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Properties

Label 44.44.1340542119...3125.1
Degree $44$
Signature $[44, 0]$
Discriminant $5^{33}\cdot 7^{22}\cdot 23^{40}$
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 for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1699532101, -31938085617, 217936624208, -507894298828, -1172684662984, 9013617025194, -14683750874453, -15803724914334, 78243526539014, -47750321297779, -138850152957509, 207300120944285, 81499856091714, -329051920771801, 71424111769606, 290016628457561, -167684548055793, -153469537128468, 147217011020842, 45568931299761, -78521093195796, -3195446926852, 28338195475365, -3395585375747, -7207408221864, 1708214965012, 1308412141053, -446721814624, -168007364622, 77186286766, 14629550053, -9363190013, -751544361, 810370867, 7249749, -49737458, 2030280, 2110262, -162828, -58706, 6141, 961, -121, -7, 1]);
 
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(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)
 

Normalized defining polynomial

\( x^{44} - 7 x^{43} - 121 x^{42} + 961 x^{41} + 6141 x^{40} - 58706 x^{39} - 162828 x^{38} + 2110262 x^{37} + 2030280 x^{36} - 49737458 x^{35} + 7249749 x^{34} + 810370867 x^{33} - 751544361 x^{32} - 9363190013 x^{31} + 14629550053 x^{30} + 77186286766 x^{29} - 168007364622 x^{28} - 446721814624 x^{27} + 1308412141053 x^{26} + 1708214965012 x^{25} - 7207408221864 x^{24} - 3395585375747 x^{23} + 28338195475365 x^{22} - 3195446926852 x^{21} - 78521093195796 x^{20} + 45568931299761 x^{19} + 147217011020842 x^{18} - 153469537128468 x^{17} - 167684548055793 x^{16} + 290016628457561 x^{15} + 71424111769606 x^{14} - 329051920771801 x^{13} + 81499856091714 x^{12} + 207300120944285 x^{11} - 138850152957509 x^{10} - 47750321297779 x^{9} + 78243526539014 x^{8} - 15803724914334 x^{7} - 14683750874453 x^{6} + 9013617025194 x^{5} - 1172684662984 x^{4} - 507894298828 x^{3} + 217936624208 x^{2} - 31938085617 x + 1699532101 \)

