↑

Properties

Label 38.0.100...687.1
Degree $38$
Signature $[0, 19]$
Discriminant $-1.008\times 10^{104}$
Root discriminant \(545.67\)
Ramified prime $647$
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 - x^37 + 9*x^36 - 105*x^35 - 2517*x^34 - 12367*x^33 + 18961*x^32 + 524484*x^31 + 2063543*x^30 + 10035685*x^29 + 68343245*x^28 + 162739626*x^27 + 853626703*x^26 - 1517430356*x^25 + 15437152669*x^24 - 38624827303*x^23 + 261377865856*x^22 - 1008245158454*x^21 + 3680056602480*x^20 - 8092055201962*x^19 + 22243202914752*x^18 - 88956425126723*x^17 + 393393581809607*x^16 - 995223440330324*x^15 + 2718528813824046*x^14 - 6567254133383172*x^13 + 17369599534798699*x^12 - 25459990107367281*x^11 - 23486845361639043*x^10 + 70552096295073850*x^9 + 302207769671571109*x^8 - 1202161182566167879*x^7 + 921854615125235345*x^6 + 3618507295866457145*x^5 - 1503510294335687605*x^4 - 10746229822177899439*x^3 + 2011567894060138443*x^2 + 8767693047911919217*x + 4654012959769314377)
 
gp: K = bnfinit(y^38 - y^37 + 9*y^36 - 105*y^35 - 2517*y^34 - 12367*y^33 + 18961*y^32 + 524484*y^31 + 2063543*y^30 + 10035685*y^29 + 68343245*y^28 + 162739626*y^27 + 853626703*y^26 - 1517430356*y^25 + 15437152669*y^24 - 38624827303*y^23 + 261377865856*y^22 - 1008245158454*y^21 + 3680056602480*y^20 - 8092055201962*y^19 + 22243202914752*y^18 - 88956425126723*y^17 + 393393581809607*y^16 - 995223440330324*y^15 + 2718528813824046*y^14 - 6567254133383172*y^13 + 17369599534798699*y^12 - 25459990107367281*y^11 - 23486845361639043*y^10 + 70552096295073850*y^9 + 302207769671571109*y^8 - 1202161182566167879*y^7 + 921854615125235345*y^6 + 3618507295866457145*y^5 - 1503510294335687605*y^4 - 10746229822177899439*y^3 + 2011567894060138443*y^2 + 8767693047911919217*y + 4654012959769314377, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - x^37 + 9*x^36 - 105*x^35 - 2517*x^34 - 12367*x^33 + 18961*x^32 + 524484*x^31 + 2063543*x^30 + 10035685*x^29 + 68343245*x^28 + 162739626*x^27 + 853626703*x^26 - 1517430356*x^25 + 15437152669*x^24 - 38624827303*x^23 + 261377865856*x^22 - 1008245158454*x^21 + 3680056602480*x^20 - 8092055201962*x^19 + 22243202914752*x^18 - 88956425126723*x^17 + 393393581809607*x^16 - 995223440330324*x^15 + 2718528813824046*x^14 - 6567254133383172*x^13 + 17369599534798699*x^12 - 25459990107367281*x^11 - 23486845361639043*x^10 + 70552096295073850*x^9 + 302207769671571109*x^8 - 1202161182566167879*x^7 + 921854615125235345*x^6 + 3618507295866457145*x^5 - 1503510294335687605*x^4 - 10746229822177899439*x^3 + 2011567894060138443*x^2 + 8767693047911919217*x + 4654012959769314377);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - x^37 + 9*x^36 - 105*x^35 - 2517*x^34 - 12367*x^33 + 18961*x^32 + 524484*x^31 + 2063543*x^30 + 10035685*x^29 + 68343245*x^28 + 162739626*x^27 + 853626703*x^26 - 1517430356*x^25 + 15437152669*x^24 - 