Properties

Label 47.47.609...241.1
Degree $47$
Signature $[47, 0]$
Discriminant $6.097\times 10^{136}$
Root discriminant \(813.43\)
Ramified prime $941$
Class number not computed
Class group not computed
Galois group $C_{47}$ (as 47T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^47 - x^46 - 460*x^45 + 327*x^44 + 94524*x^43 - 38110*x^42 - 11539413*x^41 + 809932*x^40 + 937957195*x^39 + 272017844*x^38 - 53844657814*x^37 - 36415505127*x^36 + 2255802403789*x^35 + 2452634499322*x^34 - 70139481731357*x^33 - 106937338241973*x^32 + 1626084100876542*x^31 + 3250400054052522*x^30 - 27906088812601386*x^29 - 70884683396944095*x^28 + 346512540625848199*x^27 + 1118163243198969746*x^26 - 2955220812359278078*x^25 - 12685164476223506555*x^24 + 15046746180109293999*x^23 + 101582059147167915284*x^22 - 18312922973366227046*x^21 - 554548726919125907181*x^20 - 318711846399705268031*x^19 + 1938753789920516494271*x^18 + 2414984135254760303399*x^17 - 3792697335240007382545*x^16 - 8178953624841789295386*x^15 + 2279595290530927592625*x^14 + 14681478777833447802999*x^13 + 5435523725560836559149*x^12 - 12942546360599189199808*x^11 - 11501227765200381452105*x^10 + 3183515530405052075328*x^9 + 7591082677180116655272*x^8 + 2077754691851067407232*x^7 - 1323128253866746375290*x^6 - 896927209808036028574*x^5 - 132715131473527952574*x^4 + 23448602689225942698*x^3 + 8115247135424866684*x^2 + 574328145019154892*x - 1980612833005069)
 
gp: K = bnfinit(y^47 - y^46 - 460*y^45 + 327*y^44 + 94524*y^43 - 38110*y^42 - 11539413*y^41 + 809932*y^40 + 937957195*y^39 + 272017844*y^38 - 53844657814*y^37 - 36415505127*y^36 + 2255802403789*y^35 + 2452634499322*y^34 - 70139481731357*y^33 - 106937338241973*y^32 + 1626084100876542*y^31 + 3250400054052522*y^30 - 27906088812601386*y^29 - 70884683396944095*y^28 + 346512540625848199*y^27 + 1118163243198969746*y^26 - 2955220812359278078*y^25 - 12685164476223506555*y^24 + 15046746180109293999*y^23 + 101582059147167915284*y^22 - 18312922973366227046*y^21 - 554548726919125907181*y^20 - 318711846399705268031*y^19 + 1938753789920516494271*y^18 + 2414984135254760303399*y^17 - 3792697335240007382545*y^16 - 8178953624841789295386*y^15 + 2279595290530927592625*y^14 + 14681478777833447802999*y^13 + 5435523725560836559149*y^12 - 12942546360599189199808*y^11 - 11501227765200381452105*y^10 + 3183515530405052075328*y^9 + 7591082677180116655272*y^8 + 2077754691851067407232*y^7 - 1323128253866746375290*y^6 - 896927209808036028574*y^5 - 132715131473527952574*y^4 + 23448602689225942698*y^3 + 8115247135424866684*y^2 + 574328145019154892*y - 1980612833005069, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^47 - x^46 - 460*x^45 + 327*x^44 + 94524*x^43 - 38110*x^42 - 11539413*x^41 + 809932*x^40 + 937957195*x^39 + 272017844*x^38 - 53844657814*x^37 - 36415505127*x^36 + 2255802403789*x^35 + 2452634499322*x^34 - 70139481731357*x^33 - 106937338241973*x^32 + 1626084100876542*x^31 + 3250400054052522*x^30 - 27906088812601386*x^29 - 70884683396944095*x^28 + 346512540625848199*x^27 + 1118163243198969746*x^26 - 2955220812359278078*x^25 - 12685164476223506555*x^24 + 15046746180109293999*x^23 + 101582059147167915284*x^22 - 18312922973366227046*x^21 - 554548726919125907181*x^20 - 318711846399705268031*x^19 + 1938753789920516494271*x^18 + 2414984135254760303399*x^17 - 3792697335240007382545*x^16 - 8178953624841789295386*x^15 + 2279595290530927592625*x^14 + 14681478777833447802999*x^13 + 5435523725560836559149*x^12 - 12942546360599189199808*x^11 - 11501227765200381452105*x^10 + 3183515530405052075328*x^9 + 7591082677180116655272*x^8 + 2077754691851067407232*x^7 - 1323128253866746375290*x^6 - 896927209808036028574*x^5 - 132715131473527952574*x^4 + 23448602689225942698*x^3 + 8115247135424866684*x^2 + 574328145019154892*x - 1980612833005069);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^47 - x^46 - 460*x^45 + 327*x^44 + 94524*x^43 - 38110*x^42 - 11539413*x^41 + 809932*x^40 + 937957195*x^39 + 272017844*x^38 - 53844657814*x^37 - 36415505127*x^36 + 2255802403789*x^35 + 2452634499322*x^34 - 70139481731357*x^33 - 106937338241973*x^32 + 1626084100876542*x^31 + 3250400054052522*x^30 - 27906088812601386*x^29 - 70884683396944095*x^28 + 346512540625848199*x^27 + 1118163243198969746*x^26 - 2955220812359278078*x^25 - 12685164476223506555*x^24 + 15046746180109293999*x^23 + 101582059147167915284*x^22 - 18312922973366227046*x^21 - 554548726919125907181*x^20 - 318711846399705268031*x^19 + 1938753789920516494271*x^18 + 2414984135254760303399*x^17 - 3792697335240007382545*x^16 - 8178953624841789295386*x^15 + 2279595290530927592625*x^14 + 14681478777833447802999*x^13 + 5435523725560836559149*x^12 - 12942546360599189199808*x^11 - 11501227765200381452105*x^10 + 3183515530405052075328*x^9 + 7591082677180116655272*x^8 + 2077754691851067407232*x^7 - 1323128253866746375290*x^6 - 896927209808036028574*x^5 - 132715131473527952574*x^4 + 23448602689225942698*x^3 + 8115247135424866684*x^2 + 574328145019154892*x - 1980612833005069)
 

\( x^{47} - x^{46} - 460 x^{45} + 327 x^{44} + 94524 x^{43} - 38110 x^{42} - 11539413 x^{41} + 809932 x^{40} + 937957195 x^{39} + 272017844 x^{38} + \cdots - 19\!\cdots\!69 \) Copy content Toggle raw display

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

Invariants

Degree:  $47$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[47, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(609\!\cdots\!241\) \(\medspace = 941^{46}\) 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:  \(813.43\)
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:  $941^{46/47}\approx 813.4329473077595$
Ramified primes:   \(941\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $47$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(941\)
Dirichlet character group:    $\lbrace$$\chi_{941}(1,·)$, $\chi_{941}(514,·)$, $\chi_{941}(769,·)$, $\chi_{941}(904,·)$, $\chi_{941}(660,·)$, $\chi_{941}(46,·)$, $\chi_{941}(406,·)$, $\chi_{941}(282,·)$, $\chi_{941}(796,·)$, $\chi_{941}(538,·)$, $\chi_{941}(161,·)$, $\chi_{941}(34,·)$, $\chi_{941}(93,·)$, $\chi_{941}(428,·)$, $\chi_{941}(557,·)$, $\chi_{941}(302,·)$, $\chi_{941}(797,·)$, $\chi_{941}(178,·)$, $\chi_{941}(819,·)$, $\chi_{941}(180,·)$, $\chi_{941}(437,·)$, $\chi_{941}(624,·)$, $\chi_{941}(323,·)$, $\chi_{941}(118,·)$, $\chi_{941}(631,·)$, $\chi_{941}(716,·)$, $\chi_{941}(718,·)$, $\chi_{941}(248,·)$, $\chi_{941}(339,·)$, $\chi_{941}(341,·)$, $\chi_{941}(215,·)$, $\chi_{941}(474,·)$, $\chi_{941}(858,·)$, $\chi_{941}(480,·)$, $\chi_{941}(739,·)$, $\chi_{941}(868,·)$, $\chi_{941}(743,·)$, $\chi_{941}(234,·)$, $\chi_{941}(750,·)$, $\chi_{941}(623,·)$, $\chi_{941}(752,·)$, $\chi_{941}(723,·)$, $\chi_{941}(116,·)$, $\chi_{941}(630,·)$, $\chi_{941}(887,·)$, $\chi_{941}(413,·)$, $\chi_{941}(119,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23}a^{20}+\frac{6}{23}a^{19}-\frac{8}{23}