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Properties

Label 44.44.134...125.1
Degree $44$
Signature $[44, 0]$
Discriminant $1.341\times 10^{96}$
Root discriminant \(153.01\)
Ramified primes $5,7,23$
Class number not computed
Class group not computed
Galois group $C_{44}$ (as 44T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - 7*x^43 - 121*x^42 + 961*x^41 + 6141*x^40 - 58706*x^39 - 162828*x^38 + 2110262*x^37 + 2030280*x^36 - 49737458*x^35 + 7249749*x^34 + 810370867*x^33 - 751544361*x^32 - 9363190013*x^31 + 14629550053*x^30 + 77186286766*x^29 - 168007364622*x^28 - 446721814624*x^27 + 1308412141053*x^26 + 1708214965012*x^25 - 7207408221864*x^24 - 3395585375747*x^23 + 28338195475365*x^22 - 3195446926852*x^21 - 78521093195796*x^20 + 45568931299761*x^19 + 147217011020842*x^18 - 153469537128468*x^17 - 167684548055793*x^16 + 290016628457561*x^15 + 71424111769606*x^14 - 329051920771801*x^13 + 81499856091714*x^12 + 207300120944285*x^11 - 138850152957509*x^10 - 47750321297779*x^9 + 78243526539014*x^8 - 15803724914334*x^7 - 14683750874453*x^6 + 9013617025194*x^5 - 1172684662984*x^4 - 507894298828*x^3 + 217936624208*x^2 - 31938085617*x + 1699532101)
 
gp: K = bnfinit(y^44 - 7*y^43 - 121*y^42 + 961*y^41 + 6141*y^40 - 58706*y^39 - 162828*y^38 + 2110262*y^37 + 2030280*y^36 - 49737458*y^35 + 7249749*y^34 + 810370867*y^33 - 751544361*y^32 - 9363190013*y^31 + 14629550053*y^30 + 77186286766*y^29 - 168007364622*y^28 - 446721814624*y^27 + 1308412141053*y^26 + 1708214965012*y^25 - 7207408221864*y^24 - 3395585375747*y^23 + 28338195475365*y^22 - 3195446926852*y^21 - 78521093195796*y^20 + 45568931299761*y^19 + 147217011020842*y^18 - 153469537128468*y^17 - 167684548055793*y^16 + 290016628457561*y^15 + 71424111769606*y^14 - 329051920771801*y^13 + 81499856091714*y^12 + 207300120944285*y^11 - 138850152957509*y^10 - 47750321297779*y^9 + 78243526539014*y^8 - 15803724914334*y^7 - 14683750874453*y^6 + 9013617025194*y^5 - 1172684662984*y^4 - 507894298828*y^3 + 217936624208*y^2 - 31938085617*y + 1699532101, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - 7*x^43 - 121*x^42 + 961*x^41 + 6141*x^40 - 58706*x^39 - 162828*x^38 + 2110262*x^37 + 2030280*x^36 - 49737458*x^35 + 7249749*x^34 + 810370867*x^33 - 751544361*x^32 - 9363190013*x^31 + 14629550053*x^30 + 77186286766*x^29 - 168007364622*x^28 - 446721814624*x^27 + 1308412141053*x^26 + 1708214965012*x^25 - 7207408221864*x^24 - 3395585375747*x^23 + 28338195475365*x^22 - 3195446926852*x^21 - 78521093195796*x^20 + 45568931299761*x^19 + 147217011020842*x^18 - 153469537128468*x^17 - 167684548055793*x^16 + 290016628457561*x^15 + 71424111769606*x^14 - 329051920771801*x^13 + 81499856091714*x^12 + 207300120944285*x^11 - 138850152957509*x^10 - 47750321297779*x^9 + 78243526539014*x^8 - 15803724914334*x^7 - 14683750874453*x^6 + 9013617025194*x^5 - 1172684662984*x^4 - 507894298828*x^3 + 217936624208*x^2 - 31938085617*x + 1699532101);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - 7*x^43 - 121*x^42 + 961*x^41 + 6141*x^40 - 58706*x^39 - 162828*x^38 + 2110262*x^37 + 2030280*x^36 - 49737458*x^35 + 7249749*x^34 + 810370867*x^33 - 751544361*x^32 - 9363190013*x^31 + 14629550053*x^30 + 77186286766*x^29 - 168007364622*x^28 - 446721814624*x^27 + 1308412141053*x^26 + 1708214965012*x^25 - 7207408221864*x^24 - 3395585375747*x^23 + 28338195475365*x^22 - 3195446926852*x^21 - 78521093195796*x^20 + 45568931299761*x^19 + 147217011020842*x^18 - 153469537128468*x^17 - 167684548055793*x^16 + 290016628457561*x^15 + 71424111769606*x^14 - 329051920771801*x^13 + 81499856091714*x^12 + 207300120944285*x^11 - 138850152957509*x^10 - 47750321297779*x^9 + 78243526539014*x^8 - 15803724914334*x^7 - 14683750874453*x^6 + 9013617025194*x^5 - 1172684662984*x^4 - 507894298828*x^3 + 217936624208*x^2 - 31938085617*x + 1699532101)
 

\( x^{44} - 7 x^{43} - 121 x^{42} + 961 x^{41} + 6141 x^{40} - 58706 x^{39} - 162828 x^{38} + \cdots + 1699532101 \) Copy content Toggle raw display

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(134\!\cdots\!125\) \(\medspace = 5^{33}\cdot 7^{22}\cdot 23^{40}\) 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:  \(153.01\)
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:  $5^{3/4}7^{1/2}23^{10/11}\approx 153.0068659784276$
Ramified primes:   \(5\), \(7\), \(23\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $44$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(805=5\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{805}(1,·)$, $\chi_{805}(386,·)$, $\chi_{805}(771,·)$, $\chi_{805}(519,·)$, $\chi_{805}(363,·)$, $\chi_{805}(13,·)$, $\chi_{805}(783,·)$, $\chi_{805}(657,·)$, $\chi_{805}(538,·)$, $\chi_{805}(27,·)$, $\chi_{805}(29,·)$, $\chi_{805}(36,·)$, $\chi_{805}(167,·)$, $\chi_{805}(169,·)$, $\chi_{805}(554,·)$, $\chi_{805}(48,·)$, $\chi_{805}(561,·)$, $\chi_{805}(307,·)$, $\chi_{805}(692,·)$, $\chi_{805}(694,·)$, $\chi_{805}(351,·)$, $\chi_{805}(188,·)$, $\chi_{805}(62,·)$, $\chi_{805}(64,·)$, $\chi_{805}(449,·)$, $\chi_{805}(118,·)$, $\chi_{805}(71,·)$, $\chi_{805}(328,·)$, $\chi_{805}(202,·)$, $\chi_{805}(587,·)$, $\chi_{805}(141,·)$, $\chi_{805}(211,·)$, $\chi_{805}(468,·)$, $\chi_{805}(729,·)$, $\chi_{805}(223,·)$, $\chi_{805}(484,·)$, $\chi_{805}(491,·)$, $\chi_{805}(748,·)$, $\chi_{805}(622,·)$, $\chi_{805}(239,·)$, $\chi_{805}(624,·)$, $\chi_{805}(246,·)$, $\chi_{805}(377,·)$, $\chi_{805}(762,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $\frac{1}{229}a^{38}+\frac{74}{229}a^{37}-\frac{52}{229}a^{36}-\frac{47}{229}a^{35}-\frac{14}{229}a^{34}-\frac{95}{229}a^{33}-\frac{45}{229}a^{32}+\frac