{897}(142,·)$, $\chi_{897}(86,·)$, $\chi_{897}(859,·)$, $\chi_{897}(220,·)$, $\chi_{897}(866,·)$, $\chi_{897}(356,·)$, $\chi_{897}(742,·)$, $\chi_{897}(359,·)$, $\chi_{897}(746,·)$, $\chi_{897}(625,·)$, $\chi_{897}(883,·)$, $\chi_{897}(118,·)$, $\chi_{897}(376,·)$, $\chi_{897}(122,·)$, $\chi_{897}(508,·)$, $\chi_{897}(125,·)$, $\chi_{897}(469,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{3}a^{24}-\frac{1}{3}a^{23}-\frac{1}{3}a^{22}+\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{25}+\frac{1}{3}a^{23}-\frac{1}{3}a^{21}-\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{26}-\frac{1}{9}a^{25}-\frac{1}{9}a^{24}-\frac{2}{9}a^{23}+\frac{4}{9}a^{22}-\frac{1}{9}a^{21}+\frac{4}{9}a^{20}+\frac{2}{9}a^{19}+\frac{4}{9}a^{18}+\frac{2}{9}a^{17}+\frac{1}{3}a^{16}+\frac{1}{3}a^{15}-\frac{2}{9}a^{14}+\frac{2}{9}a^{13}-\frac{1}{9}a^{12}-\frac{2}{9}a^{11}-\frac{4}{9}a^{10}-\frac{2}{9}a^{9}+\frac{4}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{3}a^{6}+\frac{4}{9}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{9}a^{27}+\frac{1}{9}a^{25}+\frac{2}{9}a^{23}+\frac{1}{3}a^{21}+\frac{1}{3}a^{19}+\frac{2}{9}a^{17}+\frac{4}{9}a^{15}+\frac{4}{9}a^{13}+\frac{1}{3}a^{11}-\frac{4}{9}a^{9}-\frac{4}{9}a^{7}-\frac{2}{9}a^{5}-\frac{2}{9}a^{4}$, $\frac{1}{27}a^{28}-\frac{1}{27}a^{27}-\frac{1}{27}a^{26}-\frac{2}{27}a^{25}+\frac{4}{27}a^{24}-\frac{10}{27}a^{23}-\frac{5}{27}a^{22}+\frac{2}{27}a^{21}+\frac{4}{27}a^{20}+\frac{2}{27}a^{19}+\frac{4}{9}a^{18}-\frac{2}{9}a^{17}-\frac{11}{27}a^{16}-\frac{7}{27}a^{15}-\frac{1}{27}a^{14}+\frac{7}{27}a^{13}-\frac{13}{27}a^{12}+\frac{7}{27}a^{11}-\frac{5}{27}a^{10}-\frac{1}{27}a^{9}-\frac{4}{9}a^{8}+\frac{1}{3}a^{7}+\frac{13}{27}a^{6}+\frac{4}{9}a^{5}+\frac{4}{9}a^{4}+\frac{1}{3}a^{3}$, $\frac{1}{27}a^{29}+\frac{1}{27}a^{27}+\frac{2}{27}a^{25}+\frac{1}{9}a^{23}-\frac{2}{9}a^{21}-\frac{7}{27}a^{19}+\frac{13}{27}a^{17}+\frac{4}{27}a^{15}+\frac{4}{9}a^{13}-\frac{4}{27}a^{11}+\frac{5}{27}a^{9}-\frac{11}{27}a^{7}-\frac{2}{27}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{81}a^{30}-\frac{1}{81}a^{29}-\frac{1}{81}a^{28}-\frac{2}{81}a^{27}+\frac{4}{81}a^{26}-\frac{10}{81}a^{25}-\frac{5}{81}a^{24}-\frac{25}{81}a^{23}-\frac{23}{81}a^{22}+\frac{29}{81}a^{21}-\frac{5}{27}a^{20}+\frac{7}{27}a^{19}-\frac{11}{81}a^{18}-\frac{7}{81}a^{17}+\frac{26}{81}a^{16}+\frac{34}{81}a^{15}+\frac{14}{81}a^{14}+\frac{7}{81}a^{13}-\frac{5}{81}a^{12}+\frac{26}{81}a^{11}-\frac{13}{27}a^{10}+\frac{4}{9}a^{9}+\frac{40}{81}a^{8}-\frac{5}{27}a^{7}+\frac{4}{27}a^{6}-\frac{2}{9}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{81}a^{31}+\frac{1}{81}a^{29}+\frac{2}{81}a^{27}+\frac{1}{27}a^{25}-\frac{2}{27}a^{23}-\frac{7}{81}a^{21}-\frac{14}{81}a^{19}+\frac{31}{81}a^{17}+\frac{4}{27}a^{15}+\frac{23}{81}a^{13}-\frac{22}{81}a^{11}+\frac{16}{81}a^{9}-\frac{2}{81}a^{8}-\frac{2}{9}a^{7}+\frac{1}{3}