38624827303*x^23 + 261377865856*x^22 - 1008245158454*x^21 + 3680056602480*x^20 - 8092055201962*x^19 + 22243202914752*x^18 - 88956425126723*x^17 + 393393581809607*x^16 - 995223440330324*x^15 + 2718528813824046*x^14 - 6567254133383172*x^13 + 17369599534798699*x^12 - 25459990107367281*x^11 - 23486845361639043*x^10 + 70552096295073850*x^9 + 302207769671571109*x^8 - 1202161182566167879*x^7 + 921854615125235345*x^6 + 3618507295866457145*x^5 - 1503510294335687605*x^4 - 10746229822177899439*x^3 + 2011567894060138443*x^2 + 8767693047911919217*x + 4654012959769314377)
 

\( x^{38} - x^{37} + 9 x^{36} - 105 x^{35} - 2517 x^{34} - 12367 x^{33} + 18961 x^{32} + \cdots + 46\!\cdots\!77 \) 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:   \(-100\!\cdots\!687\) \(\medspace = -\,647^{37}\) 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:  \(545.67\)
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:  $647^{37/38}\approx 545.6738167535524$
Ramified primes:   \(647\) 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:  \(\Q(\sqrt{-647}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(647\)
Dirichlet character group:    $\lbrace$$\chi_{647}(1,·)$, $\chi_{647}(646,·)$, $\chi_{647}(257,·)$, $\chi_{647}(269,·)$, $\chi_{647}(533,·)$, $\chi_{647}(279,·)$, $\chi_{647}(155,·)$, $\chi_{647}(158,·)$, $\chi_{647}(543,·)$, $\chi_{647}(544,·)$, $\chi_{647}(548,·)$, $\chi_{647}(390,·)$, $\chi_{647}(551,·)$, $\chi_{647}(561,·)$, $\chi_{647}(437,·)$, $\chi_{647}(183,·)$, $\chi_{647}(56,·)$, $\chi_{647}(287,·)$, $\chi_{647}(446,·)$, $\chi_{647}(447,·)$, $\chi_{647}(200,·)$, $\chi_{647}(201,·)$, $\chi_{647}(55,·)$, $\chi_{647}(591,·)$, $\chi_{647}(464,·)$, $\chi_{647}(210,·)$, $\chi_{647}(86,·)$, $\chi_{647}(96,·)$, $\chi_{647}(592,·)$, $\chi_{647}(99,·)$, $\chi_{647}(103,·)$, $\chi_{647}(104,·)$, $\chi_{647}(489,·)$, $\chi_{647}(492,·)$, $\chi_{647}(368,·)$, $\chi_{647}(360,·)$, $\chi_{647}(114,·)$, $\chi_{647}(378,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{262144}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{43}a^{25}-\frac{11}{43}a^{24}-\frac{21}{43}a^{23}-\frac{18}{43}a^{22}-\frac{13}{43}a^{21}-\frac{19}{43}a^{20}-\frac{21}{43}a^{19}-\frac{3}{43}a^{18}+\frac{4}{43}a^{17}+\frac{8}{43}a^{16}-\frac{11}{43}a^{15}-\frac{15}{43}a^{14}-\frac{20}{43}a^{13}-\frac{6}{43}a^{12}-\frac{19}{43}a^{11}+\frac{4}{43}a^{10}+\frac{2}{43}a^{9}-\frac{11}{43}a^{8}-\frac{15}{43}a^{7}-\frac{4}{43}a^{6}-\frac{16}{43}a^{5}+\frac{21}{43}a^{4}+\frac{3}{43}a^{3}+\frac{11}{43}a^{2}-\frac{3}{43}a$, $\frac{1}{43}a^{26}-\frac{13}{43}a^{24}+\frac{9}{43}a^{23}+\frac{4}{43}a^{22}+\frac{10}{43}a^{21}-\frac{15}{43}a^{20}-\frac{19}{43}a^{19}+\frac{14}{43}a^{18}+\frac{9}{43}a^{17}-\frac{9}{43}a^{16}-\frac{7}{43}a^{15}-\frac{13}{43}a^{14}-\frac{11}{43}a^{13}+\frac{1}{43}a^{12}+\frac{10}{43}a^{11}+\frac{3}{43}a^{10}+\frac{11