{32}{229}a^{31}+\frac{6}{229}a^{30}-\frac{3}{229}a^{29}-\frac{62}{229}a^{28}+\frac{82}{229}a^{27}-\frac{65}{229}a^{26}+\frac{72}{229}a^{25}+\frac{23}{229}a^{24}-\frac{75}{229}a^{23}+\frac{71}{229}a^{22}+\frac{61}{229}a^{21}+\frac{94}{229}a^{20}+\frac{7}{229}a^{19}+\frac{77}{229}a^{18}-\frac{53}{229}a^{17}+\frac{36}{229}a^{16}-\frac{76}{229}a^{15}+\frac{87}{229}a^{14}+\frac{60}{229}a^{13}+\frac{52}{229}a^{12}-\frac{43}{229}a^{11}-\frac{48}{229}a^{10}+\frac{48}{229}a^{9}-\frac{29}{229}a^{8}-\frac{15}{229}a^{7}+\frac{1}{229}a^{6}+\frac{82}{229}a^{5}+\frac{91}{229}a^{4}-\frac{75}{229}a^{3}+\frac{54}{229}a^{2}+\frac{2}{229}a+\frac{62}{229}$, $\frac{1}{31831}a^{39}+\frac{2}{31831}a^{38}-\frac{11563}{31831}a^{37}-\frac{8211}{31831}a^{36}+\frac{1538}{31831}a^{35}-\frac{7789}{31831}a^{34}+\frac{6795}{31831}a^{33}+\frac{10600}{31831}a^{32}+\frac{2740}{31831}a^{31}+\frac{11473}{31831}a^{30}-\frac{12670}{31831}a^{29}+\frac{2714}{31831}a^{28}-\frac{7572}{31831}a^{27}-\frac{6469}{31831}a^{26}-\frac{10657}{31831}a^{25}-\frac{8372}{31831}a^{24}+\frac{9135}{31831}a^{23}+\frac{8002}{31831}a^{22}-\frac{11626}{31831}a^{21}-\frac{12486}{31831}a^{20}-\frac{7068}{31831}a^{19}+\frac{586}{31831}a^{18}+\frac{9119}{31831}a^{17}+\frac{14507}{31831}a^{16}+\frac{521}{31831}a^{15}-\frac{13074}{31831}a^{14}+\frac{14281}{31831}a^{13}+\frac{2854}{31831}a^{12}+\frac{4422}{31831}a^{11}+\frac{14267}{31831}a^{10}+\frac{11629}{31831}a^{9}+\frac{9859}{31831}a^{8}+\frac{4516}{31831}a^{7}-\frac{9379}{31831}a^{6}-\frac{10393}{31831}a^{5}+\frac{243}{31831}a^{4}+\frac{12782}{31831}a^{3}-\frac{13962}{31831}a^{2}+\frac{605}{31831}a+\frac{1490}{31831}$, $\frac{1}{31831}a^{40}-\frac{30}{31831}a^{38}+\frac{9216}{31831}a^{37}-\frac{9006}{31831}a^{36}-\frac{11977}{31831}a^{35}-\frac{11821}{31831}a^{34}+\frac{15080}{31831}a^{33}+\frac{3502}{31831}a^{32}-\frac{6795}{31831}a^{31}+\frac{1775}{31831}a^{30}-\frac{6557}{31831}a^{29}+\frac{3819}{31831}a^{28}-\frac{221}{31831}a^{27}-\frac{15511}{31831}a^{26}-\frac{15831}{31831}a^{25}+\frac{4751}{31831}a^{24}+\frac{15725}{31831}a^{23}-\frac{4278}{31831}a^{22}+\frac{14241}{31831}a^{21}-\frac{11703}{31831}a^{20}-\frac{12}{31831}a^{19}+\frac{5028}{31831}a^{18}-\frac{10403}{31831}a^{17}+\frac{4867}{31831}a^{16}+\frac{340}{31831}a^{15}-\frac{6275}{31831}a^{14}-\frac{1939}{31831}a^{13}-\frac{6151}{31831}a^{12}-\frac{13203}{31831}a^{11}+\frac{2277}{31831}a^{10}-\frac{750}{31831}a^{9}+\frac{366}{31831}a^{8}-\frac{480}{31831}a^{7}-\frac{11929}{31831}a^{6}+\frac{12133}{31831}a^{5}+\frac{11740}{31831}a^{4}-\frac{13533}{31831}a^{3}+\frac{14907}{31831}a^{2}-\frac{8477}{31831}a+\frac{12032}{31831}$, $\frac{1}{31831}a^{41}-\frac{37}{31831}a^{38}+\frac{5365}{31831}a^{37}+\frac{3152}{31831}a^{36}-\frac{5435}{31831}a^{35}+\frac{7285}{31831}a^{34}+\frac{9833