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Properties

Label 38.0.105...139.1
Degree $38$
Signature $[0, 19]$
Discriminant $-1.052\times 10^{97}$
Root discriminant \(357.44\)
Ramified prime $419$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - x^37 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811)
 
gp: K = bnfinit(y^38 - y^37 + 6*y^36 - 46*y^35 - 866*y^34 - 2262*y^33 - 8218*y^32 - 21902*y^31 + 431634*y^30 + 845654*y^29 + 12392062*y^28 + 22625646*y^27 + 131347110*y^26 + 99749222*y^25 + 35665294*y^24 - 866315187*y^23 - 258426421*y^22 + 17785842992*y^21 + 90256146871*y^20 + 153552360949*y^19 + 435041961961*y^18 - 1392669637447*y^17 + 1076840441256*y^16 - 9935143209215*y^15 + 31070017846749*y^14 - 17675958023404*y^13 + 138250934902501*y^12 - 327233950365006*y^11 + 433153745352823*y^10 - 1944991946784557*y^9 + 5889532748622531*y^8 - 11577094600459131*y^7 + 25495193933767723*y^6 - 50712315827213047*y^5 + 74651226530405679*y^4 - 82308207923451862*y^3 + 77528185035084423*y^2 - 59368882440546583*y + 25758699005655811, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - x^37 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - x^37 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811)
 

\( x^{38} - x^{37} + 6 x^{36} - 46 x^{35} - 866 x^{34} - 2262 x^{33} - 8218 x^{32} - 21902 x^{31} + \cdots + 25\!\cdots\!11 \) Copy content Toggle raw display

