↑

Properties

Label 38.38.249...125.1
Degree $38$
Signature $[38, 0]$
Discriminant $2.498\times 10^{95}$
Root discriminant \(323.94\)
Ramified primes $5,191$
Class number not computed
Class group not computed
Galois group $C_{38}$ (as 38T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^38 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541)
 
gp: K = bnfinit(y^38 - 17*y^37 - 64*y^36 + 2589*y^35 - 5411*y^34 - 156768*y^33 + 733103*y^32 + 4788192*y^31 - 35074270*y^30 - 69778288*y^29 + 937716801*y^28 - 3988006*y^27 - 15662818901*y^26 + 18940301798*y^25 + 169186329709*y^24 - 366424944806*y^23 - 1167394423359*y^22 + 3780063487716*y^21 + 4693995261962*y^20 - 24515638087910*y^19 - 6315614627194*y^18 + 103828189076722*y^17 - 36155927083660*y^16 - 285274970106320*y^15 + 223944436674986*y^14 + 486329219734937*y^13 - 571966582999537*y^12 - 458764738319995*y^11 + 781485719820911*y^10 + 160103399248715*y^9 - 557488639455834*y^8 + 53135732479684*y^7 + 180915511892981*y^6 - 38184876354047*y^5 - 25486623807243*y^4 + 6173666606950*y^3 + 1214949029058*y^2 - 273982926995*y + 184740541, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541)
 

\( x^{38} - 17 x^{37} - 64 x^{36} + 2589 x^{35} - 5411 x^{34} - 156768 x^{33} + 733103 x^{32} + \cdots + 184740541 \) 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:  $[38, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(249\!\cdots\!125\) \(\medspace = 5^{19}\cdot 191^{36}\) 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:  \(323.94\)
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:  $5^{1/2}191^{18/19}\approx 323.93955530325684$
Ramified primes:   \(5\), \(191\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(955=5\cdot 191\)
Dirichlet character group:    $\lbrace$$\chi_{955}(1,·)$, $\chi_{955}(6,·)$, $\chi_{955}(769,·)$, $\chi_{955}(136,·)$, $\chi_{955}(344,·)$, $\chi_{955}(914,·)$, $\chi_{955}(579,·)$, $\chi_{955}(789,·)$, $\chi_{955}(536,·)$, $\chi_{955}(794,·)$, $\chi_{955}(796,·)$, $\chi_{955}(154,·)$, $\chi_{955}(414,·)$, $\chi_{955}(69,·)$, $\chi_{955}(816,·)$, $\chi_{955}(36,·)$, $\chi_{955}(924,·)$, $\chi_{955}(941,·)$, $\chi_{955}(559,·)$, $\chi_{955}(944,·)$, $\chi_{955}(434,·)$, $\chi_{955}(694,·)$, $\chi_{955}(316,·)$, $\chi_{955}(574,·)$, $\chi_{955}(451,·)$, $\chi_{955}(196,·)$, $\chi_{955}(709,·)$, $\chi_{955}(341,·)$, $\chi_{955}(726,·)$, $\chi_{955}(121,·)$, $\chi_{955}(216,·)$, $\chi_{955}(221,·)$, $\chi_{955}(351,·)$, $\chi_{955}(609,·)$, $\chi_{955}(871,·)$, $\chi_{955}(489,·)$, $\chi_{955}(371,·)$, $\chi_{955}(889,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7}a^{14}-\frac{2}{7}a^{8}+\frac{1}{7}a^{2}+\frac{2}{7}$, $\frac{1}{7}a^{15}-\frac{2}{7}a^{9}+\frac{1}{7}a^{3}+\frac{2}{7}a$, $\frac{1}{7}a^{16}-\frac{2}{7}a^{10}+\frac{1}{7}a^{4}+\frac{2}{7}a^{2}$, $\frac{1}{7}a^{17}-\frac{2}{7}a^{11}+\frac{1}{7}a^{5}+\frac{2}{7}a^{3}$, $\frac{1}{7}a^{18}-\frac{2}{7}a^{12}+\frac{1}{7}a^{6}+\frac{2}{7}a^{4}$, $\frac{1}{7}a^{19}-\frac{2}{7}a^{13}+\frac{1}{7}a^{7}+\frac{2}{7}a^{5}$, $\frac{1}{7}a^{20}-\frac{3}{7}a^{8}+\frac{2}{7}a^{6}+\frac{2}{7}a^{2}-\frac{3}{7}$, $\frac{1}{7}a^{21}-\frac{3}{7}a^{9}+\frac{2}{7}a^{7}+\frac{2}{7}a^{3}-\frac{3}{7}a$, $\frac{1}{7}a^{22}-\frac{3}{7}a^{10}+\frac{2}{7}a^{8}+\frac{2}{7}a^{4}-\frac{3}{7}a^{2}$, $\frac{1}{7}a^{23}-\frac{3}{7}a^{11}+\frac{2}{7}a^{9}+\frac{2}{7}a^{5}-\frac{3}{7}a^{3}$, $\frac{1}{49}a^{24}-\frac{3}{49}a^{23}+\frac{3}{49}a^{22}-\frac{1}{49}a^{21}-\frac{1}{49}a^{20}-\frac{1}{49}a^{19}+\frac{2}{49}a^{18}-\frac{2}{49}a^{17}-\frac{1}{49}a^{16}-\frac{3}{49}a^{15}+\frac{3}{49}a^{14}-\frac{19}{49}a^{13}+\frac{2}{7}a^{12}-\frac{1}{49}a^{11}-\frac{5}{49}a^{10}+\frac{17}{49}a^{9}+\frac{10}{49}a^{8}+\frac{11}{49}a^{7}+\frac{9}{49}a^{6}-\frac{10}{49}a^{5}+\frac{20}{49}a^{4}+\frac{1}{7}a^{3}+\frac{18}{49}a^{2}+\frac{4}{49}a+\frac{23}{49}$, $\frac{1}{49}a^{25}+\frac{1}{49}a^{23}+\frac{1}{49}a^{22}+\frac{3}{49}a^{21}+\frac{3}{49}a^{20}-\frac{1}{49}a^{19}-\frac{3}{49}a^{18}+\frac{1}{49}a^{16}+\frac{1}{49}a^{15}-\frac{3}{49}a^{14}+\frac{6}{49}a^{13}+\frac{6}{49}a^{12}+\frac{6}{49}a^{11}+\frac{9}{49}a^{10}-\frac{9}{49}a^{9}-\frac{8}{49}a^{8}+\frac{1}{7}a^{7}+\frac{24}{49}a^{6}+\frac{11}{49}a^{5}-\frac{3}{49}a^{4}+\frac{4}{49}a^{3}+\frac{16}{49}a^{2}-\frac{3}{7}a+\frac{13}{49}$, $\frac{1}{49}a^{26}-\frac{3}{49}a^{23}-\frac{3}{49}a^{21}-\frac{2}{49}a^{19}-\frac{2}{49}a^{18}+\frac{3}{49}a^{17}+\frac{2}{49}a^{16}+\frac{3}{49}a^{14}-\frac{24}{49}a^{13}-\frac{8}{49}a^{12}-\frac{18}{49}a^{11}-\frac{4}{49}a^{10}-\frac{18}{49}a^{9}-\frac{3}{49}a^{8}-\frac{1}{49}a^{7}+\frac{2}{49}a^{6}-\frac{1}{7}a^{5}-\frac{16}{49}a^{4}+\frac{16}{49}a^{3}+\frac{10}{49}a^{2}-\frac{19}{49}a-\frac{23}{49}$, $\frac{1}{49}a^{27}-\frac{2}{49}a^{23}-\frac{1}{49}a^{22}-\frac{3}{49}a^{21}+\frac{2}{49}a^{20}+\frac{2}{49}a^{19}+\frac{2}{49}a^{18}+\frac{3}{49}a^{17}-\frac{3}{49}a^{16}+\frac{1}{49}a^{15}-\frac{1}{49}a^{14}+\frac{19}{49}a^{13}-\frac{11}{49}a^{12}+\frac{1}{7}a^{11}-\frac{12}{49}a^{10}-\frac{