}{31831}a^{33}-\frac{1823}{31831}a^{32}+\frac{8776}{31831}a^{31}-\frac{4724}{31831}a^{30}+\frac{1799}{31831}a^{29}-\frac{9846}{31831}a^{28}+\frac{12255}{31831}a^{27}+\frac{13472}{31831}a^{26}+\frac{1266}{31831}a^{25}-\frac{4000}{31831}a^{24}+\frac{13317}{31831}a^{23}+\frac{6881}{31831}a^{22}-\frac{5477}{31831}a^{21}-\frac{8605}{31831}a^{20}+\frac{14276}{31831}a^{19}-\frac{9642}{31831}a^{18}+\frac{8082}{31831}a^{17}+\frac{4789}{31831}a^{16}-\frac{14970}{31831}a^{15}+\frac{5188}{31831}a^{14}-\frac{9177}{31831}a^{13}+\frac{1944}{31831}a^{12}-\frac{5731}{31831}a^{11}+\frac{14847}{31831}a^{10}-\frac{2295}{31831}a^{9}-\frac{7591}{31831}a^{8}+\frac{8598}{31831}a^{7}+\frac{7929}{31831}a^{6}-\frac{13293}{31831}a^{5}+\frac{5711}{31831}a^{4}+\frac{14588}{31831}a^{3}-\frac{7140}{31831}a^{2}+\frac{11556}{31831}a+\frac{8421}{31831}$, $\frac{1}{43958611}a^{42}+\frac{263}{43958611}a^{41}-\frac{39}{43958611}a^{40}+\frac{582}{43958611}a^{39}-\frac{68678}{43958611}a^{38}+\frac{5130094}{43958611}a^{37}+\frac{1439635}{43958611}a^{36}-\frac{7976174}{43958611}a^{35}+\frac{16674997}{43958611}a^{34}+\frac{6174441}{43958611}a^{33}-\frac{11638108}{43958611}a^{32}+\frac{15975874}{43958611}a^{31}+\frac{13444872}{43958611}a^{30}-\frac{20419900}{43958611}a^{29}+\frac{16618597}{43958611}a^{28}-\frac{21873522}{43958611}a^{27}+\frac{8715343}{43958611}a^{26}-\frac{10104015}{43958611}a^{25}-\frac{14561648}{43958611}a^{24}-\frac{16780401}{43958611}a^{23}+\frac{2632820}{43958611}a^{22}-\frac{2989456}{43958611}a^{21}+\frac{17406797}{43958611}a^{20}-\frac{46295}{43958611}a^{19}+\frac{1318625}{43958611}a^{18}+\frac{20725066}{43958611}a^{17}+\frac{10433765}{43958611}a^{16}-\frac{1384922}{43958611}a^{15}+\frac{6100864}{43958611}a^{14}+\frac{21663015}{43958611}a^{13}+\frac{20146569}{43958611}a^{12}-\frac{13451319}{43958611}a^{11}-\frac{18307167}{43958611}a^{10}-\frac{5912618}{43958611}a^{9}+\frac{21600851}{43958611}a^{8}-\frac{8335316}{43958611}a^{7}+\frac{17577911}{43958611}a^{6}+\frac{13830481}{43958611}a^{5}+\frac{21207207}{43958611}a^{4}-\frac{17613427}{43958611}a^{3}+\frac{69251}{316249}a^{2}-\frac{1728301}{43958611}a+\frac{3846922}{43958611}$, $\frac{1}{76\!\cdots\!39}a^{43}+\frac{39\!\cdots\!24}{76\!\cdots\!39}a^{42}-\frac{69\!\cdots\!19}{76\!\cdots\!39}a^{41}-\frac{10\!\cdots\!52}{76\!\cdots\!39}a^{40}+\frac{63\!\cdots\!06}{76\!\cdots\!39}a^{39}+\frac{18\!\cdots\!10}{76\!\cdots\!39}a^{38}-\frac{27\!\cdots\!53}{76\!\cdots\!39}a^{37}-\frac{14\!\cdots\!60}{76\!\cdots\!39}a^{36}+\frac{95\!\cdots\!62}{76\!\cdots\!39}a^{35}-\frac{26\!\cdots\!22}{76\!\cdots\!39}a^{34}-\frac{37\!\cdots\!50}{76\!\cdots\!39}a^{33}+\frac{93\!\cdots\!32}{76\!\cdots\!39}a^{32}-\frac{30\!\cdots\!28}{76\!\cdots\!39}a^{31}-\frac{31\!\cdots\!75}{76\!\cdots\!39}a^{30}+\frac{62\!\cdots\!22}{76\!