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Properties

Label 44.44.896...237.1
Degree $44$
Signature $[44, 0]$
Discriminant $8.968\times 10^{93}$
Root discriminant \(136.55\)
Ramified primes $13,23$
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 - 111*x^42 + 108*x^41 + 5516*x^40 - 5192*x^39 - 162490*x^38 + 146914*x^37 + 3169793*x^36 - 2729051*x^35 - 43378560*x^34 + 35191407*x^33 + 430978480*x^32 - 325404259*x^31 - 3178214046*x^30 + 2202001269*x^29 + 17658050940*x^28 - 11052047133*x^27 - 74656059512*x^26 + 41499918113*x^25 + 241605668931*x^24 - 117105914592*x^23 - 599724612955*x^22 + 248374887128*x^21 + 1139635785934*x^20 - 393998255255*x^19 - 1647609787093*x^18 + 462039171768*x^17 + 1792199478453*x^16 - 391870900134*x^15 - 1441857227832*x^14 + 230783287602*x^13 + 836726691516*x^12 - 86686890110*x^11 - 337712573557*x^10 + 16137410915*x^9 + 89760954010*x^8 + 763059160*x^7 - 14382578320*x^6 - 914750852*x^5 + 1175923775*x^4 + 123599834*x^3 - 31889819*x^2 - 2291321*x + 332627)
 
gp: K = bnfinit(y^44 - y^43 - 111*y^42 + 108*y^41 + 5516*y^40 - 5192*y^39 - 162490*y^38 + 146914*y^37 + 3169793*y^36 - 2729051*y^35 - 43378560*y^34 + 35191407*y^33 + 430978480*y^32 - 325404259*y^31 - 3178214046*y^30 + 2202001269*y^29 + 17658050940*y^28 - 11052047133*y^27 - 74656059512*y^26 + 41499918113*y^25 + 241605668931*y^24 - 117105914592*y^23 - 599724612955*y^22 + 248374887128*y^21 + 1139635785934*y^20 - 393998255255*y^19 - 1647609787093*y^18 + 462039171768*y^17 + 1792199478453*y^16 - 391870900134*y^15 - 1441857227832*y^14 + 230783287602*y^13 + 836726691516*y^12 - 86686890110*y^11 - 337712573557*y^10 + 16137410915*y^9 + 89760954010*y^8 + 763059160*y^7 - 14382578320*y^6 - 914750852*y^5 + 1175923775*y^4 + 123599834*y^3 - 31889819*y^2 - 2291321*y + 332627, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 111*x^42 + 108*x^41 + 5516*x^40 - 5192*x^39 - 162490*x^38 + 146914*x^37 + 3169793*x^36 - 2729051*x^35 - 43378560*x^34 + 35191407*x^33 + 430978480*x^32 - 325404259*x^31 - 3178214046*x^30 + 2202001269*x^29 + 17658050940*x^28 - 11052047133*x^27 - 74656059512*x^26 + 41499918113*x^25 + 241605668931*x^24 - 117105914592*x^23 - 599724612955*x^22 + 248374887128*x^21 + 1139635785934*x^20 - 393998255255*x^19 - 1647609787093*x^18 + 462039171768*x^17 + 1792199478453*x^16 - 391870900134*x^15 - 1441857227832*x^14 + 230783287602*x^13 + 836726691516*x^12 - 86686890110*x^11 - 337712573557*x^10 + 16137410915*x^9 + 89760954010*x^8 + 763059160*x^7 - 14382578320*x^6 - 914750852*x^5 + 1175923775*x^4 + 123599834*x^3 - 31889819*x^2 - 2291321*x + 332627);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 111*x^42 + 108*x^41 + 5516*x^40 - 5192*x^39 - 162490*x^38 + 146914*x^37 + 3169793*x^36 - 2729051*x^35 - 43378560*x^34 + 35191407*x^33 + 430978480*x^32 - 325404259*x^31 - 3178214046*x^30 + 2202001269*x^29 + 17658050940*x^28 - 11052047133*x^27 - 74656059512*x^26 + 41499918113*x^25 + 241605668931*x^24 - 117105914592*x^23 - 599724612955*x^22 + 248374887128*x^21 + 1139635785934*x^20 - 393998255255*x^19 - 1647609787093*x^18 + 462039171768*x^17 + 1792199478453*x^16 - 391870900134*x^15 - 1441857227832*x^14 + 230783287602*x^13 + 836726691516*x^12 - 86686890110*x^11 - 337712573557*x^10 + 16137410915*x^9 + 89760954010*x^8 + 763059160*x^7 - 14382578320*x^6 - 914750852*x^5 + 1175923775*x^4 + 123599834*x^3 - 31889819*x^2 - 2291321*x + 332627)
 

\( x^{44} - x^{43} - 111 x^{42} + 108 x^{41} + 5516 x^{40} - 5192 x^{39} - 162490 x^{38} + 146914 x^{37} + \cdots + 332627 \) 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:   \(896\!\cdots\!237\) \(\medspace = 13^{33}\cdot 23^{42}\) 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:  \(136.55\)
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:  $13^{3/4}23^{21/22}\approx 136.54903346772397$
Ramified primes:   \(13\), \(23\) 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:  \(299=13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{299}(1,·)$, $\chi_{299}(131,·)$, $\chi_{299}(5,·)$, $\chi_{299}(135,·)$, $\chi_{299}(12,·)$, $\chi_{299}(142,·)$, $\chi_{299}(144,·)$, $\chi_{299}(259,·)$, $\chi_{299}(148,·)$, $\chi_{299}(21,·)$, $\chi_{299}(281,·)$, $\chi_{299}(25,·)$, $\chi_{299}(27,·)$, $\chi_{299}(285,·)$, $\chi_{299}(261,·)$, $\chi_{299}(34,·)$, $\chi_{299}(291,·)$, $\chi_{299}(170,·)$, $\chi_{299}(44,·)$, $\chi_{299}(57,·)$, $\chi_{299}(60,·)$, $\chi_{299}(64,·)$, $\chi_{299}(196,·)$, $\chi_{299}(118,·)$, $\chi_{299}(268,·)$, $\chi_{299}(203,·)$, $\chi_{299}(77,·)$, $\chi_{299}(209,·)$, $\chi_{299}(83,·)$, $\chi_{299}(86,·)$, $\chi_{299}(220,·)$, $\chi_{299}(226,·)$, $\chi_{299}(99,·)$, $\chi_{299}(229,·)$, $\chi_{299}(105,·)$, $\chi_{299}(109,·)$, $\chi_{299}(112,·)$, $\chi_{299}(116,·)$, $\chi_{299}(246,·)$, $\chi_{299}(233,·)$, $\chi_{299}(248,·)$, $\chi_{299}(122,·)$, $\chi_{299}(252,·)$, $\chi_{299}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{30911}a^{33}-\frac{9297}{30911}a^{32}+\frac{14231}{30911}a^{31}-\frac{6615}{30911}a^{30}+\frac{10684}{30911}a^{29}