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

Invariants

Degree:  $38$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 19]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-105\!\cdots\!139\) \(\medspace = -\,419^{37}\) 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:  \(357.44\)
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:  $419^{37/38}\approx 357.44431219766653$
Ramified primes:   \(419\) 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:  \(\Q(\sqrt{-419}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(419\)
Dirichlet character group:    $\lbrace$$\chi_{419}(1,·)$, $\chi_{419}(7,·)$, $\chi_{419}(136,·)$, $\chi_{419}(139,·)$, $\chi_{419}(280,·)$, $\chi_{419}(283,·)$, $\chi_{419}(412,·)$, $\chi_{419}(418,·)$, $\chi_{419}(305,·)$, $\chi_{419}(40,·)$, $\chi_{419}(284,·)$, $\chi_{419}(171,·)$, $\chi_{419}(114,·)$, $\chi_{419}(47,·)$, $\chi_{419}(49,·)$, $\chi_{419}(306,·)$, $\chi_{419}(312,·)$, $\chi_{419}(60,·)$, $\chi_{419}(215,·)$, $\chi_{419}(199,·)$, $\chi_{419}(329,·)$, $\chi_{419}(330,·)$, $\chi_{419}(204,·)$, $\chi_{419}(208,·)$, $\chi_{419}(211,·)$, $\chi_{419}(220,·)$, $\chi_{419}(343,·)$, $\chi_{419}(89,·)$, $\chi_{419}(90,·)$, $\chi_{419}(135,·)$, $\chi_{419}(359,·)$, $\chi_{419}(107,·)$, $\chi_{419}(76,·)$, $\chi_{419}(113,·)$, $\chi_{419}(370,·)$, $\chi_{419}(372,·)$, $\chi_{419}(248,·)$, $\chi_{419}(379,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{262144}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13}a^{13}-\frac{1}{13}a$, $\frac{1}{13}a^{14}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{15}-\frac{1}{13}a^{3}$, $\frac{1}{13}a^{16}-\frac{1}{13}a^{4}$, $\frac{1}{13}a^{17}-\frac{1}{13}a^{5}$, $\frac{1}{13}a^{18}-\frac{1}{13}a^{6}$, $\frac{1}{13}a^{19}-\frac{1}{13}a^{7}$, $\frac{1}{13}a^{20}-\frac{1}{13}a^{8}$, $\frac{1}{13}a^{21}-\frac{1}{13}a^{9}$, $\frac{1}{13}a^{22}-\frac{1}{13}a^{10}$, $\frac{1}{169}a^{23}+\frac{1}{169}a^{22}+\frac{3}{169}a^{21}+\frac{5}{169}a^{20}-\frac{2}{169}a^{19}-\frac{5}{169}a^{18}+\frac{4}{169}a^{17}-\frac{6}{169}a^{16}+\frac{2}{169}a^{15}+\frac{3}{169}a^{14}-\frac{6}{169}a^{13}+\frac{6}{13}a^{12}-\frac{66}{169}a^{11}-\frac{40}{169}a^{10}-\frac{3}{169}a^{9}-\frac{83}{169}a^{8}-\frac{76}{169}a^{7}-\frac{60}{169}a^{6}-\frac{56}{169}a^{5}-\frac{7}{169}a^{4}+\frac{50}{169}a^{3}+\frac{23}{169}a^{2}-\frac{33}{169}a-\frac{5}{13}$, $\frac{1}{169}a^{24}+\frac{2}{169}a^{22}+\frac{2}{169}a^{21}+\frac{6}{169}a^{20}-\frac{3}{169}a^{19}-\frac{4}{169}a^{18}+\frac{3}{169}a^{17}-\frac{5}{169}a^{16}+\frac{1}{169}a^{15}+\frac{4}{169}a^{14}+\frac{6}{169}a^{13}+\frac{25}{169}a^{12}+\frac{2}{13}a^{11}+\frac{37}{169}a^{10}-\frac{80}{169}a^{9}-\frac{6}{169}a^{8}+\frac{16}{169}a^{7}+\frac{17}{169}a^{6}+\frac{36}{169}a^{5}+\frac{70}{169}a^{4}-\frac{27}{169}a^{3}-\frac{69}{169}a^{2}+\frac{46}{169}a+\frac{5}{13}$, $\frac{1}{169}a^{25}-\frac{2}{169}a^{13}+\frac{3}{13}a^{12}+\frac{1}{169}a-\frac{3}{13}$, $\frac{1}{169}a^{26}-\frac{2}{169}a^{14}+\frac{1}{169}a^{2}$, $\frac{1}{169}a^{27}-\frac{2}{169}a^{15}+\frac{1}{169}a^{3}$, $\frac{1}{9971}a^{28}-\frac{5}{9971}a^{27}-\frac{2}{9971}a^{26}-\frac{11}{9971}a^{25}-\frac{20}{9971}a^{24}-\frac{28}{9971}a^{23}-\frac{133}{9971}a^{22}+\frac{357}{9971}a^{21}-\frac{6}{767}a^{20}+\frac{337}{9971}a^{19}+\frac{324}{9971}a^{18}-\frac{6}{169}a^{17}+\frac{19}{9971}a^{16}-\frac{209}{9971}a^{15}-\frac{277}{9971}a^{14}-\frac{125}{9971}a^{13}-\frac{1085}{9971}a^{12}-\frac{3404}{9971}a^{11}-\frac{3949}{9971}a^{10}+\frac{358}{9971}a^{9}+