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

\( x^{42} - x^{41} + 62 x^{40} - 55 x^{39} + 1885 x^{38} - 1493 x^{37} + 36890 x^{36} - 26096 x^{35} + \cdots + 2097152 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-121\!\cdots\!007\) \(\medspace = -\,7^{21}\cdot 43^{40}\) 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:  \(95.11\)
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:  $7^{1/2}43^{20/21}\approx 95.11155090606901$
Ramified primes:   \(7\), \(43\) 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:  \(\Q(\sqrt{-7}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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.
Reflex fields:  unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{23}-\frac{1}{4}a^{22}-\frac{1}{2}a^{21}+\frac{1}{4}a^{20}+\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{2}a^{17}+\frac{1}{4}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{24}-\frac{1}{8}a^{23}-\frac{1}{4}a^{22}+\frac{1}{8}a^{21}-\frac{3}{8}a^{20}+\frac{3}{8}a^{19}+\frac{1}{4}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{8}a^{13}-\frac{1}{8}a^{12}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}+\frac{1}{4}a^{6}-\frac{3}{8}a^{5}-\frac{3}{8}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{25}-\frac{1}{16}a^{24}-\frac{1}{8}a^{23}+\frac{1}{16}a^{22}+\frac{5}{16}a^{21}-\frac{5}{16}a^{20}+\frac{1}{8}a^{19}-\frac{1}{2}a^{18}-\frac{1}{4}a^{17}+\frac{1}{4}a^{16}-\frac{1}{2}a^{15}+\frac{1}{16}a^{14}+\frac{7}{16}a^{13}-\frac{7}{16}a^{12}-\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}+\frac{1}{8}a^{7}-\frac{3}{16}a^{6}-\frac{3}{16}a^{5}+\frac{3}{16}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{32}a^{26}-\frac{1}{32}a^{25}-\frac{1}{16}a^{24}+\frac{1}{32}a^{23}+\frac{5}{32}a^{22}-\frac{5}{32}a^{21}-\frac{7}{16}a^{20}+\frac{1}{4}a^{19}-\frac{1}{8}a^{18}+\frac{1}{8}a^{17}+\frac{1}{4}a^{16}-\frac{15}{32}a^{15}-\frac{9}{32}a^{14}-\frac{7}{32}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{11}-\frac{1}{32}a^{10}+\frac{1}{32}a^{9}-\frac{7}{16}a^{8}-\frac{3}{32}a^{7}-\frac{3}{32}a^{6}+\frac{3}{32}a^{5}-\frac{7}{16}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{64}a^{27}-\frac{1}{64}a^{26}-\frac{1}{32}a^{25}+\frac{1}{64}a^{24}+\frac{5}{64}a^{23}-\frac{5}{64}a^{22}-\frac{7}{32}a^{21}+\frac{1}{8}a^{20}-\frac{1}{16}a^{19}+\frac{1}{16}a^{18}-\frac{3}{8}a^{17}+\frac{17}{64}a^{16}+\frac{23}{64}a^{15}+\frac{25}{64}a^{14}+\frac{15}{32}a^{13}+\frac{7}{16}a^{12}+\frac{31}{64}a^{11}+\frac{1}{64}a^{10}-\frac{7}{32}a^{9}-\frac{3}{64}a^{8}+\frac{29}{64}a^{7}+\frac{3}{64}a^{6}-\frac{7}{32}