1}{49}a^{9}+\frac{15}{49}a^{8}-\frac{1}{7}a^{7}-\frac{22}{49}a^{6}-\frac{11}{49}a^{5}-\frac{1}{49}a^{4}-\frac{18}{49}a^{3}-\frac{2}{7}a^{2}+\frac{3}{49}a-\frac{22}{49}$, $\frac{1}{49}a^{28}+\frac{3}{49}a^{22}-\frac{1}{49}a^{16}-\frac{3}{49}a^{14}-\frac{11}{49}a^{10}+\frac{20}{49}a^{8}+\frac{8}{49}a^{4}+\frac{11}{49}a^{2}-\frac{10}{49}$, $\frac{1}{49}a^{29}+\frac{3}{49}a^{23}-\frac{1}{49}a^{17}-\frac{3}{49}a^{15}-\frac{11}{49}a^{11}+\frac{20}{49}a^{9}+\frac{8}{49}a^{5}+\frac{11}{49}a^{3}-\frac{10}{49}a$, $\frac{1}{343}a^{30}-\frac{1}{343}a^{29}-\frac{2}{343}a^{28}-\frac{1}{343}a^{27}+\frac{3}{343}a^{26}+\frac{3}{343}a^{25}-\frac{19}{343}a^{23}+\frac{3}{343}a^{22}+\frac{6}{343}a^{21}+\frac{3}{343}a^{20}-\frac{15}{343}a^{19}-\frac{24}{343}a^{18}-\frac{22}{343}a^{17}+\frac{2}{49}a^{15}-\frac{23}{343}a^{14}+\frac{47}{343}a^{13}+\frac{50}{343}a^{12}+\frac{55}{343}a^{11}-\frac{18}{49}a^{10}-\frac{144}{343}a^{9}-\frac{27}{343}a^{8}-\frac{15}{343}a^{7}+\frac{18}{343}a^{6}-\frac{144}{343}a^{5}-\frac{58}{343}a^{4}-\frac{157}{343}a^{3}-\frac{1}{343}a^{2}+\frac{71}{343}a+\frac{167}{343}$, $\frac{1}{343}a^{31}-\frac{3}{343}a^{29}-\frac{3}{343}a^{28}+\frac{2}{343}a^{27}-\frac{1}{343}a^{26}+\frac{3}{343}a^{25}+\frac{2}{343}a^{24}-\frac{9}{343}a^{23}+\frac{23}{343}a^{22}+\frac{9}{343}a^{21}+\frac{16}{343}a^{20}+\frac{3}{343}a^{19}+\frac{10}{343}a^{18}+\frac{13}{343}a^{17}-\frac{3}{49}a^{16}-\frac{23}{343}a^{15}+\frac{17}{343}a^{14}+\frac{111}{343}a^{13}+\frac{16}{49}a^{12}+\frac{34}{343}a^{11}+\frac{143}{343}a^{10}-\frac{31}{343}a^{9}+\frac{6}{49}a^{8}-\frac{53}{343}a^{7}+\frac{3}{7}a^{6}-\frac{69}{343}a^{5}-\frac{124}{343}a^{4}-\frac{25}{343}a^{3}-\frac{16}{49}a^{2}-\frac{19}{49}a-\frac{120}{343}$, $\frac{1}{343}a^{32}+\frac{1}{343}a^{29}+\frac{3}{343}a^{28}+\frac{3}{343}a^{27}-\frac{2}{343}a^{26}-\frac{3}{343}a^{25}-\frac{2}{343}a^{24}-\frac{20}{343}a^{23}-\frac{10}{343}a^{22}+\frac{6}{343}a^{21}-\frac{23}{343}a^{20}+\frac{2}{49}a^{19}-\frac{10}{343}a^{18}+\frac{18}{343}a^{17}-\frac{2}{343}a^{16}+\frac{10}{343}a^{15}-\frac{2}{49}a^{14}+\frac{162}{343}a^{13}-\frac{12}{343}a^{12}+\frac{3}{7}a^{11}-\frac{38}{343}a^{10}-\frac{103}{343}a^{9}-\frac{8}{343}a^{8}+\frac{46}{343}a^{7}+\frac{167}{343}a^{6}+\frac{130}{343}a^{5}+\frac{158}{343}a^{4}+\frac{117}{343}a^{3}-\frac{101}{343}a^{2}-\frac{54}{343}a+\frac{137}{343}$, $\frac{1}{343}a^{33}-\frac{3}{343}a^{29}-\frac{2}{343}a^{28}-\frac{1}{343}a^{27}+\frac{1}{343}a^{26}+\frac{2}{343}a^{25}+\frac{1}{343}a^{24}+\frac{