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Properties

Label 38.38.290...677.1
Degree $38$
Signature $[38, 0]$
Discriminant $2.908\times 10^{93}$
Root discriminant \(288.12\)
Ramified primes $3,191$
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 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451)
 
gp: K = bnfinit(y^38 - y^37 - 188*y^36 + 180*y^35 + 15133*y^34 - 12829*y^33 - 695929*y^32 + 470210*y^31 + 20560148*y^30 - 9452291*y^29 - 414286793*y^28 + 90520176*y^27 + 5875939616*y^26 + 191303612*y^25 - 59496127537*y^24 - 16762796308*y^23 + 430644226744*y^22 + 224673611403*y^21 - 2206305371572*y^20 - 1655031822292*y^19 + 7828656065108*y^18 + 7607383848877*y^17 - 18551523651529*y^16 - 22355418258850*y^15 + 27686404325779*y^14 + 41724858463028*y^13 - 23225313872578*y^12 - 48637857116067*y^11 + 6837701957301*y^10 + 34223650023543*y^9 + 4693738691921*y^8 - 13444562674843*y^7 - 4790142874347*y^6 + 2352031466145*y^5 + 1507820068977*y^4 + 14043380009*y^3 - 154333274880*y^2 - 38062864399*y - 2836879451, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^38 - x^37 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^38 - x^37 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451)
 

\( x^{38} - x^{37} - 188 x^{36} + 180 x^{35} + 15133 x^{34} - 12829 x^{33} - 695929 x^{32} + \cdots - 2836879451 \) 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:   \(290\!\cdots\!677\) \(\medspace = 3^{19}\cdot 191^{37}\) 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:  \(288.12\)
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:  $3^{1/2}191^{37/38}\approx 288.1156178920137$
Ramified primes:   \(3\), \(191\) 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:  \(\Q(\sqrt{573}) \)
$\card{ \Gal(K/\Q) }$:  $38$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(573=3\cdot 191\)
Dirichlet character group:    $\lbrace$$\chi_{573}(1,·)$, $\chi_{573}(388,·)$, $\chi_{573}(257,·)$, $\chi_{573}(136,·)$, $\chi_{573}(521,·)$, $\chi_{573}(11,·)$, $\chi_{573}(14,·)$, $\chi_{573}(275,·)$, $\chi_{573}(532,·)$, $\chi_{573}(196,·)$, $\chi_{573}(535,·)$, $\chi_{573}(25,·)$, $\chi_{573}(154,·)$, $\chi_{573}(155,·)$, $\chi_{573}(412,·)$, $\chi_{573}(413,·)$, $\chi_{573}(160,·)$, $\chi_{573}(161,·)$, $\chi_{573}(418,·)$, $\chi_{573}(419,·)$, $\chi_{573}(548,·)$, $\chi_{573}(38,·)$, $\chi_{573}(41,·)$, $\chi_{573}(298,·)$, $\chi_{573}(559,·)$, $\chi_{573}(562,·)$, $\chi_{573}(52,·)$, $\chi_{573}(437,·)$, $\chi_{573}(185,·)$, $\chi_{573}(572,·)$, $\chi_{573}(451,·)$, $\chi_{573}(452,·)$, $\chi_{573}(121,·)$, $\chi_{573}