Properties

Label 42.0.121...007.1
Degree $42$
Signature $[0, 21]$
Discriminant $-1.218\times 10^{83}$
Root discriminant $95.11$
Ramified primes $7, 43$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 + 62*x^40 - 55*x^39 + 1885*x^38 - 1493*x^37 + 36890*x^36 - 26096*x^35 + 516884*x^34 - 325868*x^33 + 5474760*x^32 - 3063183*x^31 + 45214327*x^30 - 22319399*x^29 + 296463014*x^28 - 128129100*x^27 + 1558565095*x^26 - 584173775*x^25 + 6595257074*x^24 - 2118905803*x^23 + 22443263173*x^22 - 6093288301*x^21 + 61089146298*x^20 - 13772999112*x^19 + 131703790552*x^18 - 24134677936*x^17 + 221581028128*x^16 - 32118789120*x^15 + 284876509184*x^14 - 31567940608*x^13 + 271846162432*x^12 - 21964134400*x^11 + 184905250816*x^10 - 10296512512*x^9 + 84546772992*x^8 - 2848260096*x^7 + 23814799360*x^6 - 531300352*x^5 + 3554017280*x^4 + 28835840*x^3 + 213385216*x^2 - 11534336*x + 2097152)
 
gp: K = bnfinit(x^42 - x^41 + 62*x^40 - 55*x^39 + 1885*x^38 - 1493*x^37 + 36890*x^36 - 26096*x^35 + 516884*x^34 - 325868*x^33 + 5474760*x^32 - 3063183*x^31 + 45214327*x^30 - 22319399*x^29 + 296463014*x^28 - 128129100*x^27 + 1558565095*x^26 - 584173775*x^25 + 6595257074*x^24 - 2118905803*x^23 + 22443263173*x^22 - 6093288301*x^21 + 61089146298*x^20 - 13772999112*x^19 + 131703790552*x^18 - 24134677936*x^17 + 221581028128*x^16 - 32118789120*x^15 + 284876509184*x^14 - 31567940608*x^13 + 271846162432*x^12 - 21964134400*x^11 + 184905250816*x^10 - 10296512512*x^9 + 84546772992*x^8 - 2848260096*x^7 + 23814799360*x^6 - 531300352*x^5 + 3554017280*x^4 + 28835840*x^3 + 213385216*x^2 - 11534336*x + 2097152, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2097152, -11534336, 213385216, 28835840, 3554017280, -531300352, 23814799360, -2848260096, 84546772992, -10296512512, 184905250816, -21964134400, 271846162432, -31567940608, 284876509184, -32118789120, 221581028128, -24134677936, 131703790552, -13772999112, 61089146298, -6093288301, 22443263173, -2118905803, 6595257074, -584173775, 1558565095, -128129100, 296463014, -22319399, 45214327, -3063183, 5474760, -325868, 516884, -26096, 36890, -1493, 1885, -55, 62, -1, 1]);
 

\( x^{42} - x^{41} + 62 x^{40} - 55 x^{39} + 1885 x^{38} - 1493 x^{37} + 36890 x^{36} - 26096 x^{35} + 516884 x^{34} - 325868 x^{33} + 5474760 x^{32} - 3063183 x^{31} + 45214327 x^{30} - 22319399 x^{29} + 296463014 x^{28} - 128129100 x^{27} + 1558565095 x^{26} - 584173775 x^{25} + 6595257074 x^{24} - 2118905803 x^{23} + 22443263173 x^{22} - 6093288301 x^{21} + 61089146298 x^{20} - 13772999112 x^{19} + 131703790552 x^{18} - 24134677936 x^{17} + 221581028128 x^{16} - 32118789120 x^{15} + 284876509184 x^{14} - 31567940608 x^{13} + 271846162432 x^{12} - 21964134400 x^{11} + 184905250816 x^{10} - 10296512512 x^{9} + 84546772992 x^{8} - 2848260096 x^{7} + 23814799360 x^{6} - 531300352 x^{5} + 3554017280 x^{4} + 28835840 x^{3} + 213385216 x^{2} - 11534336 x + 2097152 \)

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:  \(-12\!\cdots\!007\)\(\medspace = -\,7^{21}\cdot 43^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $95.11$
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:  $7, 43$
