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Properties

Label 44.44.8968085474...2237.1
Degree $44$
Signature $[44, 0]$
Discriminant $13^{33}\cdot 23^{42}$
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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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![332627, -2291321, -31889819, 123599834, 1175923775, -914750852, -14382578320, 763059160, 89760954010, 16137410915, -337712573557, -86686890110, 836726691516, 230783287602, -1441857227832, -391870900134, 1792199478453, 462039171768, -1647609787093, -393998255255, 1139635785934, 248374887128, -599724612955, -117105914592, 241605668931, 41499918113, -74656059512, -11052047133, 17658050940, 2202001269, -3178214046, -325404259, 430978480, 35191407, -43378560, -2729051, 3169793, 146914, -162490, -5192, 5516, 108, -111, -1, 1]);
 
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(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)
 

Normalized defining polynomial

\( x^{44} - x^{43} - 111 x^{42} + 108 x^{41} + 5516 x^{40} - 5192 x^{39} - 162490 x^{38} + 146914 x^{37} + 3169793 x^{36} - 2729051 x^{35} - 43378560 x^{34} + 35191407 x^{33} + 430978480 x^{32} - 325404259 x^{31} - 3178214046 x^{30} + 2202001269 x^{29} + 17658050940 x^{28} - 11052047133 x^{27} - 74656059512 x^{26} + 41499918113 x^{25} + 241605668931 x^{24} - 117105914592 x^{23} - 599724612955 x^{22} + 248374887128 x^{21} + 1139635785934 x^{20} - 393998255255 x^{19} - 1647609787093 x^{18} + 462039171768 x^{17} + 1792199478453 x^{16} - 391870900134 x^{15} - 1441857227832 x^{14} + 230783287602 x^{13} + 836726691516 x^{12} - 86686890110 x^{11} - 337712573557 x^{10} + 16137410915 x^{9} + 89760954010 x^{8} + 763059160 x^{7} - 14382578320 x^{6} - 914750852 x^{5} + 1175923775 x^{4} + 123599834 x^{3} - 31889819 x^{2} - 2291321 x + 332627 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[44, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8968085474764113590945882799558127794828898423230040761200017709792699264637480790538930062237=13^{33}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.55$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $13, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
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}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{43} + \frac{15910320058666091123049086446112252926283213140079108903749815626826133047093187428040144758947423414508720638190285038326872850114}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{42} - \frac{108932617143229099247104315433661832105213306060171257288519299983489011833022752431132031607760288998625296024065355650052714272724}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{41} - \frac{13120578166283041735188923606239471586647819945531899226186652925515360299489023768699440885038428623066408141956003834799663417127873}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{40} + \frac{7514937393153040712005536718964075237155461473961621248719302049637430342001061537546321216947978163403385330548496695165592524133602}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{39} + \frac{10258377915656475465696206024704318066720624417855464494630714143001589440610849817079276386940660115174563449657172190506837771745989}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{38} - \frac{5388419747437472727626735228095890576211333493294276185387575540632031670124204624295489183831913634755423282682365529488679195982825}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{37} - \frac{15244121074526528632515644364567345798853384927147941400932397821797880280106621624080028585342866861992774376154009108860969090813590}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{36} - \frac{3497964827769317443362394532407318339868787206887374654622387810248357057766414374289223845536513824776795272068003238755657752561481}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{35} - \frac{6181119480786348514087881256175151953417898644422164998155679826992684999779556146768648006630480757540359650663132914573680788486030}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{34} - \frac{10306700679945038393795487995907142349358643944943191917875783739418016962534993695020607399038998938704702205727893650999595724295103}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{33} + \frac{443029926712428926169117605181082012561624578391763613766214246615662618997330084824808774950121996364829774143782144711875812045588449099}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{32} + \frac{164921993390298809076233757814541172293573988164493160609391111971621663989539759425190852675824047914794533972498572387944639480625049342}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{31} + \frac{93402075594765374376415817899523566950614741321898779982165291288944415826110777141293028448717147548658905353856547254697097537527078728}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{30} - \frac{174409025399343415116777635221928290114321918265673823598880543115869381826267033574019981662620834041276286823230553602082697635820156189}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{29} + \frac{390178705889806897459048441122195335598974509714454440148962681126063540877070109168014914757920416340734494335646792480369733048755610409}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{28} - \frac{410240637743470600179118630266771631112330641092187315251216660516365562345101967217788353089594642957411178639293649516148232672810213941}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{27} + \frac{140085832155173479330920419305657350061351279677185818269735575735511448231690097393029818478490575400361424489435000737710684649119270169}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{26} + \frac{334389088479609726251793626517338609966909193901744069751264255597834436202