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Properties

Label 38.0.989...491.1
Degree $38$
Signature $[0, 19]$
Discriminant $-9.898\times 10^{101}$
Root discriminant \(483.16\)
Ramified prime $571$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - x^37 + 8*x^36 + 298*x^35 + 570*x^34 + 1040*x^33 + 52682*x^32 + 320665*x^31 + 376717*x^30 + 994384*x^29 + 59327070*x^28 + 133411453*x^27 - 74052511*x^26 + 4371221504*x^25 + 16126217214*x^24 - 1486694372*x^23 + 126436359051*x^22 + 1092190917753*x^21 + 1461367679401*x^20 - 1224471417355*x^19 + 28814095113148*x^18 + 129684809968090*x^17 - 40138477207333*x^16 - 231235168601800*x^15 + 3710898858449432*x^14 + 8115639324673497*x^13 - 15016773845255805*x^12 - 10485965599249387*x^11 + 218552683964585201*x^10 + 358949096963276813*x^9 - 419776350086322541*x^8 - 407970861545640312*x^7 + 4102607542203011336*x^6 + 8187655915670745378*x^5 - 514918423725948339*x^4 - 14957650563715867722*x^3 - 7022951449740975397*x^2 + 19589758432453100807*x + 20824991381571168629)
 
gp: K = bnfinit(y^38 - y^37 + 8*y^36 + 298*y^35 + 570*y^34 + 1040*y^33 + 52682*y^32 + 320665*y^31 + 376717*y^30 + 994384*y^29 + 59327070*y^28 + 133411453*y^27 - 74052511*y^26 + 4371221504*y^25 + 16126217214*y^24 - 1486694372*y^23 + 126436359051*y^22 + 1092190917753*y^21 + 1461367679401*y^20 - 1224471417355*y^19 + 28814095113148*y^18 + 129684809968090*y^17 - 40138477207333*y^16 - 231235168601800*y^15 + 3710898858449432*y^14 + 8115639324673497*y^13 - 15016773845255805*y^12 - 10485965599249387*y^11 + 218552683964585201*y^10 + 358949096963276813*y^9 - 419776350086322541*y^8 - 407970861545640312*y^7 + 4102607542203011336*y^6 + 8187655915670745378*y^5 - 514918423725948339*y^4 - 14957650563715867722*y^3 - 7022951449740975397*y^2 + 19589758432453100807*y + 20824991381571168629, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - x^37 + 8*x^36 + 298*x^35 + 570*x^34 + 1040*x^33 + 52682*x^32 + 320665*x^31 + 376717*x^30 + 994384*x^29 + 59327070*x^28 + 133411453*x^27 - 74052511*x^26 + 4371221504*x^25 + 16126217214*x^24 - 1486694372*x^23 + 126436359051*x^22 + 1092190917753*x^21 + 1461367679401*x^20 - 1224471417355*x^19 + 28814095113148*x^18 + 129684809968090*x^17 - 40138477207333*x^16 - 231235168601800*x^15 + 3710898858449432*x^14 + 8115639324673497*x^13 - 15016773845255805*x^12 - 10485965599249387*x^11 + 218552683964585201*x^10 + 358949096963276813*x^9 - 419776350086322541*x^8 - 407970861545640312*x^7 + 4102607542203011336*x^6 + 8187655915670745378*x^5 - 514918423725948339*x^4 - 14957650563715867722*x^3 - 7022951449740975397*x^2 + 19589758432453100807*x + 20824991381571168629);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - x^37 + 8*x^36 + 298*x^35 + 570*x^34 + 1040*x^33 + 52682*x^32 + 320665*x^31 + 376717*x^30 + 994384*x^29 + 59327070*x^28 + 133411453*x^27 - 74052511*x^26 + 