9}{343}a^{23}+\frac{3}{343}a^{22}-\frac{1}{343}a^{21}+\frac{11}{343}a^{20}+\frac{12}{343}a^{19}+\frac{6}{343}a^{17}+\frac{17}{343}a^{16}-\frac{2}{49}a^{15}+\frac{24}{343}a^{14}+\frac{4}{343}a^{13}+\frac{132}{343}a^{12}-\frac{72}{343}a^{11}-\frac{166}{343}a^{10}+\frac{115}{343}a^{9}+\frac{115}{343}a^{8}-\frac{12}{49}a^{7}+\frac{13}{49}a^{6}+\frac{15}{343}a^{5}-\frac{19}{49}a^{4}+\frac{17}{49}a^{3}-\frac{11}{343}a^{2}-\frac{109}{343}a+\frac{169}{343}$, $\frac{1}{49986157291}a^{34}-\frac{8170266}{49986157291}a^{33}-\frac{126930}{145732237}a^{32}+\frac{67581368}{49986157291}a^{31}-\frac{62499945}{49986157291}a^{30}-\frac{67397843}{7140879613}a^{29}-\frac{242281610}{49986157291}a^{28}-\frac{177430845}{49986157291}a^{27}+\frac{338150588}{49986157291}a^{26}-\frac{111125508}{49986157291}a^{25}+\frac{284272119}{49986157291}a^{24}+\frac{1392718195}{49986157291}a^{23}-\frac{556681089}{49986157291}a^{22}+\frac{2240378615}{49986157291}a^{21}+\frac{2387223196}{49986157291}a^{20}-\frac{747213801}{49986157291}a^{19}-\frac{3080610577}{49986157291}a^{18}+\frac{246929174}{49986157291}a^{17}-\frac{1709246319}{49986157291}a^{16}-\frac{1448724780}{49986157291}a^{15}+\frac{801944743}{49986157291}a^{14}+\frac{13773369068}{49986157291}a^{13}-\frac{24482876779}{49986157291}a^{12}-\frac{1887801043}{7140879613}a^{11}+\frac{16057106255}{49986157291}a^{10}+\frac{1473297517}{7140879613}a^{9}-\frac{22058241308}{49986157291}a^{8}+\frac{2556859505}{49986157291}a^{7}-\frac{3143748408}{49986157291}a^{6}-\frac{8595764192}{49986157291}a^{5}+\frac{223837072}{49986157291}a^{4}+\frac{9060208362}{49986157291}a^{3}+\frac{19907801}{145732237}a^{2}-\frac{1304812191}{49986157291}a+\frac{22843977535}{49986157291}$, $\frac{1}{21544033792421}a^{35}-\frac{90}{21544033792421}a^{34}-\frac{1864649}{4033707881}a^{33}+\frac{22632392070}{21544033792421}a^{32}+\frac{6700927992}{21544033792421}a^{31}+\frac{2140340473}{3077719113203}a^{30}-\frac{212024293156}{21544033792421}a^{29}+\frac{137596888789}{21544033792421}a^{28}-\frac{66669172711}{21544033792421}a^{27}+\frac{215321422522}{21544033792421}a^{26}+\frac{100777953218}{21544033792421}a^{25}-\frac{203868943893}{21544033792421}a^{24}-\frac{295555355514}{21544033792421}a^{23}-\frac{270009964001}{21544033792421}a^{22}+\frac{726584025007}{21544033792421}a^{21}+\frac{789650244595}{21544033792421}a^{20}-\frac{221968455949}{21544033792421}a^{19}-\frac{463162995951}{21544033792421}a^{18}-\frac{980642930425}{21544033792421}a^{17}+\frac{1511780978982}{21544