Properties

Label 42.0.180...927.1
Degree $42$
Signature $[0, 21]$
Discriminant $-1.803\times 10^{86}$
Root discriminant $113.17$
Ramified prime $127$
Class number $1528865$ (GRH)
Class group $[1528865]$ (GRH)
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 + 2*x^40 + 80*x^39 - 73*x^38 + 139*x^37 + 2635*x^36 - 2181*x^35 + 3934*x^34 + 46702*x^33 - 34779*x^32 + 59013*x^31 + 489577*x^30 - 325107*x^29 + 514966*x^28 + 3148122*x^27 - 1839378*x^26 + 2684924*x^25 + 12529404*x^24 - 6257436*x^23 + 8056643*x^22 + 30745048*x^21 - 12383236*x^20 + 11933159*x^19 + 46384476*x^18 - 13840937*x^17 + 3649186*x^16 + 41025120*x^15 - 8138959*x^14 - 9679906*x^13 + 28379769*x^12 - 8357477*x^11 + 3513515*x^10 + 45545418*x^9 - 12952087*x^8 + 8843411*x^7 + 29858262*x^6 - 8807791*x^5 + 14296232*x^4 - 4154414*x^3 - 181830*x^2 + 2389098*x + 733913)
 
gp: K = bnfinit(x^42 - x^41 + 2*x^40 + 80*x^39 - 73*x^38 + 139*x^37 + 2635*x^36 - 2181*x^35 + 3934*x^34 + 46702*x^33 - 34779*x^32 + 59013*x^31 + 489577*x^30 - 325107*x^29 + 514966*x^28 + 3148122*x^27 - 1839378*x^26 + 2684924*x^25 + 12529404*x^24 - 6257436*x^23 + 8056643*x^22 + 30745048*x^21 - 12383236*x^20 + 11933159*x^19 + 46384476*x^18 - 13840937*x^17 + 3649186*x^16 + 41025120*x^15 - 8138959*x^14 - 9679906*x^13 + 28379769*x^12 - 8357477*x^11 + 3513515*x^10 + 45545418*x^9 - 12952087*x^8 + 8843411*x^7 + 29858262*x^6 - 8807791*x^5 + 14296232*x^4 - 4154414*x^3 - 181830*x^2 + 2389098*x + 733913, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![733913, 2389098, -181830, -4154414, 14296232, -8807791, 29858262, 8843411, -12952087, 45545418, 3513515, -8357477, 28379769, -9679906, -8138959, 41025120, 3649186, -13840937, 46384476, 11933159, -12383236, 30745048, 8056643, -6257436, 12529404, 2684924, -1839378, 3148122, 514966, -325107, 489577, 59013, -34779, 46702, 3934, -2181, 2635, 139, -73, 80, 2, -1, 1]);
 

\( x^{42} - x^{41} + 2 x^{40} + 80 x^{39} - 73 x^{38} + 139 x^{37} + 2635 x^{36} - 2181 x^{35} + 3934 x^{34} + 46702 x^{33} - 34779 x^{32} + 59013 x^{31} + 489577 x^{30} - 325107 x^{29} + 514966 x^{28} + 3148122 x^{27} - 1839378 x^{26} + 2684924 x^{25} + 12529404 x^{24} - 6257436 x^{23} + 8056643 x^{22} + 30745048 x^{21} - 12383236 x^{20} + 11933159 x^{19} + 46384476 x^{18} - 13840937 x^{17} + 3649186 x^{16} + 41025120 x^{15} - 8138959 x^{14} - 9679906 x^{13} + 28379769 x^{12} - 8357477 x^{11} + 3513515 x^{10} + 45545418 x^{9} - 12952087 x^{8} + 8843411 x^{7} + 29858262 x^{6} - 8807791 x^{5} + 14296232 x^{4} - 4154414 x^{3} - 181830 x^{2} + 2389098 x + 733913 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-18\!\cdots\!927\)\(\medspace = -\,127^{41}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $113.17$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $127$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(127\)
Dirichlet character group:    $\lbrace$$\chi_{127}(1,·)$, $\chi_{127}(2,·)$, $\chi_{127}(4,·)$, $\chi_{127}(5,·)$, $\chi_{127}(8,·)$, $\chi_{127}(10,·)$, $\chi_{127}(16,·)$, $\chi_{127}(19,·)$, $\chi_{127}(20,·)$, $\chi_{127}(25,·)$, $\chi_{127}(27,·)$, $\chi_{127}(32,·)$, $\chi_{127}(33,·)$, $\chi_{127}(38,·)$, $\chi_{127}(40,·)$, $\chi_{127}(47,·)$, $\chi_{127}(50,·)$, $\chi_{127}(51,·)$, $\chi_{127}(54,·)$, $\chi_{127}(61,·)$, $\chi_{127}(63,·)$, $\chi_{127}(64,·)$, $\chi_{127}(66,·)$, $\chi_{127}(73,·)$, $\chi_{127}(76,·)$, $\chi_{127}(77,·)$, $\chi_{127}(80,·)$, $\chi_{127}(87,·)$, $\chi_{127}(89,·)$, $\chi_{127}(94,·)$, $\chi_{127}(95,·)$, $\chi_{127}(100,·)$, $\chi_{127}(102,·)$, $\chi_{127}(107,·)$, $\chi_{127}(108,·)$, $\chi_{127}(111,·)$, $\chi_{127}(117,·)$, $\chi_{127}(119,·)$, $\chi_{127}(122,·)$, $\chi_{127}(123,·)$, $\chi_{127}(125,·)$$\chi_{127}(126,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{