4371221504*x^25 + 16126217214*x^24 - 1486694372*x^23 + 126436359051*x^22 + 1092190917753*x^21 + 1461367679401*x^20 - 1224471417355*x^19 + 28814095113148*x^18 + 129684809968090*x^17 - 40138477207333*x^16 - 231235168601800*x^15 + 3710898858449432*x^14 + 8115639324673497*x^13 - 15016773845255805*x^12 - 10485965599249387*x^11 + 218552683964585201*x^10 + 358949096963276813*x^9 - 419776350086322541*x^8 - 407970861545640312*x^7 + 4102607542203011336*x^6 + 8187655915670745378*x^5 - 514918423725948339*x^4 - 14957650563715867722*x^3 - 7022951449740975397*x^2 + 19589758432453100807*x + 20824991381571168629)
 

\( x^{38} - x^{37} + 8 x^{36} + 298 x^{35} + 570 x^{34} + 1040 x^{33} + 52682 x^{32} + \cdots + 20\!\cdots\!29 \) 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:   \(-989\!\cdots\!491\) \(\medspace = -\,571^{37}\) 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:  \(483.16\)
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:  $571^{37/38}\approx 483.1623142308591$
Ramified primes:   \(571\) 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:  \(\Q(\sqrt{-571}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(571\)
Dirichlet character group:    $\lbrace$$\chi_{571}(512,·)$, $\chi_{571}(1,·)$, $\chi_{571}(131,·)$, $\chi_{571}(516,·)$, $\chi_{571}(390,·)$, $\chi_{571}(8,·)$, $\chi_{571}(265,·)$, $\chi_{571}(94,·)$, $\chi_{571}(271,·)$, $\chi_{571}(401,·)$, $\chi_{571}(407,·)$, $\chi_{571}(540,·)$, $\chi_{571}(31,·)$, $\chi_{571}(164,·)$, $\chi_{571}(170,·)$, $\chi_{571}(300,·)$, $\chi_{571}(221,·)$, $\chi_{571}(306,·)$, $\chi_{571}(563,·)$, $\chi_{571}(181,·)$, $\chi_{571}(55,·)$, $\chi_{571}(440,·)$, $\chi_{571}(570,·)$, $\chi_{571}(59,·)$, $\chi_{571}(64,·)$, $\chi_{571}(323,·)$, $\chi_{571}(455,·)$, $\chi_{571}(214,·)$, $\chi_{571}(472,·)$, $\chi_{571}(218,·)$, $\chi_{571}(477,·)$, $\chi_{571}(350,·)$, $\chi_{571}(353,·)$, $\chi_{571}(99,·)$, $\chi_{571}(357,·)$, $\chi_{571}(116,·)$, $\chi_{571}(248,·)$, $\chi_{571}(507,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{262144}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{109}a^{30}-\frac{5}{109}a^{29}-\frac{3}{109}a^{28}+\frac{14}{109}a^{27}-\frac{36}{109}a^{26}+\frac{39}{109}a^{25}-\frac{54}{109}a^{24}-\frac{21}{109}a^{23}+\frac{28}{109}a^{22}+\frac{33}{109}a^{21}+\frac{32}{109}a^{20}-\frac{13}{109}a^{19}-\frac{10}{109}a^{18}-\frac{41}{109}a^{17}-\frac{7}{109}a^{16}-\frac{27}{109}a^{15}-\frac{1}{109}a^{14}-\frac{7}{109}a^{13}+\frac{29}{109}a^{12}+\frac{19}{109}a^{11}-\frac{28}{109}a^{10}+\frac{53}{109}a^{9}-\frac{22}{109}a^{8}-\frac{39}{109}a^{7}-\frac{31}{109}a^{6}+\frac{18}{109}a^{5}+\frac{23}{109}a^{4}-\frac{21}{109}a^{3}+\frac{12}{109}a^{2}+\frac{53}{109}a+\frac{12}{109}$, $\frac{1}{109}a^{31}-\frac{28}{109}a^{29}-\frac{1}{109}a^{28}+\frac{34}{109}a^{27}-\frac{32}{109}a^{26}+\frac{32}{109