}{43}a^{9}-\frac{7}{43}a^{8}+\frac{3}{43}a^{7}-\frac{17}{43}a^{6}+\frac{17}{43}a^{5}+\frac{19}{43}a^{4}+\frac{1}{43}a^{3}-\frac{11}{43}a^{2}+\frac{10}{43}a$, $\frac{1}{43}a^{27}-\frac{5}{43}a^{24}-\frac{11}{43}a^{23}-\frac{9}{43}a^{22}-\frac{12}{43}a^{21}-\frac{8}{43}a^{20}-\frac{1}{43}a^{19}+\frac{13}{43}a^{18}+\frac{11}{43}a^{16}+\frac{16}{43}a^{15}+\frac{9}{43}a^{14}-\frac{1}{43}a^{13}+\frac{18}{43}a^{12}+\frac{14}{43}a^{11}+\frac{20}{43}a^{10}+\frac{19}{43}a^{9}-\frac{11}{43}a^{8}+\frac{3}{43}a^{7}+\frac{8}{43}a^{6}-\frac{17}{43}a^{5}+\frac{16}{43}a^{4}-\frac{15}{43}a^{3}-\frac{19}{43}a^{2}+\frac{4}{43}a$, $\frac{1}{43}a^{28}+\frac{20}{43}a^{24}+\frac{15}{43}a^{23}-\frac{16}{43}a^{22}+\frac{13}{43}a^{21}-\frac{10}{43}a^{20}-\frac{6}{43}a^{19}-\frac{15}{43}a^{18}-\frac{12}{43}a^{17}+\frac{13}{43}a^{16}-\frac{3}{43}a^{15}+\frac{10}{43}a^{14}+\frac{4}{43}a^{13}-\frac{16}{43}a^{12}+\frac{11}{43}a^{11}-\frac{4}{43}a^{10}-\frac{1}{43}a^{9}-\frac{9}{43}a^{8}+\frac{19}{43}a^{7}+\frac{6}{43}a^{6}-\frac{21}{43}a^{5}+\frac{4}{43}a^{4}-\frac{4}{43}a^{3}+\frac{16}{43}a^{2}-\frac{15}{43}a$, $\frac{1}{43}a^{29}+\frac{20}{43}a^{24}+\frac{17}{43}a^{23}-\frac{14}{43}a^{22}-\frac{8}{43}a^{21}-\frac{13}{43}a^{20}+\frac{18}{43}a^{19}+\frac{5}{43}a^{18}+\frac{19}{43}a^{17}+\frac{9}{43}a^{16}+\frac{15}{43}a^{15}+\frac{3}{43}a^{14}-\frac{3}{43}a^{13}+\frac{2}{43}a^{12}-\frac{11}{43}a^{11}+\frac{5}{43}a^{10}-\frac{6}{43}a^{9}-\frac{19}{43}a^{8}+\frac{5}{43}a^{7}+\frac{16}{43}a^{6}-\frac{20}{43}a^{5}+\frac{6}{43}a^{4}-\frac{1}{43}a^{3}-\frac{20}{43}a^{2}+\frac{17}{43}a$, $\frac{1}{2881}a^{30}-\frac{22}{2881}a^{29}-\frac{8}{2881}a^{28}+\frac{12}{2881}a^{27}-\frac{2}{2881}a^{26}-\frac{25}{2881}a^{25}+\frac{738}{2881}a^{24}-\frac{444}{2881}a^{23}+\frac{649}{2881}a^{22}-\frac{208}{2881}a^{21}+\frac{12}{2881}a^{20}-\frac{533}{2881}a^{19}+\frac{894}{2881}a^{18}+\frac{994}{2881}a^{17}+\frac{965}{2881}a^{16}-\frac{935}{2881}a^{15}+\frac{574}{2881}a^{14}+\frac{22}{67}a^{13}-\frac{1077}{2881}a^{12}-\frac{472}{2881}a^{11}+\frac{400}{2881}a^{10}+\frac{280}{2881}a^{9}+\frac{356}{2881}a^{8}-\frac{659}{2881}a^{7}-\frac{669}{2881}a^{6}-\frac{452}{2881}a^{5}-\frac{827}{2881}a^{4}-\frac{1272}{2881}a^{3}+\frac{875}{2881}a^{2}-\frac{91}{2881}a$, $\frac{1}{2881}a^{31}-\frac{23}{2881}a^{29}-\frac{30}{2881}a^{28}-\frac{6}{2881}a^{27}-\frac{2}{2881}a^{26}-\frac{13}{2881}a^{25}-\frac{1159}{2881}a^{24}-\frac{7}{2881}a^{23}+\frac{2}{43}a^{22}-\frac{75}{2881}a^{21}+\frac{133}{2881}a^{20}+\frac{22}{2881}a^{19}-\frac{1113}{2881}a^{18}+\frac{1125}{2881}a^{17}+\frac{932}{2881}a^{16}+\frac{1377}{2881}a^{15}-\frac{1233}{2881}a^{14}-\frac{633}{2881}a^{13}-\frac{113}{2881}a^{12}-\frac{1408}{2881}a^{11}-\frac{836}{2881}a^{10}-\frac{1189}{2881}a^{9}-\frac{1135}{2881}a^{8}+\frac{779}{2881}a^{7}-\frac{698}{2881}a^{6}+\frac{351}{2881}a^{5}-\frac{