+\frac{11756}{30911}a^{28}-\frac{8268}{30911}a^{27}-\frac{6396}{30911}a^{26}+\frac{13375}{30911}a^{25}-\frac{1039}{30911}a^{24}+\frac{13684}{30911}a^{23}+\frac{11531}{30911}a^{22}-\frac{3317}{30911}a^{21}-\frac{9682}{30911}a^{20}-\frac{4760}{30911}a^{19}-\frac{7499}{30911}a^{18}-\frac{3362}{30911}a^{17}+\frac{10461}{30911}a^{16}-\frac{13774}{30911}a^{15}+\frac{5803}{30911}a^{14}+\frac{2166}{30911}a^{13}+\frac{993}{30911}a^{12}+\frac{6657}{30911}a^{11}-\frac{4729}{30911}a^{10}-\frac{13788}{30911}a^{9}-\frac{7432}{30911}a^{8}+\frac{11531}{30911}a^{7}+\frac{9095}{30911}a^{6}-\frac{11224}{30911}a^{5}-\frac{5533}{30911}a^{4}+\frac{7961}{30911}a^{3}+\frac{576}{30911}a^{2}+\frac{775}{30911}a-\frac{6748}{30911}$, $\frac{1}{30911}a^{34}+\frac{7178}{30911}a^{32}-\frac{88}{30911}a^{31}-\frac{6992}{30911}a^{30}-\frac{7050}{30911}a^{29}-\frac{14032}{30911}a^{28}+\frac{1665}{30911}a^{27}-\frac{8384}{30911}a^{26}-\frac{8617}{30911}a^{25}-\frac{1667}{30911}a^{24}+\frac{2003}{30911}a^{23}+\frac{1042}{30911}a^{22}+\frac{1347}{30911}a^{21}-\frac{5482}{30911}a^{20}+\frac{3333}{30911}a^{19}+\frac{13651}{30911}a^{18}+\frac{4968}{30911}a^{17}-\frac{3863}{30911}a^{16}+\frac{13198}{30911}a^{15}+\frac{12962}{30911}a^{14}+\frac{15234}{30911}a^{13}-\frac{3811}{30911}a^{12}+\frac{1578}{30911}a^{11}+\frac{7052}{30911}a^{10}-\frac{6551}{30911}a^{9}+\frac{2312}{30911}a^{8}+\frac{13454}{30911}a^{7}+\frac{3406}{30911}a^{6}+\frac{475}{30911}a^{5}+\frac{3564}{30911}a^{4}+\frac{13059}{30911}a^{3}+\frac{8244}{30911}a^{2}-\frac{3836}{30911}a+\frac{13174}{30911}$, $\frac{1}{30911}a^{35}-\frac{3071}{30911}a^{32}+\frac{3745}{30911}a^{31}-\frac{3876}{30911}a^{30}-\frac{13593}{30911}a^{29}+\frac{4127}{30911}a^{28}-\frac{9800}{30911}a^{27}-\frac{964}{30911}a^{26}+\frac{2149}{30911}a^{25}+\frac{10394}{30911}a^{24}+\frac{12448}{30911}a^{23}+\frac{11487}{30911}a^{22}+\frac{2474}{30911}a^{21}+\frac{12801}{30911}a^{20}-\frac{6635}{30911}a^{19}-\frac{14172}{30911}a^{18}-\frac{12918}{30911}a^{17}+\frac{6959}{30911}a^{16}-\frac{1555}{30911}a^{15}-\frac{1583}{30911}a^{14}-\frac{3126}{30911}a^{13}+\frac{14265}{30911}a^{12}+\frac{11512}{30911}a^{11}-\frac{2067}{30911}a^{10}-\frac{4446}{30911}a^{9}+\frac{7964}{30911}a^{8}+\frac{13546}{30911}a^{7}+\frac{597}{30911}a^{6}+\frac{15370}{30911}a^{5}+\frac{8298}{30911}a^{4}-\frac{12286}{30911}a^{3}+\frac{3710}{30911}a^{2}+\frac{14204}{30911}a-\frac{393}{30911}$, $\frac{1}{30911}a^{36}+\frac{14422}{30911}a^{32}-\frac{8629}{30911}a^{31}+\frac{11180}{30911}a^{30}