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Properties

Label 44.0.238...125.1
Degree $44$
Signature $[0, 22]$
Discriminant $2.385\times 10^{106}$
Root discriminant \(261.62\)
Ramified primes $5,11$
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 + 55*x^42 - 33*x^41 + 2200*x^40 + 3861*x^39 + 81686*x^38 + 187473*x^37 + 2797949*x^36 + 5067315*x^35 + 48269287*x^34 + 80889588*x^33 + 627196361*x^32 + 562870110*x^31 + 6666501116*x^30 + 3165816357*x^29 + 61519492375*x^28 + 18154396755*x^27 + 435915893402*x^26 + 55327492110*x^25 + 2610992250944*x^24 - 522745688124*x^23 + 13465948440286*x^22 - 5051169113364*x^21 + 60785860077503*x^20 - 28124569272129*x^19 + 233899389744845*x^18 - 131136264490176*x^17 + 790178749679506*x^16 - 517434573601242*x^15 + 2274690119726596*x^14 - 1273453323335061*x^13 + 5309434517531353*x^12 - 2092304520835266*x^11 + 8720469353666763*x^10 - 2120075052140232*x^9 + 13102352080428996*x^8 - 132063952087980*x^7 + 17079801196779882*x^6 - 477544700475315*x^5 + 14647338263254752*x^4 + 323479726646175*x^3 + 7143904676997*x^2 + 157680139023*x + 3486784401)
 
gp: K = bnfinit(y^44 + 55*y^42 - 33*y^41 + 2200*y^40 + 3861*y^39 + 81686*y^38 + 187473*y^37 + 2797949*y^36 + 5067315*y^35 + 48269287*y^34 + 80889588*y^33 + 627196361*y^32 + 562870110*y^31 + 6666501116*y^30 + 3165816357*y^29 + 61519492375*y^28 + 18154396755*y^27 + 435915893402*y^26 + 55327492110*y^25 + 2610992250944*y^24 - 522745688124*y^23 + 13465948440286*y^22 - 5051169113364*y^21 + 60785860077503*y^20 - 28124569272129*y^19 + 233899389744845*y^18 - 131136264490176*y^17 + 790178749679506*y^16 - 517434573601242*y^15 + 2274690119726596*y^14 - 1273453323335061*y^13 + 5309434517531353*y^12 - 2092304520835266*y^11 + 8720469353666763*y^10 - 2120075052140232*y^9 + 13102352080428996*y^8 - 132063952087980*y^7 + 17079801196779882*y^6 - 477544700475315*y^5 + 14647338263254752*y^4 + 323479726646175*y^3 + 7143904676997*y^2 + 157680139023*y + 3486784401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 + 55*x^42 - 33*x^41 + 2200*x^40 + 3861*x^39 + 81686*x^38 + 187473*x^37 + 2797949*x^36 + 5067315*x^35 + 48269287*x^34 + 80889588*x^33 + 627196361*x^32 + 562870110*x^31 + 6666501116*x^30 + 3165816357*x^29 + 61519492375*x^28 + 18154396755*x^27 + 435915893402*x^26 + 55327492110*x^25 + 2610992250944*x^24 - 522745688124*x^23 + 13465948440286*x^22 - 5051169113364*x^21 + 60785860077503*x^20 - 28124569272129*x^19 + 233899389744845*x^18 - 131136264490176*x^17 + 790178749679506*x^16 - 517434573601242*x^15 + 2274690119726596*x^14 - 1273453323335061*x^13 + 5309434517531353*x^12 - 2092304520835266*x^11 + 8720469353666763*x^10 - 2120075052140232*x^9 + 13102352080428996*x^8 - 132063952087980*x^7 + 17079801196779882*x^6 - 477544700475315*x^5 + 14647338263254752*x^4 + 323479726646175*x^3 + 7143904676997*x^2 + 157680139023*x + 3486784401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 + 55*x^42 - 33*x^41 + 2200*x^40 + 3861*x^39 + 81686*x^38 + 187473*x^37 + 2797949*x^36 + 5067315*x^35 + 48269287*x^34 + 80889588*x^33 + 627196361*x^32 + 562870110*x^31 + 6666501116*x^30 + 3165816357*x^29 + 61519492375*x^28 + 18154396755*x^27 + 435915893402*x^26 + 55327492110*x^25 + 2610992250944*x^24 - 522745688124*x^23 + 13465948440286*x^22 - 5051169113364*x^21 + 60785860077503*x^20 - 28124569272129*x^19 + 233899389744845*x^18 - 131136264490176*x^17 + 790178749679506*x^16 - 517434573601242*x^15 + 2274690119726596*x^14 - 1273453323335061*x^13 + 5309434517531353*x^12 - 2092304520835266*x^11 + 8720469353666763*x^10 - 2120075052140232*x^9 + 13102352080428996*x^8 - 132063952087980*x^7 + 17079801196779882*x^6 - 477544700475315*x^5 + 14647338263254752*x^4 + 323479726646175*x^3 + 7143904676997*x^2 + 157680139023*x + 3486784401)
 

\( x^{44} + 55 x^{42} - 33 x^{41} + 2200 x^{40} + 3861 x^{39} + 81686 x^{38} + 187473 x^{37} + \cdots + 3486784401 \) 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:   \(238\!\cdots\!125\) \(\medspace = 5^{33}\cdot 11^{80}\) 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:  \(261.62\)
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:  $5^{3/4}11^{20/11}\approx 261.61867933814926$