\frac{174}{767}a^{8}-\frac{3821}{9971}a^{7}-\frac{2989}{9971}a^{6}-\frac{1843}{9971}a^{5}-\frac{1632}{9971}a^{4}-\frac{4102}{9971}a^{3}+\frac{3048}{9971}a^{2}+\frac{188}{9971}a+\frac{3}{13}$, $\frac{1}{129623}a^{29}+\frac{7}{9971}a^{27}+\frac{274}{129623}a^{26}-\frac{252}{129623}a^{25}+\frac{49}{129623}a^{24}-\frac{214}{129623}a^{23}-\frac{4497}{129623}a^{22}-\frac{2364}{129623}a^{21}-\frac{4065}{129623}a^{20}+\frac{4428}{129623}a^{19}+\frac{1797}{129623}a^{18}-\frac{1751}{129623}a^{17}-\frac{4421}{129623}a^{16}-\frac{2030}{129623}a^{15}+\frac{4154}{129623}a^{14}-\frac{1415}{129623}a^{13}-\frac{36618}{129623}a^{12}-\frac{20261}{129623}a^{11}-\frac{10596}{129623}a^{10}+\frac{54143}{129623}a^{9}-\frac{3072}{129623}a^{8}-\frac{56727}{129623}a^{7}+\frac{40973}{129623}a^{6}-\frac{7012}{129623}a^{5}-\frac{37101}{129623}a^{4}+\frac{41420}{129623}a^{3}-\frac{60328}{129623}a^{2}+\frac{55}{129623}a+\frac{60}{169}$, $\frac{1}{129623}a^{30}-\frac{38}{129623}a^{27}-\frac{70}{129623}a^{26}+\frac{283}{129623}a^{25}+\frac{72}{129623}a^{24}+\frac{352}{129623}a^{23}-\frac{999}{129623}a^{22}-\frac{2804}{129623}a^{21}+\frac{3856}{129623}a^{20}+\frac{1043}{129623}a^{19}+\frac{3280}{129623}a^{18}+\frac{2482}{129623}a^{17}+\frac{76}{129623}a^{16}-\frac{2138}{129623}a^{15}+\frac{4617}{129623}a^{14}+\frac{3136}{129623}a^{13}+\frac{60066}{129623}a^{12}-\frac{22205}{129623}a^{11}+\frac{15429}{129623}a^{10}+\frac{50254}{129623}a^{9}-\frac{45508}{129623}a^{8}+\frac{29728}{129623}a^{7}+\frac{60965}{129623}a^{6}-\frac{23555}{129623}a^{5}+\frac{56474}{129623}a^{4}-\frac{19924}{129623}a^{3}+\frac{31021}{129623}a^{2}-\frac{2968}{9971}a-\frac{4}{13}$, $\frac{1}{129623}a^{31}+\frac{1}{129623}a^{28}-\frac{265}{129623}a^{27}+\frac{205}{129623}a^{26}-\frac{357}{129623}a^{25}+\frac{339}{129623}a^{24}+\frac{210}{129623}a^{23}-\frac{4156}{129623}a^{22}-\frac{3697}{129623}a^{21}+\frac{4137}{129623}a^{20}-\frac{451}{129623}a^{19}+\frac{545}{129623}a^{18}-\frac{2225}{129623}a^{17}+\frac{904}{129623}a^{16}+\frac{1835}{129623}a^{15}+\frac{2304}{129623}a^{14}-\frac{3868}{129623}a^{13}+\frac{4510}{129623}a^{12}+\frac{9995}{129623}a^{11}-\frac{37795}{129623}a^{10}+\frac{59727}{129623}a^{9}+\frac{61955}{129623}a^{8}+\frac{18559}{129623}a^{7}-\frac{5901}{129623}a^{6}+\frac{12976}{129623}a^{5}+\frac{63692}{129623}a^{4}-\frac{34616}{129623}a^{3}-\frac{3795}{9971}a^{2}-\frac{1796}{9971}a-\frac{1}{13}$, $\frac{1}{1685099}a^{32}+\frac{2}{1685099}a^{31}-\frac{5}{1685099}a^{30}+\frac{3}{1685099}a^{29}-\frac{55}{1685099}a^{28}-\frac{1760}{1685099}a^{27}-\frac{999}{1685099}a^{26}-\frac{4582}{1685099}a^{25}+\frac{4903}{1685099}a^{24}-\frac{2544}{1685099}a^{23}+\frac{42232}{1685099}a^{22}+\frac{25067}{1685099}a^{21}-\frac{18937}{1685099}a^{20}-\frac{60078}{1685099}a^{19}+\frac{53451}{1685099}a^{18}-\frac{1050}{1685099}a^{17}-\frac{49181}{1685099}a^{16}-\frac{42373}{1685099}a^{15}-\frac{27167}{1685099}a^{14}-\frac{4793}{129623}a^{13}+\frac{348606}{1685099}a^{12}-\frac{784190}{1685099}a^{11}-\frac{144048}{1685099}a^{10}+\frac{547011}{1685099}a^{9}-\frac{418917}{1685099}a^{8}+\frac{55058}{1685099}a^{7}+\frac{353103}{1685099}a^{6}-\frac{431554}{1685099}a^{5}+\frac{521162}{1685099}a^{4}+\frac{130310}{1685099}a^{3}-\frac{763963}{1685099}a^{2}+\frac{269808}{1685099}a-\frac{439}{2197}$, $\frac{1}{1685099}a^{33}+\frac{4}{1685099}a^{31}+\frac{4}{1685099}a^{29}+\frac{53}{1685099}a^{28}-\frac{2965}{1685099}a^{27}-\frac{4521}{1685099}a^{26}+\frac{690}{1685099}a^{25}+\frac{30}{129623}a^{24}+\frac{322}{129623}a^{23}