19} a^{17} + \frac{3}{19} a^{16} - \frac{8}{19} a^{14} - \frac{5}{19} a^{13} + \frac{7}{19} a^{11} + \frac{2}{19} a^{10} + \frac{1}{19} a^{8} + \frac{3}{19} a^{7} - \frac{8}{19} a^{5} - \frac{5}{19} a^{4} + \frac{7}{19} a^{2} + \frac{2}{19} a$, $\frac{1}{19} a^{18} - \frac{9}{19} a^{16} - \frac{8}{19} a^{15} - \frac{4}{19} a^{13} + \frac{7}{19} a^{12} - \frac{6}{19} a^{10} + \frac{1}{19} a^{9} - \frac{9}{19} a^{7} - \frac{8}{19} a^{6} - \frac{4}{19} a^{4} + \frac{7}{19} a^{3} - \frac{6}{19} a$, $\frac{1}{19} a^{19} - \frac{1}{19} a$, $\frac{1}{19} a^{20} - \frac{1}{19} a^{2}$, $\frac{1}{19} a^{21} - \frac{1}{19} a^{3}$, $\frac{1}{19} a^{22} - \frac{1}{19} a^{4}$, $\frac{1}{19} a^{23} - \frac{1}{19} a^{5}$, $\frac{1}{19} a^{24} - \frac{1}{19} a^{6}$, $\frac{1}{19} a^{25} - \frac{1}{19} a^{7}$, $\frac{1}{19} a^{26} - \frac{1}{19} a^{8}$, $\frac{1}{19} a^{27} - \frac{1}{19} a^{9}$, $\frac{1}{19} a^{28} - \frac{1}{19} a^{10}$, $\frac{1}{19} a^{29} - \frac{1}{19} a^{11}$, $\frac{1}{19} a^{30} - \frac{1}{19} a^{12}$, $\frac{1}{19} a^{31} - \frac{1}{19} a^{13}$, $\frac{1}{19} a^{32} - \frac{1}{19} a^{14}$, $\frac{1}{361} a^{33} + \frac{2}{361} a^{32} + \frac{1}{361} a^{31} - \frac{1}{361} a^{30} + \frac{3}{361} a^{29} - \frac{4}{361} a^{28} - \frac{1}{361} a^{27} - \frac{3}{361} a^{26} - \frac{8}{361} a^{25} - \frac{2}{361} a^{24} - \frac{6}{361} a^{23} + \frac{3}{361} a^{22} + \frac{5}{361} a^{21} + \frac{6}{361} a^{20} - \frac{4}{361} a^{19} + \frac{1}{361} a^{18} + \frac{1}{361} a^{17} - \frac{63}{361} a^{16} + \frac{48}{361} a^{15} + \frac{142}{361} a^{14} + \frac{104}{361} a^{13} - \frac{144}{361} a^{12} - \frac{129}{361} a^{11} - \frac{6}{19} a^{10} + \frac{40}{361} a^{9} - \frac{148}{361} a^{8} - \frac{169}{361} a^{7} + \frac{146}{361} a^{6} + \frac{131}{361} a^{5} + \frac{178}{361} a^{4} - \frac{93}{361} a^{3} + \frac{1}{361} a^{2} + \frac{4}{19} a$, $\frac{1}{361} a^{34} - \frac{3}{361} a^{32} - \frac{3}{361} a^{31} + \frac{5}{361} a^{30} + \frac{9}{361} a^{29} + \frac{7}{361} a^{28} - \frac{1}{361} a^{27} - \frac{2}{361} a^{26} - \frac{5}{361} a^{25} - \frac{2}{361} a^{24} - \frac{4}{361} a^{23} - \frac{1}{361} a^{22} - \frac{4}{361} a^{21} + \frac{3}{361} a^{20} + \frac{9}{361} a^{19} - \frac{1}{361} a^{18} - \frac{8}{361} a^{17} - \frac{16}{361} a^{16} + \frac{46}{361} a^{15} + \frac{86}{361} a^{14} + \frac{85}{361} a^{13} + \frac{159}{361} a^{12} + \frac{163}{361} a^{11} + \frac{21}{361} a^{10} + \frac{7}{19} a^{9} - \frac{177}{361} a^{8} - \frac{48}{361} a^{7} - \frac{161}{361} a^{6} - \frac{160}{361} a^{5} - \frac{12}{361} a^{4} - \frac{174}{361} a^{3} + \frac{93}{361} a^{2} - \frac{2}{19} a$, $\frac{1}{361} a^{35} + \frac{3}{361} a^{32} + \frac{8}{361} a^{31} + \frac{6}{361} a^{30} - \frac{3}{361} a^{29} + \frac{6}{361} a^{28} - \frac{5}{361} a^{27} + \frac{5}{361} a^{26} - \frac{7}{361} a^{25} + \frac{9}{361} a^{24} + \frac{5}{361} a^{22} - \frac{1}{361} a^{21} + \frac{8}{361} a^{20} + \frac{6}{361} a^{19} - \frac{5}{361} a^{18} + \frac{6}{361} a^{17} - \frac{86}{361} a^{16} - \frac{131}{361} a^{15} - \frac{2}{361} a^{14} + \frac{15}{361} a^{13} + \frac{92}{361} a^{12} + \frac{147}{361} a^{11} + \frac{9}{19} a^{10} - \frac{3}{19} a^{9} - \frac{131}{361} a^{8} + \frac{92}{361} a^{7} - \frac{102}{361} a^{6} - \frac{151}{361} a^{5} - \frac{96}{361} a^{4} - \frac{167}{361} a^{3} + \frac{117}{361} a^{2} - \frac{6}{19} a$, $\frac{1}{38627} a^{36} - \frac{3}{38627} a^{35} - \frac{1}{38627} a^{34} - \frac{34}{38627} a^{33} + \frac{80}{38627} a^{32} + \frac{632}{38627} a^{31} - \frac{825}{38627} a^{30} + \frac{389}{38627} a^{29} - \frac{908}{38627} a^{28} + \frac{20}{38627} a^{27} - \frac{498}{38627} a^{26} + \frac{711}{38627} a^{25} + \frac{524}{38627} a^{24} - \frac{130}{38627} a^{23} - \frac{506}{38627} a^{22} - \frac{170}{38627} a^{21} - \frac{718}{38627} a^{20} - \frac{36}{38627} a^{19} - \frac{946}{38627} a^{18} - \frac{4}{2033} a^{17} - \frac{7026}{38627} a^{16} + \frac{1685}{38627} a^{15} + \frac{10679}{38627} a^{14} - \frac{9795}{38627} a^{13} - \frac{6778}{38627} a^{12} - \frac{12000}{38627} a^{11} - \frac{11307}{38627} a^{10} - \frac{17628}{38627} a^{9} + \frac{9311}{38627} a^{8} + \frac{12649}{38627} a^{7} + \frac{9829}{38627} a^{6} + \frac{5322}{38627} a^{5} - \frac{468}{38627} a^{4} - \frac{1201}{38627} a^{3} - \frac{7663}{38627} a^{2} + \frac{997}{2033} a$, $\frac{1}{19661143} a^{37} + \frac{34}{19661143} a^{36} + \frac{2135}{19661143} a^{35} - \frac{25323}{19661143} a^{34} + \frac{16584}{19661143} a^{33} + \frac{229362}{19661143} a^{32} + \frac{453234}{19661143} a^{31} + \frac{394333}{19661143} a^{30} + \frac{7890}{1034797} a^{29} - \frac{300434}{19661143} a^{28} - \frac{251529}{19661143} a^{27} - \frac{53988}{19661143} a^{26} - \frac{47427}{19661143} a^{25} + \frac{107319}{19661143} a^{24} + \frac{7417}{19661143} a^{23} + \frac{257917}{19661143} a^{22} + \frac{491612}{19661143} a^{21} - \frac{423037}{19661143} a^{20} - \frac{226122}{19661143} a^{19} + \frac{118681}{19661143} a^{18} - \frac{339719}{19661143} a^{17} + \frac{8909055}{19661143} a^{16} - \frac{7802390}{19661143} a^{15} + \frac{7562246}{19661143} a^{14} + \frac{9176598}{19661143} a^{13} - \frac{3239847}{19661143} a^{12} + \frac{2457554}{19661143} a^{11} + \frac{6614778}{19661143} a^{10} + \frac{7689272}{19661143} a^{9} - \frac{430257}{19661143} a^{8} + \frac{271075}{1034797} a^{7} + \frac{7353634}{19661143} a^{6} + \frac{1363388}{19661143} a^{5} - \frac{4594586}{19661143} a^{4} + \frac{1969023}{19661143} a^{3} + \frac{9737558}{19661143} a^{2} - \frac{193910}{1034797} a - \frac{232}{509}$, $\frac{1}{19661143} a^{38} - \frac{39}{19661143} a^{36} + \frac{14067}{19661143} a^{35} + \frac{7176}{19661143} a^{34} + \frac{26896}{19661143} a^{33} + \frac{307232}{19661143} a^{32} - \frac{463822}{19661143} a^{31} + \frac{108928}{19661143} a^{30} + \frac{469869}{19661143} a^{29} - \frac{47}{183749} a^{28} + \frac{199262}{19661143} a^{27} + \frac{497850}{19661143} a^{26} - \frac{93221}{19661143} a^{25} - \frac{35673}{19661143} a^{24} - \frac{406551}{19661143} a^{23} - \frac{192101}{19661143} a^{22} + \frac{463375}{19661143} a^{21} - \frac{21897}{1034797} a^{20} + \frac{382046}{19661143} a^{19} + \frac{346102}{19661143} a^{18} - \frac{25571}{1034797} a^{17} - \frac{1887235}{19661143} a^{16} - \frac{8648255}{19661143} a^{15} - \frac{928683}{19661143} a^{14} - \frac{6379925}{19661143} a^{13} + \frac{8081058}{19661143} a^{12} + \frac{3025914}{19661143} a^{11} - \frac{486070}{19661143} a^{10} - \frac{7332929}{19661143} a^{9} - \frac{2607166}{19661143} a^{8} - \frac{1889932}{19661143} a^{7} + \frac{9291870}{19661143} a^{6} + \frac{9042489}{19661143} a^{5} - \frac{9792688}{19661143} a^{4} - \frac{2612866}{19661143} a^{3} + \frac{71083}{183749} a^{2} + \frac{313594}{1034797} a + \frac{253}{509}$, $\frac{1}{373561717} a^{39} + \frac{9}{373561717} a^{38} + \frac{8}{373561717} a^{37} - \frac{4537}{373561717} a^{36} + \frac{457066}{373561717} a^{35} + \frac{173799}{373561717} a^{34} - \frac{229305}{373561717} a^{33} + \frac{10414}{1034797} a^{32} + \frac{272677}{19661143} a^{31} + \frac{9619614}{373561717} a^{30} + \frac{388669}{373561717} a^{29} + \frac{1008383}{373561717} a^{28} - \frac{5625098}{373561717} a^{27} + \frac{2259229}{373561717} a^{26} - \frac{552114}{373561717} a^{25} - \frac{4723964}{373561717} a^{24} + \frac{221892}{373561717} a^{23} - \frac{4859828}{373561717} a^{22} + \frac{3547896}{373561717} a^{21} + \frac{7510227}{373561717} a^{20} + \frac{2639961}{373561