Ramified primes:   \(5\), \(11\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(605=5\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{605}(1,·)$, $\chi_{605}(386,·)$, $\chi_{605}(133,·)$, $\chi_{605}(518,·)$, $\chi_{605}(12,·)$, $\chi_{605}(397,·)$, $\chi_{605}(144,·)$, $\chi_{605}(529,·)$, $\chi_{605}(276,·)$, $\chi_{605}(23,·)$, $\chi_{605}(408,·)$, $\chi_{605}(287,·)$, $\chi_{605}(34,·)$, $\chi_{605}(419,·)$, $\chi_{605}(166,·)$, $\chi_{605}(551,·)$, $\chi_{605}(298,·)$, $\chi_{605}(177,·)$, $\chi_{605}(562,·)$, $\chi_{605}(309,·)$, $\chi_{605}(56,·)$, $\chi_{605}(441,·)$, $\chi_{605}(188,·)$, $\chi_{605}(573,·)$, $\chi_{605}(67,·)$, $\chi_{605}(452,·)$, $\chi_{605}(199,·)$, $\chi_{605}(584,·)$, $\chi_{605}(331,·)$, $\chi_{605}(78,·)$, $\chi_{605}(463,·)$, $\chi_{605}(342,·)$, $\chi_{605}(89,·)$, $\chi_{605}(474,·)$, $\chi_{605}(221,·)$, $\chi_{605}(353,·)$, $\chi_{605}(232,·)$, $\chi_{605}(364,·)$, $\chi_{605}(111,·)$, $\chi_{605}(496,·)$, $\chi_{605}(243,·)$, $\chi_{605}(122,·)$, $\chi_{605}(507,·)$, $\chi_{605}(254,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{7}$, $\frac{1}{9}a^{18}-\frac{1}{9}a^{16}+\frac{1}{9}a^{12}-\frac{1}{9}a^{10}-\frac{2}{9}a^{8}-\frac{1}{9}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}-\frac{1}{9}a^{17}+\frac{1}{9}a^{13}-\frac{1}{9}a^{11}+\frac{1}{9}a^{9}+\frac{1}{3}a^{7}+\frac{2}{9}a^{5}+\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{20}-\frac{1}{9}a^{16}+\frac{1}{9}a^{14}+\frac{1}{9}a^{8}+\frac{2}{9}a^{6}+\frac{1}{9}a^{2}$, $\frac{1}{9}a^{21}-\frac{1}{9}a^{17}+\frac{1}{9}a^{15}+\frac{1}{9}a^{9}+\frac{2}{9}a^{7}+\frac{1}{9}a^{3}$, $\frac{1}{9}a^{22}+\frac{1}{9}a^{12}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{23}+\frac{1}{9}a^{13}-\frac{2}{9}a^{3}$, $\frac{1}{9}a^{24}+\frac{1}{9}a^{14}-\frac{2}{9}a^{4}$, $\frac{1}{9}a^{25}+\frac{1}{9}a^{15}-\frac{2}{9}a^{5}$, $\frac{1}{9}a^{26}+\frac{1}{9}a^{16}-\frac{2}{9}a^{6}$, $\frac{1}{27}a^{27}+\frac{1}{27}a^{21}-\frac{1}{9}a^{17}-\frac{2}{27}a^{15}-\frac{1}{9}a^{13}+\frac{1}{9}a^{11}+\frac{4}{27}a^{9}+\frac{2}{9}a^{7}+\frac{2}{9}a^{5}+\frac{7}{27}a^{3}$, $\frac{1}{81}a^{28}+\frac{1}{27}a^{26}-\frac{1}{27}a^{24}+\frac{1}{81}a^{22}-\frac{1}{27}a^{20}+\frac{10}{81}a^{16}+\frac{1}{9}a^{14}+\frac{2}{27}a^{12}+\frac{1}{81}a^{10}-\frac{1}{27}a^{8}-\frac{5}{27}a^{6}-\frac{8}{81}a^{4}-\frac{1}{9}a^{2}$, $\frac{1}{81}a^{29}-\frac{1}{27}a^{25}+\frac{1}{81}a^{23}+\frac{1}{27}a^{21}+\frac{10}{81}a^{17}-\frac{1}{27}a^{15}-\frac{4}{27}a^{13}-\frac{8}{81}a^{11}-\frac{2}{27}a^{9}-\frac{5}{27}a^{7}+\frac{1}{81}a^{5}+\frac{2}{27}a^{3}$, $\frac{1}{81}a^{30}-\frac{1}{27}a^{26}+\frac{1}{81}a^{24}+\frac{1}{27}a^{22}+\frac{1}{81}a^{18}+\frac{2}{27}a^{16}-\frac{4}{27}a^{14}+\frac{10}{81}a^{12}+\frac{1}{27}a^{10}+\frac{1}{27}a^{8}+\frac{1}{81}a^{6}+\frac{5}{27}a^{4}-\frac{1}{9}a^{2}$, $\frac{1}{81}a^{31}+\frac{1}{81}a^{25}+\frac{1}{27}a^{23}+\frac{1}{27}a^{21}+\frac{1}{81}a^{19}-\frac{1}{27}a^{17}+\frac{1}{9}a^{15}+\frac{1}{81}a^{13}+\frac{4}{27}a^{11}-\frac{4}{27}a^{9}-\frac{8}{81}a^{7}-\frac{7}{27}a^{5}-\frac{5}{27}a^{3}-\frac{1}{3}a$, $\frac{1}{81}a^{32}+\frac{1}{81}a^{26}+\frac{1}{27}a^{24}+\frac{1}{27}a^{22}+\frac{1}{81}a^{20}-\frac{1}{27}a^{18}+\frac{1}{9}a^{16}+\frac{1}{81}a^{14}+\frac{4}{27}a^{12}-\frac{4}{27}a^{10}-\frac{8}{81}a^{8}-\frac{7}{27}a^{6}-\frac{5}{27}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{243}a^{33}+\frac{1}{243}a^{32}+\frac{1}{243}a^{31}+\frac{1}{243}a^{30}+\frac{1}{243}a^{29}+\frac{1}{243}a^{27}-\frac{11}{243}a^{26}+\frac{1}{243}a^{25}+\frac{4}{243}a^{24}-\frac{11}{243}a^{23}+\frac{2}{81}a^{22}-\frac{11}{243}a^{21}+\frac{10}{243}a^{20}-\frac{11}{243}a^{19}-\frac{2}{243}a^{18}-\frac{11}{243}a^{17}-\frac{10}{81}a^{16}-\frac{11}{243}a^{15}-\frac{29}{243}a^{14}+\frac{28}{243}a^{13}-\frac{32}{243}a^{12}+\frac{28}{243}a^{11}+\frac{2}{27}a^{10}+\frac{28}{243}a^{9}+\frac{112}{243}a^{8}+\frac{109}{243}a^{7}+\frac{70}{243}a^{6}-\frac{53}{243}a^{5}-\frac{1}{9}a^{4}+\frac{1}{3}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{243}a^{34}-\frac{1}{243}a^{29}+\frac{1}{243}a^{28}-\frac{1}{81}a^{27}+\frac{4}{81}a^{26}+\frac{1}{81}a^{25}+\frac{4}{81}a^{24}-\frac{10}{243}a^{23}+\frac{10}{243}a^{22}