033792421}a^{16}+\frac{561881905611}{21544033792421}a^{15}+\frac{17982245782}{21544033792421}a^{14}+\frac{4773148118007}{21544033792421}a^{13}-\frac{862419559766}{3077719113203}a^{12}+\frac{667362722314}{21544033792421}a^{11}-\frac{1463702880732}{3077719113203}a^{10}+\frac{7800510324567}{21544033792421}a^{9}-\frac{8917399719903}{21544033792421}a^{8}-\frac{1306585949840}{21544033792421}a^{7}+\frac{4982793864341}{21544033792421}a^{6}-\frac{9402609135979}{21544033792421}a^{5}+\frac{2022466071432}{21544033792421}a^{4}-\frac{1505278649950}{3077719113203}a^{3}-\frac{10468399747565}{21544033792421}a^{2}-\frac{3296167132215}{21544033792421}a-\frac{26336333741}{439674159029}$, $\frac{1}{64\!\cdots\!49}a^{36}+\frac{11\!\cdots\!65}{64\!\cdots\!49}a^{35}-\frac{44\!\cdots\!43}{64\!\cdots\!49}a^{34}+\frac{57\!\cdots\!64}{64\!\cdots\!49}a^{33}+\frac{13\!\cdots\!65}{92\!\cdots\!07}a^{32}+\frac{91\!\cdots\!69}{64\!\cdots\!49}a^{31}+\frac{38\!\cdots\!13}{64\!\cdots\!49}a^{30}-\frac{44\!\cdots\!41}{64\!\cdots\!49}a^{29}-\frac{79\!\cdots\!71}{92\!\cdots\!07}a^{28}-\frac{27\!\cdots\!42}{64\!\cdots\!49}a^{27}+\frac{49\!\cdots\!80}{59\!\cdots\!61}a^{26}+\frac{65\!\cdots\!76}{64\!\cdots\!49}a^{25}+\frac{37\!\cdots\!58}{64\!\cdots\!49}a^{24}+\frac{40\!\cdots\!85}{64\!\cdots\!49}a^{23}-\frac{31\!\cdots\!25}{64\!\cdots\!49}a^{22}-\frac{20\!\cdots\!05}{64\!\cdots\!49}a^{21}-\frac{83\!\cdots\!21}{13\!\cdots\!01}a^{20}+\frac{28\!\cdots\!36}{64\!\cdots\!49}a^{19}+\frac{10\!\cdots\!26}{64\!\cdots\!49}a^{18}+\frac{69\!\cdots\!97}{15\!\cdots\!69}a^{17}+\frac{38\!\cdots\!45}{64\!\cdots\!49}a^{16}+\frac{37\!\cdots\!81}{64\!\cdots\!49}a^{15}+\frac{34\!\cdots\!29}{64\!\cdots\!49}a^{14}+\frac{75\!\cdots\!06}{64\!\cdots\!49}a^{13}-\frac{21\!\cdots\!34}{64\!\cdots\!49}a^{12}+\frac{24\!\cdots\!17}{64\!\cdots\!49}a^{11}-\frac{41\!\cdots\!57}{64\!\cdots\!49}a^{10}-\frac{68\!\cdots\!68}{64\!\cdots\!49}a^{9}-\frac{16\!\cdots\!68}{64\!\cdots\!49}a^{8}-\frac{12\!\cdots\!51}{64\!\cdots\!49}a^{7}-\frac{21\!\cdots\!61}{64\!\cdots\!49}a^{6}+\frac{13\!\cdots\!28}{64\!\cdots\!49}a^{5}-\frac{23\!\cdots\!50}{64\!\cdots\!49}a^{4}-\frac{83\!\cdots\!76}{64\!\cdots\!49}a^{3}+\frac{18\!\cdots\!20}{64\!\cdots\!49}a^{2}-\frac{45\!\cdots\!99}{92\!\cdots\!07}a-\frac{32\!\cdots\!93}{64\!\cdots\!49}$, $\frac{1}{10\!\cdots\!89}a^{37}+\frac{14\!\cdots\!47}{15\!\cdots\!27}a^{36}+\frac{61\!\cdots\!88}{10\!\cdots\!89}a^{35}+\frac{47\!\cdots\!91}{10\!\cdots\!89}a^{34}+\frac{72\!\cdots\!25}{10\!\cdots\!89}a^{33}-\frac{62\!