LMFDB - Number field 36.0.29411719834995153896864925426307140281034671856927417346954345703125.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521)

gp: K = bnfinit(y^36 - y^35 + 38*y^34 - 38*y^33 + 667*y^32 - 667*y^31 + 7179*y^30 - 7179*y^29 + 53059*y^28 - 53059*y^27 + 285900*y^26 - 285900*y^25 + 1164687*y^24 - 1164687*y^23 + 3675507*y^22 - 3675507*y^21 + 9151692*y^20 - 9151692*y^19 + 18278667*y^18 - 18278667*y^17 + 29839502*y^16 - 29839502*y^15 + 40834422*y^14 - 40834422*y^13 + 48530866*y^12 - 48530866*y^11 + 52379088*y^10 - 52379088*y^9 + 53693698*y^8 - 53693698*y^7 + 53980522*y^6 - 53980522*y^5 + 54016375*y^4 - 54016375*y^3 + 54018484*y^2 - 54018484*y + 54018521, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521)

$ x^{36} - x^{35} + 38 x^{34} - 38 x^{33} + 667 x^{32} - 667 x^{31} + 7179 x^{30} - 7179 x^{29} + \cdots + 54018521 $ x^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$36$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 18]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$29411719834995153896864925426307140281034671856927417346954345703125$ 29411719834995153896864925426307140281034671856927417346954345703125 $\medspace = 5^{18}\cdot 37^{35}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$74.84$	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:	$5^{1/2}37^{35/36}\approx 74.83858494645301$
Ramified primes:	$5$, $37$ 5, 37	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{37}) $
$\card{ \Gal(K/\Q) }$:	$36$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$185=5\cdot 37$
Dirichlet character group:	$\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(134,·)$, $\chi_{185}(129,·)$, $\chi_{185}(136,·)$, $\chi_{185}(11,·)$, $\chi_{185}(141,·)$, $\chi_{185}(14,·)$, $\chi_{185}(16,·)$, $\chi_{185}(19,·)$, $\chi_{185}(21,·)$, $\chi_{185}(151,·)$, $\chi_{185}(24,·)$, $\chi_{185}(26,·)$, $\chi_{185}(154,·)$, $\chi_{185}(36,·)$, $\chi_{185}(39,·)$, $\chi_{185}(41,·)$, $\chi_{185}(46,·)$, $\chi_{185}(29,·)$, $\chi_{185}(176,·)$, $\chi_{185}(179,·)$, $\chi_{185}(181,·)$, $\chi_{185}(54,·)$, $\chi_{185}(59,·)$, $\chi_{185}(69,·)$, $\chi_{185}(71,·)$, $\chi_{185}(79,·)$, $\chi_{185}(81,·)$, $\chi_{185}(86,·)$, $\chi_{185}(89,·)$, $\chi_{185}(94,·)$, $\chi_{185}(101,·)$, $\chi_{185}(109,·)$, $\chi_{185}(119,·)$, $\chi_{185}(121,·)$, $\chi_{185}(124,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{24157817}a^{19}-\frac{9227465}{24157817}a^{18}+\frac{19}{24157817}a^{17}+\frac{3010349}{24157817}a^{16}+\frac{152}{24157817}a^{15}+\frac{10498709}{24157817}a^{14}+\frac{665}{24157817}a^{13}+\frac{10787863}{24157817}a^{12}+\frac{1729}{24157817}a^{11}+\frac{9898509}{24157817}a^{10}+\frac{2717}{24157817}a^{9}+\frac{8130747}{24157817}a^{8}+\frac{2508}{24157