+\frac{1}{81}a^{21}+\frac{2}{81}a^{20}+\frac{1}{27}a^{19}-\frac{1}{27}a^{18}-\frac{19}{243}a^{17}-\frac{8}{243}a^{16}+\frac{2}{27}a^{15}+\frac{10}{81}a^{14}-\frac{11}{81}a^{13}+\frac{2}{81}a^{12}+\frac{17}{243}a^{11}+\frac{10}{243}a^{10}+\frac{4}{81}a^{9}+\frac{8}{81}a^{8}-\frac{40}{81}a^{7}-\frac{23}{81}a^{6}-\frac{82}{243}a^{5}-\frac{4}{9}a^{4}-\frac{2}{27}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{243}a^{35}-\frac{1}{243}a^{30}+\frac{1}{243}a^{29}+\frac{1}{81}a^{27}+\frac{4}{81}a^{26}+\frac{4}{81}a^{25}+\frac{8}{243}a^{24}+\frac{10}{243}a^{23}+\frac{2}{81}a^{22}-\frac{1}{81}a^{21}-\frac{1}{27}a^{19}+\frac{8}{243}a^{18}+\frac{19}{243}a^{17}+\frac{7}{81}a^{16}-\frac{11}{81}a^{15}+\frac{7}{81}a^{14}+\frac{11}{81}a^{13}-\frac{19}{243}a^{12}-\frac{17}{243}a^{11}-\frac{4}{81}a^{10}-\frac{4}{81}a^{9}+\frac{20}{81}a^{8}+\frac{40}{81}a^{7}+\frac{116}{243}a^{6}-\frac{1}{3}a^{5}+\frac{40}{81}a^{4}+\frac{2}{27}a^{3}$, $\frac{1}{243}a^{36}-\frac{1}{243}a^{31}+\frac{1}{243}a^{30}+\frac{1}{81}a^{27}+\frac{1}{81}a^{26}+\frac{8}{243}a^{25}-\frac{8}{243}a^{24}+\frac{2}{81}a^{23}-\frac{2}{81}a^{22}-\frac{1}{27}a^{21}+\frac{8}{243}a^{19}-\frac{8}{243}a^{18}-\frac{11}{81}a^{17}-\frac{4}{27}a^{16}+\frac{13}{81}a^{15}-\frac{7}{81}a^{14}+\frac{8}{243}a^{13}+\frac{19}{243}a^{12}-\frac{13}{81}a^{11}+\frac{4}{81}a^{10}+\frac{8}{81}a^{9}-\frac{20}{81}a^{8}-\frac{100}{243}a^{7}-\frac{4}{27}a^{6}+\frac{22}{81}a^{5}-\frac{40}{81}a^{4}-\frac{7}{27}a^{3}$, $\frac{1}{729}a^{37}+\frac{1}{729}a^{35}+\frac{1}{729}a^{33}-\frac{1}{243}a^{32}-\frac{1}{729}a^{31}+\frac{2}{729}a^{29}-\frac{1}{243}a^{28}+\frac{7}{729}a^{27}+\frac{5}{243}a^{26}-\frac{25}{729}a^{25}-\frac{1}{27}a^{24}+\frac{38}{729}a^{23}+\frac{5}{243}a^{22}+\frac{4}{729}a^{21}+\frac{2}{243}a^{20}+\frac{23}{729}a^{19}+\frac{1}{81}a^{18}+\frac{62}{729}a^{17}-\frac{37}{243}a^{16}+\frac{70}{729}a^{15}-\frac{19}{243}a^{14}+\frac{104}{729}a^{13}-\frac{4}{27}a^{12}+\frac{14}{729}a^{11}+\frac{11}{243}a^{10}+\frac{73}{729}a^{9}+\frac{92}{243}a^{8}+\frac{28}{729}a^{7}-\frac{4}{27}a^{6}+\frac{295}{729}a^{5}-\frac{121}{243}a^{4}-\frac{5}{81}a^{3}-\frac{13}{27}a^{2}-\frac{2}{9}a$, $\frac{1}{1581201}a^{38}-\frac{2}{6507}a^{37}+\frac{1864}{1581201}a^{36}+\frac{997}{527067}a^{35}-\frac{2012}{1581201}a^{34}-\frac{31}{19521}a^{33}-\frac{1960}{1581201}a^{32}+\frac{310}{527067}a^{31}+\frac{1775}{1581201}a^{30}-\frac{359}{175689}a^{29}-\frac{4769}{1581201}a^{28}+\frac{1496}{175689}a^{27}-\frac{15286}{1581201}a^{26}+\frac{2800}{527067}a^{25}+\frac{20756}{1581201}a^{24}+\frac{6500}{175689}a^{23}+\frac{4930}{1581201}a^{22}-\frac{2567}{58563}a^{21}-\frac{67987}{1581201}a^{20}-\frac{2210}{527067}a^{19}+\frac{42704}{1581201}a^{18}-\frac{448}{6507}a^{17}+\frac{21997}{1581201}a^{16}-\frac{19}{58563}a^{15}-\frac{200026}{1581201}a^{14}+\frac{55399}{527067}a^{13}-\frac{184054}{1581201}a^{12}+\frac{19700}{175689}a^{11}-\frac{39461}{1581201}a^{10}+\frac{25402}{175689}a^{9}-\frac{628637}{1581201}a^{8}+\frac{210262}{527067}a^{7}+\frac{478915}{1581201}a^{6}-\frac{47590}{527067}a^{5}+\frac{19012}{175689}a^{4}+\frac{3554}{58563}a^{3}+\frac{5524}{19521}a^{2}+\frac{950}{2169}a-\frac{101}{241}$, $\frac{1}{14230809}a^{39}-\frac{6587}{14230809}a^{37}-\frac{5033}{4743603}a^{36}-\frac{12461}{14230809}a^{35}+\frac{323}{527067}a^{34}-\frac{18511}{14230809}a^{33}-\frac{11381}{4743603}a^{32}-\frac{79819}{14230809}a^{31}-\frac{3560}{1581201}a^{30}+\frac{64729}{14230809}a^{29}-\frac{7318}{1581201}a^{28}-\frac{193513}{14230809}a^{27}+\frac{51184}{4743603}a^{26}-\frac{549349}{14230809}a^{25}-\frac{34537}{1581201}a^{24}-\frac{64730}{14230809}a^{23}-\frac{2686}{175689}a^{22}-\frac{104464}{14230809}a^{21}-\frac{142907}{4743603}a^{20}-\frac{121186}{14230809}a^{19}+\frac{24529}{527067}a^{18}+\frac{1929061}{14230809}a^{17}+\frac{54430}{527067}a^{16}+\frac{1476728}{14230809}a^{15}+\frac{661198}{4743603}a^{14}-\frac{430969}{14230809}a^{13}-\frac{223063}{1581201}a^{12}-\frac{6392}{59049}a^{11}-\frac{39158}{1581201}a^{10}+\frac{1682671}{14230809}a^{9}+\frac{1526539}{4743603}a^{8}-\frac{336755}{14230809}a^{7}-\frac{2163265}{4743603}a^{6}+\frac{475717}{1581201}a^{5}+\frac{29546}{527067}a^{4}+\frac{86740}{175