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:  \(301=7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{301}(1,·)$, $\chi_{301}(6,·)$, $\chi_{301}(139,·)$, $\chi_{301}(13,·)$, $\chi_{301}(15,·)$, $\chi_{301}(272,·)$, $\chi_{301}(146,·)$, $\chi_{301}(279,·)$, $\chi_{301}(153,·)$, $\chi_{301}(111,·)$, $\chi_{301}(267,·)$, $\chi_{301}(160,·)$, $\chi_{301}(36,·)$, $\chi_{301}(293,·)$, $\chi_{301}(167,·)$, $\chi_{301}(41,·)$, $\chi_{301}(176,·)$, $\chi_{301}(181,·)$, $\chi_{301}(183,·)$, $\chi_{301}(57,·)$, $\chi_{301}(188,·)$, $\chi_{301}(64,·)$, $\chi_{301}(195,·)$, $\chi_{301}(197,·)$, $\chi_{301}(97,·)$, $\chi_{301}(78,·)$, $\chi_{301}(83,·)$, $\chi_{301}(216,·)$, $\chi_{301}(90,·)$, $\chi_{301}(92,·)$, $\chi_{301}(225,·)$, $\chi_{301}(99,·)$, $\chi_{301}(230,·)$, $\chi_{301}(232,·)$, $\chi_{301}(274,·)$, $\chi_{301}(239,·)$, $\chi_{301}(246,·)$, $\chi_{301}(169,·)$, $\chi_{301}(251,·)$, $\chi_{301}(281,·)$, $\chi_{301}(253,·)$$\chi_{301}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{23} - \frac{1}{4} a^{22} - \frac{1}{2} a^{21} + \frac{1}{4} a^{20} + \frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{2} a^{17} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{24} - \frac{1}{8} a^{23} - \frac{1}{4} a^{22} + \frac{1}{8} a^{21} - \frac{3}{8} a^{20} + \frac{3}{8} a^{19} + \frac{1}{4} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} + \frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{25} - \frac{1}{16} a^{24} - \frac{1}{8} a^{23} + \frac{1}{16} a^{22} + \frac{5}{16} a^{21} - \frac{5}{16} a^{20} + \frac{1}{8} a^{19} - \frac{1}{2} a^{18} - \frac{1}{4} a^{17} + \frac{1}{4} a^{16} - \frac{1}{2} a^{15} + \frac{1}{16} a^{14} + \frac{7}{16} a^{13} - \frac{7}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{26} - \frac{1}{32} a^{25} - \frac{1}{16} a^{24} + \frac{1}{32} a^{23} + \frac{5}{32} a^{22} - \frac{5}{32} a^{21} - \frac{7}{16} a^{20} + \frac{1}{4} a^{19} - \frac{1}{8} a^{18} + \frac{1}{8} a^{17} + \frac{1}{4} a^{16} - \frac{15}{32} a^{15} - \frac{9}{32} a^{14} - \frac{7}{32} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} - \frac{7}{16} a^{8} - \frac{3}{32} a^{7} - \frac{3}{32} a^{6} + \frac{3}{32} a^{5} - \frac{7}{16} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{64} a^{27} - \frac{1}{64} a^{26} - \frac{1}{32} a^{25} + \frac{1}{64} a^{24} + \frac{5}{64} a^{23} - \frac{5}{64} a^{22} - \frac{7}{32} a^{21} + \frac{1}{8} a^{20} - \frac{1}{16} a^{19} + \frac{1}{16} a^{18} - \frac{3}{8} a^{17} + \frac{17}{64} a^{16} + \frac{23}{64} a^{15} + \frac{25}{64} a^{14} + \frac{15}{32} a^{13} + \frac{7}{16} a^{12} + \frac{31}{64} a^{11} + \frac{1}{64} a^{10} - \frac{7}{32} a^{9} - \frac{3}{64} a^{8} + \frac{29}{64} a^{7} + \frac{3}{64} a^{6} - \frac{7}{32} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{128} a^{28} - \frac{1}{128} a^{27} - \frac{1}{64} a^{26} + \frac{1}{128} a^{25} + \frac{5}{128} a^{24} - \frac{5}{128} a^{23} - \frac{7}{64} a^{22} + \frac{1}{16} a^{21} - \frac{1}{32} a^{20} + \frac{1}{32} a^{19} + \frac{5}{16} a^{18} - \frac{47}{128} a^{17} - \frac{41}{128} a^{16} - \frac{39}{128} a^{15} - \frac{17}{64} a^{14} - \frac{9}{32} a^{13} + \frac{31}{128} a^{12} - \frac{63}{128} a^{11} + \frac{25}{64} a^{10} + \frac{61}{128} a^{9} + \frac{29}{128} a^{8} - \frac{61}{128} a^{7} - \frac{7}{64} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{29} - \frac{1}{256} a^{28} - \frac{1}{128} a^{27} + \frac{1}{256} a^{26} + \frac{5}{256} a^{25} - \frac{5}{256} a^{24} - \frac{7}{128} a^{23} + \frac{1}{32} a^{22} - \frac{1}{64} a^{21} + \frac{1}{64} a^{20} + \frac{5}{32} a^{19} - \frac{47}{256} a^{18} + \frac{87}{256} a^{17} + \frac{89}{256} a^{16} - \frac{17}{128} a^{15} - \frac{9}{64} a^{14} - \frac{97}{256} a^{13} - \frac{63}{256} a^{12} + \frac{25}{128} a^{11} + \frac{61}{256} a^{10} + \frac{29}{256} a^{9} - \frac{61}{256} a^{8} + \frac{57}{128} a^{7} + \frac{7}{16} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{30} - \frac{1}{512} a^{29} - \frac{1}{256} a^