-\frac{12791}{30911}a^{29}-\frac{11172}{30911}a^{28}-\frac{14061}{30911}a^{27}-\frac{11482}{30911}a^{26}+\frac{4300}{30911}a^{25}+\frac{5512}{30911}a^{24}-\frac{3909}{30911}a^{23}-\frac{9831}{30911}a^{22}-\frac{3987}{30911}a^{21}-\frac{3675}{30911}a^{20}-\frac{11229}{30911}a^{19}-\frac{13652}{30911}a^{18}+\frac{6531}{30911}a^{17}+\frac{7647}{30911}a^{16}-\frac{15289}{30911}a^{15}+\frac{13151}{30911}a^{14}-\frac{10725}{30911}a^{13}+\frac{826}{30911}a^{12}+\frac{9409}{30911}a^{11}+\frac{965}{30911}a^{10}+\frac{13086}{30911}a^{9}+\frac{2192}{30911}a^{8}-\frac{11708}{30911}a^{7}+\frac{2571}{30911}a^{6}+\frac{5159}{30911}a^{5}-\frac{3079}{30911}a^{4}+\frac{1340}{30911}a^{3}-\frac{9738}{30911}a^{2}-\frac{515}{30911}a-\frac{12738}{30911}$, $\frac{1}{30911}a^{37}+\frac{11698}{30911}a^{32}-\frac{10173}{30911}a^{31}-\frac{2607}{30911}a^{30}-\frac{4485}{30911}a^{29}-\frac{12258}{30911}a^{28}+\frac{5887}{30911}a^{27}+\frac{8988}{30911}a^{26}-\frac{4098}{30911}a^{25}-\frac{11286}{30911}a^{24}+\frac{6256}{30911}a^{23}-\frac{2889}{30911}a^{22}+\frac{14782}{30911}a^{21}-\frac{2412}{30911}a^{20}+\frac{12648}{30911}a^{19}-\frac{480}{30911}a^{18}-\frac{4948}{30911}a^{17}-\frac{7240}{30911}a^{16}-\frac{3218}{30911}a^{15}+\frac{5397}{30911}a^{14}+\frac{13795}{30911}a^{13}+\frac{156}{30911}a^{12}+\frac{3277}{30911}a^{11}-\frac{5853}{30911}a^{10}+\frac{2265}{30911}a^{9}+\frac{4159}{30911}a^{8}+\frac{3669}{30911}a^{7}-\frac{7558}{30911}a^{6}-\frac{11458}{30911}a^{5}-\frac{13936}{30911}a^{4}+\frac{11085}{30911}a^{3}+\frac{7472}{30911}a^{2}-\frac{6}{30911}a+\frac{11828}{30911}$, $\frac{1}{30911}a^{38}+\frac{1235}{30911}a^{32}+\frac{9801}{30911}a^{31}+\frac{7552}{30911}a^{30}+\frac{10394}{30911}a^{29}+\frac{7238}{30911}a^{28}+\frac{7533}{30911}a^{27}+\frac{11690}{30911}a^{26}-\frac{554}{30911}a^{25}+\frac{12455}{30911}a^{24}+\frac{9748}{30911}a^{23}-\frac{10163}{30911}a^{22}+\frac{6549}{30911}a^{21}+\frac{14780}{30911}a^{20}+\frac{11289}{30911}a^{19}-\frac{7064}{30911}a^{18}+\frac{2644}{30911}a^{17}+\frac{653}{30911}a^{16}-\frac{5394}{30911}a^{15}+\frac{10857}{30911}a^{14}+\frac{9308}{30911}a^{13}+\frac{9699}{30911}a^{12}-\frac{14630}{30911}a^{11}-\frac{8583}{30911}a^{10}+\frac{2585}{30911}a^{9}-\frac{9438}{30911}a^{8}-\frac{1592}{30911}a^{7}-\frac{9106}{30911}a^{6}+\frac{5399}{30911}a^{5}+\frac{8485}{30911}a^{4}+\frac{14537}{30911}a^{3}+\frac{544}{30911}a^{2}+\frac{2801}{30911}a-\frac{8590}{30911}$, $\frac{1}{30911}a^{39}-\frac{7296}{30911}a^{32}-\frac{10285}{30911}a^{31}-\