689}a^{3}+\frac{7154}{19521}a^{2}-\frac{616}{2169}a+\frac{89}{241}$, $\frac{1}{1580345570259}a^{40}-\frac{2245}{175593952251}a^{39}+\frac{55}{1580345570259}a^{38}-\frac{240240596}{526781856753}a^{37}-\frac{2616914156}{1580345570259}a^{36}-\frac{177910588}{175593952251}a^{35}-\frac{317491612}{1580345570259}a^{34}+\frac{323602846}{526781856753}a^{33}-\frac{9655450351}{1580345570259}a^{32}+\frac{362020379}{175593952251}a^{31}-\frac{7237471004}{1580345570259}a^{30}-\frac{453836723}{175593952251}a^{29}+\frac{9198552695}{1580345570259}a^{28}-\frac{6221866406}{526781856753}a^{27}+\frac{17345101016}{1580345570259}a^{26}-\frac{1045258516}{58531317417}a^{25}-\frac{77161472138}{1580345570259}a^{24}-\frac{3902431168}{175593952251}a^{23}+\frac{81804281183}{1580345570259}a^{22}-\frac{25088499452}{526781856753}a^{21}-\frac{2514786544}{1580345570259}a^{20}-\frac{8120663696}{175593952251}a^{19}+\frac{63134543980}{1580345570259}a^{18}-\frac{15106183372}{175593952251}a^{17}+\frac{237648091067}{1580345570259}a^{16}+\frac{38863914568}{526781856753}a^{15}-\frac{170217014620}{1580345570259}a^{14}+\frac{2513354132}{19510439139}a^{13}+\frac{62594641336}{1580345570259}a^{12}-\frac{3553395356}{58531317417}a^{11}-\frac{122204911028}{1580345570259}a^{10}-\frac{66693735890}{526781856753}a^{9}-\frac{163850087450}{1580345570259}a^{8}+\frac{218103691661}{526781856753}a^{7}-\frac{62540957261}{175593952251}a^{6}+\frac{3441124922}{58531317417}a^{5}+\frac{3388977910}{19510439139}a^{4}-\frac{431101589}{2167826571}a^{3}-\frac{23859254}{240869619}a^{2}+\frac{4923838}{26763291}a-\frac{936389}{2973699}$, $\frac{1}{26\!\cdots\!51}a^{41}-\frac{62\!\cdots\!56}{29\!\cdots\!39}a^{40}+\frac{87\!\cdots\!17}{26\!\cdots\!51}a^{39}-\frac{10\!\cdots\!51}{89\!\cdots\!17}a^{38}-\frac{14\!\cdots\!04}{26\!\cdots\!51}a^{37}-\frac{79\!\cdots\!38}{29\!\cdots\!39}a^{36}-\frac{16\!\cdots\!52}{26\!\cdots\!51}a^{35}+\frac{16\!\cdots\!94}{89\!\cdots\!17}a^{34}+\frac{30\!\cdots\!94}{26\!\cdots\!51}a^{33}-\frac{35\!\cdots\!32}{99\!\cdots\!13}a^{32}+\frac{33\!\cdots\!10}{26\!\cdots\!51}a^{31}-\frac{23\!\cdots\!89}{33\!\cdots\!71}a^{30}+\frac{11\!\cdots\!73}{26\!\cdots\!51}a^{29}-\frac{26\!\cdots\!08}{89\!\cdots\!17}a^{28}+\frac{96\!\cdots\!75}{26\!\cdots\!51}a^{27}-\frac{40\!\cdots\!91}{29\!\cdots\!39}a^{26}+\frac{11\!\cdots\!98}{26\!\cdots\!51}a^{25}+\frac{88\!\cdots\!91}{29\!\cdots\!39}a^{24}+\frac{37\!\cdots\!57}{26\!\cdots\!51}a^{23}-\frac{67\!\cdots\!01}{89\!\cdots\!17}a^{22}-\frac{54\!\cdots\!35}{26\!\cdots\!51}a^{21}+\frac{24\!\cdots\!44}{29\!\cdots\!39}a^{20}-\frac{13\!\cdots\!45}{26\!\cdots\!51}a^{19}+\frac{81\!\cdots\!56}{29\!\cdots\!39}a^{18}-\frac{39\!\cdots\!61}{26\!\cdots\!51}a^{17}+\frac{30\!\cdots\!48}{89\!\cdots\!17}a^{16}-\frac{39\!\cdots\!13}{26\!\cdots\!51}a^{15}+\frac{16\!\cdots\!25}{29\!\cdots\!39}a^{14}+\frac{13\!\cdots\!58}{26\!\cdots\!51}a^{13}-\frac{23\!\cdots\!90}{29\!\cdots\!39}a^{12}-\frac{11\!\cdots\!25}{26\!\cdots\!51}a^{11}-\frac{89\!\cdots\!18}{89\!\cdots\!17}a^{10}+\frac{14\!\cdots\!01}{26\!\cdots\!51}a^{9}-\frac{44\!\cdots\!75}{89\!\cdots\!17}a^{8}+\frac{97\!\cdots\!84}{29\!\cdots\!39}a^{7}+\frac{48\!\cdots\!03}{99\!\cdots\!13}a^{6}-\frac{12\!\cdots\!25}{33\!\cdots\!71}a^{5}-\frac{40\!\cdots\!29}{40\!\cdots\!91}a^{4}-\frac{18\!\cdots\!59}{45\!\cdots\!99}a^{3}-\frac{67\!\cdots\!48}{16\!\cdots\!37}a^{2}+\frac{79\!\cdots\!28}{16\!\cdots\!37}a+\frac{31\!\cdots\!25}{55\!\cdots\!79}$, $\frac{1}{24\!\cdots\!59}a^{42}+\frac{51\!\cdots\!00}{24\!\cdots\!59}a^{40}-\frac{16\!\cdots\!93}{80\!\cdots\!53}a^{39}+\frac{28\!\cdots\!75}{24\!\cdots\!59}a^{38}-\frac{15\!\cdots\!36}{89\!\cdots\!17}a^{37}-\frac{46\!\cdots\!12}{24\!\cdots\!59}a^{36}-\frac{61\!\cdots\!22}{80\!\cdots\!53}a^{35}+\frac{10\!\cdots\!70}{24\!\cdots\!59}a^{34}-\frac{41\!\cdots\!35}{26\!\cdots\!51}a^{33}+\frac{16\!\cdots\!57}{24\!\cdots\!59}a^{32}+\frac{76\!\cdots\!64}{26\!\cdots\!51}a^{31}+\frac{12\!\cdots\!07}{24\!\cdots\!59}a^{30}-\frac{16\!\cdots\!28}{80\!\cdots\!53}a^{29}+\frac{56\!\cdots\!58}{24\!\cdots\!59}a^{28}+\frac{19\!\cdots\!41}{26\!\cdots\!51}a^{27}+\frac{20\!\cdots\!17}{24\!\cdots\!59}a^{26}+\frac{15\!\cdots\!84}{29\!\cdots\!39}a^{25}+\frac{59\!