(350,·)$, $\chi_{573}(223,·)$, $\chi_{573}(316,·)$, $\chi_{573}(377,·)$, $\chi_{573}(122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{7}a^{25}-\frac{2}{7}a^{24}-\frac{2}{7}a^{23}+\frac{3}{7}a^{22}-\frac{2}{7}a^{21}+\frac{1}{7}a^{20}+\frac{3}{7}a^{19}-\frac{2}{7}a^{18}+\frac{1}{7}a^{17}+\frac{1}{7}a^{16}-\frac{1}{7}a^{15}+\frac{3}{7}a^{13}-\frac{1}{7}a^{12}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a$, $\frac{1}{7}a^{26}+\frac{1}{7}a^{24}-\frac{1}{7}a^{23}-\frac{3}{7}a^{22}-\frac{3}{7}a^{21}-\frac{2}{7}a^{20}-\frac{3}{7}a^{19}-\frac{3}{7}a^{18}+\frac{3}{7}a^{17}+\frac{1}{7}a^{16}-\frac{2}{7}a^{15}+\frac{3}{7}a^{14}-\frac{2}{7}a^{13}-\frac{2}{7}a^{12}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{2}{7}a^{7}+\frac{3}{7}a^{6}+\frac{2}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{3}+\frac{3}{7}a^{2}-\frac{3}{7}a$, $\frac{1}{7}a^{27}+\frac{1}{7}a^{24}-\frac{1}{7}a^{23}+\frac{1}{7}a^{22}+\frac{3}{7}a^{20}+\frac{1}{7}a^{19}-\frac{2}{7}a^{18}-\frac{3}{7}a^{16}-\frac{3}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{2}{7}a^{10}+\frac{1}{7}a^{9}+\frac{1}{7}a^{8}-\frac{1}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a$, $\frac{1}{7}a^{28}+\frac{1}{7}a^{24}+\frac{3}{7}a^{23}-\frac{3}{7}a^{22}-\frac{2}{7}a^{21}+\frac{2}{7}a^{19}+\frac{2}{7}a^{18}+\frac{3}{7}a^{17}+\frac{3}{7}a^{16}-\frac{1}{7}a^{15}+\frac{2}{7}a^{14}-\frac{2}{7}a^{13}-\frac{2}{7}a^{12}+\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a$, $\frac{1}{7}a^{29}-\frac{2}{7}a^{24}-\frac{1}{7}a^{23}+\frac{2}{7}a^{22}+\frac{2}{7}a^{21}+\frac{1}{7}a^{20}-\frac{1}{7}a^{19}-\frac{2}{7}a^{18}+\frac{2}{7}a^{17}-\frac{2}{7}a^{16}+\frac{3}{7}a^{15}-\frac{2}{7}a^{14}+\frac{2}{7}a^{13}+\frac{3}{7}a^{12}+\frac{1}{7}a^{11}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a$, $\frac{1}{763}a^{30}-\frac{29}{763}a^{29}+\frac{22}{763}a^{28}+\frac{4}{763}a^{27}-\frac{9}{763}a^{26}+\frac{31}{763}a^{25}-\frac{307}{763}a^{24}+\frac{323}{763}a^{23}-\frac{223}{763}a^{22}+\frac{54}{109}a^{21}+\frac{110}{763}a^{20}+\frac{292}{763}a^{19}+\frac{57}{763}a^{18}+\frac{257}{763}a^{17}-\frac{358}{763}a^{16}+\frac{72}{763}a^{15}-\frac{253}{763}a^{14}+\frac{215}{763}a^{13}-\frac{99}{763}a^{12}-\frac{50}{109}a^{11}-\frac{22}{763}a^{10}+\frac{360}{763}a^{9}-\frac{44}{763}a^{8}+\frac{40}{109}a^{7}-\frac{151}{763}a^{6}+\frac{264}{763}a^{5}+\frac{326}{763}a^{4}-\frac{355}{763}a^{3}+\frac{376}{763}a^{2}-\frac{349}{763}a-\frac{8}{109}$, $\frac{1}{763}a^{31}+\frac{53}{763}a^{29}-\frac{12}{763}a^{28}-\frac{2}{763}a^{27}-\frac{12}{763}a^{26}+\frac{47}{763}a^{25