frac{11496}{30911}a^{30}+\frac{11495}{30911}a^{29}-\frac{13868}{30911}a^{28}-\frac{8871}{30911}a^{27}-\frac{14710}{30911}a^{26}+\frac{804}{30911}a^{25}-\frac{5349}{30911}a^{24}-\frac{1586}{30911}a^{23}-\frac{15176}{30911}a^{22}+\frac{112}{30911}a^{21}+\frac{6002}{30911}a^{20}-\frac{1554}{30911}a^{19}-\frac{9391}{30911}a^{18}+\frac{10649}{30911}a^{17}-\frac{3931}{30911}a^{16}-\frac{10214}{30911}a^{15}+\frac{13955}{30911}a^{14}-\frac{6965}{30911}a^{13}-\frac{4545}{30911}a^{12}-\frac{7652}{30911}a^{11}+\frac{721}{30911}a^{10}-\frac{13219}{30911}a^{9}-\frac{3639}{30911}a^{8}+\frac{80}{30911}a^{7}-\frac{6233}{30911}a^{6}-\frac{8914}{30911}a^{5}-\frac{14450}{30911}a^{4}-\frac{1593}{30911}a^{3}+\frac{2394}{30911}a^{2}-\frac{7474}{30911}a-\frac{12190}{30911}$, $\frac{1}{30911}a^{40}+\frac{8448}{30911}a^{32}-\frac{12169}{30911}a^{31}+\frac{526}{30911}a^{30}+\frac{9965}{30911}a^{29}-\frac{15120}{30911}a^{28}+\frac{234}{30911}a^{27}+\frac{11198}{30911}a^{26}-\frac{7376}{30911}a^{25}-\frac{8935}{30911}a^{24}+\frac{11669}{30911}a^{23}-\frac{9454}{30911}a^{22}+\frac{8483}{30911}a^{21}-\frac{9791}{30911}a^{20}+\frac{5613}{30911}a^{19}+\frac{10415}{30911}a^{18}+\frac{10251}{30911}a^{17}-\frac{6017}{30911}a^{16}+\frac{10512}{30911}a^{15}+\frac{14564}{30911}a^{14}+\frac{3070}{30911}a^{13}+\frac{4102}{30911}a^{12}+\frac{9012}{30911}a^{11}+\frac{11584}{30911}a^{10}+\frac{14418}{30911}a^{9}-\frac{5898}{30911}a^{8}+\frac{15112}{30911}a^{7}+\frac{13200}{30911}a^{6}+\frac{9396}{30911}a^{5}-\frac{595}{30911}a^{4}+\frac{4081}{30911}a^{3}-\frac{8874}{30911}a^{2}-\frac{14503}{30911}a+\frac{7815}{30911}$, $\frac{1}{4296629}a^{41}-\frac{60}{4296629}a^{40}-\frac{25}{4296629}a^{39}+\frac{52}{4296629}a^{38}-\frac{26}{4296629}a^{37}+\frac{39}{4296629}a^{36}+\frac{20}{4296629}a^{35}+\frac{4}{4296629}a^{34}-\frac{62}{4296629}a^{33}-\frac{1112491}{4296629}a^{32}-\frac{1000345}{4296629}a^{31}-\frac{1194961}{4296629}a^{30}+\frac{368111}{4296629}a^{29}+\frac{2111667}{4296629}a^{28}-\frac{2065726}{4296629}a^{27}+\frac{1103311}{4296629}a^{26}+\frac{876906}{4296629}a^{25}+\frac{1916470}{4296629}a^{24}+\frac{1500730}{4296629}a^{23}+\frac{1633226}{4296629}a^{22}-\frac{196388}{4296629}a^{21}+\frac{1844732}{4296629}a^{20}+\frac{756729}{4296629}a^{19}-\frac{1757999}{4296629}a^{18}+\frac{711262}{4296629}a^{17}+\frac{619958}{4296629}a^{16}+\frac{848327}{4296629}a^{15}+\frac{1697315}{4296629}a^{14}-\frac{742944}{4296629}a^{13}-\frac{569088}{4296629}a^{12}+\frac{263095}{4296629}a^{11}-\frac{383305}{