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{128}a^{28}-\frac{1}{128}a^{27}-\frac{1}{64}a^{26}+\frac{1}{128}a^{25}+\frac{5}{128}a^{24}-\frac{5}{128}a^{23}-\frac{7}{64}a^{22}+\frac{1}{16}a^{21}-\frac{1}{32}a^{20}+\frac{1}{32}a^{19}+\frac{5}{16}a^{18}-\frac{47}{128}a^{17}-\frac{41}{128}a^{16}-\frac{39}{128}a^{15}-\frac{17}{64}a^{14}-\frac{9}{32}a^{13}+\frac{31}{128}a^{12}-\frac{63}{128}a^{11}+\frac{25}{64}a^{10}+\frac{61}{128}a^{9}+\frac{29}{128}a^{8}-\frac{61}{128}a^{7}-\frac{7}{64}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{29}-\frac{1}{256}a^{28}-\frac{1}{128}a^{27}+\frac{1}{256}a^{26}+\frac{5}{256}a^{25}-\frac{5}{256}a^{24}-\frac{7}{128}a^{23}+\frac{1}{32}a^{22}-\frac{1}{64}a^{21}+\frac{1}{64}a^{20}+\frac{5}{32}a^{19}-\frac{47}{256}a^{18}+\frac{87}{256}a^{17}+\frac{89}{256}a^{16}-\frac{17}{128}a^{15}-\frac{9}{64}a^{14}-\frac{97}{256}a^{13}-\frac{63}{256}a^{12}+\frac{25}{128}a^{11}+\frac{61}{256}a^{10}+\frac{29}{256}a^{9}-\frac{61}{256}a^{8}+\frac{57}{128}a^{7}+\frac{7}{16}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{512}a^{30}-\frac{1}{512}a^{29}-\frac{1}{256}a^{28}+\frac{1}{512}a^{27}+\frac{5}{512}a^{26}-\frac{5}{512}a^{25}-\frac{7}{256}a^{24}+\frac{1}{64}a^{23}-\frac{1}{128}a^{22}+\frac{1}{128}a^{21}+\frac{5}{64}a^{20}-\frac{47}{512}a^{19}-\frac{169}{512}a^{18}+\frac{89}{512}a^{17}-\frac{17}{256}a^{16}-\frac{9}{128}a^{15}-\frac{97}{512}a^{14}+\frac{193}{512}a^{13}+\frac{25}{256}a^{12}-\frac{195}{512}a^{11}-\frac{227}{512}a^{10}+\frac{195}{512}a^{9}-\frac{71}{256}a^{8}-\frac{9}{32}a^{7}-\frac{3}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{1024}a^{31}-\frac{1}{1024}a^{30}-\frac{1}{512}a^{29}+\frac{1}{1024}a^{28}+\frac{5}{1024}a^{27}-\frac{5}{1024}a^{26}-\frac{7}{512}a^{25}+\frac{1}{128}a^{24}-\frac{1}{256}a^{23}+\frac{1}{256}a^{22}+\frac{5}{128}a^{21}-\frac{47}{1024}a^{20}-\frac{169}{1024}a^{19}+\frac{89}{1024}a^{18}+\frac{239}{512}a^{17}+\frac{119}{256}a^{16}-\frac{97}{1024}a^{15}+\frac{193}{1024}a^{14}-\frac{231}{512}a^{13}-\frac{195}{1024}a^{12}+\frac{285}{1024}a^{11}+\frac{195}{1024}a^{10}-\frac{71}{512}a^{9}+\frac{23}{64}a^{8}+\frac{5}{16}a^{6}+\frac{1}{16}a^{5}-\frac{3}{16}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{2048}a^{32}-\frac{1}{2048}a^{31}-\frac{1}{1024}a^{30}+\frac{1}{2048}a^{29}+\frac{5}{2048}a^{28}-\frac{5}{2048}a^{27}-\frac{7}{1024}a^{26}+\frac{1}{256}a^{25}-\frac{1}{512}a^{24}+\frac{1}{512}a^{23}+\frac{5}{256}a^{22}-\frac{47}{2048}a^{21}-\frac{169}{2048}a^{20}+\frac{89}{2048}a^{19}+\frac{239}{1024}a^{18}-\frac{137}{512}a^{17}-\frac{97}{2048}a^{16}-\frac{831}{2048}a^{15}-\frac{