a^{6}+\frac{2}{9}a^{5}+\frac{2}{9}a^{4}$, $\frac{1}{243}a^{32}-\frac{1}{243}a^{31}-\frac{1}{243}a^{30}-\frac{2}{243}a^{29}+\frac{4}{243}a^{28}-\frac{10}{243}a^{27}-\frac{5}{243}a^{26}-\frac{25}{243}a^{25}-\frac{23}{243}a^{24}-\frac{52}{243}a^{23}-\frac{5}{81}a^{22}+\frac{34}{81}a^{21}-\frac{11}{243}a^{20}+\frac{74}{243}a^{19}+\frac{107}{243}a^{18}+\frac{115}{243}a^{17}-\frac{67}{243}a^{16}-\frac{74}{243}a^{15}-\frac{86}{243}a^{14}+\frac{107}{243}a^{13}-\frac{13}{81}a^{12}+\frac{13}{27}a^{11}+\frac{121}{243}a^{10}+\frac{22}{81}a^{9}-\frac{23}{81}a^{8}-\frac{2}{27}a^{7}+\frac{1}{9}a^{6}+\frac{1}{3}a^{5}+\frac{4}{9}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{243}a^{33}+\frac{1}{243}a^{31}+\frac{2}{243}a^{29}+\frac{1}{81}a^{27}-\frac{2}{81}a^{25}-\frac{7}{243}a^{23}-\frac{14}{243}a^{21}+\frac{31}{243}a^{19}+\frac{31}{81}a^{17}-\frac{58}{243}a^{15}-\frac{22}{243}a^{13}+\frac{97}{243}a^{11}+\frac{79}{243}a^{10}+\frac{7}{27}a^{9}-\frac{2}{9}a^{8}+\frac{11}{27}a^{7}+\frac{11}{27}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{15\!\cdots\!01}a^{34}+\frac{2712904409285}{15\!\cdots\!01}a^{33}-\frac{2431478225638}{15\!\cdots\!01}a^{32}+\frac{5785466392330}{15\!\cdots\!01}a^{31}+\frac{3374296086451}{15\!\cdots\!01}a^{30}-\frac{11223195848794}{15\!\cdots\!01}a^{29}-\frac{2995608004520}{15\!\cdots\!01}a^{28}-\frac{33213800348746}{15\!\cdots\!01}a^{27}+\frac{18630301119484}{15\!\cdots\!01}a^{26}-\frac{165897072019042}{15\!\cdots\!01}a^{25}-\frac{42374596167020}{514889568257067}a^{24}+\frac{24939543232330}{514889568257067}a^{23}+\frac{603907983580183}{15\!\cdots\!01}a^{22}-\frac{727747510924591}{15\!\cdots\!01}a^{21}-\frac{685228160373976}{15\!\cdots\!01}a^{20}-\frac{660291925667951}{15\!\cdots\!01}a^{19}-\frac{109049036571067}{15\!\cdots\!01}a^{18}-\frac{248182571696339}{15\!\cdots\!01}a^{17}-\frac{711915388841615}{15\!\cdots\!01}a^{16}-\frac{308977613836300}{15\!\cdots\!01}a^{15}+\frac{157636496176559}{514889568257067}a^{14}+\frac{46573222079749}{171629856085689}a^{13}-\frac{189733187078717}{15\!\cdots\!01}a^{12}+\frac{25765982393797}{514889568257067}a^{11}-\frac{72818849418926}{514889568257067}a^{10}+\frac{56628806696776}{171629856085689}a^{9}+\frac{3363475133602}{57209952028563}a^{8}+\frac{4457874706472}{19069984009521}a^{7}+\frac{515923937260}{57209952028563}a^{6}+\frac{1472864461546}{6356661336507}a^{5}-\frac{575563587046}{6356661336507}a^{4}+\frac{1150608181999}{6356661336507}a^{3}-\frac{1358812642016}{6356661336507}a^{2}+\frac{367475295482}{2118887112169}a+\frac{638971366622}{2118887112169}$, $\frac{1}{15\!\cdots\!01}a^{35}+\frac{3050717031166}{15\!\cdots\!01}a^{33}+\frac{62483902669}{171629856085689}a^{32}-\frac{4941238501570}{15\!\cdots\!01}a^{31}-\frac{111314757802}{171629856085689}a^{30}+\frac{56205017740}{19069984009521}a^{29}-\frac{472900760876}{171629856085689}a^{28}-\frac{19811885582252}{514889568257067}a^{27}-\frac{8763138079046}{171629856085689}a^{26}+\frac{35467541311880}{15\!