}+\frac{20}{109}a^{24}-\frac{339}{763}a^{23}-\frac{29}{109}a^{22}+\frac{9}{109}a^{21}-\frac{6}{763}a^{20}+\frac{132}{763}a^{19}+\frac{275}{763}a^{18}+\frac{17}{109}a^{17}-\frac{282}{763}a^{16}+\frac{200}{763}a^{15}-\frac{146}{763}a^{14}+\frac{32}{763}a^{13}-\frac{60}{763}a^{12}-\frac{362}{763}a^{11}+\frac{158}{763}a^{10}+\frac{41}{763}a^{9}+\frac{312}{763}a^{8}-\frac{45}{109}a^{7}+\frac{354}{763}a^{6}+\frac{25}{763}a^{5}-\frac{275}{763}a^{4}+\frac{13}{109}a^{2}+\frac{69}{763}a-\frac{14}{109}$, $\frac{1}{763}a^{32}-\frac{1}{763}a^{29}+\frac{31}{763}a^{28}-\frac{6}{763}a^{27}-\frac{3}{109}a^{26}+\frac{23}{763}a^{25}+\frac{18}{763}a^{24}+\frac{48}{109}a^{23}+\frac{219}{763}a^{22}-\frac{202}{763}a^{21}-\frac{139}{763}a^{20}-\frac{268}{763}a^{19}-\frac{68}{763}a^{18}+\frac{267}{763}a^{17}+\frac{208}{763}a^{16}-\frac{21}{109}a^{15}+\frac{34}{763}a^{14}-\frac{17}{109}a^{13}-\frac{20}{763}a^{12}-\frac{149}{763}a^{11}-\frac{101}{763}a^{10}-\frac{347}{763}a^{9}-\frac{163}{763}a^{8}+\frac{229}{763}a^{7}-\frac{38}{763}a^{6}+\frac{12}{763}a^{5}-\frac{165}{763}a^{4}-\frac{60}{763}a^{3}+\frac{306}{763}a^{2}-\frac{349}{763}a-\frac{12}{109}$, $\frac{1}{763}a^{33}+\frac{2}{763}a^{29}+\frac{16}{763}a^{28}-\frac{17}{763}a^{27}+\frac{2}{109}a^{26}+\frac{7}{109}a^{25}+\frac{29}{763}a^{24}-\frac{221}{763}a^{23}+\frac{338}{763}a^{22}+\frac{239}{763}a^{21}-\frac{158}{763}a^{20}+\frac{32}{109}a^{19}+\frac{324}{763}a^{18}-\frac{298}{763}a^{17}+\frac{258}{763}a^{16}+\frac{106}{763}a^{15}-\frac{372}{763}a^{14}+\frac{195}{763}a^{13}-\frac{248}{763}a^{12}+\frac{312}{763}a^{11}-\frac{369}{763}a^{10}+\frac{197}{763}a^{9}+\frac{185}{763}a^{8}+\frac{242}{763}a^{7}-\frac{139}{763}a^{6}+\frac{99}{763}a^{5}+\frac{38}{109}a^{4}-\frac{7}{109}a^{3}+\frac{27}{763}a^{2}+\frac{330}{763}a-\frac{8}{109}$, $\frac{1}{763}a^{34}-\frac{5}{109}a^{29}+\frac{48}{763}a^{28}+\frac{6}{763}a^{27}-\frac{6}{109}a^{26}-\frac{33}{763}a^{25}-\frac{152}{763}a^{24}+\frac{237}{763}a^{23}-\frac{296}{763}a^{22}-\frac{260}{763}a^{21}+\frac{113}{763}a^{20}-\frac{369}{763}a^{19}+\frac{351}{763}a^{18}+\frac{289}{763}a^{17}-\frac{268}{763}a^{16}+\frac{29}{763}a^{15}+\frac{47}{763}a^{14}-\frac{19}{109}a^{13}+\frac{183}{763}a^{12}-\frac{323}{763}a^{11}+\frac{241}{763}a^{10}+\frac{228}{763}a^{9}+\frac{221}{763}a^{8}-\frac{45}{763}a^{7}-\frac{362}{763}a^{6}-\frac{153}{763}a^{5}-\frac{156}{763}a^{4}+\frac{43}{109}a^{3}+\frac{2}{109}a^{2}+\frac{206}{763}a+\frac{16}{109}$, $\frac{1}{763}a^{35}+\frac{2}{109}a^{29}+\frac{13}{763}a^{28}-\frac{11}{763}a^{27}-\frac{3}{109}a^{26}-\frac{48}{763}a^{25