{28} + \frac{1}{512} a^{27} + \frac{5}{512} a^{26} - \frac{5}{512} a^{25} - \frac{7}{256} a^{24} + \frac{1}{64} a^{23} - \frac{1}{128} a^{22} + \frac{1}{128} a^{21} + \frac{5}{64} a^{20} - \frac{47}{512} a^{19} - \frac{169}{512} a^{18} + \frac{89}{512} a^{17} - \frac{17}{256} a^{16} - \frac{9}{128} a^{15} - \frac{97}{512} a^{14} + \frac{193}{512} a^{13} + \frac{25}{256} a^{12} - \frac{195}{512} a^{11} - \frac{227}{512} a^{10} + \frac{195}{512} a^{9} - \frac{71}{256} a^{8} - \frac{9}{32} a^{7} - \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1024} a^{31} - \frac{1}{1024} a^{30} - \frac{1}{512} a^{29} + \frac{1}{1024} a^{28} + \frac{5}{1024} a^{27} - \frac{5}{1024} a^{26} - \frac{7}{512} a^{25} + \frac{1}{128} a^{24} - \frac{1}{256} a^{23} + \frac{1}{256} a^{22} + \frac{5}{128} a^{21} - \frac{47}{1024} a^{20} - \frac{169}{1024} a^{19} + \frac{89}{1024} a^{18} + \frac{239}{512} a^{17} + \frac{119}{256} a^{16} - \frac{97}{1024} a^{15} + \frac{193}{1024} a^{14} - \frac{231}{512} a^{13} - \frac{195}{1024} a^{12} + \frac{285}{1024} a^{11} + \frac{195}{1024} a^{10} - \frac{71}{512} a^{9} + \frac{23}{64} a^{8} + \frac{5}{16} a^{6} + \frac{1}{16} a^{5} - \frac{3}{16} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{2048} a^{32} - \frac{1}{2048} a^{31} - \frac{1}{1024} a^{30} + \frac{1}{2048} a^{29} + \frac{5}{2048} a^{28} - \frac{5}{2048} a^{27} - \frac{7}{1024} a^{26} + \frac{1}{256} a^{25} - \frac{1}{512} a^{24} + \frac{1}{512} a^{23} + \frac{5}{256} a^{22} - \frac{47}{2048} a^{21} - \frac{169}{2048} a^{20} + \frac{89}{2048} a^{19} + \frac{239}{1024} a^{18} - \frac{137}{512} a^{17} - \frac{97}{2048} a^{16} - \frac{831}{2048} a^{15} - \frac{231}{1024} a^{14} - \frac{195}{2048} a^{13} - \frac{739}{2048} a^{12} + \frac{195}{2048} a^{11} - \frac{71}{1024} a^{10} + \frac{23}{128} a^{9} + \frac{5}{32} a^{7} + \frac{1}{32} a^{6} - \frac{3}{32} a^{5} - \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4096} a^{33} - \frac{1}{4096} a^{32} - \frac{1}{2048} a^{31} + \frac{1}{4096} a^{30} + \frac{5}{4096} a^{29} - \frac{5}{4096} a^{28} - \frac{7}{2048} a^{27} + \frac{1}{512} a^{26} - \frac{1}{1024} a^{25} + \frac{1}{1024} a^{24} + \frac{5}{512} a^{23} - \frac{47}{4096} a^{22} - \frac{169}{4096} a^{21} + \frac{89}{4096} a^{20} - \frac{785}{2048} a^{19} + \frac{375}{1024} a^{18} - \frac{97}{4096} a^{17} + \frac{1217}{4096} a^{16} + \frac{793}{2048} a^{15} - \frac{195}{4096} a^{14} + \frac{1309}{4096} a^{13} - \frac{1853}{4096} a^{12} - \frac{71}{2048} a^{11} + \frac{23}{256} a^{10} - \frac{27}{64} a^{8} + \frac{1}{64} a^{7} - \frac{3}{64} a^{6} + \frac{13}{32} a^{5} + \frac{5}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{8192} a^{34} - \frac{1}{8192} a^{33} - \frac{1}{4096} a^{32} + \frac{1}{8192} a^{31} + \frac{5}{8192} a^{30} - \frac{5}{8192} a^{29} - \frac{7}{4096} a^{28} + \frac{1}{1024} a^{27} - \frac{1}{2048} a^{26} + \frac{1}{2048} a^{25} + \frac{5}{1024} a^{24} - \frac{47}{8192} a^{23} - \frac{169}{8192} a^{22} + \frac{89}{8192} a^{21} + \frac{1263}{4096} a^{20} - \frac{649}{2048} a^{19} - \frac{97}{8192} a^{18} - \frac{2879}{8192} a^{17} - \frac{1255}{4096} a^{16} + \frac{3901}{8192} a^{15} - \frac{2787}{8192} a^{14} + \frac{2243}{8192} a^{13} + \frac{1977}{4096} a^{12} + \frac{23}{512} a^{11} - \frac{1}{2} a^{10} - \frac{27}{128} a^{9} - \frac{63}{128} a^{8} - \frac{3}{128} a^{7} - \frac{19}{64} a^{6} - \frac{11}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{16384} a^{35} - \frac{1}{16384} a^{34} - \frac{1}{8192} a^{33} + \frac{1}{16384} a^{32} + \frac{5}{16384} a^{31} - \frac{5}{16384} a^{30} - \frac{7}{8192} a^{29} + \frac{1}{2048} a^{28} - \frac{1}{4096} a^{27} + \frac{1}{4096} a^{26} + \frac{5}{2048} a^{25} - \frac{47}{16384} a^{24} - \frac{169}{16384} a^{23} + \frac{89}{16384} a^{22} + \frac{1263}{8192} a^{21} - \frac{649}{4096} a^{20} - \frac{97}{16384} a^{19} - \frac{2879}{16384} a^{18} + \frac{2841}{8192} a^{17} + \frac{3901}{16384} a^{16} - \frac{2787}{16384} a^{15} - \frac{5949}{16384} a^{14} - \frac{2119}{8192} a^{13} + \frac{23}{1024} a^{12} + \frac{1}{4} a^{11} + \frac{101}{256} a^{10} + \frac{65}{256} a^{9} + \frac{125}{256} a^{8} - \frac{19}{128} a^{7} + \frac{21}{64} a^{6} + \frac{15}{32} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{32768} a^{36} - \frac{1}{32768} a^{35} - \frac{1}{16384} a^{34} + \frac{1}{32768} a^{33} + \frac{5}{32768} a^{32} - \frac{5}{32768} a^{31} - \frac{7}{16384} a^{30} + \frac{1}{4096} a^{29} - \frac{1}{8192} a^{28} + \frac{1}{8192} a^{27} + \frac{5}{4096} a^{26} - \frac{47}{32768} a^{25} - \frac{169}{32768} a^{24} + \frac{89}{32768} a^{23} + \frac{1263}{16384} a^{22} - \frac{649}{8192} a^{21} - \frac{97}{32768} a^{20} - \frac{2879}{32768} a^{19} + \frac{2841}{16384} a^{18} - \frac{12483}{32768} a^{17} + \frac{13597}{32768} a^{16} - \frac{5949}{32768} a^{15} - \frac{2119}{16384} a^{14} - \frac{1001}{2048} a^{13} - \frac{3}{8} a^{12} - \frac{155}{512} a^{11} - \frac{191}{512} a^{10} - \frac{131}{512} a^{9} - \frac{19}{256} a^{8} - \frac{43}{128} a^{7} + \frac{15}{64} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{65536} a^{37} - \frac{1}{65536} a^{36} - \frac{1}{32768} a^{35} + \frac{1}{65536} a^{34} + \frac{5}{65536} a^{33} - \frac{5}{65536} a^{32} - \frac{7}{32768} a^{31} + \frac{1}{8192} a^{30} - \frac{1}{16384} a^{29} + \frac{1}{16384} a^{28} + \frac{5}{8192} a^{27} - \frac{47}{65536} a^{26} - \frac{169}{65536} a^{25} + \frac{89}{65536} a^{24} + \frac{1263}{32768} a^{23} - \frac{649}{16384} a^{22} - \frac{97}{65536} a^{21} - \frac{2879}{65536} a^{20} - \frac{13543}{32768} a^{19} + \frac{20285}{65536} a^{18} + \frac{13597}{65536} a^{17} - \frac{5949}{65536} a^{16} + \frac{14265}{32768} a^{15} + \frac{1047}{4096} a^{14} - \frac{3}{16} a^{13} + \frac{357}{1024} a^{12} - \frac{191}{1024} a^{11} - \frac{131}{1024} a^{10} - \frac{19}{512} a^{9} - \frac{43}{256} a^{8} + \frac{15}{128} a^{7} - \frac{3}{16} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4}$, $\frac{1}{131072} a^{38} - \frac{1}{131072} a^{37} - \frac{1}{65536} a^{36} + \frac{1}{131072} a^{35} + \frac{5}{131072} a^{34} - \frac{5}{131072} a^{33} - \frac{7}{65536} a^{32} + \frac{1}{16384} a^{31} - \frac{1}{32768} a^{30} + \frac{1}{32768} a^{29} + \frac{5}{16384} a^{28} - \frac{47}{131072} a^{27} - \frac{169}{131072} a^{26} + \frac{89}{131072} a^{25} + \frac{1263}{65536} a^{24} - \frac{649}{32768} a^{23} - \frac{97}{131072} a^{22} - \frac{2879}{131072} a^{21} + \frac{19225}{65536} a^{20} - \frac{45251}{131072} a^{19} + \frac{13597}{131072} a^{18} + \frac{59587}{131072} a^{17} + \frac{14265}{65536} a^{16} - \frac{3049}{8192} a^{15} - \frac{3}{32} a^{14} - \frac{667}{2048} a^{13} - \frac{191}{2048} a^{12} - \frac{131}{2048} a^{11} + \frac{493}{1024} a^{10} - \frac{43}{512} a^{9} + \frac{15}{256} a^{8} + \frac{13}{32} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{262144} a^{39} - \frac{1}{262144} a^{38} - \frac{1}{131072} a^{37} + \frac{1}{262144} a^{36} + \frac{5}{262144} a^{35} - \frac{5}{262144} a^{34} - \frac{7}{131072} a^{33} + \frac{1}{32768} a^{32} - \frac{1}{65536} a^{31} + \frac{1}{65536} a^{30} + \frac{5}{32768} a^{29} - \frac{47}{262144} a^{28} - \frac{169}{262144} a^{27} + \frac{89}{262144} a^{26} + \frac{1263}{131072} a^{25} - \frac{649}{65536} a^{24} - \frac{97}{262144} a^{23} - \frac{2879}{262144} a^{22} - \frac{46311}{131072} a^{21} + \frac{85821}{262144} a^{20} - \frac{117475}{262144} a^{19} - \frac{71485}{262144} a^{18} - \frac{51271}{131072} a^{17} + \frac{5143}{16384} a^{16} + \frac{29}{64} a^{15} - \frac{667}{4096} a^{14} + \frac{1857}{4096} a^{13} + \frac{1917}{4096} a^{12} + \frac{493}{2048} a^{11} + \frac{469}{1024} a^{10} + \frac{15}{512} a^{9} - \frac{19}{64} a^{8} - \frac{7}{16} a^{7} - \frac{1}{16} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{232973139968} a^{40} + \frac{287065}{232973139968} a^{39} - \frac{103341}{29121642496} a^{38} + \frac{192249}{232973139968} a^{37} + \frac{1838903}{232973139968} a^{36} + \frac{425457}{232973139968} a^{35} - \frac{148579}{58243284992} a^{34} + \frac{410835}{29121642496} a^{33} - \frac{8629243}{58243284992} a^{32} + \frac{25752031}{58243284992} a^{31} + \frac{3669509}{7280410624} a^{30} - \frac{229095311}{232973139968} a^{29} - \frac{23625231}{232973139968} a^{28} + \frac{1503769635}{232973139968} a^{27} - \frac{384862707}{58243284992} a^{26} - \frac{1785088033}{58243284992} a^{25} - \frac{8779215185}{232973139968} a^{24} - \frac{11778264041}{232973139968} a^{23} - \frac{6435252649}{29121642496} a^{22} + \frac{3702629989}{232973139968} a^{21} - \frac{8920138249}{232973139968} a^{20} - \frac{49593138983}{232973139968} a^{19} - \frac{19361964231}{58243284992} a^{18} - \frac{30382771}{7280410624} a^{17} + \frac{276835557}{29121642496} a^{16} + \frac{766175957}{3640205312} a^{15} + \frac{3127092445}{7280410624} a^{14} + \frac{228242695}{910051328} a^{13} - \frac{128435369}{455025664} a^{12} + \frac{33628559}{455025664} a^{11} - \frac{37289043}{113756416} a^{10} - \frac{107104027}{227512832} a^{9} - \frac{8817353}{113756416} a^{8} + \frac{6903775}{28439104} a^{7} - \frac{3089785}{14219552} a^{6} + \frac{2550683}{14219552} a^{5} - \frac{3479141}{7109776} a^{4} + \frac{1327151}{3554888} a^{3} + \frac{433295}{888722} a^{2} - \frac{149359}{888722} a + \frac{217820}{444361}$, $\frac{1}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{41} + \frac{645772548989304297564030882421054838664064787558576078221035927759437467801098956743465038621823759863464270755463844645701688934505}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{40} + \frac{86786019713339668906600029904924762648368488321565934328103597358424846293974823610036285994413452985955308575841023478964681241325459977}{137946954156678638424732518820242317596375247972381524298858688484984540019608856296473557011633894306187931075174430255271301740372384131842048} a^{39} + \frac{19229438053003942474659791446814082879780415758721402113060646005976864390145591308369466248103767784829697744925312129313834064118536477}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{38} - \frac{1122219812865929706622991742133690676451288286472087378198634580381829331099435251490422369799107775030336824291852502462737818994364003409}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{37} + \frac{8249105952595353035190533417998628579237937491391826178423906223765229709387309775327228874899334143635578785074408188461960151798347570525}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{36} + \frac{511499935109032952859189825934336746832228329165892506300271596079670961494555082658314330305613508576356210614702981946380059161354473381}{34486738539169659606183129705060579399093811993095381074714672121246135004902