\cdots\!01}a^{25}+\frac{26852771911600}{171629856085689}a^{24}+\frac{739040525434252}{15\!\cdots\!01}a^{23}+\frac{231222817253}{2118887112169}a^{22}+\frac{115923013970821}{15\!\cdots\!01}a^{21}-\frac{41175028156313}{171629856085689}a^{20}-\frac{123382309576849}{514889568257067}a^{19}+\frac{44261304314342}{171629856085689}a^{18}+\frac{110483740134734}{15\!\cdots\!01}a^{17}+\frac{30075533612000}{171629856085689}a^{16}+\frac{572162322055898}{15\!\cdots\!01}a^{15}-\frac{35319303748253}{171629856085689}a^{14}+\frac{692962564321894}{15\!\cdots\!01}a^{13}-\frac{701783368556132}{15\!\cdots\!01}a^{12}+\frac{107799559267507}{514889568257067}a^{11}+\frac{160827885569590}{514889568257067}a^{10}-\frac{23118026672659}{171629856085689}a^{9}-\frac{470827732561}{171629856085689}a^{8}-\frac{692319098015}{2118887112169}a^{7}+\frac{1445504876933}{6356661336507}a^{6}-\frac{8292311653169}{19069984009521}a^{5}+\frac{6551075536705}{19069984009521}a^{4}-\frac{1138856687503}{6356661336507}a^{3}-\frac{862386593905}{2118887112169}a^{2}+\frac{537566129052}{2118887112169}a-\frac{489171958932}{2118887112169}$, $\frac{1}{46\!\cdots\!03}a^{36}-\frac{1}{46\!\cdots\!03}a^{35}-\frac{1}{46\!\cdots\!03}a^{34}-\frac{708248922752}{46\!\cdots\!03}a^{33}+\frac{8952122964793}{46\!\cdots\!03}a^{32}+\frac{10516670999459}{46\!\cdots\!03}a^{31}+\frac{6555660761740}{46\!\cdots\!03}a^{30}+\frac{39962508288212}{46\!\cdots\!03}a^{29}+\frac{52149211241647}{46\!\cdots\!03}a^{28}+\frac{64287376273055}{46\!\cdots\!03}a^{27}+\frac{22756889291629}{15\!\cdots\!01}a^{26}+\frac{187544432208625}{15\!\cdots\!01}a^{25}+\frac{123530971840171}{46\!\cdots\!03}a^{24}+\frac{11\!\cdots\!94}{46\!\cdots\!03}a^{23}-\frac{22\!\cdots\!43}{46\!\cdots\!03}a^{22}+\frac{923920499297086}{46\!\cdots\!03}a^{21}+\frac{10\!\cdots\!39}{46\!\cdots\!03}a^{20}-\frac{665290337383460}{46\!\cdots\!03}a^{19}+\frac{10\!\cdots\!15}{46\!\cdots\!03}a^{18}+\frac{992402845537301}{46\!\cdots\!03}a^{17}-\frac{203990582610490}{15\!\cdots\!01}a^{16}-\frac{114665679380260}{514889568257067}a^{15}-\frac{16\!\cdots\!47}{46\!\cdots\!03}a^{14}-\frac{560558021818547}{15\!\cdots\!01}a^{13}-\frac{761779516151687}{15\!\cdots\!01}a^{12}-\frac{1300956870884}{6356661336507}a^{11}-\frac{176734604605798}{514889568257067}a^{10}-\frac{83408275842401}{171629856085689}a^{9}+\frac{75073499036743}{171629856085689}a^{8}+\frac{892577052649}{57209952028563}a^{7}-\frac{1852216317248}{6356661336507}a^{6}-\frac{5308678755935}{19069984009521}a^{5}-\frac{8533510189190}{19069984009521}a^{4}+\frac{2789318772988}{6356661336507}a^{3}+\frac{646717766017}{6356661336507}a^{2}+\frac{817912530199}{2118887112169}a+\frac{888515428252}{2118887112169}$, $\frac{1}{46\!\cdots\!03}a^{37}+\frac{1}{46\!\cdots\!03}a^{35}+\frac{5754585116378}{46\!