\cdots\!81}{24\!\cdots\!59}a^{24}+\frac{36\!\cdots\!54}{80\!\cdots\!53}a^{23}-\frac{36\!\cdots\!70}{20\!\cdots\!59}a^{22}-\frac{36\!\cdots\!05}{89\!\cdots\!17}a^{21}+\frac{17\!\cdots\!83}{24\!\cdots\!59}a^{20}-\frac{41\!\cdots\!56}{89\!\cdots\!17}a^{19}+\frac{45\!\cdots\!20}{24\!\cdots\!59}a^{18}-\frac{28\!\cdots\!50}{80\!\cdots\!53}a^{17}+\frac{10\!\cdots\!44}{24\!\cdots\!59}a^{16}-\frac{49\!\cdots\!42}{26\!\cdots\!51}a^{15}-\frac{19\!\cdots\!99}{24\!\cdots\!59}a^{14}-\frac{13\!\cdots\!99}{26\!\cdots\!51}a^{13}+\frac{33\!\cdots\!38}{24\!\cdots\!59}a^{12}-\frac{13\!\cdots\!46}{80\!\cdots\!53}a^{11}+\frac{33\!\cdots\!00}{24\!\cdots\!59}a^{10}-\frac{66\!\cdots\!06}{80\!\cdots\!53}a^{9}+\frac{49\!\cdots\!32}{26\!\cdots\!51}a^{8}-\frac{39\!\cdots\!09}{89\!\cdots\!17}a^{7}-\frac{16\!\cdots\!97}{29\!\cdots\!39}a^{6}-\frac{13\!\cdots\!41}{33\!\cdots\!71}a^{5}-\frac{99\!\cdots\!63}{36\!\cdots\!19}a^{4}-\frac{44\!\cdots\!35}{40\!\cdots\!91}a^{3}+\frac{10\!\cdots\!32}{45\!\cdots\!99}a^{2}+\frac{18\!\cdots\!21}{50\!\cdots\!11}a+\frac{24\!\cdots\!91}{20\!\cdots\!77}$, $\frac{1}{21\!\cdots\!31}a^{43}-\frac{26}{21\!\cdots\!31}a^{41}+\frac{49\!\cdots\!88}{72\!\cdots\!77}a^{40}+\frac{74\!\cdots\!48}{21\!\cdots\!31}a^{39}+\frac{77\!\cdots\!72}{80\!\cdots\!53}a^{38}-\frac{20\!\cdots\!93}{21\!\cdots\!31}a^{37}-\frac{48\!\cdots\!20}{72\!\cdots\!77}a^{36}+\frac{23\!\cdots\!56}{21\!\cdots\!31}a^{35}-\frac{21\!\cdots\!44}{24\!\cdots\!59}a^{34}+\frac{53\!\cdots\!47}{21\!\cdots\!31}a^{33}+\frac{18\!\cdots\!45}{24\!\cdots\!59}a^{32}-\frac{24\!\cdots\!25}{21\!\cdots\!31}a^{31}+\frac{28\!\cdots\!57}{72\!\cdots\!77}a^{30}-\frac{64\!\cdots\!62}{21\!\cdots\!31}a^{29}+\frac{12\!\cdots\!78}{24\!\cdots\!59}a^{28}-\frac{21\!\cdots\!73}{21\!\cdots\!31}a^{27}+\frac{59\!\cdots\!93}{26\!\cdots\!51}a^{26}-\frac{11\!\cdots\!41}{21\!\cdots\!31}a^{25}-\frac{30\!\cdots\!91}{72\!\cdots\!77}a^{24}-\frac{51\!\cdots\!43}{21\!\cdots\!31}a^{23}-\frac{69\!\cdots\!67}{80\!\cdots\!53}a^{22}+\frac{71\!\cdots\!58}{21\!\cdots\!31}a^{21}+\frac{29\!\cdots\!44}{80\!\cdots\!53}a^{20}-\frac{89\!\cdots\!15}{21\!\cdots\!31}a^{19}+\frac{19\!\cdots\!36}{72\!\cdots\!77}a^{18}-\frac{18\!\cdots\!72}{21\!\cdots\!31}a^{17}-\frac{95\!\cdots\!02}{24\!\cdots\!59}a^{16}-\frac{16\!\cdots\!06}{21\!\cdots\!31}a^{15}-\frac{16\!\cdots\!43}{24\!\cdots\!59}a^{14}+\frac{33\!\cdots\!45}{21\!\cdots\!31}a^{13}+\frac{58\!\cdots\!45}{72\!\cdots\!77}a^{12}+\frac{16\!\cdots\!88}{21\!\cdots\!31}a^{11}+\frac{70\!\cdots\!25}{72\!\cdots\!77}a^{10}-\frac{31\!\cdots\!31}{24\!\cdots\!59}a^{9}+\frac{21\!\cdots\!50}{80\!\cdots\!53}a^{8}+\frac{64\!\cdots\!17}{26\!\cdots\!51}a^{7}+\frac{35\!\cdots\!51}{29\!\cdots\!39}a^{6}+\frac{28\!\cdots\!83}{33\!\cdots\!71}a^{5}-\frac{13\!\cdots\!19}{36\!\cdots\!19}a^{4}-\frac{16\!\cdots\!50}{40\!\cdots\!91}a^{3}+\frac{67\!\cdots\!67}{15\!\cdots\!33}a^{2}+\frac{21\!\cdots\!40}{50\!\cdots\!11}a-\frac{26\!\cdots\!14}{55\!\cdots\!79}$ 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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:   \( -\frac{206453937129450474090235194780836103762465925883325361075495171416915649249475957274748320728649842170}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{43} - \frac{272786577700807540266756763941442590817242706823631293869101630768389530530892401209320707755881380082}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{42} - \frac{11300556066796447488541641333948298560229641314860899119897856649626454181569937613081162090473253516634}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{41} - \frac{42738792492235598883360868983776844349763476558626250714869492569324437020588692229970329770004650877832}{5859751792578364514994178235432098553024437114039787989338068634967118244697145461503065423091052071270806019043443} a^{40} - \frac{370188800569930822646064249526491528902587131731214423476789495353901068732369262370341433234544572259384}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{39} - \frac{688898771171632931815496195416883048425704193942280496038728755168341929879255281765051526378888331799904}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{38} - \frac{26223771595922832499