\cdots\!39}a^{29}+\frac{21\!\cdots\!27}{76\!\cdots\!39}a^{28}+\frac{22\!\cdots\!87}{76\!\cdots\!39}a^{27}-\frac{83\!\cdots\!35}{76\!\cdots\!39}a^{26}-\frac{27\!\cdots\!99}{76\!\cdots\!39}a^{25}+\frac{24\!\cdots\!43}{76\!\cdots\!39}a^{24}-\frac{26\!\cdots\!11}{76\!\cdots\!39}a^{23}+\frac{14\!\cdots\!98}{76\!\cdots\!39}a^{22}+\frac{59\!\cdots\!96}{76\!\cdots\!39}a^{21}+\frac{10\!\cdots\!08}{76\!\cdots\!39}a^{20}-\frac{19\!\cdots\!46}{76\!\cdots\!39}a^{19}+\frac{31\!\cdots\!35}{76\!\cdots\!39}a^{18}+\frac{15\!\cdots\!46}{76\!\cdots\!39}a^{17}+\frac{23\!\cdots\!82}{76\!\cdots\!39}a^{16}-\frac{29\!\cdots\!86}{76\!\cdots\!39}a^{15}-\frac{13\!\cdots\!11}{76\!\cdots\!39}a^{14}-\frac{14\!\cdots\!07}{76\!\cdots\!39}a^{13}-\frac{32\!\cdots\!16}{76\!\cdots\!39}a^{12}+\frac{18\!\cdots\!60}{76\!\cdots\!39}a^{11}-\frac{20\!\cdots\!48}{76\!\cdots\!39}a^{10}-\frac{10\!\cdots\!98}{76\!\cdots\!39}a^{9}-\frac{92\!\cdots\!75}{76\!\cdots\!39}a^{8}+\frac{36\!\cdots\!47}{76\!\cdots\!39}a^{7}+\frac{12\!\cdots\!31}{76\!\cdots\!39}a^{6}+\frac{16\!\cdots\!38}{76\!\cdots\!39}a^{5}+\frac{37\!\cdots\!01}{76\!\cdots\!39}a^{4}+\frac{11\!\cdots\!80}{76\!\cdots\!39}a^{3}+\frac{16\!\cdots\!56}{76\!\cdots\!39}a^{2}+\frac{15\!\cdots\!33}{76\!\cdots\!39}a+\frac{23\!\cdots\!07}{76\!\cdots\!39}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $43$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

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

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 - 7*x^43 - 121*x^42 + 961*x^41 + 6141*x^40 - 58706*x^39 - 162828*x^38 + 2110262*x^37 + 2030280*x^36 - 49737458*x^35 + 7249749*x^34 + 810370867*x^33 - 751544361*x^32 - 9363190013*x^31 + 14629550053*x^30 + 77186286766*x^29 - 168007364622*x^28 - 446721814624*x^27 + 1308412141053*x^26 + 1708214965012*x^25 - 7207408221864*x^24 - 3395585375747*x^23 + 28338195475365*x^22 - 3195446926852*x^21 - 78521093195796*x^20 + 45568931299761*x^19 + 147217011020842*x^18 - 153469537128468*x^17 - 167684548055793*x^16 + 290016628457561*x^15 + 71424111769606*x^14 - 329051920771801*x^13 + 81499856091714*x^12 + 207300120944285*x^11 - 138850152957509*x^10 - 47750321297779*x^9 + 78243526539014*x^8 - 15803724914334*x^7 - 14683750874453*x^6 + 9013617025194*x^5 - 1172684662984*x^4 - 507894298828*x^3 + 217936624208*x^2 - 31938085617*x + 1699532101)
 
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 - 7*x^43 - 121*x^42 + 961*x^41 + 6141*x^40 - 58706*x^39 - 162828*x^38 + 2110262*x^37 + 2030280*x^36 - 49737458*x^35 + 7249749*x^34 + 810370867*x^33 - 751544361*x^32 - 9363190013*x^31 + 14629550053*x^30 + 77186286766*x^29 - 168007364622*x^28 - 446721814624*x^27 + 1308412141053*x^26 + 1708214965012*x^25 - 7207408221864*x^24 - 3395585375747*x^23 + 28338195475365*x^22 - 3195446926852*x^21 - 