\cdots\!03}a^{33}+\frac{408513680818}{514889568257067}a^{32}+\frac{9000077818741}{15\!\cdots\!01}a^{31}-\frac{2959636122883}{514889568257067}a^{30}+\frac{20404976634622}{15\!\cdots\!01}a^{29}-\frac{6290998785248}{514889568257067}a^{28}+\frac{30746075878334}{46\!\cdots\!03}a^{27}+\frac{22763925881521}{514889568257067}a^{26}-\frac{713559614582552}{46\!\cdots\!03}a^{25}+\frac{80467018498069}{514889568257067}a^{24}+\frac{287850847598869}{46\!\cdots\!03}a^{23}+\frac{15604231059350}{57209952028563}a^{22}+\frac{64048564670008}{15\!\cdots\!01}a^{21}+\frac{170048849786977}{514889568257067}a^{20}-\frac{11\!\cdots\!42}{46\!\cdots\!03}a^{19}-\frac{85033235903338}{514889568257067}a^{18}+\frac{339035817730766}{46\!\cdots\!03}a^{17}-\frac{254708392747741}{514889568257067}a^{16}-\frac{43734060732242}{46\!\cdots\!03}a^{15}+\frac{669954390948313}{46\!\cdots\!03}a^{14}-\frac{11500879418183}{57209952028563}a^{13}+\frac{19380342728096}{171629856085689}a^{12}-\frac{223555007097046}{514889568257067}a^{11}+\frac{1374234359401}{171629856085689}a^{10}-\frac{3865236049276}{19069984009521}a^{9}+\frac{49143508334894}{171629856085689}a^{8}-\frac{24363437327002}{57209952028563}a^{7}+\frac{1925696423744}{19069984009521}a^{6}+\frac{5576441270447}{19069984009521}a^{5}+\frac{36409827093}{2118887112169}a^{4}+\frac{878431798822}{2118887112169}a^{3}-\frac{1022365072299}{2118887112169}a^{2}-\frac{171041452727}{2118887112169}a+\frac{374554525808}{2118887112169}$, $\frac{1}{13\!\cdots\!09}a^{38}-\frac{1}{13\!\cdots\!09}a^{37}-\frac{1}{13\!\cdots\!09}a^{36}-\frac{2}{13\!\cdots\!09}a^{35}+\frac{4}{13\!\cdots\!09}a^{34}-\frac{4264991716303}{13\!\cdots\!09}a^{33}-\frac{14481242569589}{13\!\cdots\!09}a^{32}-\frac{16728108520840}{13\!\cdots\!09}a^{31}-\frac{51524012722604}{13\!\cdots\!09}a^{30}+\frac{212912948531828}{13\!\cdots\!09}a^{29}+\frac{20672779749070}{46\!\cdots\!03}a^{28}+\frac{156778554026086}{46\!\cdots\!03}a^{27}+\frac{265420707642655}{13\!\cdots\!09}a^{26}+\frac{78386631167486}{13\!\cdots\!09}a^{25}+\frac{18\!\cdots\!41}{13\!\cdots\!09}a^{24}-\frac{64\!\cdots\!18}{13\!\cdots\!09}a^{23}-\frac{57\!\cdots\!28}{13\!\cdots\!09}a^{22}-\frac{24\!\cdots\!28}{13\!\cdots\!09}a^{21}+\frac{24\!\cdots\!18}{13\!\cdots\!09}a^{20}+\frac{68\!\cdots\!65}{13\!\cdots\!09}a^{19}+\frac{20\!\cdots\!91}{46\!\cdots\!03}a^{18}-\frac{179362662530918}{15\!\cdots\!01}a^{17}+\frac{11\!\cdots\!87}{13\!\cdots\!09}a^{16}-\frac{732602038597178}{46\!\cdots\!03}a^{15}+\frac{657815413424044}{46\!\cdots\!03}a^{14}+\frac{647638006877950}{15\!\cdots\!01}a^{13}+\frac{136022108864875}{514889568257067}a^{12}+\frac{209819406410473}{514889568257067}a^{11}+\frac{124798597966799}{514889568257067}a^{10}-\frac{58181633423267}{171629856085689}a^{9}+\frac{5225570105531}{57209952028563}a^{8}-\frac{8835268081217}{57209952028563}a^{7}+\frac{1382955461524}{19069984009521}a^{6}-\frac{957324824750}{6356661336507}a^{5}-\frac{3388139839108}{19069984009521}a^{4}+\frac{981036351016}{6356661336507}a^{3}+\frac{1003158996755}{6356661336507}a^{2}+\frac{437569804938}{2118887112169}a-\frac{943820865010}{2118887112169}$, $\frac{1}{13\!