a^{18}+\frac{4}{23}a^{17}-\frac{5}{23}a^{16}+\frac{3}{23}a^{15}+\frac{7}{23}a^{14}+\frac{9}{23}a^{13}+\frac{4}{23}a^{12}-\frac{1}{23}a^{9}-\frac{6}{23}a^{8}+\frac{8}{23}a^{7}-\frac{4}{23}a^{6}+\frac{5}{23}a^{5}-\frac{3}{23}a^{4}-\frac{7}{23}a^{3}-\frac{9}{23}a^{2}-\frac{4}{23}a$, $\frac{1}{23}a^{21}+\frac{2}{23}a^{19}+\frac{6}{23}a^{18}-\frac{6}{23}a^{17}+\frac{10}{23}a^{16}-\frac{11}{23}a^{15}-\frac{10}{23}a^{14}-\frac{4}{23}a^{13}-\frac{1}{23}a^{12}-\frac{1}{23}a^{10}-\frac{2}{23}a^{8}-\frac{6}{23}a^{7}+\frac{6}{23}a^{6}-\frac{10}{23}a^{5}+\frac{11}{23}a^{4}+\frac{10}{23}a^{3}+\frac{4}{23}a^{2}+\frac{1}{23}a$, $\frac{1}{23}a^{22}-\frac{6}{23}a^{19}+\frac{10}{23}a^{18}+\frac{2}{23}a^{17}-\frac{1}{23}a^{16}+\frac{7}{23}a^{15}+\frac{5}{23}a^{14}+\frac{4}{23}a^{13}-\frac{8}{23}a^{12}-\frac{1}{23}a^{11}+\frac{6}{23}a^{8}-\frac{10}{23}a^{7}-\frac{2}{23}a^{6}+\frac{1}{23}a^{5}-\frac{7}{23}a^{4}-\frac{5}{23}a^{3}-\frac{4}{23}a^{2}+\frac{8}{23}a$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{23}a^{24}-\frac{1}{23}a^{2}$, $\frac{1}{23}a^{25}-\frac{1}{23}a^{3}$, $\frac{1}{23}a^{26}-\frac{1}{23}a^{4}$, $\frac{1}{23}a^{27}-\frac{1}{23}a^{5}$, $\frac{1}{23}a^{28}-\frac{1}{23}a^{6}$, $\frac{1}{23}a^{29}-\frac{1}{23}a^{7}$, $\frac{1}{23}a^{30}-\frac{1}{23}a^{8}$, $\frac{1}{23}a^{31}-\frac{1}{23}a^{9}$, $\frac{1}{23}a^{32}-\frac{1}{23}a^{10}$, $\frac{1}{23}a^{33}-\frac{1}{23}a^{11}$, $\frac{1}{23}a^{34}-\frac{1}{23}a^{12}$, $\frac{1}{529}a^{35}-\frac{8}{529}a^{34}-\frac{4}{529}a^{33}-\frac{9}{529}a^{32}-\frac{10}{529}a^{31}+\frac{2}{529}a^{30}+\frac{4}{529}a^{29}-\frac{1}{529}a^{28}-\frac{11}{529}a^{27}+\frac{6}{529}a^{26}+\frac{8}{529}a^{25}-\frac{5}{529}a^{24}+\frac{6}{529}a^{23}-\frac{1}{529}a^{22}+\frac{8}{529}a^{21}+\frac{6}{529}a^{20}+\frac{104}{529}a^{19}+\frac{36}{529}a^{18}-\frac{95}{529}a^{17}-\frac{225}{529}a^{16}+\frac{38}{529}a^{15}+\frac{256}{529}a^{14}+\frac{224}{529}a^{13}+\frac{55}{529}a^{12}+\frac{5}{529}a^{11}+\frac{231}{529}a^{10}-\frac{226}{529}a^{9}-\frac{60}{529}a^{8}+\frac{52}{529}a^{7}-\frac{157}{529}a^{6}+\frac{190}{529}a^{5}+\frac{209}{529}a^{4}+\frac{127}{529}a^{3}+\frac{194}{529}a^{2}-\frac{76}{529}a-\frac{8}{23}$, $\frac{1}{529}a^{36}+\frac{1}{529}a^{34}+\frac{5}{529}a^{33}+\frac{10}{529}a^{32}-\frac{9}{529}a^{31}-\frac{3}{529}a^{30}+\frac{8}{529}a^{29}+\frac{4}{529}a^{28}+\frac{10}{529}a^{27}+\frac{10}{529}a^{26}-\frac{10}{529}a^{25}-\frac{11}{529}a^{24}+\frac{1}{529}a^{23}+\frac{1}{529}a^{21}-\frac{9}{529}a^{20}-\frac{236}{529}a^{19}+\frac{9}{529}a^{18}-\frac{157}{529}a^{17}-\frac{60}{529}a^{16}-\frac{222}{529}a^{15}+\frac{248}{529}a^{14}+\frac{145}{529}a^{13}-\frac{199}{529}a^{12}+\frac{225}{529}a^{11}+\frac{12}{529}a^{10}-\frac{189}{529}a^{9}+\frac{170}{529}a^{8}-\frac{63}{529}a^{7}+\frac{199}{529}a^{6}-\frac{65}{529}a^{5}-\frac{18}{529}a^{4}+\frac{129}{529}a^{3}-\frac{19}{529}a^{2}-\frac{171}{529}a+\frac{5}{23}$, $\frac{1}{529}a^{37}-\frac{10}{529}a^{34}-\frac{9}{529}a^{33}+\frac{7}{529}a^{31}+\frac{6}{529}a^{30}+\frac{11}{529}a^{28}-\frac{2}{529}a^{27}+\frac{7}{529}a^{26}+\frac{4}{529}a^{25}+\frac{6}{529}a^{24}-\frac{6}{529}a^{23}+\frac{2}{529}a^{22}+\frac{6}{529}a^{21}+\frac{11}{529}a^{20}-\frac{118}{529}a^{19}+\frac{37}{529}a^{18}-\frac{149}{529}a^{17}+\frac{26}{529}a^{16}+\frac{187}{529}a^{15}-\frac{157}{529}a^{14}+\frac{175}{529}a^{13}+\frac{124}{529}a^{12}+\frac{30}{529}a^{11}+\frac{86}{529}a^{10}+\frac{143}{529}a^{9}