}a^{25}+\frac{36}{109}a^{24}+\frac{32}{109}a^{23}-\frac{45}{109}a^{22}-\frac{21}{109}a^{21}+\frac{38}{109}a^{20}+\frac{34}{109}a^{19}+\frac{18}{109}a^{18}+\frac{6}{109}a^{17}+\frac{47}{109}a^{16}-\frac{27}{109}a^{15}-\frac{12}{109}a^{14}-\frac{6}{109}a^{13}-\frac{54}{109}a^{12}-\frac{42}{109}a^{11}+\frac{22}{109}a^{10}+\frac{25}{109}a^{9}-\frac{40}{109}a^{8}-\frac{8}{109}a^{7}-\frac{28}{109}a^{6}+\frac{4}{109}a^{5}-\frac{15}{109}a^{4}+\frac{16}{109}a^{3}+\frac{4}{109}a^{2}-\frac{50}{109}a-\frac{49}{109}$, $\frac{1}{21037}a^{32}-\frac{57}{21037}a^{31}+\frac{45}{21037}a^{30}+\frac{8424}{21037}a^{29}+\frac{3469}{21037}a^{28}-\frac{1166}{21037}a^{27}+\frac{3370}{21037}a^{26}-\frac{4936}{21037}a^{25}-\frac{621}{21037}a^{24}-\frac{2748}{21037}a^{23}+\frac{3716}{21037}a^{22}+\frac{2336}{21037}a^{21}-\frac{8625}{21037}a^{20}-\frac{2869}{21037}a^{19}+\frac{7188}{21037}a^{18}-\frac{8}{109}a^{17}+\frac{8664}{21037}a^{16}+\frac{8930}{21037}a^{15}+\frac{387}{21037}a^{14}-\frac{3602}{21037}a^{13}-\frac{7709}{21037}a^{12}-\frac{1211}{21037}a^{11}+\frac{869}{21037}a^{10}+\frac{1096}{21037}a^{9}-\frac{2604}{21037}a^{8}+\frac{6410}{21037}a^{7}-\frac{2734}{21037}a^{6}+\frac{9137}{21037}a^{5}+\frac{7019}{21037}a^{4}+\frac{8350}{21037}a^{3}-\frac{8885}{21037}a^{2}+\frac{8523}{21037}a+\frac{3451}{21037}$, $\frac{1}{21037}a^{33}+\frac{77}{21037}a^{31}-\frac{12}{21037}a^{30}+\frac{4997}{21037}a^{29}-\frac{5118}{21037}a^{28}-\frac{367}{21037}a^{27}-\frac{5653}{21037}a^{26}+\frac{21}{109}a^{25}+\frac{841}{21037}a^{24}-\frac{6240}{21037}a^{23}-\frac{10118}{21037}a^{22}+\frac{8148}{21037}a^{21}-\frac{6590}{21037}a^{20}-\frac{6963}{21037}a^{19}+\frac{9241}{21037}a^{18}-\frac{8320}{21037}a^{17}-\frac{2303}{21037}a^{16}+\frac{2579}{21037}a^{15}-\frac{9914}{21037}a^{14}-\frac{8443}{21037}a^{13}+\frac{9838}{21037}a^{12}+\frac{5761}{21037}a^{11}+\frac{10099}{21037}a^{10}+\frac{617}{21037}a^{9}-\frac{10199}{21037}a^{8}+\frac{8095}{21037}a^{7}-\frac{2723}{21037}a^{6}+\frac{6342}{21037}a^{5}+\frac{1010}{21037}a^{4}-\frac{6750}{21037}a^{3}-\frac{6737}{21037}a^{2}-\frac{5397}{21037}a+\frac{9111}{21037}$, $\frac{1}{21037}a^{34}-\frac{62}{21037}a^{31}-\frac{12}{21037}a^{30}+\frac{4171}{21037}a^{29}-\frac{5965}{21037}a^{28}-\frac{4265}{21037}a^{27}+\frac{5306}{21037}a^{26}+\frac{10353}{21037}a^{25}+\frac{7223}{21037}a^{24}+\frac{7706}{21037}a^{23}+\frac{4761}{21037}a^{22}+\frac{3064}{21037}a^{21}-\frac{2705}{21037}a^{20}-\frac{5885}{21037}a^{19}+\frac{4852}{21037}a^{18}+\frac{5996}{21037}a^{17}+\frac{143}{21037}a^{16}-\frac{10058}{21037}a^{15}-\frac{4467}{21037}a^{14}+\frac{9079}{21037}a^{13}-\frac{5122}{21037}a^{12}+\frac{8004}{21037}a^{11}+\frac{5500}{21037}a^{10}+\frac{7120}{21037}a^{9}-\frac{609}{21037}a^{8}-\frac{862}{21037}a^{7}+\frac{10350}{21037}a^{6}+\frac{9245}{21037}a^{5}+\frac{9785}{21037}a^{4}+\frac