\cdots\!09}a^{39}+\frac{1}{13\!\cdots\!09}a^{37}+\frac{2}{13\!\cdots\!09}a^{35}+\frac{7894942750198}{46\!\cdots\!03}a^{33}+\frac{693742598104}{514889568257067}a^{32}+\frac{23985747693412}{46\!\cdots\!03}a^{31}+\frac{310833915908}{514889568257067}a^{30}-\frac{196689435975358}{13\!\cdots\!09}a^{29}-\frac{8608882265327}{514889568257067}a^{28}-\frac{561383933668925}{13\!\cdots\!09}a^{27}-\frac{28586049840785}{514889568257067}a^{26}-\frac{239246540920049}{13\!\cdots\!09}a^{25}+\frac{48718048513633}{514889568257067}a^{24}-\frac{20\!\cdots\!64}{46\!\cdots\!03}a^{23}+\frac{28326609231883}{171629856085689}a^{22}+\frac{57\!\cdots\!79}{13\!\cdots\!09}a^{21}+\frac{175327040539495}{514889568257067}a^{20}-\frac{27\!\cdots\!42}{13\!\cdots\!09}a^{19}+\frac{156711544646909}{514889568257067}a^{18}-\frac{22\!\cdots\!11}{13\!\cdots\!09}a^{17}-\frac{56\!\cdots\!93}{13\!\cdots\!09}a^{16}+\frac{337740616592746}{15\!\cdots\!01}a^{15}+\frac{187978512455213}{514889568257067}a^{14}+\frac{514318884710657}{15\!\cdots\!01}a^{13}-\frac{586899615563629}{15\!\cdots\!01}a^{12}+\frac{195315743061514}{514889568257067}a^{11}+\frac{60364846039232}{514889568257067}a^{10}-\frac{76934745384244}{171629856085689}a^{9}-\frac{69919507947562}{171629856085689}a^{8}-\frac{2953273150940}{57209952028563}a^{7}-\frac{2412456403819}{6356661336507}a^{6}+\frac{1229133290365}{6356661336507}a^{5}+\frac{9346938234137}{19069984009521}a^{4}-\frac{1347087820661}{6356661336507}a^{3}-\frac{1900711327805}{6356661336507}a^{2}+\frac{539551885305}{2118887112169}a+\frac{885920996734}{2118887112169}$, $\frac{1}{41\!\cdots\!27}a^{40}-\frac{1}{41\!\cdots\!27}a^{39}-\frac{1}{41\!\cdots\!27}a^{38}-\frac{2}{41\!\cdots\!27}a^{37}+\frac{4}{41\!\cdots\!27}a^{36}-\frac{10}{41\!\cdots\!27}a^{35}-\frac{5}{41\!\cdots\!27}a^{34}-\frac{34444236924034}{41\!\cdots\!27}a^{33}-\frac{52948252615316}{41\!\cdots\!27}a^{32}+\frac{38933404091804}{41\!\cdots\!27}a^{31}+\frac{83838614682775}{13\!\cdots\!09}a^{30}+\frac{236531739549031}{13\!\cdots\!09}a^{29}+\frac{700930631767390}{41\!\cdots\!27}a^{28}+\frac{733545464710511}{41\!\cdots\!27}a^{27}+\frac{18\!\cdots\!94}{41\!\cdots\!27}a^{26}-\frac{67\!\cdots\!56}{41\!\cdots\!27}a^{25}-\frac{55\!\cdots\!17}{41\!\cdots\!27}a^{24}+\frac{15\!\cdots\!79}{41\!\cdots\!27}a^{23}-\frac{13\!\cdots\!03}{41\!\cdots\!27}a^{22}+\frac{59\!\cdots\!97}{41\!\cdots\!27}a^{21}+\frac{29\!\cdots\!50}{13\!\cdots\!09}a^{20}-\frac{17\!\cdots\!50}{46\!\cdots\!03}a^{19}+\frac{20\!\cdots\!99}{41\!\cdots\!27}a^{18}-\frac{990665770410977}{13\!\cdots\!09}a^{17}+\frac{30\!\cdots\!86}{13\!\cdots\!09}a^{16}+\frac{52766902893670}{46\!