+\frac{53}{129623}a^{22}-\frac{9908}{1685099}a^{21}+\frac{4157}{129623}a^{20}-\frac{55362}{1685099}a^{19}+\frac{14}{9971}a^{18}+\frac{27188}{1685099}a^{17}+\frac{10918}{1685099}a^{16}-\frac{36918}{1685099}a^{15}+\frac{23082}{1685099}a^{14}+\frac{9150}{1685099}a^{13}+\frac{51687}{129623}a^{12}-\frac{34668}{129623}a^{11}-\frac{24571}{129623}a^{10}+\frac{2471}{1685099}a^{9}+\frac{39900}{129623}a^{8}+\frac{93552}{1685099}a^{7}+\frac{2372}{9971}a^{6}-\frac{355221}{1685099}a^{5}-\frac{483833}{1685099}a^{4}-\frac{739376}{1685099}a^{3}-\frac{330366}{1685099}a^{2}-\frac{705951}{1685099}a-\frac{461}{2197}$, $\frac{1}{1685099}a^{34}+\frac{5}{1685099}a^{31}-\frac{2}{1685099}a^{30}+\frac{2}{1685099}a^{29}-\frac{28}{1685099}a^{28}+\frac{2935}{1685099}a^{27}+\frac{3048}{1685099}a^{26}-\frac{3226}{1685099}a^{25}+\frac{915}{1685099}a^{24}-\frac{3097}{1685099}a^{23}-\frac{62096}{1685099}a^{22}+\frac{58982}{1685099}a^{21}-\frac{8669}{1685099}a^{20}-\frac{28905}{1685099}a^{19}-\frac{37116}{1685099}a^{18}-\frac{79}{1685099}a^{17}-\frac{55318}{1685099}a^{16}-\frac{64384}{1685099}a^{15}-\frac{55693}{1685099}a^{14}+\frac{2148}{129623}a^{13}+\frac{261191}{1685099}a^{12}-\frac{283033}{1685099}a^{11}-\frac{408258}{1685099}a^{10}+\frac{615445}{1685099}a^{9}+\frac{202577}{1685099}a^{8}+\frac{543128}{1685099}a^{7}-\frac{346759}{1685099}a^{6}-\frac{672361}{1685099}a^{5}-\frac{537480}{1685099}a^{4}-\frac{568362}{1685099}a^{3}-\frac{747804}{1685099}a^{2}-\frac{147184}{1685099}a+\frac{261}{2197}$, $\frac{1}{21906287}a^{35}-\frac{1}{21906287}a^{34}-\frac{4}{21906287}a^{33}+\frac{1}{21906287}a^{32}+\frac{60}{21906287}a^{31}+\frac{11}{21906287}a^{30}-\frac{45}{21906287}a^{29}-\frac{487}{21906287}a^{28}-\frac{45506}{21906287}a^{27}-\frac{13795}{21906287}a^{26}+\frac{39248}{21906287}a^{25}-\frac{396}{371293}a^{24}-\frac{44182}{21906287}a^{23}-\frac{32172}{21906287}a^{22}+\frac{8967}{21906287}a^{21}+\frac{379693}{21906287}a^{20}+\frac{646469}{21906287}a^{19}-\frac{717502}{21906287}a^{18}-\frac{596773}{21906287}a^{17}-\frac{158888}{21906287}a^{16}-\frac{90158}{21906287}a^{15}-\frac{32949}{1685099}a^{14}-\frac{139825}{21906287}a^{13}+\frac{8199246}{21906287}a^{12}+\frac{939010}{21906287}a^{11}-\frac{7465226}{21906287}a^{10}-\frac{7519089}{21906287}a^{9}-\frac{8368025}{21906287}a^{8}+\frac{137916}{1685099}a^{7}+\frac{5362807}{21906287}a^{6}+\frac{8376928}{21906287}a^{5}+\frac{3450871}{21906287}a^{4}-\frac{4761852}{21906287}a^{3}+\frac{9572227}{21906287}a^{2}-\frac{9709964}{21906287}a-\frac{8894}{28561}$, $\frac{1}{134572752638857}a^{36}+\frac{1248128}{134572752638857}a^{35}-\frac{1927240}{134572752638857}a^{34}+\frac{10735639}{134572752638857}a^{33}-\frac{34420093}{134572752638857}a^{32}-\frac{101921531}{134572752638857}a^{31}+\frac{268330474}{134572752638857}a^{30}-\frac{611545}{175453393271}a^{29}-\frac{282027124}{10351750202989}a^{28}+\frac{173090539899}{134572752638857}a^{27}-\frac{254445217010}{134572752638857}a^{26}+\frac{372238826618}{134572752638857}a^{25}+\frac{276265922889}{134572752638857}a^{24}-\frac{314952566619}{134572752638857}a^{23}-\frac{2369295696839}{134572752638857}a^{22}+\frac{1310286653304}{134572752638857}a^{21}+\frac{1901541152501}{134572752638857}a^{20}-\frac{4879535581412}{134572752638857}a^{19}-\frac{4698647154554}{134572752638857}a^{18}+\frac{5023042831098}{134572752638857}a^{17}-\frac{448992435620}{134572752638857}a^{16}+\frac{624459860846}{134572752638857}a^{15}-\frac{1562885053324}{134572752638857}a^{14}+\frac{90854860470}{10351750202989}a^{13}-\frac