214074118389252908473576546982768793607563817825435093096032960512} a^{35} + \frac{6465830844175771709899834159586004798373555706780685978624222038241281769607801166673909399386517174402817305204893936317797873112861901091}{137946954156678638424732518820242317596375247972381524298858688484984540019608856296473557011633894306187931075174430255271301740372384131842048} a^{34} - \frac{9689178235702315570119304068616190612568682328836667985312647543764592131340828761843241924282330174438459380546246817941179409734382206501}{137946954156678638424732518820242317596375247972381524298858688484984540019608856296473557011633894306187931075174430255271301740372384131842048} a^{33} - \frac{32427968379907604701856806359932543548264045048047137391153291750540325517791776377167718385636448163915180310114033087543192731268563143289}{137946954156678638424732518820242317596375247972381524298858688484984540019608856296473557011633894306187931075174430255271301740372384131842048} a^{32} + \frac{16003282318614371506054748523300006195224544772428100345376198118209993412596557821878762826691480794689966371282964681408882947093941459043}{34486738539169659606183129705060579399093811993095381074714672121246135004902214074118389252908473576546982768793607563817825435093096032960512} a^{31} + \frac{314844413292037683483923297553682577628190098899530920798678748915501973559618054508875805829470849448152674906473040978070267401457345085153}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{30} + \frac{148690710018113449724490164163905465796332365066274044649862072097144680003069921600672219520418792422277715291836292819895472745977532819297}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{29} + \frac{2128656017003824578466317211065785508404360252725328659279366692537727384197888656258990306322280599944584393687934847399276306364578390925039}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{28} - \frac{114721463400547737860076115484950294035697409129340236000188790020032173135071478317056932283230872351291512695511626889759918531203565049291}{68973477078339319212366259410121158798187623986190762149429344242492270009804428148236778505816947153093965537587215127635650870186192065921024} a^{27} - \frac{797196206270258231283564581437297853462987904266112199277163006628121335749173633046731355541720901508612835536422204095214134626354657373657}{68973477078339319212366259410121158798187623986190762149429344242492270009804428148236778505816947153093965537587215127635650870186192065921024} a^{26} + \frac{15237272866325818425971367311936816287636105243238694987623389534773599847568995683436144042545510153396626177982522441800828050373297037295831}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{25} + \frac{1197021787579582587099595332388047659900011998662103268503114412986620122010278343079637184791672857379838842003553178241930607236488683830039}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{24} - \frac{16674222074040845239679297423279147496700019879187707594359761505301283479589452976843745787792458525084567272025154297135537616871079670924517}{137946954156678638424732518820242317596375247972381524298858688484984540019608856296473557011633894306187931075174430255271301740372384131842048} a^{23} - \frac{61863104869176758149365858223491772629308462772074282118066961968942629031580238279381276530837019134439181138566095866917183872903998054560959}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{22} - \frac{11706955136197898409885275122969361120675543838858563613407571504102986687208289621256872319909