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:  \(1340542119957152709801283008679613456161693073520931071709778240989995846641249954700469970703125=5^{33}\cdot 7^{22}\cdot 23^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.01$
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:  $5, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(805=5\cdot 7\cdot 23\)
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}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{43} + \frac{39631007111206546002637587480741815916293057159168706025690799733965703667392901724253790439938287443410620430235707202561691057560106955388422377655611464824450530858571760890514352033674410336953046406740015047254797782924}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{42} - \frac{69456341816644070089515132006186634657561402304782922425237773219803849546460483991770405205342643218834661891351102881470948486498232002392353182410439626860765132206025218388993077827003555874313250597293744178177971316016419}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{41} - \frac{101779410053836465730935121047340682733884321918081975393509839512730406847736501506991498525785065298568762260408886539838969596315303220725871900663518736851491536549459616610067793938870951436643238861334414642376324141728252}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{40} + \frac{63242247434566965976162637063017953250003750382218919844028153236360180030921412693422350907299529058104160792326442431708600308984312509342814219758081617328242810638217884884596622573006654238054120009058463078609498957042506}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{39} + \frac{1872112805817630502250663620962216682783183110971683576439479269505564842929887435194941224186946086119955786434478089389857373009369100922334407380603911503612797078682167187329507167078167715690676762110209212064160168874532910}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{38} - \frac{2751844673045942983426614067413899774187859444519326573441248644560321068088777450349428890859178705537871365193956775159358813151583907061406586126718585923522396556724803042152116307161364183462752355093378789629760664786447656153}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{37} - \frac{1479266861891426643275197804625905454052495853262943399144415371395115934389925503091941703303072318904749194279062768200430824793715443286412402609647721544818975262701906164482086410007133325564039192352395678862197130098319579360}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{36} + \frac{959460265149980127550693585128574006649213190856079063292731377936561481505592270111624281863105427028119007494149277795976061100635530393597222594879660510661486137049590905768368354207584415809428104399502745626520405964047747062}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{35} - \frac{266125403919119248063052895234599488369669488862839230824621296878653369671956969105023511264046716593928703429828346251051424282837600664096976623934150431903529749225120153284355076234928654313901003361426218240904418999897294222}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{34} - \frac{3765139693122723993671714543437767359172515630872299818365398680135148913964796982833464260700913016646003940891409652675038303586314457850526076030399292254697376541365929423527751002425712234533110453677699857552014205008237624850}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{33} + \frac{939786446495040771472087015356513989892228123362160163694556269378613520027265614723635989018973392160236107903163971525389322486288003022959630885348832136293029110423850265362100214905853513384280838745041056531950330044416061732}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{32} - \frac{302612279185609699749819494600246998579835616859395300871967332651931822201078703355683726127537139982193479714665043496135202812589196764024761288703720171488153828109342439314612184442209062842264599561575008430352573807719616628}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{31} - \frac{3193882612001578178743306439755460494993680028997098399840469415946931749304171643790733572310132822300697722212085249809963892441471237519683730349638933538971827861173546924213145303967170469195817268544545409753153420195303084875}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{30} + \frac{624935815517817593093859469076808527895667151296913555393796329455255798365486127936780745517092266248437876862729359761719766365157805372740728944773850984085564594572785846297328417146072394763940211581039314398863648854396492622}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{29} + \frac{2160219394022878947951928842115181769028878383583737364475914111413578381097624900603545036650507039614355337101970360752948020221628420242222279835194980180823980251210459649625448584446568510660916719419848574264753268784980924727}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{28} + \frac{222906896595471722482179447222742104449920986030986431486957918382243379022250046586592748186294001301174920687844675347564178170440372871144463816259356709247822182573317859834267138791863575770196705068970896454190358639401840787}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{27} - \frac{83877539942361396269956811171168446139021953121458013129160744454560548577195470213505088993251156652703981380697025148366550442071983437554845850049250146270889724434065761189100501855241035305445107213986873737212283020757484935}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{26} - \frac{2733536573674269179830214582255035632767134306556518402822682310393619435961655668917638994004555255244321999806982376526135730490635944696032603012634037591056218881341983516142138029998344665704207993984941964402788745230154772799}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{25} + \frac{2407918967442613045814363295865441224725332931701996183968739749436535239859836258241820350504939367320827585650216049220466380530461027483837503385037104365215497036396855924350138805102758474754888460081013103386477601891967016443}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{24} - \frac{2649568977042357137450493049281218154962492759416296444477746739272318506312678226784850879552957638140471911364536409816636726584171694440631702173677683898447484858863104276700565753741141158522940672140806610625349162030038823111}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{23} + \frac{1488667662517568777722397059213824754922704222105807784661656647546559980685654498330212743990001160533719263360441844374194721426968398509199909619998534823875069925982133179601529992809168429114898591423662986078894565041995665698}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{22} + \frac{59145140072119538856611977344951250439820467812462771794167897456731574740946297435450165871702099394749460887168762028241708029577863316478797883327471616716291178404820074