504042698697219472375458744259078413795832724614927509380970744558955064580220823492437}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{37} - \frac{79680024212739045073644757510655824675039743936291923472794923971380004470461059749699692604375034315024100}{5859751792578364514994178235432098553024437114039787989338068634967118244697145461503065423091052071270806019043443} a^{36} - \frac{1033464764163788691795739792820739142898854513879312712962641067352077844027890480227615466424112959937763382}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{35} - \frac{292783526887822707763957347260964123892905245365960339761190572713572206377242991643942853918041757061342666}{651083532508707168332686470603566505891604123782198665482007626107457582744127273500340602565672452363422891004827} a^{34} - \frac{22253834083111143437542872847260775876344591594643327734274598855065836808957694532142476704845474179850536310}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{33} - \frac{1665793510752240929308867327179583350767537221325911217678839051444225311819825838208672388844465802966699918}{217027844169569056110895490201188835297201374594066221827335875369152527581375757833446867521890817454474297001609} a^{32} - \frac{325452106172239887812880394065246092664895884510967636155532110986279070475193102777279980938514570075464900650}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{31} - \frac{550541627457835008548367973723796970287776462618001055275439817252768885544832652531412855168981793104563846972}{5859751792578364514994178235432098553024437114039787989338068634967118244697145461503065423091052071270806019043443} a^{30} - \frac{2724106412090925434621175042124562638019574323026560969768208987531237625445977875698473142858059099461976139416}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{29} - \frac{1887755130796036146722382693761565599682857711357919561299395687478642341688626700989797880449308589089635195720}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{28} - \frac{20110751313038722315364122181108248714525921086966568555083879217364202290822812052785441431595616424958330168714}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{27} - \frac{17179037116589224438694530442706241529441836471402304499108045521928961860810669062916327744355782512726539580776}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{26} - \frac{131221709890062581623246389006086438882061844560388065858607644587175872360815988759512497286264872332660275412672}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{25} - \frac{360215069724809074372534669766587949915617670004030318951720807365124612084163953874414918611687662190825491448292}{5859751792578364514994178235432098553024437114039787989338068634967118244697145461503065423091052071270806019043443} a^{24} - \frac{651182413525847453868938376897308892879707724265360151666721130131848326103305137218177505929413944009752419570502}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{23} - \frac{699919690237890812738672605099437256133181267191569930595716229850536064482954131431466072100352841064997529510595}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{22} - \frac{1354771271651774708017190788139592823046325279122886455574405278074577884766457935514070608476531867966047509124336}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{21} - \frac{3560628487708851282441215566195567642995773347156130015462610297414906829670809292141889115524739024396081795760284}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{20} + \frac{583548991337176490444823415825656534078725263258313171908561656872690802688709150206277257632450993949495127508052}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{19} - \frac{47900888230918942392449931471764932259041606720681018644133035990434635749588087943479912550988316380543130431894002}{5859751792578364514994178235432098553024437114039787989338068634967118244697145461503065423091052071270806019043443} a^{18} + \frac{24062810438305830411272088906345664182503992259349905482739146619672959697665332549125505887306748074230559001243823}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{17} - \frac{60966473363594