186417237335839472094470222858656397755843667425}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{21} + \frac{125404604309210372475753279969697989772277781966696946430665272096565914715185419755345223103848120199098743802093806184152563889650627082807893}{551787816626714553698930075280969270385500991889526097195434753939938160078435425185894228046535577224751724300697721021085206961489536527368192} a^{20} + \frac{4886883981192904070408133828501130604478125151915429075951136023219687156142822022318806981739014147599271790317115141095252907799042666438447}{17243369269584829803091564852530289699546905996547690537357336060623067502451107037059194626454236788273491384396803781908912717546548016480256} a^{19} - \frac{7291971119296082701818005465105083544713035208446007285079157710379134674031784244928960075402319228646538072924373414252830447436319601449443}{137946954156678638424732518820242317596375247972381524298858688484984540019608856296473557011633894306187931075174430255271301740372384131842048} a^{18} - \frac{168975444944257240211613477955136512958437549754145249742960342684048052382197493544511248382262105955460018183725317356477604601648899929217}{538855289674525931346611401641571553110840812392115329292416751894470859451597094908099832076694899633546605762400118184653522423329625515008} a^{17} + \frac{2121565907073623737230189688861115739420401592342682724541806790389108247984943512237163025709466881792990091219007433811548639132119919968407}{4310842317396207450772891213132572424886726499136922634339334015155766875612776759264798656613559197068372846099200945477228179386637004120064} a^{16} + \frac{3114391049885585061345284605824115351583263904818301391076638306193059629755208045598186630543561421144050086299540684303147314999643156530867}{17243369269584829803091564852530289699546905996547690537357336060623067502451107037059194626454236788273491384396803781908912717546548016480256} a^{15} - \frac{415087244482663293356475083586970775588484537972238005032566557219024978973271933559588973581151046798066048224674679914763668536521729297965}{2155421158698103725386445606566286212443363249568461317169667007577883437806388379632399328306779598534186423049600472738614089693318502060032} a^{14} - \frac{1950821845481276607106318078875886298366651739881372376677160229635496739256597800192077917948523368406538744903281974261853760692960350323411}{4310842317396207450772891213132572424886726499136922634339334015155766875612776759264798656613559197068372846099200945477228179386637004120064} a^{13} - \frac{130697821241555538617719082969566587402588402188636743929100309562721076064424285498201658397348142375207900832399966157573864658235936285473}{269427644837262965673305700820785776555420406196057664646208375947235429725798547454049916038347449816773302881200059092326761211664812757504} a^{12} - \frac{169656230326275058057760604605858608005795609789839268996402169762721260508558156537388369903800385361346803424516650568462182611513221224757}{538855289674525931346611401641571553110840812392115329292416751894470859451597094908099832076694899633546605762400118184653522423329625515008} a^{11} - \frac{230939921292539101410499705439003786080125150735816669061694324985854634272115427032351035741090479753366156481615469098972743820936993644127}{538855289674525931346611401641571553110840812392115329292416751894470859451597094908099832076694899633546605762400118184653522423329625515008} a^{10} - \frac{110626783903916973949959421685368547293415764437583237827418906693549369180681112226089546361907104172783