\cdots\!03}a^{15}-\frac{179924587426505}{15\!\cdots\!01}a^{14}+\frac{255937483684151}{514889568257067}a^{13}+\frac{632595072550531}{15\!\cdots\!01}a^{12}+\frac{27224077866970}{514889568257067}a^{11}-\frac{13635186556253}{57209952028563}a^{10}-\frac{78045629422829}{171629856085689}a^{9}+\frac{64194946514642}{171629856085689}a^{8}-\frac{2139332297330}{19069984009521}a^{7}-\frac{550490940779}{57209952028563}a^{6}+\frac{3588954768095}{19069984009521}a^{5}-\frac{1030922901590}{2118887112169}a^{4}-\frac{702801064855}{2118887112169}a^{3}-\frac{874007728697}{6356661336507}a^{2}+\frac{417085948870}{2118887112169}a+\frac{583951187756}{2118887112169}$, $\frac{1}{57\!\cdots\!53}a^{41}-\frac{49}{57\!\cdots\!53}a^{40}-\frac{67}{57\!\cdots\!53}a^{39}+\frac{151}{57\!\cdots\!53}a^{38}+\frac{574}{57\!\cdots\!53}a^{37}-\frac{271}{57\!\cdots\!53}a^{36}+\frac{613}{57\!\cdots\!53}a^{35}+\frac{842}{57\!\cdots\!53}a^{34}-\frac{11\!\cdots\!58}{57\!\cdots\!53}a^{33}+\frac{79\!\cdots\!96}{57\!\cdots\!53}a^{32}-\frac{24\!\cdots\!36}{64\!\cdots\!17}a^{31}+\frac{17\!\cdots\!02}{64\!\cdots\!17}a^{30}+\frac{27\!\cdots\!47}{57\!\cdots\!53}a^{29}+\frac{12\!\cdots\!94}{57\!\cdots\!53}a^{28}+\frac{11\!\cdots\!20}{57\!\cdots\!53}a^{27}+\frac{22\!\cdots\!73}{57\!\cdots\!53}a^{26}-\frac{29\!\cdots\!18}{57\!\cdots\!53}a^{25}+\frac{66\!\cdots\!50}{57\!\cdots\!53}a^{24}+\frac{26\!\cdots\!26}{57\!\cdots\!53}a^{23}-\frac{58\!\cdots\!10}{57\!\cdots\!53}a^{22}+\frac{82\!\cdots\!60}{19\!\cdots\!51}a^{21}-\frac{49\!\cdots\!21}{19\!\cdots\!51}a^{20}+\frac{97\!\cdots\!96}{57\!\cdots\!53}a^{19}-\frac{91\!\cdots\!19}{21\!\cdots\!39}a^{18}-\frac{44\!\cdots\!15}{19\!\cdots\!51}a^{17}-\frac{25\!\cdots\!53}{71\!\cdots\!13}a^{16}-\frac{51\!\cdots\!37}{64\!\cdots\!17}a^{15}-\frac{14\!\cdots\!72}{64\!\cdots\!17}a^{14}+\frac{28\!\cdots\!45}{71\!\cdots\!13}a^{13}-\frac{15\!\cdots\!15}{71\!\cdots\!13}a^{12}+\frac{22\!\cdots\!23}{71\!\cdots\!13}a^{11}-\frac{26\!\cdots\!95}{71\!\cdots\!13}a^{10}+\frac{99997816900667}{79\!\cdots\!57}a^{9}-\frac{10\!\cdots\!39}{23\!\cdots\!71}a^{8}-\frac{533696890959815}{26\!\cdots\!19}a^{7}-\frac{26\!\cdots\!54}{79\!\cdots\!57}a^{6}+\frac{351784095381974}{26\!\cdots\!19}a^{5}-\frac{965413908435437}{26\!\cdots\!19}a^{4}+\frac{110321555896202}{294525308591491}a^{3}+\frac{257242029094978}{883575925774473}a^{2}+\frac{66720695516764}{294525308591491}a+\frac{33966289761}{2118887112169}$, $\frac{1}{17\!\cdots\!59}a^{42}-\frac{1}{17\!\cdots\!59}a^{41}+\frac{83}{17\!\cdots\!59}a^{40}-\frac{563}{17\!\cdots\!59}a^{39}+\frac{316}{17\!\cdots\!59}a^{38}-\frac{241}{17\!\cdots\!59}a^{37}+\frac{1366}{17\!\cdots\!59}a^{36}-\frac{2260}{17\!\cdots\!59}a^{35}-\frac{5}{12\!\cdots\!81}a^{34}+\frac{30\!\cdots\!33}{17\!\cdots\!59}a^{33}+\frac{39\!\cdots\!59}{57\!\cdots\!53}a^{32}+\frac{14\!\cdots\!10}{19\!\cdots\!51}a^{31}-\frac{10\!\cdots\!71}{17\!