304}{2881}a^{4}+\frac{1299}{2881}a^{3}-\frac{338}{2881}a^{2}+\frac{1281}{2881}a$, $\frac{1}{152693}a^{32}+\frac{20}{152693}a^{31}-\frac{26}{152693}a^{30}+\frac{112}{152693}a^{29}-\frac{381}{152693}a^{28}+\frac{579}{152693}a^{27}+\frac{824}{152693}a^{26}+\frac{666}{152693}a^{25}-\frac{16088}{152693}a^{24}+\frac{598}{3551}a^{23}+\frac{68730}{152693}a^{22}-\frac{74778}{152693}a^{21}-\frac{60602}{152693}a^{20}-\frac{55890}{152693}a^{19}+\frac{43451}{152693}a^{18}+\frac{55625}{152693}a^{17}+\frac{46669}{152693}a^{16}-\frac{34605}{152693}a^{15}-\frac{617}{152693}a^{14}+\frac{506}{2279}a^{13}-\frac{35076}{152693}a^{12}+\frac{15836}{152693}a^{11}+\frac{71542}{152693}a^{10}-\frac{18854}{152693}a^{9}-\frac{19438}{152693}a^{8}-\frac{62469}{152693}a^{7}-\frac{36057}{152693}a^{6}+\frac{55441}{152693}a^{5}-\frac{74660}{152693}a^{4}+\frac{49893}{152693}a^{3}+\frac{11728}{152693}a^{2}+\frac{8808}{152693}a$, $\frac{1}{6565799}a^{33}+\frac{14}{6565799}a^{32}-\frac{411}{6565799}a^{31}-\frac{633}{6565799}a^{30}+\frac{21313}{6565799}a^{29}-\frac{420}{97997}a^{28}-\frac{291}{123883}a^{27}+\frac{26462}{6565799}a^{26}+\frac{16539}{6565799}a^{25}-\frac{3225503}{6565799}a^{24}+\frac{703404}{6565799}a^{23}-\frac{2961039}{6565799}a^{22}-\frac{10408}{123883}a^{21}+\frac{16646}{6565799}a^{20}-\frac{34554}{123883}a^{19}+\frac{2416352}{6565799}a^{18}-\frac{2222959}{6565799}a^{17}-\frac{1509186}{6565799}a^{16}+\frac{1465763}{6565799}a^{15}+\frac{646150}{6565799}a^{14}-\frac{2840629}{6565799}a^{13}-\frac{584396}{6565799}a^{12}+\frac{338145}{6565799}a^{11}-\frac{668639}{6565799}a^{10}+\frac{2017215}{6565799}a^{9}-\frac{981408}{6565799}a^{8}+\frac{2647172}{6565799}a^{7}+\frac{2156781}{6565799}a^{6}+\frac{2295}{152693}a^{5}+\frac{2587696}{6565799}a^{4}-\frac{558195}{6565799}a^{3}+\frac{1039992}{6565799}a^{2}-\frac{1272643}{6565799}a-\frac{6}{43}$, $\frac{1}{1923779107}a^{34}+\frac{118}{1923779107}a^{33}+\frac{74}{28713121}a^{32}+\frac{21209}{1923779107}a^{31}+\frac{1878}{1923779107}a^{30}-\frac{15587100}{1923779107}a^{29}+\frac{1968875}{1923779107}a^{28}+\frac{19343991}{1923779107}a^{27}-\frac{18706903}{1923779107}a^{26}+\frac{8874280}{1923779107}a^{25}-\frac{364450642}{1923779107}a^{24}-\frac{255372536}{1923779107}a^{23}-\frac{49083131}{1923779107}a^{22}+\frac{884400253}{1923779107}a^{21}-\frac{316371800}{1923779107}a^{20}+\frac{909767064}{1923779107}a^{19}-\frac{923835802}{1923779107}a^{18}-\frac{296207922}{1923779107}a^{17}+\frac{214286812}{1923779107}a^{16}+\frac{286354284}{1923779107}a^{15}+\frac{955242001}{1923779107}a^{14}+\frac{591478583}{1923779107}a^{13}+\frac{490384253}{1923779107}a^{12}-\frac{738738653}{1923779107}a^{11}+\frac{85370634}{1923779107}a^{10}-\frac{319515579}{1923779107}a^{9}+\frac{669677880}{1923779107}a^{8}+\frac{705009691}{1923779107}a^{7}+\frac{887925213}{1923779107}a^{6}-\frac{4277684}{1923779