814250330281036626770105251016371029}{269427644837262965673305700820785776555420406196057664646208375947235429725798547454049916038347449816773302881200059092326761211664812757504} a^{9} - \frac{28661923149573616136228282898076513499008535414513691349643804139506412418475775190650521376192399105503119969200539308752447794392065161473}{67356911209315741418326425205196444138855101549014416161552093986808857431449636863512479009586862454193325720300014773081690302916203189376} a^{8} - \frac{12588838027639905582156372614647234565002258249236839938382302963541092412778611792983086969205150645202334991429946462780169199647703816685}{67356911209315741418326425205196444138855101549014416161552093986808857431449636863512479009586862454193325720300014773081690302916203189376} a^{7} - \frac{1245001600499623405178768421381413147051084076646546734564566003080433460276325312154164392848441726784407482669575009837625874719089161495}{16839227802328935354581606301299111034713775387253604040388023496702214357862409215878119752396715613548331430075003693270422575729050797344} a^{6} - \frac{6169450509940588734504835567725892505021303981080956860054126468171887218022820063213513071609295741584018173589782286768515462229360182121}{16839227802328935354581606301299111034713775387253604040388023496702214357862409215878119752396715613548331430075003693270422575729050797344} a^{5} + \frac{550720352581392069592687225800141427883832874506499193107000916338039790538054939435595916635337482088718092425600227324505599964645402817}{2104903475291116919322700787662388879339221923406700505048502937087776794732801151984764969049589451693541428759375461658802821966131349668} a^{4} - \frac{520781647833989346168552526533341438396758641475111860763913339598554250514296185415572932910075118278996194174191163281105125694429639871}{4209806950582233838645401575324777758678443846813401010097005874175553589465602303969529938099178903387082857518750923317605643932262699336} a^{3} - \frac{923232291220332389609563904568627419026072659478274935529564204461125366583570385695132058698153933985515223478458001602131815421676233035}{2104903475291116919322700787662388879339221923406700505048502937087776794732801151984764969049589451693541428759375461658802821966131349668} a^{2} - \frac{78227528103480132354651044452483543837339653762960765077503161200255183087279069143867282041777195092463132465050138073186812865039230167}{526225868822779229830675196915597219834805480851675126262125734271944198683200287996191242262397362923385357189843865414700705491532837417} a + \frac{230299922922104299698452157721084856637966516577014726778526430077109899549651292238708940114933320498887507652368759465058478016532295972}{526225868822779229830675196915597219834805480851675126262125734271944198683200287996191242262397362923385357189843865414700705491532837417}$

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

Class group and class number

not computed

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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{-7}) \), 3.3.1849.1, 6.0.1172648743.2, 7.7.6321363049.1, 14.0.32908474225670013957008743.1, \(\Q(\zeta_{43})^+\)

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{6}$ $42$ $42$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{6}$ $42$ $42$ $42$ $21^{2}$ $21^{2}$ $42$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{3}$

In the table, R denotes a ramified prime. 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
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43Data not computed