{5934}{21037}a^{3}-\frac{9683}{21037}a^{2}-\frac{2154}{21037}a-\frac{3633}{21037}$, $\frac{1}{536079496789}a^{35}-\frac{9994298}{536079496789}a^{34}+\frac{7705736}{536079496789}a^{33}+\frac{2609654}{536079496789}a^{32}+\frac{1041453955}{536079496789}a^{31}-\frac{1809805076}{536079496789}a^{30}+\frac{48414738482}{536079496789}a^{29}+\frac{202572054375}{536079496789}a^{28}+\frac{148510824659}{536079496789}a^{27}+\frac{52451247252}{536079496789}a^{26}-\frac{31616597687}{536079496789}a^{25}-\frac{186329434250}{536079496789}a^{24}-\frac{220952813615}{536079496789}a^{23}+\frac{12986825630}{536079496789}a^{22}+\frac{257569583421}{536079496789}a^{21}+\frac{39241486520}{536079496789}a^{20}+\frac{70413099379}{536079496789}a^{19}-\frac{99873740828}{536079496789}a^{18}+\frac{1137050762}{536079496789}a^{17}-\frac{183425535233}{536079496789}a^{16}+\frac{207365613526}{536079496789}a^{15}+\frac{89765303259}{536079496789}a^{14}-\frac{83331530965}{536079496789}a^{13}-\frac{78530415247}{536079496789}a^{12}+\frac{171826465360}{536079496789}a^{11}+\frac{611297402}{3210056867}a^{10}-\frac{215827744999}{536079496789}a^{9}+\frac{205233818172}{536079496789}a^{8}+\frac{235840625278}{536079496789}a^{7}+\frac{13259723138}{536079496789}a^{6}-\frac{83940011708}{536079496789}a^{5}+\frac{67867971688}{536079496789}a^{4}+\frac{191614802101}{536079496789}a^{3}-\frac{33116648974}{536079496789}a^{2}+\frac{76052957915}{536079496789}a-\frac{35384260485}{536079496789}$, $\frac{1}{11\!\cdots\!57}a^{36}-\frac{201}{11\!\cdots\!57}a^{35}-\frac{10711874344}{11\!\cdots\!57}a^{34}-\frac{23566033107}{11\!\cdots\!57}a^{33}+\frac{22678819798}{11\!\cdots\!57}a^{32}-\frac{1911910363724}{11\!\cdots\!57}a^{31}-\frac{3420138786616}{11\!\cdots\!57}a^{30}-\frac{295325018217804}{11\!\cdots\!57}a^{29}+\frac{590219770781312}{11\!\cdots\!57}a^{28}+\frac{565755812991610}{11\!\cdots\!57}a^{27}+\frac{172885360366989}{11\!\cdots\!57}a^{26}-\frac{219858610583761}{11\!\cdots\!57}a^{25}+\frac{178522373034512}{11\!\cdots\!57}a^{24}+\frac{572315743631293}{11\!\cdots\!57}a^{23}-\frac{255563320280031}{11\!\cdots\!57}a^{22}-\frac{535879173295744}{11\!\cdots\!57}a^{21}-\frac{318266486642711}{11\!\cdots\!57}a^{20}+\frac{136737257288183}{11\!\cdots\!57}a^{19}-\frac{3397302052803}{10883889232973}a^{18}+\frac{157786380917089}{11\!\cdots\!57}a^{17}-\frac{127435545806904}{11\!\cdots\!57}a^{16}-\frac{327102704864916}{11\!\cdots\!57}a^{15}+\frac{160985885588775}{11\!\cdots\!57}a^{14}-\frac{498838736842243}{11\!\cdots\!57}a^{13}-\frac{258948361400042}{11\!\cdots\!57}a^{12}+\frac{264018407776314}{11\!\cdots\!57}a^{11}-\frac{557137438704788}{11\!\cdots\!57}a^{10}-\frac{329541314244559}{11\!\cdots\!57}a^{9}+\frac{539132555533321}{11\!\cdots\!57}a^{8}+\frac{286597417219138}{11\!\cdots\!57}a^{7}-\frac{185537310706693}{11\!\cdots\!57}a^{6}+\frac{45158425663619}{11\!