\cdots\!59}a^{30}-\frac{53\!\cdots\!16}{17\!\cdots\!59}a^{29}-\frac{47\!\cdots\!70}{17\!\cdots\!59}a^{28}+\frac{74\!\cdots\!19}{17\!\cdots\!59}a^{27}-\frac{71\!\cdots\!77}{17\!\cdots\!59}a^{26}+\frac{53\!\cdots\!40}{17\!\cdots\!59}a^{25}+\frac{85\!\cdots\!91}{17\!\cdots\!59}a^{24}+\frac{46\!\cdots\!48}{17\!\cdots\!59}a^{23}-\frac{59\!\cdots\!33}{19\!\cdots\!51}a^{22}+\frac{10\!\cdots\!95}{57\!\cdots\!53}a^{21}-\frac{35\!\cdots\!33}{17\!\cdots\!59}a^{20}-\frac{88\!\cdots\!93}{57\!\cdots\!53}a^{19}+\frac{21\!\cdots\!56}{57\!\cdots\!53}a^{18}+\frac{78\!\cdots\!74}{64\!\cdots\!17}a^{17}+\frac{79\!\cdots\!27}{19\!\cdots\!51}a^{16}-\frac{12\!\cdots\!27}{71\!\cdots\!13}a^{15}+\frac{27\!\cdots\!16}{64\!\cdots\!17}a^{14}+\frac{42\!\cdots\!89}{21\!\cdots\!39}a^{13}-\frac{10\!\cdots\!61}{21\!\cdots\!39}a^{12}+\frac{22\!\cdots\!72}{71\!\cdots\!13}a^{11}+\frac{27\!\cdots\!75}{79\!\cdots\!57}a^{10}-\frac{12\!\cdots\!99}{79\!\cdots\!57}a^{9}-\frac{879732052762972}{26\!\cdots\!19}a^{8}-\frac{10\!\cdots\!07}{26\!\cdots\!19}a^{7}-\frac{768494441546863}{79\!\cdots\!57}a^{6}+\frac{787330338869234}{26\!\cdots\!19}a^{5}+\frac{307104300993725}{26\!\cdots\!19}a^{4}+\frac{65032365876010}{883575925774473}a^{3}+\frac{51013144340542}{294525308591491}a^{2}+\frac{52573247060809}{294525308591491}a+\frac{521653465295}{2118887112169}$, $\frac{1}{19\!\cdots\!13}a^{43}+\frac{44\!\cdots\!36}{19\!\cdots\!13}a^{42}-\frac{47\!\cdots\!63}{66\!\cdots\!71}a^{41}-\frac{12\!\cdots\!58}{19\!\cdots\!13}a^{40}+\frac{12\!\cdots\!99}{66\!\cdots\!71}a^{39}-\frac{39\!\cdots\!78}{19\!\cdots\!13}a^{38}+\frac{20\!\cdots\!13}{19\!\cdots\!13}a^{37}-\frac{16\!\cdots\!46}{19\!\cdots\!13}a^{36}+\frac{40\!\cdots\!45}{19\!\cdots\!13}a^{35}-\frac{62\!\cdots\!13}{19\!\cdots\!13}a^{34}+\frac{15\!\cdots\!80}{19\!\cdots\!13}a^{33}-\frac{46\!\cdots\!76}{66\!\cdots\!71}a^{32}+\frac{19\!\cdots\!62}{19\!\cdots\!13}a^{31}+\frac{88\!\cdots\!98}{19\!\cdots\!13}a^{30}-\frac{29\!\cdots\!93}{22\!\cdots\!57}a^{29}-\frac{15\!\cdots\!09}{19\!\cdots\!13}a^{28}+\frac{10\!\cdots\!28}{19\!\cdots\!13}a^{27}+\frac{55\!\cdots\!51}{19\!\cdots\!13}a^{26}-\frac{24\!\cdots\!41}{24\!\cdots\!73}a^{25}-\frac{19\!\cdots\!13}{19\!\cdots\!13}a^{24}-\frac{85\!\cdots\!56}{19\!\cdots\!13}a^{23}+\frac{10\!\cdots\!60}{22\!\cdots\!57}a^{22}-\frac{61\!\cdots\!53}{19\!\cdots\!13}a^{21}+\frac{95\!\cdots\!49}{19\!\cdots\!13}a^{20}-\frac{92\!\cdots\!26}{22\!\cdots\!57}a^{19}+\frac{17\!\cdots\!33}{66\!\cdots\!71}a^{18}-\frac{22\!\cdots\!97}{22\!\cdots\!57}a^{17}-\frac{11\!\cdots\!97}{27\!\cdots\!97}a^{16}-\frac{28\!\cdots\!95}{73\!\cdots\!19}a^{15}-\frac{17\!\cdots\!81}{73\!\cdots\!19}a^{14}-\frac{16\!\cdots\!76}{24\!\cdots\!73}a^{13}-\frac{57\!\cdots\!60}{24\!\cdots\!73}a^{12}+\frac{18\!\cdots\!78}{82\!\cdots\!91}a^{11}+\frac{23\!\cdots\!01}{82\!\cdots\!91}a^{10}+\frac{12\!\cdots\!03}{33\!