107}a^{5}-\frac{642308101}{1923779107}a^{4}-\frac{94081728}{1923779107}a^{3}-\frac{315355759}{1923779107}a^{2}-\frac{294405059}{1923779107}a-\frac{8}{43}$, $\frac{1}{1821818814329}a^{35}-\frac{306}{1821818814329}a^{34}-\frac{74960}{1821818814329}a^{33}-\frac{42582}{1821818814329}a^{32}-\frac{36381843}{1821818814329}a^{31}+\frac{281211747}{1821818814329}a^{30}+\frac{18695905197}{1821818814329}a^{29}+\frac{19727546855}{1821818814329}a^{28}+\frac{20854856120}{1821818814329}a^{27}-\frac{17192492902}{1821818814329}a^{26}+\frac{1058678046}{1821818814329}a^{25}-\frac{293988572841}{1821818814329}a^{24}-\frac{408708648707}{1821818814329}a^{23}-\frac{792266451895}{1821818814329}a^{22}-\frac{547663511940}{1821818814329}a^{21}-\frac{287829184962}{1821818814329}a^{20}+\frac{427205862097}{1821818814329}a^{19}+\frac{847275370393}{1821818814329}a^{18}+\frac{8582741346}{42367879403}a^{17}-\frac{649828371070}{1821818814329}a^{16}+\frac{192538975562}{1821818814329}a^{15}+\frac{82062965824}{1821818814329}a^{14}+\frac{31070219724}{1821818814329}a^{13}+\frac{108593815968}{1821818814329}a^{12}+\frac{290885178658}{1821818814329}a^{11}+\frac{817709695709}{1821818814329}a^{10}+\frac{787064781807}{1821818814329}a^{9}+\frac{859084521304}{1821818814329}a^{8}+\frac{472508561500}{1821818814329}a^{7}+\frac{483107176171}{1821818814329}a^{6}+\frac{564703319412}{1821818814329}a^{5}+\frac{832863835659}{1821818814329}a^{4}+\frac{301466621098}{1821818814329}a^{3}-\frac{150054974996}{1821818814329}a^{2}+\frac{690650379419}{1821818814329}a-\frac{11863}{40721}$, $\frac{1}{18\!\cdots\!51}a^{36}+\frac{166}{18\!\cdots\!51}a^{35}-\frac{475082}{18\!\cdots\!51}a^{34}-\frac{16205284}{18\!\cdots\!51}a^{33}-\frac{3508165808}{18\!\cdots\!51}a^{32}+\frac{194556276357}{18\!\cdots\!51}a^{31}-\frac{270336832810}{18\!\cdots\!51}a^{30}+\frac{20524398767474}{18\!\cdots\!51}a^{29}-\frac{20949072706302}{18\!\cdots\!51}a^{28}+\frac{233048082645}{27707960773153}a^{27}+\frac{4886798305851}{18\!\cdots\!51}a^{26}-\frac{15455177299034}{18\!\cdots\!51}a^{25}+\frac{270193773016179}{18\!\cdots\!51}a^{24}-\frac{688260733614760}{18\!\cdots\!51}a^{23}+\frac{343138728102737}{18\!\cdots\!51}a^{22}+\frac{207422523944287}{18\!\cdots\!51}a^{21}-\frac{675827947742021}{18\!\cdots\!51}a^{20}-\frac{758714617491071}{18\!\cdots\!51}a^{19}+\frac{113908348208413}{18\!\cdots\!51}a^{18}-\frac{166950074672635}{18\!\cdots\!51}a^{17}-\frac{148189296931847}{18\!\cdots\!51}a^{16}+\frac{445306775750750}{18\!\cdots\!51}a^{15}-\frac{885847347056752}{18\!\cdots\!51}a^{14}+\frac{5424220380711}{35027044750967}a^{13}-\frac{229659636905301}{18\!\cdots\!51}a^{12}-\frac{373278863594034}{18\!\cdots\!51}a^{11}-\frac{626936117711356}{18\!\cdots\!51}a^{10}+\frac{263256511721785}{18\!\cdots\!51}a^{9}-\frac{354750552177539}{18\!\cdots\!51}a^{8}+\frac{467558174878791}{18\!