817}a^{7}-\frac{9781297}{24157817}a^{6}+\frac{1254}{24157817}a^{5}-\frac{6320798}{24157817}a^{4}+\frac{285}{24157817}a^{3}+\frac{1467662}{24157817}a^{2}+\frac{19}{24157817}a+\frac{5702887}{24157817}$, $\frac{1}{24157817}a^{20}+\frac{20}{24157817}a^{18}+\frac{9227465}{24157817}a^{17}+\frac{170}{24157817}a^{16}+\frac{11920003}{24157817}a^{15}+\frac{800}{24157817}a^{14}+\frac{10966570}{24157817}a^{13}+\frac{2275}{24157817}a^{12}-\frac{4131543}{24157817}a^{11}+\frac{4004}{24157817}a^{10}+\frac{3339106}{24157817}a^{9}+\frac{4290}{24157817}a^{8}-\frac{10487763}{24157817}a^{7}+\frac{2640}{24157817}a^{6}-\frac{6674031}{24157817}a^{5}+\frac{825}{24157817}a^{4}-\frac{1906866}{24157817}a^{3}+\frac{100}{24157817}a^{2}+\frac{11920003}{24157817}a+\frac{2}{24157817}$, $\frac{1}{24157817}a^{21}+\frac{514229}{24157817}a^{18}-\frac{210}{24157817}a^{17}+\frac{28657}{24157817}a^{16}-\frac{2240}{24157817}a^{15}-\frac{5745074}{24157817}a^{14}-\frac{11025}{24157817}a^{13}-\frac{2468450}{24157817}a^{12}-\frac{30576}{24157817}a^{11}-\frac{1368538}{24157817}a^{10}-\frac{50050}{24157817}a^{9}-\frac{3997984}{24157817}a^{8}-\frac{47520}{24157817}a^{7}-\frac{4310627}{24157817}a^{6}-\frac{24255}{24157817}a^{5}+\frac{3720009}{24157817}a^{4}-\frac{5600}{24157817}a^{3}+\frac{6724580}{24157817}a^{2}-\frac{378}{24157817}a+\frac{6731345}{24157817}$, $\frac{1}{24157817}a^{22}-\frac{231}{24157817}a^{18}-\frac{9741694}{24157817}a^{17}-\frac{2618}{24157817}a^{16}-\frac{11434431}{24157817}a^{15}-\frac{13860}{24157817}a^{14}-\frac{6221297}{24157817}a^{13}-\frac{42042}{24157817}a^{12}+\frac{3368750}{24157817}a^{11}-\frac{77077}{24157817}a^{10}-\frac{4791}{24157817}a^{9}-\frac{84942}{24157817}a^{8}+\frac{10525159}{24157817}a^{7}-\frac{53361}{24157817}a^{6}+\frac{152574}{330929}a^{5}-\frac{16940}{24157817}a^{4}+\frac{5116217}{24157817}a^{3}-\frac{2079}{24157817}a^{2}-\frac{3039006}{24157817}a-\frac{42}{24157817}$, $\frac{1}{24157817}a^{23}+\frac{8759604}{24157817}a^{18}+\frac{1771}{24157817}a^{17}+\frac{7537312}{24157817}a^{16}+\frac{21252}{24157817}a^{15}+\frac{3198782}{24157817}a^{14}+\frac{111573}{24157817}a^{13}+\frac{7109952}{24157817}a^{12}+\frac{322322}{24157817}a^{11}-\frac{8441827}{24157817}a^{10}+\frac{542685}{24157817}a^{9}+\frac{4417990}{24157817}a^{8}+\frac{525987}{24157817}a^{7}-\frac{1664724}{24157817}a^{6}+\frac{272734}{24157817}a^{5}-\frac{5519101}{24157817}a^{4}+\frac{63756}{24157817}a^{3}-\frac{2218522}{24157817}a^{2}+\frac{4347}{24157817}a-\frac{11313038}{24157817}$, $\frac{1}{24157817}a^{24}+\frac{2024}{24157817}a^{18}+\frac{10209555}{24157817}a^{17}+\frac{25806}{24157817}a^{16}+\frac{418909}{24157817}a^{15}+\frac{145728}{24157817}a^{14}+\frac{54893}{330929}a^{13}+\frac{460460}{24157817}a^{12}-\frac{6845884}{24157