\cdots\!37}a^{9}+\frac{12\!\cdots\!77}{27\!\cdots\!97}a^{8}-\frac{85\!\cdots\!11}{30\!\cdots\!33}a^{7}-\frac{38\!\cdots\!42}{91\!\cdots\!99}a^{6}+\frac{12\!\cdots\!55}{30\!\cdots\!33}a^{5}+\frac{45\!\cdots\!90}{10\!\cdots\!11}a^{4}+\frac{40\!\cdots\!37}{10\!\cdots\!11}a^{3}-\frac{15\!\cdots\!28}{33\!\cdots\!37}a^{2}-\frac{15\!\cdots\!65}{33\!\cdots\!37}a+\frac{21\!\cdots\!16}{24\!\cdots\!83}$ 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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 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809)
 
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 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809, 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 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809);
 
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 + 188*x^42 - 191*x^41 + 15682*x^40 - 16255*x^39 + 767101*x^38 - 815866*x^37 + 24539323*x^36 - 26986921*x^35 + 543368874*x^34 - 624329637*x^33 + 8632506022*x^32 - 10505494933*x^31 + 101311538399*x^30 - 132828023198*x^29 + 907079143955*x^28 - 1305563213549*x^27 + 6469389498694*x^26 - 10386079139341*x^25 + 38961713404851*x^24 - 70119950822874*x^23 + 212107487819539*x^22 - 267306271081404*x^21 + 954204965731533*x^20 - 348452808851934*x^19 + 3286452427237125*x^18 + 1220729003449623*x^17 + 8721817319211705*x^16 + 7424451290225763*x^15 + 18808628124207888*x^14 + 21088254372919965*x^13 + 35377638646744746*x^12 + 44100561928942881*x^11 + 62049209117584005*x^10 + 79953228494360262*x^9 + 106199279503507242*x^8 + 136817944804830789*x^7 + 181780810904989953*x^6 + 226679666465622069*x^5 + 318662867727691812*x^4 + 354165989590339128*x^3 + 601822619090434404*x^2 + 452107998974921166*x + 1353359858390092809);
 
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

\(\Q(\sqrt{13}) \), 4.0.10459917.4, \(\Q(\zeta_{23})^+\), 22.22.3075626510913487571920886830127053316437.1

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 $44$ R $44$ $44$ $44$ R $22^{2}$ $44$ R $22^{2}$ $44$ $44$ $44$ ${\href{/padicField/43.11.0.1}{11} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{11}$ ${\href{/padicField/53.11.0.1}{11} }^{4}$ $44$

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
\(3\) Copy content Toggle raw display 3.22.11.2$x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$$2$$11$$11$22T1$[\ ]_{2}^{11}$
3.22.11.2$x^{22} + 33 x^{20} + 495 x^{18} + 4455 x^{16} + 26730 x^{14} + 4 x^{13} + 112266 x^{12} - 406 x^{11} + 336798 x^{10} + 1650 x^{9} + 721710 x^{8} + 20196 x^{7} + 1082565 x^{6} - 35640 x^{5} + 1082569 x^{4} - 37422 x^{3} + 649567 x^{2} + 20898 x + 177172$$2$$11$$11$22T1$[\ ]_{2}^{11}$
\(13\) Copy content Toggle raw display Deg $44$$4$$11$$33$
\(23\) Copy content Toggle raw display 23.22.21.17$x^{22} + 23$$22$$1$$21$22T1$[\ ]_{22}$
23.22.21.17$x^{22} + 23$$22$$1$$21$22T1$[\ ]_{22}$