+\frac{20}{529}a^{8}-\frac{83}{529}a^{7}-\frac{11}{23}a^{6}-\frac{208}{529}a^{5}-\frac{80}{529}a^{4}-\frac{123}{529}a^{3}+\frac{95}{529}a^{2}+\frac{260}{529}a+\frac{8}{23}$, $\frac{1}{51313}a^{38}+\frac{11}{51313}a^{37}+\frac{11}{51313}a^{36}+\frac{17}{51313}a^{35}-\frac{439}{51313}a^{34}+\frac{308}{51313}a^{33}-\frac{149}{51313}a^{32}-\frac{700}{51313}a^{31}+\frac{248}{51313}a^{30}+\frac{34}{2231}a^{29}-\frac{301}{51313}a^{28}+\frac{557}{51313}a^{27}-\frac{452}{51313}a^{26}-\frac{1017}{51313}a^{25}+\frac{379}{51313}a^{24}-\frac{236}{51313}a^{23}+\frac{990}{51313}a^{22}-\frac{1053}{51313}a^{21}-\frac{992}{51313}a^{20}+\frac{14108}{51313}a^{19}-\frac{9090}{51313}a^{18}+\frac{10563}{51313}a^{17}+\frac{20441}{51313}a^{16}+\frac{17044}{51313}a^{15}+\frac{16023}{51313}a^{14}-\frac{196}{529}a^{13}+\frac{394}{2231}a^{12}+\frac{20621}{51313}a^{11}-\frac{20257}{51313}a^{10}-\frac{21515}{51313}a^{9}+\frac{3584}{51313}a^{8}-\frac{662}{51313}a^{7}-\frac{17898}{51313}a^{6}+\frac{160}{2231}a^{5}-\frac{73}{529}a^{4}-\frac{5288}{51313}a^{3}-\frac{22140}{51313}a^{2}-\frac{19381}{51313}a+\frac{778}{2231}$, $\frac{1}{51313}a^{39}-\frac{13}{51313}a^{37}-\frac{7}{51313}a^{36}-\frac{44}{51313}a^{35}-\frac{392}{51313}a^{34}+\frac{440}{51313}a^{33}-\frac{1098}{51313}a^{32}-\frac{297}{51313}a^{31}-\frac{491}{51313}a^{30}+\frac{894}{51313}a^{29}+\frac{279}{51313}a^{28}-\frac{1050}{51313}a^{27}+\frac{172}{51313}a^{26}+\frac{1}{2231}a^{25}-\frac{1107}{51313}a^{24}-\frac{100}{51313}a^{23}+\frac{1055}{51313}a^{22}+\frac{309}{51313}a^{21}-\frac{297}{51313}a^{20}-\frac{13152}{51313}a^{19}-\frac{2355}{51313}a^{18}-\frac{22323}{51313}a^{17}-\frac{7405}{51313}a^{16}-\frac{25573}{51313}a^{15}-\frac{17367}{51313}a^{14}-\frac{21978}{51313}a^{13}-\frac{228}{2231}a^{12}+\frac{741}{2231}a^{11}-\frac{545}{51313}a^{10}-\frac{18450}{51313}a^{9}+\frac{23740}{51313}a^{8}-\frac{16824}{51313}a^{7}+\frac{1320}{51313}a^{6}-\frac{1389}{51313}a^{5}+\frac{15179}{51313}a^{4}+\frac{3436}{51313}a^{3}+\frac{20944}{51313}a^{2}-\frac{842}{51313}a-\frac{798}{2231}$, $\frac{1}{51313}a^{40}+\frac{39}{51313}a^{37}+\frac{2}{51313}a^{36}+\frac{1}{2231}a^{35}+\frac{747}{51313}a^{34}+\frac{287}{51313}a^{33}-\frac{488}{51313}a^{32}-\frac{182}{51313}a^{31}-\frac{247}{51313}a^{30}-\frac{710}{51313}a^{29}+\frac{81}{51313}a^{28}+\frac{41}{51313}a^{27}+\frac{355}{51313}a^{26}-\frac{1039}{51313}a^{25}-\frac{120}{51313}a^{24}-\frac{364}{51313}a^{23}-\frac{595}{51313}a^{22}+\frac{273}{51313}a^{21}-\frac{537}{51313}a^{20}-\frac{23233}{51313}a^{19}+\frac{22661}{51313}a^{18}-\frac{21697}{51313}a^{17}+\frac{19097}{51313}a^{16}-\frac{3666}{51313}a^{15}-\frac{7097}{51313}a^{14}-\frac{23577}{51313}a^{13}-\frac{21224}{51313}a^{12}+\frac{2815}{51313}a^{11}-\frac{7766}{51313}a^{10}-\frac{23155}{51313}a^{9}+\frac{6391}{51313}a^{8}+\frac{21426}{51313}a^{7}-\frac{487}{51313}a^{6}+\frac{3655}{51313}a^{5}+\frac{19441}{51313}a^{4}-\frac{5896}{51313}a^{3}-\frac{8526}{51313}a^{2}-\frac{23733}{51313}a+\frac{608}{2231}$, $\frac{1}{51313}a^{41}-\frac{39}{51313}a^{37}-\frac{18}{51313}a^{36}-\frac{13}{51313}a^{35}-\frac{925}{51313}a^{34}-\frac{278}{51313}a^{33}-\frac{773}{51313}a^{32}+\frac{475}{51313}a^{31}-\frac{488}{51313}a^{30}-\frac{929}{51313}a^{29}-\frac{151}{51313}a^{28}+\frac{651}{51313}a^{27}+\frac{293}{51313}a^{26}+\frac{743}{51313}a^{25}-\frac{983}{51313}a^{24}-\frac{606}{51313}a^{23}+\frac{463}{51313}a^{22}+\frac{81}{51313}a^{21}+\frac{32}{51313}a^{20}-\frac{3460}{51313}a^{19}-\frac{18715}{51313}a^{18}+\frac{21912}{51313}a^{17}+\frac{4235}{51313}a^{16}-\frac{4162}{51313}a^{15}+\frac{20147}{51313}a^{14}-\frac{22873}{51313}a^{13}+\frac{25466}{51313}a^{12}+\frac{4852}{51313}a^{11}+\frac{15021}{51313}a^{10}+\frac{19618}{51313}a^{9}-\frac{883}{51313}a^{8}-\frac{5127}{51313}a^{7}-\frac{15735}{51313}a^{6}+\frac{14825}{51313}a^{5}+\frac{11176}{51313}a^{4}-\frac{24230}{51313}a^{3}-\frac{13000}{51313}a^{2}+\frac{13049}{51313}a+\frac{19}{2231}$, $\frac{1}{51313}a^{42}+\frac{1}{2231}a^{37}+\frac{28}{51313}a^{36}+\frac{29}{51313}a^{35}-\frac{618}{51313}a^{34}+\frac{472}{51313}a^{33}-\frac{680}{51313}a^{32}-\frac{919}{51313}a^{31}-\frac{763}{51313}a^{30}-\frac{596}{51313}a^{29}+\frac{649}{51313}a^{28}+\frac{94}{51313}a^{27}+\frac{25}{2231}a^{26}-\frac{294}{51313}a^{25}-\frac{957}{51313}a^{24}-\frac{593}{51313}a^{23}-\frac{303}{51313}a^{22}+\frac{42}{2231}a^{21}-\frac{1020}{51313}a^{20}+\frac{20889}{51313}a^{19}-\frac{16475}{51313}a^{18}-\frac{21472}{51313}a^{17}-\frac{24479}{51313}a^{16}-\frac{4419}{51313}a^{15}+\frac{915}{51313}a^{14}-\frac{16438}{51313}a^{13}-\frac{11591}{51313}a^{12}+\frac{1142}{51313}a^{11}-\frac{25057}{51313}a^{10}+\frac{17900}{51313}a^{9}+\frac{12235}{51313}a^{8}-\frac{7700}{51313}a^{7}+\frac{8219}{51313}a^{6}+\frac{19187}{51313}a^{5}-\frac{23066}{51313}a^{4}+\frac{9494}{51313}a^{3}+\frac{24335}{51313}a^{2}-\frac{6679}{51313}a+\frac{660}{2231}$, $\frac{1}{29607601}a^{43}+\frac{202}{29607601}a^{42}-\frac{275}{29607601}a^{41}+\frac{221}{29607601}a^{40}-\frac{169}{29607601}a^{39}-\frac{90}{29607601}a^{38}+\frac{10325}{29607601}a^{37}+\frac{24209}{29607601}a^{36}+\frac{1856}{29607601}a^{35}-\frac{152326}{29607601}a^{34}-\frac{369614}{29607601}a^{33}+\frac{128592}{29607601}a^{32}+\frac{598679}{29607601}a^{31}+\frac{439723}{29607601}a^{30}+\frac{638075}{29607601}a^{29}-\frac{450983}{29607601}a^{28}-\frac{9508}{29607601}a^{27}-\frac{455568}{29607601}a^{26}+\frac{20648}{1287287}a^{25}-\frac{424027}{29607601}a^{24}+\frac{429307}{29607601}a^{23}+\frac{307134}{29607601}a^{22}+\frac{818}{51313}a^{21}-\frac{490906}{29607601}a^{20}+\frac{7383498}{29607601}a^{19}-\frac{5740073}{29607601}a^{18}-\frac{7475787}{29607601}a^{17}+\frac{6456762}{29607601}a^{16}-\frac{12090597}{29607601}a^{15}+\frac{693042}{29607601}a^{14}+\frac{8207541}{29607601}a^{13}-\frac{6235124}{29607601}a^{12}-\frac{3851591}{29607601}a^{11}-\frac{5771025}{29607601}a^{10}+\frac{1735}{29607601}a^{9}-\frac{14072046}{29607601}a^{8}+\frac{12522704}{29607601}a^{7}-\frac{12092399}{29607601}a^{6}-\frac{684008}{29607601}a^{5}+\frac{2543099}{29607601}a^{4}-\frac{1979882}{29607601}a^{3}+\frac{8329051}{29607601}a^{2}-\frac{2634634}{29607601}a-\frac{348734}{1287287}$, $\frac{1}{10333052749}a^{44}+\frac{8}{10333052749}a^{43}-\frac{39463}{10333052749}a^{42}-\frac{62406}{10333052749}a^{41}-\frac{94396}{10333052749}a^{40}+\frac{57507}{10333052749}a^{39}+\frac{73}{29607601}a^{38}-\frac{869847}{10333052749}a^{37}-\frac{6658798}{10333052749}a^{36}-\frac{152153}{449263163}a^{35}+\frac{205130856}{10333052749}a^{34}-\frac{173809617}{10333052749}a^{33}+\frac{111261256}{10333052749}a^{32}-\frac{90746445}{10333052749}a^{31}+\frac{47024562}{10333052749}a^{30}-\frac{82062295}{10333052749}a^{29}-\frac{6704844}{449263163}a^{28}-\frac{98181598}{10333052749}a^{27}+\frac{867757}{449263163}a^{26}-\frac{8107878}{449263163}a^{25}+\frac{68400563}{10333052749