817}a^{11}+\frac{868296}{24157817}a^{10}+\frac{23667}{24157817}a^{9}+\frac{976833}{24157817}a^{8}-\frac{11295903}{24157817}a^{7}+\frac{623392}{24157817}a^{6}+\frac{1744218}{24157817}a^{5}+\frac{200376}{24157817}a^{4}-\frac{10450511}{24157817}a^{3}+\frac{24840}{24157817}a^{2}-\frac{8640795}{24157817}a+\frac{506}{24157817}$, $\frac{1}{24157817}a^{25}-\frac{11551643}{24157817}a^{18}-\frac{12650}{24157817}a^{17}-\frac{4757583}{24157817}a^{16}-\frac{161920}{24157817}a^{15}-\frac{10658684}{24157817}a^{14}-\frac{885500}{24157817}a^{13}-\frac{2814028}{24157817}a^{12}-\frac{2631200}{24157817}a^{11}-\frac{7728256}{24157817}a^{10}-\frac{4522375}{24157817}a^{9}+\frac{7703363}{24157817}a^{8}-\frac{4452800}{24157817}a^{7}-\frac{141378}{330929}a^{6}-\frac{2337720}{24157817}a^{5}+\frac{3359448}{24157817}a^{4}-\frac{552000}{24157817}a^{3}-\frac{7777192}{24157817}a^{2}-\frac{37950}{24157817}a+\frac{4793238}{24157817}$, $\frac{1}{24157817}a^{26}-\frac{14950}{24157817}a^{18}-\frac{2696719}{24157817}a^{17}-\frac{203320}{24157817}a^{16}+\frac{5828228}{24157817}a^{15}-\frac{1196000}{24157817}a^{14}-\frac{3157239}{24157817}a^{13}-\frac{3887000}{24157817}a^{12}+\frac{10705649}{24157817}a^{11}-\frac{7482475}{24157817}a^{10}-\frac{11644706}{24157817}a^{9}-\frac{8551400}{24157817}a^{8}-\frac{4022533}{24157817}a^{7}-\frac{5525520}{24157817}a^{6}-\frac{5570430}{24157817}a^{5}-\frac{1794000}{24157817}a^{4}-\frac{1022049}{24157817}a^{3}-\frac{224250}{24157817}a^{2}+\frac{6854102}{24157817}a-\frac{4600}{24157817}$, $\frac{1}{24157817}a^{27}+\frac{11994418}{24157817}a^{18}+\frac{80730}{24157817}a^{17}+\frac{4532707}{24157817}a^{16}+\frac{1076400}{24157817}a^{15}-\frac{794738}{24157817}a^{14}+\frac{6054750}{24157817}a^{13}+\frac{11671207}{24157817}a^{12}-\frac{5791742}{24157817}a^{11}+\frac{4435719}{24157817}a^{10}+\frac{7909933}{24157817}a^{9}-\frac{11490027}{24157817}a^{8}+\frac{7811263}{24157817}a^{7}-\frac{8694279}{24157817}a^{6}-\frac{7204517}{24157817}a^{5}+\frac{8427955}{24157817}a^{4}+\frac{4036500}{24157817}a^{3}-\frac{11054651}{24157817}a^{2}+\frac{279450}{24157817}a+\frac{5224457}{24157817}$, $\frac{1}{24157817}a^{28}+\frac{98280}{24157817}a^{18}-\frac{5940882}{24157817}a^{17}+\frac{1392300}{24157817}a^{16}+\frac{12047818}{24157817}a^{15}+\frac{8424000}{24157817}a^{14}+\frac{7462847}{24157817}a^{13}+\frac{3790558}{24157817}a^{12}-\frac{6506017}{24157817}a^{11}+\frac{6338966}{24157817}a^{10}-\frac{11428600}{24157817}a^{9}-\frac{9230271}{24157817}a^{8}+\frac{9945359}{24157817}a^{7}-\frac{7038034}{24157817}a^{6}-\frac{6410043}{24157817}a^{5}-\frac{10644317}{24157817}a^{4}+\frac{946233}{24157817}a^{3}+\frac{1701000}{24157817}a^{2}-\frac{5249132}{24157817}a+\frac{35100}{24157817}$, $\frac{1}{24157817}a^{29}+\frac{9026955