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Properties

Label 46.46.1970164844...0625.1
Degree $46$
Signature $[46, 0]$
Discriminant $3^{23}\cdot 5^{23}\cdot 47^{45}$
Root discriminant $167.41$
Ramified primes $3, 5, 47$
Class number Not computed
Class group Not computed
Galois group $C_{46}$ (as 46T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![179255809888549, -3486586786238757, 3486586786238757, 72582025669816027, -72582025669816027, -450389684965560613, 450389684965560613, 1249268374599413467, -1249268374599413467, -1937590487084912933, 1937590487084912933, 1915611591133408987, -1915611591133408987, -1307740147376148773, 1307740147376148773, 649294836718939867, -649294836718939867, -242882876618527013, 242882876618527013, 70161935078829787, -70161935078829787, -15925388137943333, 15925388137943333, 2874313868881627, -2874313868881627, -415633982312741, 415633982312741, 48333022342875, -48333022342875, -4520361808165, 4520361808165, 338739637979, -338739637979, -20171264293, 20171264293, 941141723, -941141723, -33642789, 33642789, 889051, -889051, -16357, 16357, 187, -187, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 - 187*x^44 + 187*x^43 + 16357*x^42 - 16357*x^41 - 889051*x^40 + 889051*x^39 + 33642789*x^38 - 33642789*x^37 - 941141723*x^36 + 941141723*x^35 + 20171264293*x^34 - 20171264293*x^33 - 338739637979*x^32 + 338739637979*x^31 + 4520361808165*x^30 - 4520361808165*x^29 - 48333022342875*x^28 + 48333022342875*x^27 + 415633982312741*x^26 - 415633982312741*x^25 - 2874313868881627*x^24 + 2874313868881627*x^23 + 15925388137943333*x^22 - 15925388137943333*x^21 - 70161935078829787*x^20 + 70161935078829787*x^19 + 242882876618527013*x^18 - 242882876618527013*x^17 - 649294836718939867*x^16 + 649294836718939867*x^15 + 1307740147376148773*x^14 - 1307740147376148773*x^13 - 1915611591133408987*x^12 + 1915611591133408987*x^11 + 1937590487084912933*x^10 - 1937590487084912933*x^9 - 1249268374599413467*x^8 + 1249268374599413467*x^7 + 450389684965560613*x^6 - 450389684965560613*x^5 - 72582025669816027*x^4 + 72582025669816027*x^3 + 3486586786238757*x^2 - 3486586786238757*x + 179255809888549)
 
gp: K = bnfinit(x^46 - x^45 - 187*x^44 + 187*x^43 + 16357*x^42 - 16357*x^41 - 889051*x^40 + 889051*x^39 + 33642789*x^38 - 33642789*x^37 - 941141723*x^36 + 941141723*x^35 + 20171264293*x^34 - 20171264293*x^33 - 338739637979*x^32 + 338739637979*x^31 + 4520361808165*x^30 - 4520361808165*x^29 - 48333022342875*x^28 + 48333022342875*x^27 + 415633982312741*x^26 - 415633982312741*x^25 - 2874313868881627*x^24 + 2874313868881627*x^23 + 15925388137943333*x^22 - 15925388137943333*x^21 - 70161935078829787*x^20 + 70161935078829787*x^19 + 242882876618527013*x^18 - 242882876618527013*x^17 - 649294836718939867*x^16 + 649294836718939867*x^15 + 1307740147376148773*x^14 - 1307740147376148773*x^13 - 1915611591133408987*x^12 + 1915611591133408987*x^11 + 1937590487084912933*x^10 - 1937590487084912933*x^9 - 1249268374599413467*x^8 + 1249268374599413467*x^7 + 450389684965560613*x^6 - 450389684965560613*x^5 - 72582025669816027*x^4 + 72582025669816027*x^3 + 3486586786238757*x^2 - 3486586786238757*x + 179255809888549, 1)
 

Normalized defining polynomial

\( x^{46} - x^{45} - 187 x^{44} + 187 x^{43} + 16357 x^{42} - 16357 x^{41} - 889051 x^{40} + 889051 x^{39} + 33642789 x^{38} - 33642789 x^{37} - 941141723 x^{36} + 941141723 x^{35} + 20171264293 x^{34} - 20171264293 x^{33} - 338739637979 x^{32} + 338739637979 x^{31} + 4520361808165 x^{30} - 4520361808165 x^{29} - 48333022342875 x^{28} + 48333022342875 x^{27} + 415633982312741 x^{26} - 415633982312741 x^{25} - 2874313868881627 x^{24} + 2874313868881627 x^{23} + 15925388137943333 x^{22} - 15925388137943333 x^{21} - 70161935078829787 x^{20} + 70161935078829787 x^{19} + 242882876618527013 x^{18} - 242882876618527013 x^{17} - 649294836718939867 x^{16} + 649294836718939867 x^{15} + 1307740147376148773 x^{14} - 1307740147376148773 x^{13} - 1915611591133408987 x^{12} + 1915611591133408987 x^{11} + 1937590487084912933 x^{10} - 1937590487084912933 x^{9} - 1249268374599413467 x^{8} + 1249268374599413467 x^{7} + 450389684965560613 x^{6} - 450389684965560613 x^{5} - 72582025669816027 x^{4} + 72582025669816027 x^{3} + 3486586786238757 x^{2} - 3486586786238757 x + 179255809888549 \)