}a^{24}+\frac{176388269}{10333052749}a^{23}-\frac{114948300}{10333052749}a^{22}+\frac{8636080}{10333052749}a^{21}-\frac{177301325}{10333052749}a^{20}-\frac{4747014}{449263163}a^{19}-\frac{3977152186}{10333052749}a^{18}-\frac{4037721329}{10333052749}a^{17}+\frac{3603919715}{10333052749}a^{16}-\frac{3535923597}{10333052749}a^{15}+\frac{4516624244}{10333052749}a^{14}-\frac{3194772617}{10333052749}a^{13}+\frac{3293767944}{10333052749}a^{12}+\frac{144195868}{449263163}a^{11}-\frac{216357072}{449263163}a^{10}-\frac{4265202256}{10333052749}a^{9}-\frac{3832014357}{10333052749}a^{8}+\frac{3276649766}{10333052749}a^{7}-\frac{1368437924}{10333052749}a^{6}+\frac{22324968}{106526317}a^{5}+\frac{48589427}{449263163}a^{4}-\frac{4819321725}{10333052749}a^{3}+\frac{4306430892}{10333052749}a^{2}+\frac{2802020173}{10333052749}a-\frac{60238231}{449263163}$, $\frac{1}{14\!\cdots\!33}a^{45}-\frac{50843566}{14\!\cdots\!33}a^{44}-\frac{8219797658}{14\!\cdots\!33}a^{43}+\frac{8853321985640}{14\!\cdots\!33}a^{42}+\frac{12461523134527}{14\!\cdots\!33}a^{41}+\frac{8270961724154}{14\!\cdots\!33}a^{40}-\frac{6660767441529}{14\!\cdots\!33}a^{39}-\frac{602732747360}{14\!\cdots\!33}a^{38}-\frac{346618820570517}{14\!\cdots\!33}a^{37}+\frac{11\!\cdots\!50}{14\!\cdots\!33}a^{36}+\frac{646982985906245}{14\!\cdots\!33}a^{35}-\frac{24\!\cdots\!11}{14\!\cdots\!33}a^{34}+\frac{24\!\cdots\!35}{14\!\cdots\!33}a^{33}+\frac{20\!\cdots\!07}{14\!\cdots\!33}a^{32}+\frac{22\!\cdots\!40}{14\!\cdots\!33}a^{31}+\frac{82\!\cdots\!45}{14\!\cdots\!33}a^{30}-\frac{17\!\cdots\!55}{14\!\cdots\!33}a^{29}-\frac{19\!\cdots\!32}{14\!\cdots\!33}a^{28}-\frac{26\!\cdots\!04}{14\!\cdots\!33}a^{27}-\frac{28\!\cdots\!24}{14\!\cdots\!33}a^{26}+\frac{19\!\cdots\!50}{14\!\cdots\!33}a^{25}+\frac{18\!\cdots\!25}{14\!\cdots\!33}a^{24}+\frac{14\!\cdots\!88}{14\!\cdots\!33}a^{23}+\frac{28\!\cdots\!88}{14\!\cdots\!33}a^{22}-\frac{91\!\cdots\!60}{14\!\cdots\!33}a^{21}-\frac{29\!\cdots\!06}{14\!\cdots\!33}a^{20}-\frac{55\!\cdots\!52}{14\!\cdots\!33}a^{19}-\frac{62\!\cdots\!46}{14\!\cdots\!33}a^{18}+\frac{50\!\cdots\!92}{14\!\cdots\!33}a^{17}-\frac{87\!\cdots\!33}{14\!\cdots\!33}a^{16}+\frac{51\!\cdots\!59}{14\!\cdots\!33}a^{15}-\frac{67\!\cdots\!23}{14\!\cdots\!33}a^{14}-\frac{42\!\cdots\!12}{14\!\cdots\!33}a^{13}+\frac{30\!\cdots\!84}{14\!\cdots\!33}a^{12}-\frac{49\!\cdots\!28}{14\!\cdots\!33}a^{11}+\frac{19\!\cdots\!71}{14\!\cdots\!33}a^{10}+\frac{44\!\cdots\!58}{14\!\cdots\!33}a^{9}-\frac{23\!\cdots\!69}{14\!\cdots\!33}a^{8}+\frac{19\!\cdots\!02}{14\!\cdots\!33}a^{7}-\frac{15\!\cdots\!71}{14\!\cdots\!33}a^{6}-\frac{33\!\cdots\!96}{14\!\cdots\!33}a^{5}-\frac{62\!\cdots\!32}{14\!\cdots\!33}a^{4}+\frac{11\!\cdots\!90}{14\!\cdots\!33}a^{3}-\frac{52\!\cdots\!38}{14\!\cdots\!33}a^{2}+\frac{32\!\cdots\!05}{14\!\cdots\!33}a+\frac{28\!\cdots\!75}{60\!\cdots\!71}$, $\frac{1}{24\!\cdots\!99}a^{46}+\frac{77\!\cdots\!85}{24\!\cdots\!99}a^{45}+\frac{93\!\cdots\!02}{24\!\cdots\!99}a^{44}-\frac{40\!\cdots\!95}{24\!\cdots\!99}a^{43}+\frac{10\!\cdots\!45}{24\!\cdots\!99}a^{42}-\frac{10\!\cdots\!72}{24\!\cdots\!99}a^{41}+\frac{22\!\cdots\!35}{24\!\cdots\!99}a^{40}+\frac{12\!\cdots\!00}{24\!\cdots\!99}a^{39}-\frac{16\!\cdots\!83}{24\!\cdots\!99}a^{38}-\frac{89\!\cdots\!28}{10\!\cdots\!13}a^{37}+\frac{79\!\cdots\!98}{24\!\cdots\!99}a^{36}-\frac{66\!\cdots\!80}{24\!\cdots\!99}a^{35}-\frac{34\!\cdots\!92}{24\!\cdots\!99}a^{34}+\frac{16\!\cdots\!50}{24\!