\cdots\!51}a^{7}+\frac{270324631708810}{18\!\cdots\!51}a^{6}-\frac{50514789184749}{18\!\cdots\!51}a^{5}-\frac{920335611235885}{18\!\cdots\!51}a^{4}+\frac{644065608892948}{18\!\cdots\!51}a^{3}+\frac{14654789184143}{43172869111657}a^{2}+\frac{339831985416660}{18\!\cdots\!51}a-\frac{2488441}{41494699}$, $\frac{1}{28\!\cdots\!33}a^{37}-\frac{41\!\cdots\!00}{28\!\cdots\!33}a^{36}-\frac{74\!\cdots\!05}{28\!\cdots\!33}a^{35}-\frac{54\!\cdots\!68}{28\!\cdots\!33}a^{34}-\frac{47\!\cdots\!72}{28\!\cdots\!33}a^{33}-\frac{12\!\cdots\!78}{66\!\cdots\!31}a^{32}+\frac{30\!\cdots\!61}{28\!\cdots\!33}a^{31}-\frac{20\!\cdots\!27}{28\!\cdots\!33}a^{30}-\frac{33\!\cdots\!73}{28\!\cdots\!33}a^{29}-\frac{41\!\cdots\!34}{28\!\cdots\!33}a^{28}-\frac{10\!\cdots\!42}{28\!\cdots\!33}a^{27}+\frac{18\!\cdots\!87}{28\!\cdots\!33}a^{26}-\frac{16\!\cdots\!10}{28\!\cdots\!33}a^{25}-\frac{86\!\cdots\!81}{28\!\cdots\!33}a^{24}-\frac{30\!\cdots\!87}{28\!\cdots\!33}a^{23}+\frac{10\!\cdots\!18}{28\!\cdots\!33}a^{22}-\frac{98\!\cdots\!77}{28\!\cdots\!33}a^{21}-\frac{12\!\cdots\!95}{28\!\cdots\!33}a^{20}+\frac{65\!\cdots\!24}{28\!\cdots\!33}a^{19}-\frac{13\!\cdots\!52}{28\!\cdots\!33}a^{18}+\frac{11\!\cdots\!07}{28\!\cdots\!33}a^{17}-\frac{72\!\cdots\!19}{28\!\cdots\!33}a^{16}+\frac{13\!\cdots\!57}{54\!\cdots\!61}a^{15}-\frac{57\!\cdots\!59}{28\!\cdots\!33}a^{14}+\frac{85\!\cdots\!68}{28\!\cdots\!33}a^{13}-\frac{83\!\cdots\!85}{28\!\cdots\!33}a^{12}-\frac{13\!\cdots\!10}{28\!\cdots\!33}a^{11}-\frac{10\!\cdots\!66}{28\!\cdots\!33}a^{10}-\frac{12\!\cdots\!96}{28\!\cdots\!33}a^{9}-\frac{67\!\cdots\!89}{28\!\cdots\!33}a^{8}+\frac{28\!\cdots\!96}{28\!\cdots\!33}a^{7}-\frac{41\!\cdots\!08}{28\!\cdots\!33}a^{6}-\frac{10\!\cdots\!78}{28\!\cdots\!33}a^{5}-\frac{40\!\cdots\!30}{28\!\cdots\!33}a^{4}-\frac{13\!\cdots\!05}{28\!\cdots\!33}a^{3}+\frac{11\!\cdots\!50}{28\!\cdots\!33}a^{2}+\frac{31\!\cdots\!34}{28\!\cdots\!33}a-\frac{30\!\cdots\!87}{64\!\cdots\!17}$ 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 + 9*x^36 - 105*x^35 - 2517*x^34 - 12367*x^33 + 18961*x^32 + 524484*x^31 + 2063543*x^30 + 10035685*x^29 + 68343245*x^28 + 162739626*x^27 + 853626703*x^26 - 1517430356*x^25 + 15437152669*x^24 - 38624827303*x^23 + 261377865856*x^22 - 1008245158454*x^21 + 3680056602480*x^20 - 8092055201962*x^19 + 22243202914752*x^18 - 88956425126723*x^17 + 393393581809607*x^16 - 995223440330324*x^15 + 2718528813824046*x^14 - 6567254133383172*x^13 + 17369599534798699*x^12 - 25459990107367281*x^11 - 23486845361639043*x^10 + 70552096295073850*x^9 + 302207769671571109*x^8 - 1202161182566167879*x^7 + 921854615125235345*x^6 + 3618507295866457145*x^5 - 1503510294335687605*x^4 - 10746229822177899439*x^3 + 2011567894060138443*x^2 + 8767693047911919217*x + 4654012959769314377)
 