}-\frac{53}{109}a^{24}+\frac{3}{7}a^{23}-\frac{31}{109}a^{22}+\frac{263}{763}a^{21}+\frac{211}{763}a^{20}+\frac{216}{763}a^{19}-\frac{5}{763}a^{18}+\frac{1}{109}a^{17}-\frac{293}{763}a^{16}+\frac{60}{763}a^{15}+\frac{24}{109}a^{14}-\frac{249}{763}a^{13}+\frac{136}{763}a^{12}-\frac{19}{763}a^{11}-\frac{106}{763}a^{10}-\frac{41}{763}a^{9}+\frac{159}{763}a^{8}-\frac{45}{763}a^{7}+\frac{12}{763}a^{6}+\frac{37}{763}a^{5}-\frac{170}{763}a^{4}-\frac{312}{763}a^{3}-\frac{41}{763}a^{2}-\frac{4}{763}a+\frac{47}{109}$, $\frac{1}{1370639588843}a^{36}+\frac{833090371}{1370639588843}a^{35}-\frac{556156977}{1370639588843}a^{34}+\frac{704988761}{1370639588843}a^{33}+\frac{648807200}{1370639588843}a^{32}+\frac{750279711}{1370639588843}a^{31}-\frac{712377978}{1370639588843}a^{30}+\frac{7689553381}{1370639588843}a^{29}-\frac{83682178585}{1370639588843}a^{28}-\frac{88840402270}{1370639588843}a^{27}+\frac{504445485}{195805655549}a^{26}+\frac{85862574660}{1370639588843}a^{25}+\frac{381333541629}{1370639588843}a^{24}+\frac{802953904}{1370639588843}a^{23}+\frac{204823429009}{1370639588843}a^{22}-\frac{149150188562}{1370639588843}a^{21}+\frac{74670012202}{195805655549}a^{20}+\frac{7003252694}{195805655549}a^{19}-\frac{604737816362}{1370639588843}a^{18}-\frac{345010856341}{1370639588843}a^{17}+\frac{426709361642}{1370639588843}a^{16}-\frac{639083872715}{1370639588843}a^{15}+\frac{91079744481}{195805655549}a^{14}+\frac{479531287259}{1370639588843}a^{13}+\frac{217261920087}{1370639588843}a^{12}+\frac{5196343346}{195805655549}a^{11}+\frac{613471234725}{1370639588843}a^{10}+\frac{610227836935}{1370639588843}a^{9}-\frac{111995596197}{1370639588843}a^{8}-\frac{540066546438}{1370639588843}a^{7}-\frac{394756406594}{1370639588843}a^{6}-\frac{92916546647}{1370639588843}a^{5}+\frac{518409606985}{1370639588843}a^{4}-\frac{469238922909}{1370639588843}a^{3}+\frac{239234853605}{1370639588843}a^{2}-\frac{79324356280}{195805655549}a-\frac{5044819}{66442367}$, $\frac{1}{67\!\cdots\!89}a^{37}+\frac{35\!\cdots\!50}{96\!\cdots\!27}a^{36}-\frac{43\!\cdots\!66}{67\!\cdots\!89}a^{35}-\frac{27\!\cdots\!45}{67\!\cdots\!89}a^{34}-\frac{19\!\cdots\!61}{67\!\cdots\!89}a^{33}+\frac{45\!\cdots\!26}{96\!\cdots\!27}a^{32}-\frac{32\!\cdots\!54}{67\!\cdots\!89}a^{31}-\frac{25\!\cdots\!16}{67\!\cdots\!89}a^{30}-\frac{24\!\cdots\!01}{67\!\cdots\!89}a^{29}-\frac{40\!\cdots\!39}{67\!\cdots\!89}a^{28}+\frac{28\!\cdots\!79}{67\!\cdots\!89}a^{27}+\frac{21\!\cdots\!72}{67\!\cdots\!89}a^{26}-\frac{16\!\cdots\!04}{67\!\cdots\!89}a^{25}-\frac{76\!\cdots\!77}{67\!\cdots\!89}a^{24}-\frac{58\!