717} a^{19} + \frac{952808}{373561717} a^{18} + \frac{1121670}{373561717} a^{17} - \frac{29329430}{373561717} a^{16} + \frac{36840367}{373561717} a^{15} - \frac{50974776}{373561717} a^{14} - \frac{4282122}{373561717} a^{13} + \frac{84665112}{373561717} a^{12} - \frac{83233788}{373561717} a^{11} - \frac{109217796}{373561717} a^{10} - \frac{160603565}{373561717} a^{9} - \frac{36466423}{373561717} a^{8} + \frac{73393423}{373561717} a^{7} - \frac{46622985}{373561717} a^{6} + \frac{119831414}{373561717} a^{5} - \frac{156452088}{373561717} a^{4} - \frac{177177155}{373561717} a^{3} + \frac{113395}{19661143} a^{2} - \frac{17420}{54463} a - \frac{79}{509}$, $\frac{1}{21640803827527} a^{40} - \frac{10381}{21640803827527} a^{39} - \frac{45318}{21640803827527} a^{38} + \frac{537139}{21640803827527} a^{37} + \frac{62308633}{21640803827527} a^{36} - \frac{8050177856}{21640803827527} a^{35} + \frac{17272512348}{21640803827527} a^{34} + \frac{25868273044}{21640803827527} a^{33} + \frac{21653540436}{1138989675133} a^{32} - \frac{517296799242}{21640803827527} a^{31} - \frac{346859790347}{21640803827527} a^{30} + \frac{323333946633}{21640803827527} a^{29} + \frac{55420110529}{21640803827527} a^{28} + \frac{382100248000}{21640803827527} a^{27} - \frac{99584067552}{21640803827527} a^{26} - \frac{60181367649}{21640803827527} a^{25} + \frac{146468088430}{21640803827527} a^{24} + \frac{46642974945}{21640803827527} a^{23} - \frac{151419273681}{21640803827527} a^{22} - \frac{485697147037}{21640803827527} a^{21} + \frac{66978822185}{21640803827527} a^{20} + \frac{146735341751}{21640803827527} a^{19} + \frac{556938728590}{21640803827527} a^{18} + \frac{391826411786}{21640803827527} a^{17} + \frac{387349179937}{21640803827527} a^{16} - \frac{2397838978859}{21640803827527} a^{15} + \frac{3110757281309}{21640803827527} a^{14} - \frac{452963130987}{1138989675133} a^{13} + \frac{6496913622368}{21640803827527} a^{12} + \frac{6225980942897}{21640803827527} a^{11} - \frac{3557749040526}{21640803827527} a^{10} - \frac{6226025891295}{21640803827527} a^{9} - \frac{8589454833696}{21640803827527} a^{8} - \frac{8669512855797}{21640803827527} a^{7} + \frac{3560630388858}{21640803827527} a^{6} + \frac{2903015360036}{21640803827527} a^{5} - \frac{6029210500183}{21640803827527} a^{4} - \frac{2560236032782}{21640803827527} a^{3} + \frac{449891967806}{1138989675133} a^{2} - \frac{2712650494}{59946825007} a - \frac{7736935}{29486879}$, $\frac{1}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{41} + \frac{147269237418794573127785910739047813131772637981088690700268800731561994854530159598678836100266497}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{40} + \frac{34322952148173241118263686025152407107628371614177858733451611551956984435138428328542799112072883924392}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{39} + \frac{34374936051048472910832804569440343783364272442330150769991003482877573234626062433800285230645875401375}{1798650348156993847788206255430980033281476123006948188855856976883603085888548043820829534522673500935407189509} a^{38} + \frac{340407247969555205122269055172984900573217583671972829347856427441501541695616247663657974026196430725565}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{37} - \frac{70829006341392041609190432358025501207132593794208503840382167155434974980065575835626260841458026932805080}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{36} + \frac{8235921563934427751945783584994946292076066804116734174968880811789083447784076892658375997310138439751057475}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{35} + \frac{11609866316199563526106611245489812696873414020500894050963404648394227177949099413073078376534644553875320842}