4296629}a^{10}-\frac{257485}{4296629}a^{9}+\frac{241131}{4296629}a^{8}-\frac{882849}{4296629}a^{7}-\frac{187947}{4296629}a^{6}+\frac{470703}{4296629}a^{5}+\frac{994900}{4296629}a^{4}-\frac{1578722}{4296629}a^{3}-\frac{2131184}{4296629}a^{2}+\frac{273479}{4296629}a+\frac{13885}{30911}$, $\frac{1}{4296629}a^{42}-\frac{11}{4296629}a^{40}-\frac{58}{4296629}a^{39}+\frac{36}{4296629}a^{38}+\frac{8}{4296629}a^{37}-\frac{3}{4296629}a^{36}-\frac{47}{4296629}a^{35}+\frac{39}{4296629}a^{34}-\frac{41}{4296629}a^{33}-\frac{709688}{4296629}a^{32}+\frac{660884}{4296629}a^{31}-\frac{1746149}{4296629}a^{30}+\frac{1839621}{4296629}a^{29}-\frac{1558663}{4296629}a^{28}+\frac{679182}{4296629}a^{27}+\frac{2044416}{4296629}a^{26}-\frac{2133632}{4296629}a^{25}+\frac{308838}{4296629}a^{24}+\frac{821483}{4296629}a^{23}-\frac{1571009}{4296629}a^{22}+\frac{870509}{4296629}a^{21}-\frac{1866869}{4296629}a^{20}+\frac{1284240}{4296629}a^{19}+\frac{1341142}{4296629}a^{18}+\frac{585426}{4296629}a^{17}-\frac{376017}{4296629}a^{16}-\frac{1186196}{4296629}a^{15}-\frac{909194}{4296629}a^{14}+\frac{434735}{4296629}a^{13}-\frac{1194528}{4296629}a^{12}+\frac{142836}{4296629}a^{11}-\frac{246698}{4296629}a^{10}-\frac{1000779}{4296629}a^{9}-\frac{210183}{4296629}a^{8}+\frac{1564440}{4296629}a^{7}+\frac{1488016}{4296629}a^{6}-\frac{742301}{4296629}a^{5}+\frac{697436}{4296629}a^{4}+\frac{2082638}{4296629}a^{3}-\frac{2037193}{4296629}a^{2}+\frac{1097751}{4296629}a-\frac{11887}{30911}$, $\frac{1}{95\!\cdots\!57}a^{43}+\frac{15\!\cdots\!14}{95\!\cdots\!57}a^{42}-\frac{10\!\cdots\!24}{95\!\cdots\!57}a^{41}-\frac{13\!\cdots\!73}{95\!\cdots\!57}a^{40}+\frac{75\!\cdots\!02}{95\!\cdots\!57}a^{39}+\frac{10\!\cdots\!89}{95\!\cdots\!57}a^{38}-\frac{53\!\cdots\!25}{95\!\cdots\!57}a^{37}-\frac{15\!\cdots\!90}{95\!\cdots\!57}a^{36}-\frac{34\!\cdots\!81}{95\!\cdots\!57}a^{35}-\frac{61\!\cdots\!30}{95\!\cdots\!57}a^{34}-\frac{10\!\cdots\!03}{95\!\cdots\!57}a^{33}+\frac{44\!\cdots\!99}{95\!\cdots\!57}a^{32}+\frac{16\!\cdots\!42}{95\!\cdots\!57}a^{31}+\frac{93\!\cdots\!28}{95\!\cdots\!57}a^{30}-\frac{17\!\cdots\!89}{95\!\cdots\!57}a^{29}+\frac{39\!\cdots\!09}{95\!\cdots\!57}a^{28}-\frac{41\!\cdots\!41}{95\!\cdots\!57}a^{27}+\frac{14\!\cdots\!69}{95\!\cdots\!57}a^{26}+\frac{33\!\cdots\!14}{95\!\cdots\!57}a^{25}-\frac{23\!\cdots\!45}{95\!\cdots\!57}a^{24}+\frac{26\!\cdots\!28}{95\!\cdots\!57}a^{23}-\frac{44\!\cdots\!11}{95\!\cdots\!57}a^{22}-\frac{22\!\cdots\!12}{95\!\cdots\!57}a^{21}-\frac{40\!\cdots\!50}{95\!