}{24157817}a^{18}-\frac{475020}{24157817}a^{17}-\frac{8424920}{24157817}a^{16}-\frac{6514560}{24157817}a^{15}-\frac{1135786}{24157817}a^{14}+\frac{10907809}{24157817}a^{13}+\frac{590839}{24157817}a^{12}+\frac{5517565}{24157817}a^{11}-\frac{1602530}{24157817}a^{10}-\frac{10521044}{24157817}a^{9}-\frac{11756892}{24157817}a^{8}-\frac{11946104}{24157817}a^{7}+\frac{11605053}{24157817}a^{6}+\frac{11059465}{24157817}a^{5}-\frac{9290482}{24157817}a^{4}-\frac{2150983}{24157817}a^{3}-\frac{745185}{24157817}a^{2}-\frac{1832220}{24157817}a+\frac{5777857}{24157817}$, $\frac{1}{24157817}a^{30}-\frac{593775}{24157817}a^{18}-\frac{10832346}{24157817}a^{17}-\frac{8652150}{24157817}a^{16}+\frac{3762623}{24157817}a^{15}-\frac{5124116}{24157817}a^{14}-\frac{11195620}{24157817}a^{13}-\frac{11007031}{24157817}a^{12}-\frac{3257943}{24157817}a^{11}+\frac{5745990}{24157817}a^{10}+\frac{86965}{330929}a^{9}-\frac{6147161}{24157817}a^{8}+\frac{7876442}{24157817}a^{7}-\frac{8588063}{24157817}a^{6}+\frac{924121}{24157817}a^{5}+\frac{6194768}{24157817}a^{4}+\frac{11459059}{24157817}a^{3}-\frac{11451375}{24157817}a^{2}+\frac{3370431}{24157817}a-\frac{237510}{24157817}$, $\frac{1}{24157817}a^{31}-\frac{7651487}{24157817}a^{18}+\frac{2629575}{24157817}a^{17}-\frac{11455366}{24157817}a^{16}-\frac{11501584}{24157817}a^{15}+\frac{7537456}{24157817}a^{14}-\frac{2671728}{24157817}a^{13}-\frac{5871753}{24157817}a^{12}-\frac{6403166}{24157817}a^{11}-\frac{6714912}{24157817}a^{10}-\frac{11434225}{24157817}a^{9}-\frac{919815}{24157817}a^{8}+\frac{6972800}{24157817}a^{7}-\frac{11285816}{24157817}a^{6}+\frac{1896291}{24157817}a^{5}-\frac{10239905}{24157817}a^{4}-\frac{11330219}{24157817}a^{3}-\frac{4715977}{24157817}a^{2}+\frac{11044215}{24157817}a+\frac{6361718}{24157817}$, $\frac{1}{24157817}a^{32}+\frac{3365856}{24157817}a^{18}-\frac{11024015}{24157817}a^{17}+\frac{1751474}{24157817}a^{16}+\frac{10988264}{24157817}a^{15}+\frac{94939}{24157817}a^{14}+\frac{9225532}{24157817}a^{13}+\frac{9081188}{24157817}a^{12}+\frac{8380212}{24157817}a^{11}-\frac{5995441}{24157817}a^{10}-\frac{11710073}{24157817}a^{9}-\frac{9654393}{24157817}a^{8}-\frac{2663118}{24157817}a^{7}+\frac{7753443}{24157817}a^{6}-\frac{5928556}{24157817}a^{5}-\frac{263551}{24157817}a^{4}+\frac{1754288}{24157817}a^{3}-\frac{1790475}{24157817}a^{2}+\frac{6793069}{24157817}a+\frac{1472562}{24157817}$, $\frac{1}{24157817}a^{33}+\frac{3247511}{24157817}a^{18}+\frac{10273661}{24157817}a^{17}+\frac{2139745}{24157817}a^{16}-\frac{4201016}{24157817}a^{15}+\frac{3256182}{24157817}a^{14}-\frac{6693888}{24157817}a^{13}-\frac{2341483}{24157817}a^{12}-\frac{3526568}{24157817}a^{11}-\frac{4197031}{24157817}a^{10}+\frac{1127498}{24157817}a^{9}-\frac{8985087}{24157817}a^{8}-\frac{2735