78521093195796*x^20 + 45568931299761*x^19 + 147217011020842*x^18 - 153469537128468*x^17 - 167684548055793*x^16 + 290016628457561*x^15 + 71424111769606*x^14 - 329051920771801*x^13 + 81499856091714*x^12 + 207300120944285*x^11 - 138850152957509*x^10 - 47750321297779*x^9 + 78243526539014*x^8 - 15803724914334*x^7 - 14683750874453*x^6 + 9013617025194*x^5 - 1172684662984*x^4 - 507894298828*x^3 + 217936624208*x^2 - 31938085617*x + 1699532101, 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 - 7*x^43 - 121*x^42 + 961*x^41 + 6141*x^40 - 58706*x^39 - 162828*x^38 + 2110262*x^37 + 2030280*x^36 - 49737458*x^35 + 7249749*x^34 + 810370867*x^33 - 751544361*x^32 - 9363190013*x^31 + 14629550053*x^30 + 77186286766*x^29 - 168007364622*x^28 - 446721814624*x^27 + 1308412141053*x^26 + 1708214965012*x^25 - 7207408221864*x^24 - 3395585375747*x^23 + 28338195475365*x^22 - 3195446926852*x^21 - 78521093195796*x^20 + 45568931299761*x^19 + 147217011020842*x^18 - 153469537128468*x^17 - 167684548055793*x^16 + 290016628457561*x^15 + 71424111769606*x^14 - 329051920771801*x^13 + 81499856091714*x^12 + 207300120944285*x^11 - 138850152957509*x^10 - 47750321297779*x^9 + 78243526539014*x^8 - 15803724914334*x^7 - 14683750874453*x^6 + 9013617025194*x^5 - 1172684662984*x^4 - 507894298828*x^3 + 217936624208*x^2 - 31938085617*x + 1699532101);
 
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 - 7*x^43 - 121*x^42 + 961*x^41 + 6141*x^40 - 58706*x^39 - 162828*x^38 + 2110262*x^37 + 2030280*x^36 - 49737458*x^35 + 7249749*x^34 + 810370867*x^33 - 751544361*x^32 - 9363190013*x^31 + 14629550053*x^30 + 77186286766*x^29 - 168007364622*x^28 - 446721814624*x^27 + 1308412141053*x^26 + 1708214965012*x^25 - 7207408221864*x^24 - 3395585375747*x^23 + 28338195475365*x^22 - 3195446926852*x^21 - 78521093195796*x^20 + 45568931299761*x^19 + 147217011020842*x^18 - 153469537128468*x^17 - 167684548055793*x^16 + 290016628457561*x^15 + 71424111769606*x^14 - 329051920771801*x^13 + 81499856091714*x^12 + 207300120944285*x^11 - 138850152957509*x^10 - 47750321297779*x^9 + 78243526539014*x^8 - 15803724914334*x^7 - 14683750874453*x^6 + 9013617025194*x^5 - 1172684662984*x^4 - 507894298828*x^3 + 217936624208*x^2 - 31938085617*x + 1699532101);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.6125.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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 $44$ $44$ R R ${\href{/padicField/11.11.0.1}{11} }^{4}$ $44$ $44$ ${\href{/padicField/19.11.0.1}{11} }^{4}$ R $22^{2}$ $22^{2}$ $44$ $22^{2}$ $44$ ${\href{/padicField/47.4.0.1}{4} }^{11}$ $44$ ${\href{/padicField/59.11.0.1}{11} }^{4}$

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$
\(7\) Copy content Toggle raw display Deg $44$$2$$22$$22$
\(23\) Copy content Toggle raw display Deg $44$$11$$4$$40$