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Properties

Label 46.46.197...625.1
Degree $46$
Signature $[46, 0]$
Discriminant $1.970\times 10^{102}$
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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Normalized defining polynomial

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

\( x^{46} - x^{45} - 187 x^{44} + 187 x^{43} + 16357 x^{42} - 16357 x^{41} - 889051 x^{40} + \cdots + 179255809888549 \) Copy content Toggle raw display

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

Invariants

Degree:  $46$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[46, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(197\!\cdots\!625\) \(\medspace = 3^{23}\cdot 5^{23}\cdot 47^{45}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(167.41\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}5^{1/2}47^{45/46}\approx 167.4146788277953$
Ramified primes:   \(3\), \(5\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{705}) \)
$\card{ \Gal(K/\Q) }$:  $46$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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 \(a\))

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

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

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $45$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

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

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$

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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $23^{2}$ 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.

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display Deg $46$$2$$23$$23$
\(5\) Copy content Toggle raw display Deg $46$$2$$23$$23$
\(47\) Copy content Toggle raw display Deg $46$$46$$1$$45$