\cdots\!35}{10\!\cdots\!89}a^{32}-\frac{34\!\cdots\!51}{10\!\cdots\!89}a^{31}-\frac{48\!\cdots\!82}{10\!\cdots\!89}a^{30}+\frac{82\!\cdots\!82}{10\!\cdots\!89}a^{29}-\frac{70\!\cdots\!69}{10\!\cdots\!89}a^{28}-\frac{82\!\cdots\!85}{10\!\cdots\!89}a^{27}-\frac{14\!\cdots\!43}{15\!\cdots\!27}a^{26}+\frac{69\!\cdots\!44}{10\!\cdots\!89}a^{25}-\frac{86\!\cdots\!51}{10\!\cdots\!89}a^{24}+\frac{66\!\cdots\!81}{10\!\cdots\!89}a^{23}-\frac{73\!\cdots\!32}{10\!\cdots\!89}a^{22}-\frac{63\!\cdots\!04}{10\!\cdots\!89}a^{21}+\frac{18\!\cdots\!54}{10\!\cdots\!89}a^{20}-\frac{61\!\cdots\!46}{10\!\cdots\!89}a^{19}-\frac{35\!\cdots\!28}{10\!\cdots\!89}a^{18}-\frac{51\!\cdots\!38}{15\!\cdots\!27}a^{17}-\frac{78\!\cdots\!61}{15\!\cdots\!27}a^{16}+\frac{11\!\cdots\!09}{10\!\cdots\!89}a^{15}+\frac{57\!\cdots\!15}{10\!\cdots\!89}a^{14}-\frac{45\!\cdots\!93}{10\!\cdots\!89}a^{13}+\frac{25\!\cdots\!29}{10\!\cdots\!89}a^{12}+\frac{44\!\cdots\!79}{10\!\cdots\!89}a^{11}+\frac{36\!\cdots\!95}{10\!\cdots\!89}a^{10}-\frac{11\!\cdots\!49}{10\!\cdots\!89}a^{9}+\frac{33\!\cdots\!07}{15\!\cdots\!27}a^{8}+\frac{73\!\cdots\!95}{15\!\cdots\!27}a^{7}+\frac{39\!\cdots\!79}{10\!\cdots\!89}a^{6}+\frac{40\!\cdots\!09}{10\!\cdots\!89}a^{5}-\frac{38\!\cdots\!33}{10\!\cdots\!89}a^{4}-\frac{10\!\cdots\!35}{10\!\cdots\!89}a^{3}-\frac{44\!\cdots\!24}{10\!\cdots\!89}a^{2}-\frac{39\!\cdots\!26}{10\!\cdots\!89}a-\frac{51\!\cdots\!20}{10\!\cdots\!89}$ 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:  $37$
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 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541)
 
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 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541, 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 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541);
 
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 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541);
 
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

\(\Q(\sqrt{5}) \), 19.19.114445997944945591651333831028437092270721.1

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 $38$ $38$ R ${\href{/padicField/7.2.0.1}{2} }^{19}$ $19^{2}$ $38$ $38$ $19^{2}$ $38$ $19^{2}$ $19^{2}$ $38$ $19^{2}$ $38$ $38$ $38$ $19^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $38$$2$$19$$19$
\(191\) Copy content Toggle raw display 191.19.18.1$x^{19} + 191$$19$$1$$18$$C_{19}$$[\ ]_{19}$
191.19.18.1$x^{19} + 191$$19$$1$$18$$C_{19}$$[\ ]_{19}$