477637044413889762993189220372827264498251330875613020596}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{21} + \frac{105158528663922125167387198462157710598202896895884269278322720265333436849723215473720799733717292956272068504567621705730468546511353442524616638029053424536202209512103942952927525009420733081016038102837241709366639163900532608}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{20} - \frac{198141505925265840474999138530301869607224639577770060662654086162467201748818040240062241260545220551892071632006032766056399310364900720762229163163490891294079835980923601792000128789172984830845576331047378384705310890870811646}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{19} + \frac{3171111389446206555144816743413077768766919622279951668012512203184233765555036752202911027513324368442935377537434030992205361055167837349483379638806369613558265166792793762100447552441124349242212938674445718075459635349538287235}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{18} + \frac{1554000946175803310397124251258655316367667696692618902298346737632377525234133647093538281961709156684888990661643557304314488405832441686375474208680294692303695197630929461419615874963165776179907318684685354548208602222258394946}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{17} + \frac{2300034001640510645677975936685815002466658273672763698186222044171574031172499207384335170235433564093113801571344956222611334441129088278529025808274090367399873709518011139804114303149578516234287687871352056135690137164590658482}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{16} - \frac{2966843481315155683340177898947829178619404697366949540516928787050648203903516420739801178067389018994689808286475469354183061403285420921712504762348056053814385957769498064437420048254222842205958596877503163635063480260900252386}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{15} - \frac{1381162980375801858261117316055467296082384704172809664676092588316273866485310002755634732954693984637867408685345431036218798011969113009016016201873149291741652908702661209803975847681127263590710058564092559797188123882785129911}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{14} - \frac{1460139641567840558267082825151002484037319646195212372362209482247059700268037604405182181621792993339680179419348312453298401640961593831887382109671498122388126411005400130480238161973131386159838133006053888808286211345374346807}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{13} - \frac{3264926966662615452760549929137903435476896958403166477453700766409153950795262598221364191871536975453426775888561480997397919532377045222227926154179045939537401217949241651455295847136047693837073419253301055202734486565260461916}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{12} + \frac{1869257665329702507814722608954399959569516902743351149377450069115516112471479389153483265197794597427377880908177821127691352151049554225183591832962581671983869942521692050029574666675175079031870187983562488734857612649308202260}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{11} - \frac{2081853532447792656675865634119707861651008245371449200223132011094403238265848993071810023818785595879904568127119212163696331874030017774465202172122449756798345072354892491425098078449762463153201590893478338012036705707349496548}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{10} - \frac{1024909312865146096134151117888965487752008444351760796505227923991483817860372784014980137680850330304324937205718751205820124740556742967849881016650672706587796307610891362371596044462687186277708592572640858960616054036709980298}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{9} - \frac{928995755944336166720482467202296884431957297297443272259095499891247389224877575047547988938473266180980024533565559466464228628321056828823384511231622496355554942601323996985616452840965889001611954797703407072591855349777183575}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{8} + \frac{3668027320605514176154100562646149770888791079554947088915027192012512946878614153325850737195732829441019807657275196411399195431356688577298745431123086535690345171163976223375463679209086087571565364645273394491869347503750221447}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{7} + \frac{1245891200366465366382100227202208045451509336875060256098514806691951030372963605754568719659316226240822328662664753719120487349641936895679170923300766719568037490858455167215292043018629285453017199851361533298549406200809761531}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{6} + \frac{166298747102471064158471455236725985647746032084216645925032703294894544933937867670379057670248637520723198382934576861948770986444669314772785767939090096737232342384560112595804247397937420486797976611207671744509314242439605838}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{5} + \frac{3763488842443713163158503221999851490428834404195621543030112149533403874250059582611105689350791180741234612329826032882651772332250500202378200362139834938679697038433548575134848362640524521625690118572121707513708812993545866501}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{4} + \frac{1189865313481112269228158893791616627283490855631844835559215064856350558287360825226549569356854660398450793791909133073831480168725113828980544936908801715744630689476073363829414709810784904267956863081108071300493754335064012680}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{3} + \frac{1600494890340960069745437158004710928183918634509033651145931997170378001184510794556627951615236830331446872979203812795675834579744641591548317934957916532895503718231679314992340572053543306948449840646762255504341132285865219956}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a^{2} + \frac{1522907157641723427137819031675369761275967915447355461550063513726340548295999580991556523048244258123323306657497568764585543097383801406370038600460738661697443429110271511164331918347523294857549572917036927295055873406996729033}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339} a + \frac{2300311121849420668259003090566231456839751273841801610208407732502142017780642840801743540928215993901083952339300314266569087965444228113926983431538654606801745787651692808080508290808130911158979078962761840330800356647204476407}{7609337974402430717852508914095642813102855709597626713550426661213539055013491413710443740881673750658643079357720878351788017013404466473538492237575920305979290745733209888653374080222909680457365547884837580834117943655465374339}$

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

\(\Q(\sqrt{5}) \), 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.

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

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

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
5Data not computed
7Data not computed
23Data not computed