915983072121033882410424408255740459051707220960871859328158851033332408576873523818875119589509088946}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{16} + \frac{171526396314866662548383585582824194469291200298946981029242791349469531593705094188741334679286080916370921511600546}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{15} - \frac{204473157723115373320477416194427689308670686251863906810743728711863422866180923758459684686817377744454068677887046}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{14} + \frac{843664952191286023003404721017433950150329782423453818983847768278446313535415664279236995213511426337701918801067990}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{13} - \frac{1783256559671146031324766383319651913221963285315170843542632470703991489757640109594339081897961320772386599108337300}{5859751792578364514994178235432098553024437114039787989338068634967118244697145461503065423091052071270806019043443} a^{12} + \frac{2153871909893559041519316717646173228282093219323885681567679648899204133632937174987309481453628624735781761769651374}{17579255377735093544982534706296295659073311342119363968014205904901354734091436384509196269273156213812418057130329} a^{11} - \frac{4224111090604384436592822100908335027022842621451952648327022361868657569171054815966070278737606756584343469715189086}{5859751792578364514994178235432098553024437114039787989338068634967118244697145461503065423091052071270806019043443} a^{10} + \frac{402757730113455972123045597737228886107970736379589608401961665512515270869643376792066048991227742962553653608693672}{1953250597526121504998059411810699517674812371346595996446022878322372748232381820501021807697017357090268673014481} a^{9} - \frac{780942455910726303055027337633143583278367596666164730038161140623668445906609836370664943111439428923156518406651118}{651083532508707168332686470603566505891604123782198665482007626107457582744127273500340602565672452363422891004827} a^{8} + \frac{36706142004757585554976759801876409208413016251665108057902710812879773189827917629461897499634552828957639705783058}{217027844169569056110895490201188835297201374594066221827335875369152527581375757833446867521890817454474297001609} a^{7} - \frac{1638736578106999867993445727512937341184830641462435986587005754324899979822862801512399192108578254577570037336044}{893118700286292411978993786836168046490540636189572929330600310161121512680558674211715504205312005985490934163} a^{6} - \frac{126521432088888821821734057658239957596194182333172716785054485775169915405634554584436891476573971032403073137508}{893118700286292411978993786836168046490540636189572929330600310161121512680558674211715504205312005985490934163} a^{5} - \frac{236193652647733993977479019060575925675682205933102564766782852454661083399365928276369580600831676286402702732724}{99235411142921379108777087426240894054504515132174769925622256684569056964506519356857278245034667331721214907} a^{4} - \frac{579580959753465970889172939733471674220486077374728314080833283321652211443727136049748455937053605327678457734}{11026156793657931012086343047360099339389390570241641102846917409396561884945168817428586471670518592413468323} a^{3} - \frac{7540008425315503495341957830701049981693906530043755188952012432244383270192041392072319026080527916932170258560}{3675385597885977004028781015786699779796463523413880367615639136465520628315056272476195490556839530804489441} a^{2} - \frac{3488532072710772047971612865642027116755252075499125569899062643915462430971388547668894774079135833225356}{136125392514295444593658556140248139992461611978291865467245893943167430678335417499118351502105167807573683} a - \frac{25656771370299017815859574210949179137683599821878422184962812612016327836443480706628840950457818404374}{45375130838098481531219518713416046664153870659430621822415297981055810226111805833039450500701722602524561} \)  (order $10$) 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 + 55*x^42 - 33*x^41 + 2200*x^40 + 3861*x^39 + 81686*x^38 + 187473*x^37 + 2797949*x^36 + 5067315*x^35 + 48269287*x^34 + 80889588*x^33 + 627196361*x^32 + 562870110*x^31 + 6666501116*x^30 + 3165816357*x^29 + 