\cdots\!70}{67\!\cdots\!89}a^{23}-\frac{12\!\cdots\!98}{96\!\cdots\!27}a^{22}-\frac{17\!\cdots\!40}{67\!\cdots\!89}a^{21}-\frac{16\!\cdots\!74}{13\!\cdots\!61}a^{20}-\frac{21\!\cdots\!40}{67\!\cdots\!89}a^{19}+\frac{30\!\cdots\!80}{67\!\cdots\!89}a^{18}+\frac{91\!\cdots\!80}{67\!\cdots\!89}a^{17}+\frac{68\!\cdots\!77}{67\!\cdots\!89}a^{16}-\frac{26\!\cdots\!58}{67\!\cdots\!89}a^{15}+\frac{23\!\cdots\!00}{67\!\cdots\!89}a^{14}+\frac{16\!\cdots\!34}{67\!\cdots\!89}a^{13}+\frac{14\!\cdots\!42}{67\!\cdots\!89}a^{12}-\frac{60\!\cdots\!03}{67\!\cdots\!89}a^{11}+\frac{28\!\cdots\!71}{67\!\cdots\!89}a^{10}-\frac{24\!\cdots\!92}{67\!\cdots\!89}a^{9}+\frac{18\!\cdots\!32}{67\!\cdots\!89}a^{8}+\frac{13\!\cdots\!30}{67\!\cdots\!89}a^{7}-\frac{37\!\cdots\!72}{67\!\cdots\!89}a^{6}-\frac{28\!\cdots\!87}{67\!\cdots\!89}a^{5}-\frac{26\!\cdots\!18}{67\!\cdots\!89}a^{4}-\frac{43\!\cdots\!73}{96\!\cdots\!27}a^{3}+\frac{13\!\cdots\!88}{67\!\cdots\!89}a^{2}+\frac{36\!\cdots\!67}{13\!\cdots\!61}a-\frac{46\!\cdots\!99}{32\!\cdots\!41}$ 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 - x^37 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451)
 
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 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451, 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 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451);
 
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 - 188*x^36 + 180*x^35 + 15133*x^34 - 12829*x^33 - 695929*x^32 + 470210*x^31 + 20560148*x^30 - 9452291*x^29 - 414286793*x^28 + 90520176*x^27 + 5875939616*x^26 + 191303612*x^25 - 59496127537*x^24 - 16762796308*x^23 + 430644226744*x^22 + 224673611403*x^21 - 2206305371572*x^20 - 1655031822292*x^19 + 7828656065108*x^18 + 7607383848877*x^17 - 18551523651529*x^16 - 22355418258850*x^15 + 27686404325779*x^14 + 41724858463028*x^13 - 23225313872578*x^12 - 48637857116067*x^11 + 6837701957301*x^10 + 34223650023543*x^9 + 4693738691921*x^8 - 13444562674843*x^7 - 4790142874347*x^6 + 2352031466145*x^5 + 1507820068977*x^4 + 14043380009*x^3 - 154333274880*x^2 - 38062864399*x - 2836879451);
 
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

\(\Q(\sqrt{573}) \), 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 $38$ R $38$ ${\href{/padicField/7.2.0.1}{2} }^{19}$ $19^{2}$ $19^{2}$ $38$ $38$ $38$ $19^{2}$ $38$ $38$ $19^{2}$ $19^{2}$ $19^{2}$ $19^{2}$ $38$

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
\(3\) Copy content Toggle raw display Deg $38$$2$$19$$19$
\(191\) Copy content Toggle raw display Deg $38$$38$$1$$37$