\cdots\!57}a^{5}-\frac{528745773097009}{11\!\cdots\!57}a^{4}+\frac{34223286014576}{11\!\cdots\!57}a^{3}+\frac{11557676136758}{11\!\cdots\!57}a^{2}-\frac{394033856796899}{11\!\cdots\!57}a+\frac{136090950505364}{11\!\cdots\!57}$, $\frac{1}{10\!\cdots\!17}a^{37}+\frac{17\!\cdots\!79}{10\!\cdots\!17}a^{36}+\frac{90\!\cdots\!09}{10\!\cdots\!17}a^{35}+\frac{20\!\cdots\!22}{10\!\cdots\!17}a^{34}+\frac{21\!\cdots\!16}{10\!\cdots\!17}a^{33}-\frac{16\!\cdots\!75}{10\!\cdots\!17}a^{32}-\frac{37\!\cdots\!23}{10\!\cdots\!17}a^{31}-\frac{61\!\cdots\!61}{10\!\cdots\!17}a^{30}-\frac{14\!\cdots\!67}{10\!\cdots\!17}a^{29}+\frac{50\!\cdots\!60}{10\!\cdots\!17}a^{28}-\frac{47\!\cdots\!40}{10\!\cdots\!17}a^{27}+\frac{41\!\cdots\!87}{10\!\cdots\!17}a^{26}+\frac{36\!\cdots\!99}{10\!\cdots\!17}a^{25}-\frac{34\!\cdots\!05}{10\!\cdots\!17}a^{24}-\frac{24\!\cdots\!40}{10\!\cdots\!17}a^{23}-\frac{20\!\cdots\!51}{10\!\cdots\!17}a^{22}+\frac{49\!\cdots\!34}{10\!\cdots\!17}a^{21}+\frac{51\!\cdots\!57}{10\!\cdots\!17}a^{20}-\frac{47\!\cdots\!85}{10\!\cdots\!17}a^{19}+\frac{12\!\cdots\!36}{10\!\cdots\!17}a^{18}-\frac{22\!\cdots\!11}{10\!\cdots\!17}a^{17}-\frac{18\!\cdots\!23}{10\!\cdots\!17}a^{16}+\frac{20\!\cdots\!43}{10\!\cdots\!17}a^{15}-\frac{40\!\cdots\!30}{10\!\cdots\!17}a^{14}-\frac{30\!\cdots\!31}{10\!\cdots\!17}a^{13}-\frac{97\!\cdots\!41}{10\!\cdots\!17}a^{12}+\frac{32\!\cdots\!14}{10\!\cdots\!17}a^{11}-\frac{19\!\cdots\!58}{10\!\cdots\!17}a^{10}-\frac{44\!\cdots\!71}{10\!\cdots\!17}a^{9}+\frac{44\!\cdots\!74}{10\!\cdots\!17}a^{8}-\frac{26\!\cdots\!50}{10\!\cdots\!17}a^{7}+\frac{39\!\cdots\!12}{10\!\cdots\!17}a^{6}-\frac{39\!\cdots\!15}{10\!\cdots\!17}a^{5}+\frac{35\!\cdots\!68}{10\!\cdots\!17}a^{4}+\frac{29\!\cdots\!89}{10\!\cdots\!17}a^{3}-\frac{30\!\cdots\!36}{10\!\cdots\!17}a^{2}+\frac{82\!\cdots\!55}{10\!\cdots\!17}a-\frac{47\!\cdots\!39}{10\!\cdots\!17}$ 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:  $18$
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 + 8*x^36 + 298*x^35 + 570*x^34 + 1040*x^33 + 52682*x^32 + 320665*x^31 + 376717*x^30 + 994384*x^29 + 59327070*x^28 + 133411453*x^27 - 74052511*x^26 + 4371221504*x^25 + 16126217214*x^24 - 1486694372*x^23 + 126436359051*x^22 + 1092190917753*x^21 + 1461367679401*x^20 - 1224471417355*x^19 + 28814095113148*x^18 + 129684809968090*x^17 - 40138477207333*x^16 - 231235168601800*x^15 + 3710898858449432*x^14 + 8115639324673497*x^13 - 15016773845255805*x^12 - 10485965599249387*x^11 + 218552683964585201*x^10 + 358949096963276813*x^9 - 419776350086322541*x^8 - 407970861545640312*x^7 + 4102607542203011336*x^6 + 8187655915670745378*x^5 - 514918423725948339*x^4 - 14957650563715867722*x^3 - 7022951449740975397*x^2 + 19589758432453100807*x + 20824991381571168629)
 