\cdots\!99}a^{33}-\frac{11\!\cdots\!39}{24\!\cdots\!99}a^{32}-\frac{33\!\cdots\!24}{24\!\cdots\!99}a^{31}+\frac{49\!\cdots\!51}{24\!\cdots\!99}a^{30}+\frac{27\!\cdots\!19}{24\!\cdots\!99}a^{29}-\frac{65\!\cdots\!83}{24\!\cdots\!99}a^{28}+\frac{40\!\cdots\!79}{24\!\cdots\!99}a^{27}-\frac{38\!\cdots\!22}{24\!\cdots\!99}a^{26}-\frac{72\!\cdots\!23}{24\!\cdots\!99}a^{25}-\frac{45\!\cdots\!63}{24\!\cdots\!99}a^{24}+\frac{19\!\cdots\!95}{24\!\cdots\!99}a^{23}-\frac{21\!\cdots\!73}{24\!\cdots\!99}a^{22}-\frac{37\!\cdots\!23}{24\!\cdots\!99}a^{21}-\frac{15\!\cdots\!27}{24\!\cdots\!99}a^{20}+\frac{43\!\cdots\!18}{10\!\cdots\!13}a^{19}-\frac{13\!\cdots\!63}{24\!\cdots\!99}a^{18}+\frac{95\!\cdots\!59}{24\!\cdots\!99}a^{17}-\frac{10\!\cdots\!38}{24\!\cdots\!99}a^{16}+\frac{59\!\cdots\!82}{24\!\cdots\!99}a^{15}-\frac{11\!\cdots\!76}{24\!\cdots\!99}a^{14}-\frac{76\!\cdots\!13}{24\!\cdots\!99}a^{13}-\frac{75\!\cdots\!01}{24\!\cdots\!99}a^{12}+\frac{53\!\cdots\!70}{24\!\cdots\!99}a^{11}-\frac{49\!\cdots\!08}{24\!\cdots\!99}a^{10}-\frac{89\!\cdots\!28}{24\!\cdots\!99}a^{9}-\frac{52\!\cdots\!67}{24\!\cdots\!99}a^{8}-\frac{79\!\cdots\!84}{24\!\cdots\!99}a^{7}-\frac{11\!\cdots\!36}{24\!\cdots\!99}a^{6}-\frac{38\!\cdots\!82}{24\!\cdots\!99}a^{5}+\frac{12\!\cdots\!88}{10\!\cdots\!13}a^{4}+\frac{22\!\cdots\!52}{24\!\cdots\!99}a^{3}-\frac{16\!\cdots\!05}{24\!\cdots\!99}a^{2}-\frac{77\!\cdots\!18}{24\!\cdots\!99}a-\frac{19\!\cdots\!93}{10\!\cdots\!13}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $23$

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:  $46$
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^47 - x^46 - 460*x^45 + 327*x^44 + 94524*x^43 - 38110*x^42 - 11539413*x^41 + 809932*x^40 + 937957195*x^39 + 272017844*x^38 - 53844657814*x^37 - 36415505127*x^36 + 2255802403789*x^35 + 2452634499322*x^34 - 70139481731357*x^33 - 106937338241973*x^32 + 1626084100876542*x^31 + 3250400054052522*x^30 - 27906088812601386*x^29 - 70884683396944095*x^28 + 346512540625848199*x^27 + 1118163243198969746*x^26 - 2955220812359278078*x^25 - 12685164476223506555*x^24 + 15046746180109293999*x^23 + 101582059147167915284*x^22 - 18312922973366227046*x^21 - 554548726919125907181*x^20 - 318711846399705268031*x^19 + 1938753789920516494271*x^18 + 2414984135254760303399*x^17 - 3792697335240007382545*x^16 - 8178953624841789295386*x^15 + 2279595290530927592625*x^14 + 14681478777833447802999*x^13 + 5435523725560836559149*x^12 - 12942546360599189199808*x^11 - 11501227765200381452105*x^10 + 3183515530405052075328*x^9 + 7591082677180116655272*x^8 + 2077754691851067407232*x^7 - 1323128253866746375290*x^6 - 896927209808036028574*x^5 - 132715131473527952574*x^4 + 23448602689225942698*x^3 + 8115247135424866684*x^2 + 574328145019154892*x - 1980612833005069)
 
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^47 - x^46 - 460*x^45 + 327*x^44 + 94524*x^43 - 38110*x^42 - 11539413*x^41 + 809932*x^40 + 937957195*x^39 + 272017844*x^38 - 53844657814*x^37 - 36415505127*x^36 + 2255802403789*x^35 + 2452634499322*x^34 - 70139481731357*x^33 - 106937338241973*x^32 + 1626084100876542*x^31 + 3250400054052522*x^30 - 27906088812601386*x^29 - 70884683396944095*x^28 + 346512540625848199*x^27 + 1118163243198969746*x^26 - 2955220812359278078*x^25 - 12685164476223506555*x^24 + 15046746180109293999*x^23 + 101582059147167915284*x^22 - 18312922973366227046*x^21 - 554548726919125907181*x^20 - 318711846399705268031*x^19 + 1938753789920516494271*x^18 + 2414984135254760303399*x^17 - 3792697335240007382545*x^16 - 8178953624841789295386*x^15 + 2279595290530927592625*x^14 + 14681478777833447802999*x^13 + 5435523725560836559149*x^12 - 12942546360599189199808*x^11 - 11501227765200381452105*x^10 + 3183515530405052075328*x^9 + 7591082677180116655272*x^8 + 2077754691851067407232*x^7 - 1323128253866746375290*x^6 - 896927209808036028574*x^5 - 132715131473527952574*x^4 + 23448602689225942698*x^3 + 8115247135424866684*x^2 + 574328145019154892*x - 1980612833005069, 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^47 - x^46 - 460*x^45 + 327*x^44 + 94524*x^43 - 38110*x^42 - 11539413*x^41 + 809932*x^40 + 937957195*x^39 + 272017844*x^38 - 53844657814*x^37 - 36415505127*x^36 + 2255802403789*x^35 + 2452634499322*x^34 - 70139481731357*x^33 - 106937338241973*x^32 + 1626084100876542*x^31 + 3250400054052522*x^30 - 27906088812601386*x^29 - 70884683396944095*x^28 + 346512540625848199*x^27 + 1118163243198969746*x^26 - 2955220812359278078*x^25 - 12685164476223506555*x^24 + 15046746180109293999*x^23 + 101582059147167915284*x^22 - 18312922973366227046*x^21 - 554548726919125907181*x^20 - 318711846399705268031*x^19 + 1938753789920516494271*x^18 + 2414984135254760303399*x^17 - 3792697335240007382545*x^16 - 8178953624841789295386*x^15 + 2279595290530927592625*x^14 + 14681478777833447802999*x^13 + 5435523725560836559149*x^12 - 12942546360599189199808*x^11 - 11501227765200381452105*x^10 + 3183515530405052075328*x^9 + 7591082677180116655272*x^8 + 2077754691851067407232*x^7 - 1323128253866746375290*x^6 - 896927209808036028574*x^5 - 132715131473527952574*x^4 + 23448602689225942698*x^3 + 8115247135424866684*x^2 + 574328145019154892*x - 1980612833005069);
 
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^47 - x^46 - 460*x^45 + 327*x^44 + 94524*x^43 - 38110*x^42 - 11539413*x^41 + 809932*x^40 + 937957195*x^39 + 272017844*x^38 - 53844657814*x^37 - 36415505127*x^36 + 2255802403789*x^35 + 2452634499322*x^34 - 70139481731357*x^33 - 106937338241973*x^32 + 1626084100876542*x^31 + 3250400054052522*x^30 - 27906088812601386*x^29 - 70884683396944095*x^28 + 346512540625848199*x^27 + 1118163243198969746*x^26 - 2955220812359278078*x^25 - 12685164476223506555*x^24 + 15046746180109293999*x^23 + 101582059147167915284*x^22 - 18312922973366227046*x^21 - 554548726919125907181*x^20 - 318711846399705268031*x^19 + 1938753789920516494271*x^18 + 2414984135254760303399*x^17 - 3792697335240007382545*x^16 - 8178953624841789295386*x^15 + 2279595290530927592625*x^14 + 14681478777833447802999*x^13 + 5435523725560836559149*x^12 - 12942546360599189199808*x^11 - 11501227765200381452105*x^10 + 3183515530405052075328*x^9 + 7591082677180116655272*x^8 + 2077754691851067407232*x^7 - 1323128253866746375290*x^6 - 896927209808036028574*x^5 - 132715131473527952574*x^4 + 23448602689225942698*x^3 + 8115247135424866684*x^2 + 574328145019154892*x - 1980612833005069);
 
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

$C_{47}$ (as 47T1):

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 47
The 47 conjugacy class representatives for $C_{47}$
Character table for $C_{47}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ ${\href{/padicField/23.1.0.1}{1} }^{47}$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$

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
\(941\) Copy content Toggle raw display Deg $47$$47$$1$$46$