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 + 9*x^36 - 105*x^35 - 2517*x^34 - 12367*x^33 + 18961*x^32 + 524484*x^31 + 2063543*x^30 + 10035685*x^29 + 68343245*x^28 + 162739626*x^27 + 853626703*x^26 - 1517430356*x^25 + 15437152669*x^24 - 38624827303*x^23 + 261377865856*x^22 - 1008245158454*x^21 + 3680056602480*x^20 - 8092055201962*x^19 + 22243202914752*x^18 - 88956425126723*x^17 + 393393581809607*x^16 - 995223440330324*x^15 + 2718528813824046*x^14 - 6567254133383172*x^13 + 17369599534798699*x^12 - 25459990107367281*x^11 - 23486845361639043*x^10 + 70552096295073850*x^9 + 302207769671571109*x^8 - 1202161182566167879*x^7 + 921854615125235345*x^6 + 3618507295866457145*x^5 - 1503510294335687605*x^4 - 10746229822177899439*x^3 + 2011567894060138443*x^2 + 8767693047911919217*x + 4654012959769314377, 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 + 9*x^36 - 105*x^35 - 2517*x^34 - 12367*x^33 + 18961*x^32 + 524484*x^31 + 2063543*x^30 + 10035685*x^29 + 68343245*x^28 + 162739626*x^27 + 853626703*x^26 - 1517430356*x^25 + 15437152669*x^24 - 38624827303*x^23 + 261377865856*x^22 - 1008245158454*x^21 + 3680056602480*x^20 - 8092055201962*x^19 + 22243202914752*x^18 - 88956425126723*x^17 + 393393581809607*x^16 - 995223440330324*x^15 + 2718528813824046*x^14 - 6567254133383172*x^13 + 17369599534798699*x^12 - 25459990107367281*x^11 - 23486845361639043*x^10 + 70552096295073850*x^9 + 302207769671571109*x^8 - 1202161182566167879*x^7 + 921854615125235345*x^6 + 3618507295866457145*x^5 - 1503510294335687605*x^4 - 10746229822177899439*x^3 + 2011567894060138443*x^2 + 8767693047911919217*x + 4654012959769314377);
 
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 + 9*x^36 - 105*x^35 - 2517*x^34 - 12367*x^33 + 18961*x^32 + 524484*x^31 + 2063543*x^30 + 10035685*x^29 + 68343245*x^28 + 162739626*x^27 + 853626703*x^26 - 1517430356*x^25 + 15437152669*x^24 - 38624827303*x^23 + 261377865856*x^22 - 1008245158454*x^21 + 3680056602480*x^20 - 8092055201962*x^19 + 22243202914752*x^18 - 88956425126723*x^17 + 393393581809607*x^16 - 995223440330324*x^15 + 2718528813824046*x^14 - 6567254133383172*x^13 + 17369599534798699*x^12 - 25459990107367281*x^11 - 23486845361639043*x^10 + 70552096295073850*x^9 + 302207769671571109*x^8 - 1202161182566167879*x^7 + 921854615125235345*x^6 + 3618507295866457145*x^5 - 1503510294335687605*x^4 - 10746229822177899439*x^3 + 2011567894060138443*x^2 + 8767693047911919217*x + 4654012959769314377);
 
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

\(\Q(\sqrt{-647}) \), 19.19.394709020826813768130190448054073575320458976755889.1

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 $19^{2}$ $19^{2}$ $38$ $19^{2}$ $38$ $19^{2}$ $19^{2}$ $38$ $38$ $19^{2}$ $19^{2}$ $38$ $19^{2}$ ${\href{/padicField/43.1.0.1}{1} }^{38}$ $38$ ${\href{/padicField/53.1.0.1}{1} }^{38}$ $38$

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
\(647\) Copy content Toggle raw display Deg $38$$38$$1$$37$