61519492375*x^28 + 18154396755*x^27 + 435915893402*x^26 + 55327492110*x^25 + 2610992250944*x^24 - 522745688124*x^23 + 13465948440286*x^22 - 5051169113364*x^21 + 60785860077503*x^20 - 28124569272129*x^19 + 233899389744845*x^18 - 131136264490176*x^17 + 790178749679506*x^16 - 517434573601242*x^15 + 2274690119726596*x^14 - 1273453323335061*x^13 + 5309434517531353*x^12 - 2092304520835266*x^11 + 8720469353666763*x^10 - 2120075052140232*x^9 + 13102352080428996*x^8 - 132063952087980*x^7 + 17079801196779882*x^6 - 477544700475315*x^5 + 14647338263254752*x^4 + 323479726646175*x^3 + 7143904676997*x^2 + 157680139023*x + 3486784401)
 
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 + 55*x^42 - 33*x^41 + 2200*x^40 + 3861*x^39 + 81686*x^38 + 187473*x^37 + 2797949*x^36 + 5067315*x^35 + 48269287*x^34 + 80889588*x^33 + 627196361*x^32 + 562870110*x^31 + 6666501116*x^30 + 3165816357*x^29 + 61519492375*x^28 + 18154396755*x^27 + 435915893402*x^26 + 55327492110*x^25 + 2610992250944*x^24 - 522745688124*x^23 + 13465948440286*x^22 - 5051169113364*x^21 + 60785860077503*x^20 - 28124569272129*x^19 + 233899389744845*x^18 - 131136264490176*x^17 + 790178749679506*x^16 - 517434573601242*x^15 + 2274690119726596*x^14 - 1273453323335061*x^13 + 5309434517531353*x^12 - 2092304520835266*x^11 + 8720469353666763*x^10 - 2120075052140232*x^9 + 13102352080428996*x^8 - 132063952087980*x^7 + 17079801196779882*x^6 - 477544700475315*x^5 + 14647338263254752*x^4 + 323479726646175*x^3 + 7143904676997*x^2 + 157680139023*x + 3486784401, 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 + 55*x^42 - 33*x^41 + 2200*x^40 + 3861*x^39 + 81686*x^38 + 187473*x^37 + 2797949*x^36 + 5067315*x^35 + 48269287*x^34 + 80889588*x^33 + 627196361*x^32 + 562870110*x^31 + 6666501116*x^30 + 3165816357*x^29 + 61519492375*x^28 + 18154396755*x^27 + 435915893402*x^26 + 55327492110*x^25 + 2610992250944*x^24 - 522745688124*x^23 + 13465948440286*x^22 - 5051169113364*x^21 + 60785860077503*x^20 - 28124569272129*x^19 + 233899389744845*x^18 - 131136264490176*x^17 + 790178749679506*x^16 - 517434573601242*x^15 + 2274690119726596*x^14 - 1273453323335061*x^13 + 5309434517531353*x^12 - 2092304520835266*x^11 + 8720469353666763*x^10 - 2120075052140232*x^9 + 13102352080428996*x^8 - 132063952087980*x^7 + 17079801196779882*x^6 - 477544700475315*x^5 + 14647338263254752*x^4 + 323479726646175*x^3 + 7143904676997*x^2 + 157680139023*x + 3486784401);
 
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 + 55*x^42 - 33*x^41 + 2200*x^40 + 3861*x^39 + 81686*x^38 + 187473*x^37 + 2797949*x^36 + 5067315*x^35 + 48269287*x^34 + 80889588*x^33 + 627196361*x^32 + 562870110*x^31 + 6666501116*x^30 + 3165816357*x^29 + 61519492375*x^28 + 18154396755*x^27 + 435915893402*x^26 + 55327492110*x^25 + 2610992250944*x^24 - 522745688124*x^23 + 13465948440286*x^22 - 5051169113364*x^21 + 60785860077503*x^20 - 28124569272129*x^19 + 233899389744845*x^18 - 131136264490176*x^17 + 790178749679506*x^16 - 517434573601242*x^15 + 2274690119726596*x^14 - 1273453323335061*x^13 + 5309434517531353*x^12 - 2092304520835266*x^11 + 8720469353666763*x^10 - 2120075052140232*x^9 + 13102352080428996*x^8 - 132063952087980*x^7 + 17079801196779882*x^6 - 477544700475315*x^5 + 14647338263254752*x^4 + 323479726646175*x^3 + 7143904676997*x^2 + 157680139023*x + 3486784401);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 11.11.672749994932560009201.1, 22.22.22099245882898413967719412126414511946210986328125.1

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 $44$ ${\href{/padicField/3.4.0.1}{4} }^{11}$ R $44$ R $44$ $44$ $22^{2}$ $44$ $22^{2}$ ${\href{/padicField/31.11.0.1}{11} }^{4}$ $44$ ${\href{/padicField/41.11.0.1}{11} }^{4}$ $44$ $44$ $44$ $22^{2}$

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
\(5\) Copy content Toggle raw display Deg $44$$4$$11$$33$
\(11\) Copy content Toggle raw display 11.11.20.9$x^{11} + 110 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$
11.11.20.9$x^{11} + 110 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$
11.11.20.9$x^{11} + 110 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$
11.11.20.9$x^{11} + 110 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$