272}{24157817}a^{7}-\frac{10845643}{24157817}a^{6}+\frac{6571000}{24157817}a^{5}+\frac{2034705}{24157817}a^{4}+\frac{5253245}{24157817}a^{3}+\frac{3211459}{24157817}a^{2}+\frac{9994749}{24157817}a+\frac{4385235}{24157817}$, $\frac{1}{24157817}a^{34}+\frac{6001613}{24157817}a^{18}-\frac{11247330}{24157817}a^{17}-\frac{8624429}{24157817}a^{16}-\frac{7209150}{24157817}a^{15}-\frac{3632760}{24157817}a^{14}-\frac{11890585}{24157817}a^{13}+\frac{7243473}{24157817}a^{12}+\frac{9627811}{24157817}a^{11}+\frac{10983998}{24157817}a^{10}+\frac{9288548}{24157817}a^{9}-\frac{5812453}{24157817}a^{8}+\frac{9738915}{24157817}a^{7}+\frac{4177637}{24157817}a^{6}-\frac{11830833}{24157817}a^{5}-\frac{6660060}{24157817}a^{4}-\frac{4332130}{24157817}a^{3}+\frac{2168299}{24157817}a^{2}-\frac{9001840}{24157817}a-\frac{8544096}{24157817}$, $\frac{1}{24157817}a^{35}-\frac{6930889}{24157817}a^{18}-\frac{1865991}{24157817}a^{17}-\frac{1986663}{24157817}a^{16}+\frac{2119110}{24157817}a^{15}-\frac{8958658}{24157817}a^{14}+\frac{2210633}{24157817}a^{13}-\frac{1061469}{24157817}a^{12}-\frac{2101386}{24157817}a^{11}+\frac{8250205}{24157817}a^{10}-\frac{5668499}{24157817}a^{9}-\frac{4706846}{24157817}a^{8}+\frac{2452224}{24157817}a^{7}+\frac{3775594}{24157817}a^{6}+\frac{4556142}{24157817}a^{5}+\frac{7395578}{24157817}a^{4}+\frac{6913601}{24157817}a^{3}+\frac{2420443}{24157817}a^{2}-\frac{1785658}{24157817}a+\frac{8632882}{24157817}$ 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, 1/24157817*a^19 - 9227465/24157817*a^18 + 19/24157817*a^17 + 3010349/24157817*a^16 + 152/24157817*a^15 + 10498709/24157817*a^14 + 665/24157817*a^13 + 10787863/24157817*a^12 + 1729/24157817*a^11 + 9898509/24157817*a^10 + 2717/24157817*a^9 + 8130747/24157817*a^8 + 2508/24157817*a^7 - 9781297/24157817*a^6 + 1254/24157817*a^5 - 6320798/24157817*a^4 + 285/24157817*a^3 + 1467662/24157817*a^2 + 19/24157817*a + 5702887/24157817, 1/24157817*a^20 + 20/24157817*a^18 + 9227465/24157817*a^17 + 170/24157817*a^16 + 11920003/24157817*a^15 + 800/24157817*a^14 + 10966570/24157817*a^13 + 2275/24157817*a^12 - 4131543/24157817*a^11 + 4004/24157817*a^10 + 3339106/24157817*a^9 + 4290/24157817*a^8 - 10487763/24157817*a^7 + 2640/24157817*a^6 - 6674031/24157817*a^5 + 825/24157817*a^4 - 1906866/24157817*a^3 + 100/24157817*a^2 + 11920003/24157817*a + 2/24157817, 1/24157817*a^21 + 514229/24157817*a^18 - 210/24157817*a^17 + 28657/24157817*a^16 - 2240/24157817*a^15 - 5745074/24157817*a^14 - 11025/24157817*a^13 - 2468450/24157817*a^12 - 30576/24157817*a^11 - 1368538/24157817*a^10 - 50050/24157817*a^9 - 3997984/24157817*a^8 - 47520/24157817*a^7 - 4310627/24157817*a^6 - 24255/24157817*a^5 + 3720009/24157817*a^4 - 5600/24157817*a^3 + 6724580/24157817*a^2 - 378/24157817*a + 