\cdots\!57}a^{20}+\frac{52\!\cdots\!61}{95\!\cdots\!57}a^{19}+\frac{45\!\cdots\!65}{95\!\cdots\!57}a^{18}+\frac{20\!\cdots\!46}{95\!\cdots\!57}a^{17}+\frac{56\!\cdots\!05}{95\!\cdots\!57}a^{16}+\frac{43\!\cdots\!73}{95\!\cdots\!57}a^{15}+\frac{12\!\cdots\!30}{95\!\cdots\!57}a^{14}+\frac{84\!\cdots\!06}{95\!\cdots\!57}a^{13}+\frac{41\!\cdots\!88}{95\!\cdots\!57}a^{12}+\frac{14\!\cdots\!68}{95\!\cdots\!57}a^{11}+\frac{25\!\cdots\!78}{95\!\cdots\!57}a^{10}-\frac{28\!\cdots\!64}{95\!\cdots\!57}a^{9}-\frac{12\!\cdots\!22}{95\!\cdots\!57}a^{8}+\frac{38\!\cdots\!60}{95\!\cdots\!57}a^{7}-\frac{26\!\cdots\!37}{95\!\cdots\!57}a^{6}+\frac{29\!\cdots\!10}{95\!\cdots\!57}a^{5}-\frac{22\!\cdots\!89}{95\!\cdots\!57}a^{4}+\frac{19\!\cdots\!56}{95\!\cdots\!57}a^{3}+\frac{21\!\cdots\!04}{95\!\cdots\!57}a^{2}-\frac{32\!\cdots\!22}{95\!\cdots\!57}a+\frac{70\!\cdots\!03}{68\!\cdots\!63}$ 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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:  $43$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

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

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 111*x^42 + 108*x^41 + 5516*x^40 - 5192*x^39 - 162490*x^38 + 146914*x^37 + 3169793*x^36 - 2729051*x^35 - 43378560*x^34 + 35191407*x^33 + 430978480*x^32 - 325404259*x^31 - 3178214046*x^30 + 2202001269*x^29 + 17658050940*x^28 - 11052047133*x^27 - 74656059512*x^26 + 41499918113*x^25 + 241605668931*x^24 - 117105914592*x^23 - 599724612955*x^22 + 248374887128*x^21 + 1139635785934*x^20 - 393998255255*x^19 - 1647609787093*x^18 + 462039171768*x^17 + 1792199478453*x^16 - 391870900134*x^15 - 1441857227832*x^14 + 230783287602*x^13 + 836726691516*x^12 - 86686890110*x^11 - 337712573557*x^10 + 16137410915*x^9 + 89760954010*x^8 + 763059160*x^7 - 14382578320*x^6 - 914750852*x^5 + 1175923775*x^4 + 123599834*x^3 - 31889819*x^2 - 2291321*x + 332627)
 
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 - 111*x^42 + 108*x^41 + 5516*x^40 - 5192*x^39 - 162490*x^38 + 146914*x^37 + 3169793*x^36 - 2729051*x^35 - 43378560*x^34 + 35191407*x^33 + 430978480*x^32 - 325404259*x^31 - 3178214046*x^30 + 2202001269*x^29 + 17658050940*x^28 - 11052047133*x^27 - 74656059512*x^26 + 41499918113*x^25 + 241605668931*x^24 - 117105914592*x^23 - 599724612955*x^22 + 248374887128*x^21 + 1139635785934*x^20 - 393998255255*x^19 - 1647609787093*x^18 + 462039171768*x^17 + 1792199478453*x^16 - 391870900134*x^15 - 1441857227832*x^14 + 230783287602*x^13 + 836726691516*x^12 - 86686890110*x^11 - 337712573557*x^10 + 16137410915*x^9 + 89760954010*x^8 + 763059160*x^7 - 14382578320*x^6 - 914750852*x^5 + 1175923775*x^4 + 123599834*x^3 - 31889819*x^2 - 2291321*x + 332627, 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 - 111*x^42 + 108*x^41 + 5516*x^40 - 5192*x^39 - 162490*x^38 + 146914*x^37 + 3169793*x^36 - 2729051*x^35 - 43378560*x^34 + 35191407*x^33 + 430978480*x^32 - 325404259*x^31 - 3178214046*x^30 + 2202001269*x^29 + 17658050940*x^28 - 11052047133*x^27 - 74656059512*x^26 + 41499918113*x^25 + 241605668931*x^24 - 117105914592*x^23 - 599724612955*x^22 + 248374887128*x^21 + 1139635785934*x^20 - 393998255255*x^19 - 1647609787093*x^18 + 462039171768*x^17 + 1792199478453*x^16 - 391870900134*x^15 - 1441857227832*x^14 + 230783287602*x^13 + 836726691516*x^12 - 86686890110*x^11 - 337712573557*x^10 + 16137410915*x^9 + 89760954010*x^8 + 763059160*x^7 - 14382578320*x^6 - 914750852*x^5 + 1175923775*x^4 + 123599834*x^3 - 31889819*x^2 - 2291321*x + 332627);
 
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 - 111*x^42 + 108*x^41 + 5516*x^40 - 5192*x^39 - 162490*x^38 + 146914*x^37 + 3169793*x^36 - 2729051*x^35 - 43378560*x^34 + 35191407*x^33 + 430978480*x^32 - 325404259*x^31 - 3178214046*x^30 + 2202001269*x^29 + 17658050940*x^28 - 11052047133*x^27 - 74656059512*x^26 + 41499918113*x^25 + 241605668931*x^24 - 117105914592*x^23 - 599724612955*x^22 + 248374887128*x^21 + 1139635785934*x^20 - 393998255255*x^19 - 1647609787093*x^18 + 462039171768*x^17 + 1792199478453*x^16 - 391870900134*x^15 - 1441857227832*x^14 + 230783287602*x^13 + 836726691516*x^12 - 86686890110*x^11 - 337712573557*x^10 + 16137410915*x^9 + 89760954010*x^8 + 763059160*x^7 - 14382578320*x^6 - 914750852*x^5 + 1175923775*x^4 + 123599834*x^3 - 31889819*x^2 - 2291321*x + 332627);
 
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

\(\Q(\sqrt{13}) \), 4.4.1162213.1, \(\Q(\zeta_{23})^+\), 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 $44$ ${\href{/padicField/3.11.0.1}{11} }^{4}$ $44$ $44$ $44$ R ${\href{/padicField/17.11.0.1}{11} }^{4}$ $44$ R ${\href{/padicField/29.11.0.1}{11} }^{4}$ $44$ $44$ $44$ ${\href{/padicField/43.11.0.1}{11} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{11}$ $22^{2}$ $44$

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
\(13\) Copy content Toggle raw display Deg $44$$4$$11$$33$
\(23\) Copy content Toggle raw display 23.22.21.1$x^{22} + 506$$22$$1$$21$22T1$[\ ]_{22}$
23.22.21.1$x^{22} + 506$$22$$1$$21$22T1$[\ ]_{22}$