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 + 8*x^36 + 298*x^35 + 570*x^34 + 1040*x^33 + 52682*x^32 + 320665*x^31 + 376717*x^30 + 994384*x^29 + 59327070*x^28 + 133411453*x^27 - 74052511*x^26 + 4371221504*x^25 + 16126217214*x^24 - 1486694372*x^23 + 126436359051*x^22 + 1092190917753*x^21 + 1461367679401*x^20 - 1224471417355*x^19 + 28814095113148*x^18 + 129684809968090*x^17 - 40138477207333*x^16 - 231235168601800*x^15 + 3710898858449432*x^14 + 8115639324673497*x^13 - 15016773845255805*x^12 - 10485965599249387*x^11 + 218552683964585201*x^10 + 358949096963276813*x^9 - 419776350086322541*x^8 - 407970861545640312*x^7 + 4102607542203011336*x^6 + 8187655915670745378*x^5 - 514918423725948339*x^4 - 14957650563715867722*x^3 - 7022951449740975397*x^2 + 19589758432453100807*x + 20824991381571168629, 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 + 8*x^36 + 298*x^35 + 570*x^34 + 1040*x^33 + 52682*x^32 + 320665*x^31 + 376717*x^30 + 994384*x^29 + 59327070*x^28 + 133411453*x^27 - 74052511*x^26 + 4371221504*x^25 + 16126217214*x^24 - 1486694372*x^23 + 126436359051*x^22 + 1092190917753*x^21 + 1461367679401*x^20 - 1224471417355*x^19 + 28814095113148*x^18 + 129684809968090*x^17 - 40138477207333*x^16 - 231235168601800*x^15 + 3710898858449432*x^14 + 8115639324673497*x^13 - 15016773845255805*x^12 - 10485965599249387*x^11 + 218552683964585201*x^10 + 358949096963276813*x^9 - 419776350086322541*x^8 - 407970861545640312*x^7 + 4102607542203011336*x^6 + 8187655915670745378*x^5 - 514918423725948339*x^4 - 14957650563715867722*x^3 - 7022951449740975397*x^2 + 19589758432453100807*x + 20824991381571168629);
 
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 + 8*x^36 + 298*x^35 + 570*x^34 + 1040*x^33 + 52682*x^32 + 320665*x^31 + 376717*x^30 + 994384*x^29 + 59327070*x^28 + 133411453*x^27 - 74052511*x^26 + 4371221504*x^25 + 16126217214*x^24 - 1486694372*x^23 + 126436359051*x^22 + 1092190917753*x^21 + 1461367679401*x^20 - 1224471417355*x^19 + 28814095113148*x^18 + 129684809968090*x^17 - 40138477207333*x^16 - 231235168601800*x^15 + 3710898858449432*x^14 + 8115639324673497*x^13 - 15016773845255805*x^12 - 10485965599249387*x^11 + 218552683964585201*x^10 + 358949096963276813*x^9 - 419776350086322541*x^8 - 407970861545640312*x^7 + 4102607542203011336*x^6 + 8187655915670745378*x^5 - 514918423725948339*x^4 - 14957650563715867722*x^3 - 7022951449740975397*x^2 + 19589758432453100807*x + 20824991381571168629);
 
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 $C_{38}$

Intermediate fields

\(\Q(\sqrt{-571}) \), 19.19.41634173570364661205169708858211372543325791407961.1

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 $38$ $38$ $19^{2}$ $38$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $19^{2}$ $19^{2}$ $19^{2}$ $38$ $19^{2}$ $38$ $38$ $19^{2}$

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
\(571\) Copy content Toggle raw display Deg $38$$38$$1$$37$