6731345/24157817, 1/24157817*a^22 - 231/24157817*a^18 - 9741694/24157817*a^17 - 2618/24157817*a^16 - 11434431/24157817*a^15 - 13860/24157817*a^14 - 6221297/24157817*a^13 - 42042/24157817*a^12 + 3368750/24157817*a^11 - 77077/24157817*a^10 - 4791/24157817*a^9 - 84942/24157817*a^8 + 10525159/24157817*a^7 - 53361/24157817*a^6 + 152574/330929*a^5 - 16940/24157817*a^4 + 5116217/24157817*a^3 - 2079/24157817*a^2 - 3039006/24157817*a - 42/24157817, 1/24157817*a^23 + 8759604/24157817*a^18 + 1771/24157817*a^17 + 7537312/24157817*a^16 + 21252/24157817*a^15 + 3198782/24157817*a^14 + 111573/24157817*a^13 + 7109952/24157817*a^12 + 322322/24157817*a^11 - 8441827/24157817*a^10 + 542685/24157817*a^9 + 4417990/24157817*a^8 + 525987/24157817*a^7 - 1664724/24157817*a^6 + 272734/24157817*a^5 - 5519101/24157817*a^4 + 63756/24157817*a^3 - 2218522/24157817*a^2 + 4347/24157817*a - 11313038/24157817, 1/24157817*a^24 + 2024/24157817*a^18 + 10209555/24157817*a^17 + 25806/24157817*a^16 + 418909/24157817*a^15 + 145728/24157817*a^14 + 54893/330929*a^13 + 460460/24157817*a^12 - 6845884/24157817*a^11 + 868296/24157817*a^10 + 23667/24157817*a^9 + 976833/24157817*a^8 - 11295903/24157817*a^7 + 623392/24157817*a^6 + 1744218/24157817*a^5 + 200376/24157817*a^4 - 10450511/24157817*a^3 + 24840/24157817*a^2 - 8640795/24157817*a + 506/24157817, 1/24157817*a^25 - 11551643/24157817*a^18 - 12650/24157817*a^17 - 4757583/24157817*a^16 - 161920/24157817*a^15 - 10658684/24157817*a^14 - 885500/24157817*a^13 - 2814028/24157817*a^12 - 2631200/24157817*a^11 - 7728256/24157817*a^10 - 4522375/24157817*a^9 + 7703363/24157817*a^8 - 4452800/24157817*a^7 - 141378/330929*a^6 - 2337720/24157817*a^5 + 3359448/24157817*a^4 - 552000/24157817*a^3 - 7777192/24157817*a^2 - 37950/24157817*a + 4793238/24157817, 1/24157817*a^26 - 14950/24157817*a^18 - 2696719/24157817*a^17 - 203320/24157817*a^16 + 5828228/24157817*a^15 - 1196000/24157817*a^14 - 3157239/24157817*a^13 - 3887000/24157817*a^12 + 10705649/24157817*a^11 - 7482475/24157817*a^10 - 11644706/24157817*a^9 - 8551400/24157817*a^8 - 4022533/24157817*a^7 - 5525520/24157817*a^6 - 5570430/24157817*a^5 - 1794000/24157817*a^4 - 1022049/24157817*a^3 - 224250/24157817*a^2 + 6854102/24157817*a - 4600/24157817, 1/24157817*a^27 + 11994418/24157817*a^18 + 80730/24157817*a^17 + 4532707/24157817*a^16 + 1076400/24157817*a^15 - 794738/24157817*a^14 + 6054750/24157817*a^13 + 11671207/24157817*a^12 - 5791742/24157817*a^11 + 4435719/24157817*a^10 + 7909933/24157817*a^9 - 11490027/24157817*a^8 + 7811263/24157817*a^7 - 8694279/24157817*a^6 - 7204517/24157817*a^5 + 8427955/24157817*a^4 + 4036500/24157817*a^3 - 11054651/24157817*a^2 + 279450/24157817*a + 5224457/24157817, 1/24157817*a^28 + 98280/24157817*a^18 - 5940882/24157817*a^17 + 1392300/24157817*a^16 + 12047818/24157817*a^15 + 8424000/24157817*a^14 + 7462847/24157817*a^13 + 3790558/24157817*a^12 - 6506017/24157817*a^11 + 6338966/24157817*a^10 - 11428600/24157817*a^9 - 9230271/24157817*a^8 + 9945359/24157817*a^7 - 7038034/24157817*a^6 - 6410043/24157817*a^5 - 10644317/24157817*a^4 + 946233/24157817*a^3 + 1701000/24157817*a^2 - 5249132/24157817*a + 35100/24157817, 1/24157817*a^29 + 9026955/24157817*a^18 - 475020/24157817*a^17 - 8424920/24157817*a^16 - 6514560/24157817*a^15 - 1135786/24157817*a^14 + 10907809/24157817*a^13 + 590839/24157817*a^12 + 5517565/24157817*a^11 - 1602530/24157817*a^10 - 10521044/24157817*a^9 - 11756892/24157817*a^8 - 11946104/24157817*a^7 + 11605053/24157817*a^6 + 11059465/24157817*a^5 - 9290482/24157817*a^4 - 2150983/24157817*a^3 - 745185/24157817*a^2 - 1832220/24157817*a + 5777857/24157817, 1/24157817*a^30 - 593775/24157817*a^18 - 10832346/24157817*a^17 - 8652150/24157817*a^16 + 3762623/24157817*a^15 - 5124116/24157817*a^14 - 11195620/24157817*a^13 - 11007031/24157817*a^12 - 3257943/24157817*a^11 + 5745990/24157817*a^10 + 86965/330929*a^9 - 6147161/24157817*a^8 + 7876442/24157817*a^7 - 8588063/24157817*a^6 + 924121/24157817*a^5 + 6194768/24157817*a^4 + 11459059/24157817*a^3 - 11451375/24157817*a^2 + 3370431/24157817*a - 237510/24157817, 1/24157817*a^31 - 7651487/24157817*a^18 + 2629575/24157817*a^17 - 11455366/24157817*a^16 - 11501584/24157817*a^15 + 7537456/24157817*a^14 - 2671728/24157817*a^13 - 5871753/24157817*a^12 - 6403166/24157817*a^11 - 6714912/24157817*a^10 - 11434225/24157817*a^9 - 919815/24157817*a^8 + 6972800/24157817*a^7 - 11285816/24157817*a^6 + 1896291/24157817*a^5 - 10239905/24157817*a^4 - 11330219/24157817*a^3 - 4715977/24157817*a^2 + 11044215/24157817*a + 6361718/24157817, 1/24157817*a^32 + 3365856/24157817*a^18 - 11024015/24157817*a^17 + 1751474/24157817*a^16 + 10988264/24157817*a^15 + 94939/24157817*a^14 + 9225532/24157817*a^13 + 9081188/24157817*a^12 + 8380212/24157817*a^11 - 5995441/24157817*a^10 - 11710073/24157817*a^9 - 9654393/24157817*a^8 - 2663118/24157817*a^7 + 7753443/24157817*a^6 - 5928556/24157817*a^5 - 263551/24157817*a^4 + 1754288/24157817*a^3 - 1790475/24157817*a^2 + 6793069/24157817*a + 1472562/24157817, 1/24157817*a^33 + 3247511/24157817*a^18 + 10273661/24157817*a^17 + 2139745/24157817*a^16 - 4201016/24157817*a^15 + 3256182/24157817*a^14 - 6693888/24157817*a^13 - 2341483/24157817*a^12 - 3526568/24157817*a^11 - 4197031/24157817*a^10 + 1127498/24157817*a^9 - 8985087/24157817*a^8 - 2735272/24157817*a^7 - 10845643/24157817*a^6 + 6571000/24157817*a^5 + 2034705/24157817*a^4 + 5253245/24157817*a^3 + 3211459/24157817*a^2 + 9994749/24157817*a + 4385235/24157817, 1/24157817*a^34 + 6001613/24157817*a^18 - 