452843075069421922533446965873086039623239652972046306758656714}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{25} - \frac{239837743746127050581702628095814663640422193777142224587680429880942374617113859189550839667141202239934539906465329918499732098882869145}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{24} + \frac{262799546540271347593694841336350692629360287533199951658783115353784191244685937115548949798160763515725230623324071783120976610415676228}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{23} - \frac{445633641206106000732275126745438944659936805160394358323815566591434331552500704622514959837750173731349749155982740817266818393812231711}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{22} - \frac{222547718953726194245869562574335055969575383042096651314035725980226036619040876117658372962620065933523991031561020413810669172810204912}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{21} - \frac{40862856294557735361617807347351266831383210681229866068273162800378434885691714298155785983451206574650806542318351360755689919110382850}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{20} + \frac{52973340919428052205463806089198990170445418154839575267641913446386973856728655040568544846975030112474547848263509456230681207141549961}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{19} + \frac{459467474440114058034352960796106026629266729371593612302865851399091333281906252785080889061851679390886318264694754603069259198907920665}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{18} + \frac{205021071960078251067577867306770771433007339751560031472556932945812819620545759740917363046008978093735387424196972194028550207741159946}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{17} + \frac{5650155088182809255802081477821626927205205713647460933453226689798983528655603500290335515576328410382098320501898675985871744917713205}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{16} + \frac{439219912518016740209229235321079443866370185626033366488375642864093810308497759271631198197875920998937154229223737933681321081177199373}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{15} + \frac{127307177709051532604297667765495745726930867829135777176577451835103706220295330165617466594363179863722268932471148541109819791364872530}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{14} + \frac{84943330999699096394548801242742849880413010366532954743913817074007728237417057009643933674337999161355204976120240145252961454148916306}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{13} + \frac{415447431869897906037546672075591327140707174242712161232494516804983789056036341766587931720358438825565375480380903392331606189833840388}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{12} + \frac{148700644602280581285279211096063822677259300192374275781338106908675903712099357847972570000360310943485734412168006074906120641410221568}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{11} + \frac{256022339298187546125850956103479309752099125588588690427347728877208400477633372055474831399139305710131913093516633399453692933208713378}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{10} - \frac{284199582626506214030718250024673579065929076741095996309319195587292333556459339026158293045437968669561954599451309329398363916216393564}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{9} - \frac{12707122472120994003162181867411353133222023987305238769928067761771388558623017004740179466852227275003626084030657541970453236984363922}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{8} + \frac{38661838661562711329935344965079173856102707626725687258066551243329888052373393018012733145737081973882430603319859296901098187099531160}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{7} - \frac{269174830309697638978633646070703824521155277261519301585642294004918692198498829501179288467522387507900099418619277136502216441157722937}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{6} + \frac{291844822819742902724188267204974848603354161318169439415555518192628276392685839027323960274699571639968321484599811745157121300279153410}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{5} - \frac{227945038953710566613270939822198867168616909596907839074326131311481828359843488292307500182136205422691177761123840131604958645533402889}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{4} + \frac{190847528247524792457689325584839679864487443285881252112265307171887208294966185203197035664046247534918225201405058176538237076159305356}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{3} + \frac{216934395380446835021825015282416744065370079112282269643658809333225384267655657479952014020694341882294602888882559480492897424224764104}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a^{2} - \frac{326313852419525824038090728679177375584043928251235359280476430951819453079523014011329800599137099334255101913770723119751044035906363522}{957460178671688698839575215456318116286196921405599093477027888770476814947070080909516292121001508669961013298896280711033200842644153557} a + \frac{706866501515563714341989711416133443640274261395159760695803574511158592356533151485728888066622464638757350030227253999948580200819103}{6888202724256753229061692197527468462490625333853230888323941645830768452856619287118822245474830997625618800711484033892325185918303263}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $43$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $44$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{4}$ $44$ $44$ $44$ R ${\href{/LocalNumberField/17.11.0.1}{11} }^{4}$ $44$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ $44$ $44$ $44$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{11}$ $22^{2}$ $44$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
23Data not computed