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{34} + \frac{23356998031764409135261654707725826792505331971484168999612410499695782364553718912291244187878374706240450241}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{33} - \frac{420655169503977544792885909176916072240626634672393079175867574274235618349227150868198262992547867060335973769}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{32} + \frac{456106768306698025971596598744745630277209291765010910232604555731560800062947661473718887091801750272825512659}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{31} + \frac{340179409113570452272005340862098313092797206395733196862388373713938377157538965129099943506959246188037197390}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{30} + \frac{78637848653195065936719418687446557503456735457818418830930554640048690376353957126152742152533952077323143697}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{29} - \frac{8867271035907678335640283035973192453487450473334658057670665078484669181406157866014595805891059708268303557}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{28} + \frac{40361270791538287738509305939268742385452778595108998215927346975074298770076068483270095738916733101333506975}{1798650348156993847788206255430980033281476123006948188855856976883603085888548043820829534522673500935407189509} a^{27} - \frac{298041242182073444090320994755684901906048126540395180574331326050764338929101841976577121987167734076624692545}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{26} - \frac{684816008559372306918654278835269317096869450711276549923203808940907304752573419682951082899438792877017007530}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{25} - \frac{307547227179110026851941269861130258721490105273664931585968883310709444805088757394310604947016396919566249889}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{24} + \frac{601490865383399696858615036852182446398704990768677631691951678712281229997426146757500370110903023998134725684}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{23} + \frac{797905523778882821413566219779936802242146250161424187999241018459016157830224006709118568715520327997792952642}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{22} + \frac{39456670120856662180660286987422350670769731782098444697171806785218411417954669064490802700462409456923798038}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{21} + \frac{170543766089304602308138059350330170438581677034946901067877615196784903295390732046716041158324910508703484467}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{20} + \frac{488209232129230421114331796044130176521809358455214997630717175597431408049117727672499501586023655675088797086}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{19} - \frac{493113662547247215684171105014117090449233447910712160533883892655955533622839401072696260594197639036458034205}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{18} + \frac{748183993060273678632389804391971096073033536122893653395854657161773185040373206714654095183257105676996289479}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{17} + \frac{10968391795601521886581225556239610185176900498552612565880168276149282970568706663706874530919586973658680906492}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{16} + \frac{6590683778713370087370062157837024756983603934165862224991495212650437937517536589544377179093258067078769571966}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{15} + \frac{12938411171284910642058143984809417255040433718593234144358222715446009410132615876648580420547742247995891911602}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{14} - \frac{7896112138070272738329133333920591323908300974638849527362036911657622