11247330/24157817*a^17 - 8624429/24157817*a^16 - 7209150/24157817*a^15 - 3632760/24157817*a^14 - 11890585/24157817*a^13 + 7243473/24157817*a^12 + 9627811/24157817*a^11 + 10983998/24157817*a^10 + 9288548/24157817*a^9 - 5812453/24157817*a^8 + 9738915/24157817*a^7 + 4177637/24157817*a^6 - 11830833/24157817*a^5 - 6660060/24157817*a^4 - 4332130/24157817*a^3 + 2168299/24157817*a^2 - 9001840/24157817*a - 8544096/24157817, 1/24157817*a^35 - 6930889/24157817*a^18 - 1865991/24157817*a^17 - 1986663/24157817*a^16 + 2119110/24157817*a^15 - 8958658/24157817*a^14 + 2210633/24157817*a^13 - 1061469/24157817*a^12 - 2101386/24157817*a^11 + 8250205/24157817*a^10 - 5668499/24157817*a^9 - 4706846/24157817*a^8 + 2452224/24157817*a^7 + 3775594/24157817*a^6 + 4556142/24157817*a^5 + 7395578/24157817*a^4 + 6913601/24157817*a^3 + 2420443/24157817*a^2 - 1785658/24157817*a + 8632882/24157817

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:	$17$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521)

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^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521, 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^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521);

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^36 - x^35 + 38*x^34 - 38*x^33 + 667*x^32 - 667*x^31 + 7179*x^30 - 7179*x^29 + 53059*x^28 - 53059*x^27 + 285900*x^26 - 285900*x^25 + 1164687*x^24 - 1164687*x^23 + 3675507*x^22 - 3675507*x^21 + 9151692*x^20 - 9151692*x^19 + 18278667*x^18 - 18278667*x^17 + 29839502*x^16 - 29839502*x^15 + 40834422*x^14 - 40834422*x^13 + 48530866*x^12 - 48530866*x^11 + 52379088*x^10 - 52379088*x^9 + 53693698*x^8 - 53693698*x^7 + 53980522*x^6 - 53980522*x^5 + 54016375*x^4 - 54016375*x^3 + 54018484*x^2 - 54018484*x + 54018521);

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

$C_{36}$ (as 36T1):

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 36

The 36 conjugacy class representatives for $C_{36}$

Character table for $C_{36}$ is not computed

Intermediate fields

$\Q(\sqrt{37}) $, 3.3.1369.1, 4.0.1266325.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.0.2779962840304068953125.1, $\Q(\zeta_{37})^+$

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	$36$	${\href{/padicField/3.9.0.1}{9} }^{4}$	R	$18^{2}$	${\href{/padicField/11.6.0.1}{6} }^{6}$	$36$	$36$	$36$	${\href{/padicField/23.12.0.1}{12} }^{3}$	${\href{/padicField/29.12.0.1}{12} }^{3}$	${\href{/padicField/31.4.0.1}{4} }^{9}$	R	$18^{2}$	${\href{/padicField/43.4.0.1}{4} }^{9}$	${\href{/padicField/47.6.0.1}{6} }^{6}$	$18^{2}$	$36$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$5$ 5		Deg $36$	$2$	$18$	$18$
$37$ 37		Deg $36$	$36$	$1$	$35$

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