{39210557506415}{134572752638857}a^{12}-\frac{596720300909}{2280894112523}a^{11}-\frac{3542064171858}{134572752638857}a^{10}-\frac{25083869119891}{134572752638857}a^{9}-\frac{3817952153662}{134572752638857}a^{8}+\frac{11685510995361}{134572752638857}a^{7}-\frac{58433865046538}{134572752638857}a^{6}-\frac{59924972027524}{134572752638857}a^{5}-\frac{54481581119575}{134572752638857}a^{4}-\frac{195332700674}{134572752638857}a^{3}+\frac{39531220223216}{134572752638857}a^{2}-\frac{43230768250419}{134572752638857}a+\frac{21368912073}{175453393271}$, $\frac{1}{60\!\cdots\!23}a^{37}-\frac{44\!\cdots\!01}{46\!\cdots\!71}a^{36}-\frac{14\!\cdots\!78}{11\!\cdots\!63}a^{35}-\frac{14\!\cdots\!01}{60\!\cdots\!23}a^{34}+\frac{13\!\cdots\!65}{60\!\cdots\!23}a^{33}-\frac{52\!\cdots\!23}{60\!\cdots\!23}a^{32}+\frac{42\!\cdots\!45}{60\!\cdots\!23}a^{31}+\frac{45\!\cdots\!42}{60\!\cdots\!23}a^{30}+\frac{16\!\cdots\!19}{46\!\cdots\!71}a^{29}+\frac{16\!\cdots\!51}{60\!\cdots\!23}a^{28}-\frac{19\!\cdots\!17}{60\!\cdots\!23}a^{27}-\frac{84\!\cdots\!19}{60\!\cdots\!23}a^{26}+\frac{10\!\cdots\!97}{46\!\cdots\!71}a^{25}-\frac{15\!\cdots\!56}{60\!\cdots\!23}a^{24}-\frac{10\!\cdots\!97}{60\!\cdots\!23}a^{23}-\frac{85\!\cdots\!92}{60\!\cdots\!23}a^{22}+\frac{16\!\cdots\!79}{46\!\cdots\!71}a^{21}+\frac{17\!\cdots\!55}{60\!\cdots\!23}a^{20}+\frac{22\!\cdots\!81}{60\!\cdots\!23}a^{19}-\frac{14\!\cdots\!87}{60\!\cdots\!23}a^{18}+\frac{55\!\cdots\!77}{60\!\cdots\!23}a^{17}-\frac{17\!\cdots\!67}{60\!\cdots\!23}a^{16}+\frac{21\!\cdots\!47}{60\!\cdots\!23}a^{15}-\frac{40\!\cdots\!82}{60\!\cdots\!23}a^{14}+\frac{10\!\cdots\!60}{60\!\cdots\!23}a^{13}-\frac{13\!\cdots\!66}{60\!\cdots\!23}a^{12}+\frac{26\!\cdots\!51}{60\!\cdots\!23}a^{11}+\frac{15\!\cdots\!76}{60\!\cdots\!23}a^{10}+\frac{29\!\cdots\!32}{60\!\cdots\!23}a^{9}+\frac{97\!\cdots\!68}{60\!\cdots\!23}a^{8}-\frac{77\!\cdots\!50}{60\!\cdots\!23}a^{7}+\frac{18\!\cdots\!12}{60\!\cdots\!23}a^{6}+\frac{80\!\cdots\!28}{60\!\cdots\!23}a^{5}-\frac{31\!\cdots\!00}{60\!\cdots\!23}a^{4}-\frac{23\!\cdots\!38}{10\!\cdots\!97}a^{3}+\frac{21\!\cdots\!07}{60\!\cdots\!23}a^{2}+\frac{17\!\cdots\!99}{60\!\cdots\!23}a+\frac{25\!\cdots\!30}{78\!\cdots\!69}$ 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:  $13$

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:  $18$
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^38 - x^37 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811)
 
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^38 - x^37 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811, 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^38 - x^37 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811);
 
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^38 - x^37 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811);
 
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

$C_{38}$ (as 38T1):

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 38
The 38 conjugacy class representatives for $C_{38}$
Character table for $C_{38}$ is not computed

Intermediate fields

\(\Q(\sqrt{-419}) \), 19.19.158435468857090504879482314342339950574787173641.1

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 $38$ $19^{2}$ $19^{2}$ $19^{2}$ $38$ ${\href{/padicField/13.1.0.1}{1} }^{38}$ $38$ $38$ $19^{2}$ $19^{2}$ $38$ $19^{2}$ $19^{2}$ $19^{2}$ $19^{2}$ $38$ ${\href{/padicField/59.1.0.1}{1} }^{38}$

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
\(419\) Copy content Toggle raw display Deg $38$$38$$1$$37$