579091384882639000068823053193653992587949}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{13} + \frac{691446252487096631807406391281705937017132376789154745443486384727513592266778432320242745012687703816398582376}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{12} + \frac{5623239133783445255229713081742554927985763320182351128729701018293271734805072912193761033872114427365491695307}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{11} + \frac{7464768007291367129163498922647487620056117471786472648411593346420096905768668058727824648667930702672127888035}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{10} + \frac{8852562676393264925657729664484910519609029029659067292119161567689795859370258863050152291595938301903691228460}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{9} - \frac{11588029612561029764666458860046865212252547053594981047078781691468265278125830658413234600229022212779472183523}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{8} + \frac{15140152347250051324715647914109958036279580584895541840339989725102983005914374428881253196933381907090141863103}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{7} + \frac{15447987046516314207156494585757278107504493429025689349673202073115429273677539989842647353340003306801790384048}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{6} + \frac{447424027012369804261020128639034394203293963494974955420615472201153909698181109393861021415890581211606704127}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{5} - \frac{10129767636816961698151275196188304465911266133754832014485473255443509616824952213017901346460557557104785812710}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{4} + \frac{9722357934450129153812941093806617969394034932408523036777208433129554321843502350621324164480252055114370586175}{34174356614982883107975918853188620632348046337132015588261282560788458631882412832595761155930796517772736600671} a^{3} + \frac{35687650997114039687236435502739521550985211645928688136885864651292563582821707515326345329037714526635251}{589914840326990438762940720049517885628558912104607474206578214786357194453443110469278299285888324347460541} a^{2} - \frac{15553569598404704193928397819434207053319250486782570379590794037548594880738586270431427626485052542746289468}{94665807797736518304642434496367370172709269631944641518729314572821215046765686516885764974877552680810904711} a + \frac{21810700274739501565516817119316180877579803667767483570399058645320988742888944351469034202329566725285218}{46564588193672660258063174862945091083477260025550733654072461668874183495703731685629987690544787349144567}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{1528865}$, which has order $1528865$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 418513668789608050000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{21}\cdot 418513668789608050000 \cdot 1528865}{2\sqrt{180282079628321418522579756639824623344453525380673158224385384254625263736344592716927}}\approx 1.37673307318290$ (assuming GRH)

Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-127}) \), 3.3.16129.1, 6.0.33038369407.1, 7.7.4195872914689.1, 14.0.2235879388560037062539773567.1, 21.21.1191446152405248657777607437681912764659201.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{6}$ $42$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{3}$ $42$ $21^{2}$ $21^{2}$ $21^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{42}$ $42$ $42$ $21^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{14}$ $21^{2}$ $42$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{6}$ $42$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{7}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
127Data not computed