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

Invariants

Degree:  $46$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[46, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1970164844993531780799086603323019623048928218135963528794760377066463986137444118428099155426025390625=3^{23}\cdot 5^{23}\cdot 47^{45}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.41$
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:  $3, 5, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(705=3\cdot 5\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{705}(256,·)$, $\chi_{705}(1,·)$, $\chi_{705}(644,·)$, $\chi_{705}(389,·)$, $\chi_{705}(134,·)$, $\chi_{705}(136,·)$, $\chi_{705}(526,·)$, $\chi_{705}(271,·)$, $\chi_{705}(16,·)$, $\chi_{705}(661,·)$, $\chi_{705}(601,·)$, $\chi_{705}(539,·)$, $\chi_{705}(29,·)$, $\chi_{705}(286,·)$, $\chi_{705}(419,·)$, $\chi_{705}(164,·)$, $\chi_{705}(166,·)$, $\chi_{705}(44,·)$, $\chi_{705}(541,·)$, $\chi_{705}(689,·)$, $\chi_{705}(434,·)$, $\chi_{705}(179,·)$, $\chi_{705}(569,·)$, $\chi_{705}(571,·)$, $\chi_{705}(316,·)$, $\chi_{705}(61,·)$, $\chi_{705}(704,·)$, $\chi_{705}(449,·)$, $\chi_{705}(451,·)$, $\chi_{705}(196,·)$, $\chi_{705}(584,·)$, $\chi_{705}(331,·)$, $\chi_{705}(464,·)$, $\chi_{705}(599,·)$, $\chi_{705}(344,·)$, $\chi_{705}(676,·)$, $\chi_{705}(346,·)$, $\chi_{705}(359,·)$, $\chi_{705}(104,·)$, $\chi_{705}(361,·)$, $\chi_{705}(106,·)$, $\chi_{705}(241,·)$, $\chi_{705}(374,·)$, $\chi_{705}(121,·)$, $\chi_{705}(509,·)$, $\chi_{705}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{56033036169427} a^{24} + \frac{9412640218401}{56033036169427} a^{23} - \frac{96}{56033036169427} a^{22} - \frac{25467357551487}{56033036169427} a^{21} + \frac{4032}{56033036169427} a^{20} + \frac{10099651009794}{56033036169427} a^{19} - \frac{97280}{56033036169427} a^{18} + \frac{27479923356061}{56033036169427} a^{17} + \frac{1488384}{56033036169427} a^{16} + \frac{1783562834454}{56033036169427} a^{15} - \frac{15040512}{56033036169427} a^{14} - \frac{16646586454904}{56033036169427} a^{13} + \frac{101384192}{56033036169427} a^{12} - \frac{23412375478130}{56033036169427} a^{11} - \frac{449839104}{56033036169427} a^{10} + \frac{11929994850615}{56033036169427} a^{9} + \frac{1265172480}{56033036169427} a^{8} + \frac{27401048527951}{56033036169427} a^{7} - \frac{2099249152}{56033036169427} a^{6} - \frac{17857052647459}{56033036169427} a^{5} + \frac{1799356416}{56033036169427} a^{4} + \frac{6678721923865}{56033036169427} a^{3} - \frac{603979776}{56033036169427} a^{2} - \frac{11402137835144}{56033036169427} a + \frac{33554432}{56033036169427}$, $\frac{1}{56033036169427} a^{25} - \frac{100}{56033036169427} a^{23} - \frac{18382475295823}{56033036169427} a^{22} + \frac{4400}{56033036169427} a^{21} - \frac{7300222880959}{56033036169427} a^{20} - \frac{112000}{56033036169427} a^{19} - \frac{2756711370693}{56033036169427} a^{18} + \frac{1824000}{56033036169427} a^{17} - \frac{17480036903282}{56033036169427} a^{16} - \frac{19845120}{56033036169427} a^{15} + \frac{27612692755861}{56033036169427} a^{14} + \frac{146227200}{56033036169427} a^{13} - \frac{22909842439676}{56033036169427} a^{12} - \frac{724172800}{56033036169427} a^{11} + \frac{13938061079364}{56033036169427} a^{10} + \frac{2342912000}{56033036169427} a^{9} - \frac{23773479628065}{56033036169427} a^{8} - \frac{4685824000}{56033036169427} a^{7} - \frac{8486076913297}{56033036169427} a^{6} + \frac{5248122880}{56033036169427} a^{5} - \frac{1353542991131}{56033036169427} a^{4} - \frac{2726297600}{56033036169427} a^{3} - \frac{10665190037433}{56033036169427} a^{2} + \frac{419430400}{56033036169427} a + \frac{15523790724968}{56033036169427}$, $\frac{1}{56033036169427} a^{26} + \frac{26352967833445}{56033036169427} a^{23} - \frac{5200}{56033036169427} a^{22} + \frac{23483685763983}{56033036169427} a^{21} + \frac{291200}{56033036169427} a^{20} - \frac{1386261440979}{56033036169427} a^{19} - \frac{7904000}{56033036169427} a^{18} - \frac{15106473599105}{56033036169427} a^{17} + \frac{128993280}{56033036169427} a^{16} - \frac{18163168476447}{56033036169427} a^{15} - \frac{1357824000}{56033036169427} a^{14} - \frac{6577402847266}{56033036169427} a^{13} + \frac{9414246400}{56033036169427} a^{12} + \frac{26088032382298}{56033036169427} a^{11} - \frac{42640998400}{56033036169427} a^{10} - \frac{7467754124532}{56033036169427} a^{9} + \frac{121831424000}{56033036169427} a^{8} - \frac{13999996420120}{56033036169427} a^{7} - \frac{204676792320}{56033036169427} a^{6} + \frac{5998349684633}{56033036169427} a^{5} + \frac{177209344000}{56033036169427} a^{4} - \frac{15189431684057}{56033036169427} a^{3} - \frac{59978547200}{56033036169427} a^{2} - \frac{4029269400892}{56033036169427} a + \frac{3355443200}{56033036169427}$, $\frac{1}{56033036169427} a^{27} - \frac{5616}{56033036169427} a^{23} - \frac{24151066018939}{56033036169427} a^{22} + \frac{329472}{56033036169427} a^{21} - \frac{17915988657627}{56033036169427} a^{20} - \frac{9434880}{56033036169427} a^{19} - \frac{21866459693609}{56033036169427} a^{18} + \frac{163897344}{56033036169427} a^{17} - \frac{4388239101619}{56033036169427} a^{16} - \frac{1857503232}{56033036169427} a^{15} + \frac{1678669179280}{56033036169427} a^{14} + \frac{14077919232}{56033036169427} a^{13} + \frac{22318432212087}{56033036169427} a^{12} - \frac{71171702784}{56033036169427} a^{11} - \frac{22761011012344}{56033036169427} a^{10} + \frac{233916334080}{56033036169427} a^{9} - \frac{13012488550219}{56033036169427} a^{8} - \frac{473680576512}{56033036169427} a^{7} - \frac{28014825012205}{56033036169427} a^{6} + \frac{535881056256}{56033036169427} a^{5} - \frac{10049440302236}{56033036169427} a^{4} - \frac{280699600896}{56033036169427} a^{3} + \frac{16580265391625}{56033036169427} a^{2} + \frac{43486543872}{56033036169427} a - \frac{10483813516711}{56033036169427}$, $\frac{1}{56033036169427} a^{28} - \frac{1916707248584}{56033036169427} a^{23} - \frac{209664}{56033036169427} a^{22} + \frac{9745342738512}{56033036169427} a^{21} + \frac{13208832}{56033036169427} a^{20} - \frac{7658992150629}{56033036169427} a^{19} - \frac{382427136}{56033036169427} a^{18} + \frac{7879717934999}{56033036169427} a^{17} + \frac{6501261312}{56033036169427} a^{16} - \frac{11745926854489}{56033036169427} a^{15} - \frac{70389596160}{56033036169427} a^{14} - \frac{1806767924541}{56033036169427} a^{13} + \frac{498201919488}{56033036169427} a^{12} + \frac{2874193454745}{56033036169427} a^{11} - \frac{2292380073984}{56033036169427} a^{10} + \frac{26360370038356}{56033036169427} a^{9} + \frac{6631528071168}{56033036169427} a^{8} - \frac{10443613285931}{56033036169427} a^{7} - \frac{11253502181376}{56033036169427} a^{6} + \frac{3877634842350}{56033036169427} a^{5} + \frac{9824486031360}{56033036169427} a^{4} - \frac{17851643698625}{56033036169427} a^{3} - \frac{3348463878144}{56033036169427} a^{2} + \frac{870445969646}{56033036169427} a + \frac{188441690112}{56033036169427}$, $\frac{1}{56033036169427} a^{29} - \frac{233856}{56033036169427} a^{23} - \frac{6159444617271}{56033036169427} a^{22} + \frac{15434496}{56033036169427} a^{21} - \frac{12054357240867}{56033036169427} a^{20} - \frac{471453696}{56033036169427} a^{19} - \frac{27490088632892}{56033036169427} a^{18} + \frac{8531066880}{56033036169427} a^{17} - \frac{25314944443084}{56033036169427} a^{16} - \frac{99447865344}{56033036169427} a^{15} + \frac{8500117429400}{56033036169427} a^{14} + \frac{769412431872}{56033036169427} a^{13} + \frac{16410056875479}{56033036169427} a^{12} - \frac{3951550267392}{56033036169427} a^{11} + \frac{11976072746649}{56033036169427} a^{10} + \frac{13149696688128}{56033036169427} a^{9} + \frac{44490743929}{199405822667} a^{8} - \frac{26897106862080}{56033036169427} a^{7} - \frac{25893728405998}{56033036169427} a^{6} - \frac{25350410563795}{56033036169427} a^{5} - \frac{16347490701224}{56033036169427} a^{4} - \frac{16184242077696}{56033036169427} a^{3} - \frac{24712167028386}{56033036169427} a^{2} + \frac{2522219544576}{56033036169427} a - \frac{23482315353188}{56033036169427}$, $\frac{1}{56033036169427} a^{30} - \frac{5561410003283}{56033036169427} a^{23} - \frac{7015680}{56033036169427} a^{22} - \frac{11040505558336}{56033036169427} a^{21} + \frac{471453696}{56033036169427} a^{20} - \frac{12011119764705}{56033036169427} a^{19} - \frac{14218444800}{56033036169427} a^{18} + \frac{2789211314356}{56033036169427} a^{17} + \frac{248619663360}{56033036169427} a^{16} - \frac{4550913710564}{56033036169427} a^{15} - \frac{2747901542400}{56033036169427} a^{14} + \frac{7475929786480}{56033036169427} a^{13} + \frac{19757751336960}{56033036169427} a^{12} - \frac{12473553771607}{56033036169427} a^{11} + \frac{20018195521958}{56033036169427} a^{10} - \frac{27526227474268}{56033036169427} a^{9} - \frac{11194112226335}{56033036169427} a^{8} - \frac{8272475399235}{56033036169427} a^{7} - \frac{11975094729064}{56033036169427} a^{6} - \frac{21164815987099}{56033036169427} a^{5} + \frac{12374798756411}{56033036169427} a^{4} + \frac{25664909906283}{56033036169427} a^{3} - \frac{26656002612826}{56033036169427} a^{2} + \frac{18297339903624}{56033036169427} a + \frac{7846905249792}{56033036169427}$, $\frac{1}{56033036169427} a^{31} - \frac{8055040}{56033036169427} a^{23} + \frac{15394495820766}{56033036169427} a^{22} + \frac{567074816}{56033036169427} a^{21} - \frac{1620454298449}{56033036169427} a^{20} - \frac{18043289600}{56033036169427} a^{19} - \frac{12211692238199}{56033036169427} a^{18} + \frac{335826124800}{56033036169427} a^{17} - \frac{27185752118894}{56033036169427} a^{16} - \frac{3996330885120}{56033036169427} a^{15} + \frac{6176569427746}{56033036169427} a^{14} - \frac{24623277633747}{56033036169427} a^{13} + \frac{8736883350843}{56033036169427} a^{12} + \frac{4768364122745}{56033036169427} a^{11} + \frac{24802877131095}{56033036169427} a^{10} - \frac{11319456196670}{56033036169427} a^{9} + \frac{2144832124695}{56033036169427} a^{8} - \frac{11674269200260}{56033036169427} a^{7} + \frac{17583206216696}{56033036169427} a^{6} + \frac{11973698051339}{56033036169427} a^{5} - \frac{15552400006888}{56033036169427} a^{4} - \frac{17788704306716}{56033036169427} a^{3} - \frac{14104529971622}{56033036169427} a^{2} - \frac{3953155563942}{56033036169427} a - \frac{4425789025756}{56033036169427}$, $\frac{1}{56033036169427} a^{32} + \frac{11089894227274}{56033036169427} a^{23} - \frac{206209024}{56033036169427} a^{22} + \frac{2171639116826}{56033036169427} a^{21} + \frac{14434631680}{56033036169427} a^{20} + \frac{4203679659082}{56033036169427} a^{19} - \frac{447768166400}{56033036169427} a^{18} + \frac{13188236276859}{56033036169427} a^{17} + \frac{7992661770240}{56033036169427} a^{16} - \frac{26224122798613}{56033036169427} a^{15} + \frac{22323905094054}{56033036169427} a^{14} - \frac{6377194163226}{56033036169427} a^{13} - \frac{19073456490980}{56033036169427} a^{12} - \frac{7984847161555}{56033036169427} a^{11} + \frac{7355918531925}{56033036169427} a^{10} - \frac{13131201161278}{56033036169427} a^{9} - \frac{18671918736774}{56033036169427} a^{8} + \frac{20625864401375}{56033036169427} a^{7} + \frac{24414731892213}{56033036169427} a^{6} - \frac{19609277586460}{56033036169427} a^{5} + \frac{19575869117758}{56033036169427} a^{4} - \frac{15918223474149}{56033036169427} a^{3} + \frac{5839736305167}{56033036169427} a^{2} + \frac{21373535324443}{56033036169427} a - \frac{9882888909855}{56033036169427}$, $\frac{1}{56033036169427} a^{33} - \frac{243032064}{56033036169427} a^{23} + \frac{2173797716017}{56033036169427} a^{22} + \frac{17822351360}{56033036169427} a^{21} + \frac{4113018493060}{56033036169427} a^{20} - \frac{583276953600}{56033036169427} a^{19} - \frac{21979740655879}{56033036169427} a^{18} + \frac{11082262118400}{56033036169427} a^{17} - \frac{3658008486450}{56033036169427} a^{16} - \frac{21906163048026}{56033036169427} a^{15} - \frac{4383358727436}{56033036169427} a^{14} + \frac{1509259649111}{56033036169427} a^{13} - \frac{8972989098489}{56033036169427} a^{12} + \frac{3392380867180}{56033036169427} a^{11} + \frac{15738183910246}{56033036169427} a^{10} - \frac{15107957090153}{56033036169427} a^{9} - \frac{18544503985216}{56033036169427} a^{8} + \frac{27067831174208}{56033036169427} a^{7} - \frac{15629231648011}{56033036169427} a^{6} - \frac{2639275314711}{56033036169427} a^{5} - \frac{19329691421391}{56033036169427} a^{4} + \frac{23820827968950}{56033036169427} a^{3} - \frac{13922422034222}{56033036169427} a^{2} + \frac{12317383591924}{56033036169427} a + \frac{11299725881497}{56033036169427}$, $\frac{1}{56033036169427} a^{34} + \frac{25988431702801}{56033036169427} a^{23} - \frac{5508726784}{56033036169427} a^{22} - \frac{1544897683415}{56033036169427} a^{21} + \frac{396628328448}{56033036169427} a^{20} - \frac{19795238274587}{56033036169427} a^{19} - \frac{12559897067520}{56033036169427} a^{18} - \frac{8638233844366}{56033036169427} a^{17} + \frac{3620655479988}{56033036169427} a^{16} + \frac{13828570025378}{56033036169427} a^{15} - \frac{11670064314902}{56033036169427} a^{14} + \frac{26503475163146}{56033036169427} a^{13} - \frac{11534094948412}{56033036169427} a^{12} + \frac{3061475633448}{56033036169427} a^{11} - \frac{19980303568732}{56033036169427} a^{10} + \frac{15247411053797}{56033036169427} a^{9} - \frac{4755536242448}{56033036169427} a^{8} - \frac{7076969870559}{56033036169427} a^{7} - \frac{6699213491604}{56033036169427} a^{6} - \frac{12055028401518}{56033036169427} a^{5} - \frac{12722822286161}{56033036169427} a^{4} - \frac{27019604841748}{56033036169427} a^{3} - \frac{23612464216427}{56033036169427} a^{2} - \frac{21976354806770}{56033036169427} a - \frac{26020415428694}{56033036169427}$, $\frac{1}{56033036169427} a^{35} - \frac{6648463360}{56033036169427} a^{23} + \frac{27890954330693}{56033036169427} a^{22} + \frac{501484093440}{56033036169427} a^{21} - \frac{23374227139729}{56033036169427} a^{20} - \frac{16754127667200}{56033036169427} a^{19} - \frac{8561113739899}{56033036169427} a^{18} - \frac{12816959186162}{56033036169427} a^{17} - \frac{26574492891566}{56033036169427} a^{16} + \frac{20158972185221}{56033036169427} a^{15} + \frac{1268304802487}{56033036169427} a^{14} - \frac{9762242712616}{56033036169427} a^{13} - \frac{13611570648348}{56033036169427} a^{12} - \frac{20931687334291}{56033036169427} a^{11} - \frac{20957311860973}{56033036169427} a^{10} + \frac{18993841468472}{56033036169427} a^{9} - \frac{10313129490444}{56033036169427} a^{8} - \frac{7664683828385}{56033036169427} a^{7} - \frac{12911293495497}{56033036169427} a^{6} + \frac{7242150219356}{56033036169427} a^{5} + \frac{5733214478478}{56033036169427} a^{4} + \frac{20134435026528}{56033036169427} a^{3} + \frac{6213633232194}{56033036169427} a^{2} - \frac{13598747593076}{56033036169427} a + \frac{4659782461105}{56033036169427}$, $\frac{1}{56033036169427} a^{36} - \frac{753382524123}{56033036169427} a^{23} - \frac{136768389120}{56033036169427} a^{22} - \frac{21647936459726}{56033036169427} a^{21} + \frac{10052476600320}{56033036169427} a^{20} - \frac{21579704770592}{56033036169427} a^{19} + \frac{12816959186162}{56033036169427} a^{18} - \frac{21122405472786}{56033036169427} a^{17} - \frac{2221940193118}{56033036169427} a^{16} - \frac{11902273207027}{56033036169427} a^{15} + \frac{12914372074259}{56033036169427} a^{14} - \frac{2752248231738}{56033036169427} a^{13} + \frac{6762025833446}{56033036169427} a^{12} - \frac{11581411827480}{56033036169427} a^{11} - \frac{12534490764270}{56033036169427} a^{10} - \frac{68080086813}{199405822667} a^{9} - \frac{10044933199117}{56033036169427} a^{8} + \frac{19457013508010}{56033036169427} a^{7} - \frac{9146315804777}{56033036169427} a^{6} - \frac{69170746580}{56033036169427} a^{5} - \frac{21851344550785}{56033036169427} a^{4} - \frac{20387303028231}{56033036169427} a^{3} + \frac{494381216092}{56033036169427} a^{2} + \frac{4402403947598}{56033036169427} a + \frac{17894727122633}{56033036169427}$, $\frac{1}{56033036169427} a^{37} - \frac{168681013248}{56033036169427} a^{23} + \frac{18093413563320}{56033036169427} a^{22} + \frac{12988438020096}{56033036169427} a^{21} - \frac{9725320655714}{56033036169427} a^{20} + \frac{7444574733976}{56033036169427} a^{19} - \frac{18963042547710}{56033036169427} a^{18} - \frac{14210861489902}{56033036169427} a^{17} - \frac{22527311492919}{56033036169427} a^{16} + \frac{7598815366871}{56033036169427} a^{15} + \frac{19092451872361}{56033036169427} a^{14} + \frac{340852266307}{56033036169427} a^{13} + \frac{25870617807075}{56033036169427} a^{12} + \frac{16859799969505}{56033036169427} a^{11} - \frac{24268094958327}{56033036169427} a^{10} - \frac{17458791326481}{56033036169427} a^{9} - \frac{15122419679343}{56033036169427} a^{8} + \frac{19853550934782}{56033036169427} a^{7} - \frac{24044593377992}{56033036169427} a^{6} - \frac{23162476090579}{56033036169427} a^{5} + \frac{163059227265}{56033036169427} a^{4} - \frac{7319235708562}{56033036169427} a^{3} + \frac{9259562666612}{56033036169427} a^{2} - \frac{26529826666052}{56033036169427} a + \frac{18407836572086}{56033036169427}$, $\frac{1}{56033036169427} a^{38} - \frac{5839964970913}{56033036169427} a^{23} - \frac{3204939251712}{56033036169427} a^{22} + \frac{6857093886069}{56033036169427} a^{21} + \frac{15169986116788}{56033036169427} a^{20} - \frac{9863190185906}{56033036169427} a^{19} - \frac{5820232613231}{56033036169427} a^{18} + \frac{27748021037314}{56033036169427} a^{17} - \frac{14315037724284}{56033036169427} a^{16} - \frac{25293083149986}{56033036169427} a^{15} + \frac{15348602879037}{56033036169427} a^{14} - \frac{26806356729578}{56033036169427} a^{13} - \frac{13743436391841}{56033036169427} a^{12} - \frac{21726868877298}{56033036169427} a^{11} - \frac{12102843596570}{56033036169427} a^{10} + \frac{18920149135450}{56033036169427} a^{9} - \frac{19724505379717}{56033036169427} a^{8} - \frac{16093900662412}{56033036169427} a^{7} + \frac{20509625372148}{56033036169427} a^{6} + \frac{8239814329634}{56033036169427} a^{5} + \frac{27290009541221}{56033036169427} a^{4} - \frac{17261915923087}{56033036169427} a^{3} - \frac{8367120533976}{56033036169427} a^{2} - \frac{7333550685838}{56033036169427} a - \frac{13460825044988}{56033036169427}$, $\frac{1}{56033036169427} a^{39} - \frac{4032020348928}{56033036169427} a^{23} + \frac{6550818372691}{56033036169427} a^{22} - \frac{20804625278194}{56033036169427} a^{21} + \frac{3000381375970}{56033036169427} a^{20} - \frac{23694717219053}{56033036169427} a^{19} - \frac{21123663728400}{56033036169427} a^{18} + \frac{12198565336354}{56033036169427} a^{17} + \frac{16427438030658}{56033036169427} a^{16} - \frac{25067232542512}{56033036169427} a^{15} + \frac{8708408511345}{56033036169427} a^{14} + \frac{8595131802912}{56033036169427} a^{13} + \frac{27343677556123}{56033036169427} a^{12} + \frac{25928910799037}{56033036169427} a^{11} - \frac{16108678453698}{56033036169427} a^{10} + \frac{26980657516250}{56033036169427} a^{9} + \frac{12383381720283}{56033036169427} a^{8} - \frac{13136514940975}{56033036169427} a^{7} - \frac{5806617967918}{56033036169427} a^{6} - \frac{2559184933851}{56033036169427} a^{5} + \frac{10787322094651}{56033036169427} a^{4} + \frac{1381147424618}{56033036169427} a^{3} - \frac{6169015359103}{56033036169427} a^{2} - \frac{15233237401096}{56033036169427} a - \frac{9403535768612}{56033036169427}$, $\frac{1}{56033036169427} a^{40} - \frac{23278944305972}{56033036169427} a^{23} - \frac{15647325589293}{56033036169427} a^{22} + \frac{1719775511043}{56033036169427} a^{21} - \frac{16169159475187}{56033036169427} a^{20} - \frac{25932465300747}{56033036169427} a^{19} + \frac{8512207609514}{56033036169427} a^{18} + \frac{25208648775201}{56033036169427} a^{17} - \frac{24698995491287}{56033036169427} a^{16} - \frac{25579233660416}{56033036169427} a^{15} + \frac{16670428163044}{56033036169427} a^{14} - \frac{26191903303621}{56033036169427} a^{13} + \frac{19227277256548}{56033036169427} a^{12} + \frac{4869441879047}{56033036169427} a^{11} + \frac{17633771031979}{56033036169427} a^{10} - \frac{17999986001033}{56033036169427} a^{9} + \frac{9649538535448}{56033036169427} a^{8} - \frac{126178840072}{56033036169427} a^{7} - \frac{8822631080831}{56033036169427} a^{6} + \frac{3213403257523}{56033036169427} a^{5} - \frac{17264342807725}{56033036169427} a^{4} - \frac{27369543537529}{56033036169427} a^{3} + \frac{26682775344151}{56033036169427} a^{2} + \frac{9732842855491}{56033036169427} a - \frac{5441613828593}{56033036169427}$, $\frac{1}{56033036169427} a^{41} + \frac{20225608835494}{56033036169427} a^{23} + \frac{8262568914811}{56033036169427} a^{22} - \frac{10529992559398}{56033036169427} a^{21} - \frac{20564607411868}{56033036169427} a^{20} + \frac{7077675655662}{56033036169427} a^{19} + \frac{24663351211246}{56033036169427} a^{18} + \frac{19214940290424}{56033036169427} a^{17} + \frac{10780337152809}{56033036169427} a^{16} + \frac{19116702248938}{56033036169427} a^{15} - \frac{22027431661063}{56033036169427} a^{14} - \frac{3604614211796}{56033036169427} a^{13} - \frac{10750395970175}{56033036169427} a^{12} - \frac{899700934181}{56033036169427} a^{11} - \frac{3315597860397}{56033036169427} a^{10} - \frac{21601745912145}{56033036169427} a^{9} - \frac{27417761443254}{56033036169427} a^{8} - \frac{23472406277691}{56033036169427} a^{7} - \frac{2970166978745}{56033036169427} a^{6} - \frac{16408987935054}{56033036169427} a^{5} + \frac{25958976238781}{56033036169427} a^{4} + \frac{26156403114815}{56033036169427} a^{3} - \frac{10910515203592}{56033036169427} a^{2} - \frac{20627646924012}{56033036169427} a + \frac{22937478402504}{56033036169427}$, $\frac{1}{56033036169427} a^{42} + \frac{1203297489500}{56033036169427} a^{23} + \frac{26005225887508}{56033036169427} a^{22} + \frac{12524965853047}{56033036169427} a^{21} - \frac{14509522539861}{56033036169427} a^{20} - \frac{10159747488533}{56033036169427} a^{19} + \frac{22410403887066}{56033036169427} a^{18} - \frac{13335340565864}{56033036169427} a^{17} + \frac{15052538155857}{56033036169427} a^{16} - \frac{24787112109133}{56033036169427} a^{15} - \frac{12669988370149}{56033036169427} a^{14} + \frac{15006621545145}{56033036169427} a^{13} + \frac{20776980902898}{56033036169427} a^{12} + \frac{13803290103506}{56033036169427} a^{11} + \frac{14532667143580}{56033036169427} a^{10} + \frac{21397896478083}{56033036169427} a^{9} - \frac{2528176376296}{56033036169427} a^{8} + \frac{14351540816671}{56033036169427} a^{7} + \frac{2796843206524}{56033036169427} a^{6} + \frac{2863365844893}{56033036169427} a^{5} - \frac{6315176769231}{56033036169427} a^{4} - \frac{26869466744737}{56033036169427} a^{3} + \frac{25145612191465}{56033036169427} a^{2} + \frac{10557368818323}{56033036169427} a - \frac{18107691109034}{56033036169427}$, $\frac{1}{56033036169427} a^{43} + \frac{6718064680364}{56033036169427} a^{23} + \frac{15975452506193}{56033036169427} a^{22} + \frac{22885187265150}{56033036169427} a^{21} + \frac{13018921587616}{56033036169427} a^{20} - \frac{16519047908272}{56033036169427} a^{19} - \frac{9568119938867}{56033036169427} a^{18} - \frac{5091146855154}{56033036169427} a^{17} - \frac{9582640681932}{56033036169427} a^{16} - \frac{22157058376314}{56033036169427} a^{15} + \frac{2918580603561}{56033036169427} a^{14} + \frac{15934574011393}{56033036169427} a^{13} + \frac{20574925261614}{56033036169427} a^{12} + \frac{21084376763023}{56033036169427} a^{11} + \frac{5949163350110}{56033036169427} a^{10} + \frac{23763640847503}{56033036169427} a^{9} + \frac{19278390372614}{56033036169427} a^{8} - \frac{15477089195052}{56033036169427} a^{7} + \frac{24157671530064}{56033036169427} a^{6} - \frac{627612290272}{56033036169427} a^{5} - \frac{21107570402272}{56033036169427} a^{4} + \frac{25955806063600}{56033036169427} a^{3} - \frac{3523716116576}{56033036169427} a^{2} - \frac{13161086706747}{56033036169427} a - \frac{14782449772902}{56033036169427}$, $\frac{1}{56033036169427} a^{44} - \frac{19360866843834}{56033036169427} a^{23} - \frac{4577037453030}{56033036169427} a^{22} + \frac{14044203963334}{56033036169427} a^{21} + \frac{16233666866748}{56033036169427} a^{20} + \frac{5453621201840}{56033036169427} a^{19} + \frac{14940114927665}{56033036169427} a^{18} - \frac{27086728907375}{56033036169427} a^{17} + \frac{13166156980060}{56033036169427} a^{16} + \frac{8727527722406}{56033036169427} a^{15} + \frac{6978832936055}{56033036169427} a^{14} + \frac{3713442932012}{56033036169427} a^{13} - \frac{1559199943109}{56033036169427} a^{12} - \frac{22740676914506}{56033036169427} a^{11} - \frac{10435013300551}{56033036169427} a^{10} + \frac{3912965882058}{56033036169427} a^{9} - \frac{10347058178969}{56033036169427} a^{8} - \frac{6204953095914}{56033036169427} a^{7} + \frac{26555455583566}{56033036169427} a^{6} + \frac{19375929099032}{56033036169427} a^{5} - \frac{5348685852465}{56033036169427} a^{4} + \frac{18975930179829}{56033036169427} a^{3} + \frac{27126510222257}{56033036169427} a^{2} + \frac{21878004478244}{56033036169427} a + \frac{3987693078325}{56033036169427}$, $\frac{1}{56033036169427} a^{45} + \frac{10422461067880}{56033036169427} a^{23} + \frac{4491180546361}{56033036169427} a^{22} - \frac{18927328990724}{56033036169427} a^{21} + \frac{14449351528717}{56033036169427} a^{20} + \frac{20554044990870}{56033036169427} a^{19} - \frac{13768534129144}{56033036169427} a^{18} - \frac{2765376122160}{56033036169427} a^{17} + \frac{23487988676237}{56033036169427} a^{16} - \frac{9724588024924}{56033036169427} a^{15} + \frac{10857780502753}{56033036169427} a^{14} - \frac{13759712815586}{56033036169427} a^{13} - \frac{19016363345587}{56033036169427} a^{12} + \frac{1578260318184}{56033036169427} a^{11} + \frac{1514255971920}{56033036169427} a^{10} + \frac{18502773120223}{56033036169427} a^{9} - \frac{20470970037186}{56033036169427} a^{8} - \frac{20270925066724}{56033036169427} a^{7} + \frac{8181949518388}{56033036169427} a^{6} + \frac{19731130901637}{56033036169427} a^{5} - \frac{27462266349046}{56033036169427} a^{4} - \frac{14168624473849}{56033036169427} a^{3} - \frac{25228655337996}{56033036169427} a^{2} - \frac{19067574887525}{56033036169427} a + \frac{15068822471886}{56033036169427}$

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:  $45$
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

$C_{46}$ (as 46T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{705}) \), \(\Q(\zeta_{47})^+\)

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 $23^{2}$ R R $46$ $23^{2}$ $23^{2}$ $23^{2}$ $46$ $46$ $23^{2}$ $46$ $46$ $23^{2}$ $23^{2}$ R $23^{2}$ $46$

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
3Data not computed
5Data not computed
47Data not computed