231}{1024}a^{14}-\frac{195}{2048}a^{13}-\frac{739}{2048}a^{12}+\frac{195}{2048}a^{11}-\frac{71}{1024}a^{10}+\frac{23}{128}a^{9}+\frac{5}{32}a^{7}+\frac{1}{32}a^{6}-\frac{3}{32}a^{5}-\frac{3}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{4096}a^{33}-\frac{1}{4096}a^{32}-\frac{1}{2048}a^{31}+\frac{1}{4096}a^{30}+\frac{5}{4096}a^{29}-\frac{5}{4096}a^{28}-\frac{7}{2048}a^{27}+\frac{1}{512}a^{26}-\frac{1}{1024}a^{25}+\frac{1}{1024}a^{24}+\frac{5}{512}a^{23}-\frac{47}{4096}a^{22}-\frac{169}{4096}a^{21}+\frac{89}{4096}a^{20}-\frac{785}{2048}a^{19}+\frac{375}{1024}a^{18}-\frac{97}{4096}a^{17}+\frac{1217}{4096}a^{16}+\frac{793}{2048}a^{15}-\frac{195}{4096}a^{14}+\frac{1309}{4096}a^{13}-\frac{1853}{4096}a^{12}-\frac{71}{2048}a^{11}+\frac{23}{256}a^{10}-\frac{27}{64}a^{8}+\frac{1}{64}a^{7}-\frac{3}{64}a^{6}+\frac{13}{32}a^{5}+\frac{5}{16}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{8192}a^{34}-\frac{1}{8192}a^{33}-\frac{1}{4096}a^{32}+\frac{1}{8192}a^{31}+\frac{5}{8192}a^{30}-\frac{5}{8192}a^{29}-\frac{7}{4096}a^{28}+\frac{1}{1024}a^{27}-\frac{1}{2048}a^{26}+\frac{1}{2048}a^{25}+\frac{5}{1024}a^{24}-\frac{47}{8192}a^{23}-\frac{169}{8192}a^{22}+\frac{89}{8192}a^{21}+\frac{1263}{4096}a^{20}-\frac{649}{2048}a^{19}-\frac{97}{8192}a^{18}-\frac{2879}{8192}a^{17}-\frac{1255}{4096}a^{16}+\frac{3901}{8192}a^{15}-\frac{2787}{8192}a^{14}+\frac{2243}{8192}a^{13}+\frac{1977}{4096}a^{12}+\frac{23}{512}a^{11}-\frac{1}{2}a^{10}-\frac{27}{128}a^{9}-\frac{63}{128}a^{8}-\frac{3}{128}a^{7}-\frac{19}{64}a^{6}-\frac{11}{32}a^{5}-\frac{1}{16}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{16384}a^{35}-\frac{1}{16384}a^{34}-\frac{1}{8192}a^{33}+\frac{1}{16384}a^{32}+\frac{5}{16384}a^{31}-\frac{5}{16384}a^{30}-\frac{7}{8192}a^{29}+\frac{1}{2048}a^{28}-\frac{1}{4096}a^{27}+\frac{1}{4096}a^{26}+\frac{5}{2048}a^{25}-\frac{47}{16384}a^{24}-\frac{169}{16384}a^{23}+\frac{89}{16384}a^{22}+\frac{1263}{8192}a^{21}-\frac{649}{4096}a^{20}-\frac{97}{16384}a^{19}-\frac{2879}{16384}a^{18}+\frac{2841}{8192}a^{17}+\frac{3901}{16384}a^{16}-\frac{2787}{16384}a^{15}-\frac{5949}{16384}a^{14}-\frac{2119}{8192}a^{13}+\frac{23}{1024}a^{12}+\frac{1}{4}a^{11}+\frac{101}{256}a^{10}+\frac{65}{256}a^{9}+\frac{125}{256}a^{8}-\frac{19}{128}a^{7}+\frac{21}{64}a^{6}+\frac{15}{32}a^{5}+\frac{1}{4}a^{4}$, $\frac{1}{32768}a^{36}-\frac{1}{32768}a^{35}-\frac{1}{16384}a^{34}+\frac{1}{32768}a^{33}+\frac{5}{32768}a^{32}-\frac{5}{32768}a^{31}-\frac{7}{16384}a^{30}+\frac{1}{4096}a^{29}-\frac{1}{8192}a^{28}+\frac{1}{8192}a^{27}+\frac{5}{4096}a^{26}-\frac{47}{32768}a^{25}-\frac{169}{32768}a^{24}+\frac{89}{32768}a^{23}+\frac{1263}{16384}a^{22}-\frac{649}{8192}a^{21}-\frac{97}{32768}a^{20}-\frac{2879}{32768}a^{19}+\frac{2841}{16384}a^{18}-\frac{12483}{32768}a^{17}+\frac{13597}{32768}a^{16}-\frac{5949}{32768}a^{15}-\frac{2119}{16384}a^{14}-\frac{1001}{2048}a^{13}-\frac{3}{8}a^{12}-\frac{155}{512}a^{11}-\frac{191}{512}a^{10}-\frac{131}{512}a^{9}-\frac{19}{256}a^{8}-\frac{43}{128}a^{7}+\frac{15}{64}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{65536}a^{37}-\frac{1}{65536}a^{36}-\frac{1}{32768}a^{35}+\frac{1}{65536}a^{34}+\frac{5}{65536}a^{33}-\frac{5}{65536}a^{32}-\frac{7}{32768}a^{31}+\frac{1}{8192}a^{30}-\frac{1}{16384}a^{29}+\frac{1}{16384}a^{28}+\frac{5}{8192}a^{27}-\frac{47}{65536}a^{26}-\frac{169}{65536}a^{25}+\frac{89}{65536}a^{24}+\frac{1263}{32768}a^{23}-\frac{649}{16384}a^{22}-\frac{97}{65536}a^{21}-\frac{2879}{65536}a^{20}-\frac{13543}{32768}a^{19}+\frac{20285}{65536}a^{18}+\frac{13597}{65536}a^{17}-\frac{5949}{65536}a^{16}+\frac{14265}{32768}a^{15}+\frac{1047}{4096}a^{14}-\frac{3}{16}a^{13}+\frac{357}{1024}a^{12}-\frac{191}{1024}a^{11}-\frac{131}{1024}a^{10}-\frac{19}{512}a^{9}-\frac{43}{256}a^{8}+\frac{15}{128}a^{7}-\frac{3}{16}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{131072}a^{38}-\frac{1}{131072}a^{37}-\frac{1}{65536}a^{36}+\frac{1}{131072}a^{35}+\frac{5}{131072}a^{34}-\frac{5}{131072}a^{33}-\frac{7}{65536}a^{32}+\frac{1}{16384}a^{31}-\frac{1}{32768}a^{30}+\frac{1}{32768}a^{29}+\frac{5}{16384}a^{28}-\frac{47}{131072}a^{27}-\frac{169}{131072}a^{26}+\frac{89}{131072}a^{25}+\frac{1263}{65536}a^{24}-\frac{649}{32768}a^{23}-\frac{97}{131072}a^{22}-\frac{2879}{131072}a^{21}+\frac{19225}{65536}a^{20}-\frac{45251}{131072}a^{19}+\frac{13597}{131072}a^{18}+\frac{59587}{131072}a^{17}+\frac{14265}{65536}a^{16}-\frac{3049}{8192}a^{15}-\frac{3}{32}a^{14}-\frac{667}{2048}a^{13}-\frac{191}{2048}a^{12}-\frac{131}{2048}a^{11}+\frac{493}{1024}a^{10}-\frac{43}{512}a^{9}+\frac{15}{256}a^{8}+\frac{13}{32}a^{7}+\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{262144}a^{39}-\frac{1}{262144}a^{38}-\frac{1}{131072}a^{37}+\frac{1}{262144}a^{36}+\frac{5}{262144}a^{35}-\frac{5}{262144}a^{34}-\frac{7}{131072}a^{33}+\frac{1}{32768}a^{32}-\frac{1}{65536}a^{31}+\frac{1}{65536}a^{30}+\frac{5}{32768}a^{29}-\frac{47}{262144}a^{28}-\frac{169}{262144}a^{27}+\frac{89}{262144}a^{26}+\frac{1263}{131072}a^{25}-\frac{649}{65536}a^{24}-\frac{97}{262144}a^{23}-\frac{2879}{262144}a^{22}-\frac{46311}{131072}a^{21}+\frac{85821}{262144}a^{20}-\frac{117475}{262144}a^{19}-\frac{71485}{262144}a^{18}-\frac{51271}{131072}a^{17}+\frac{5143}{16384}a^{16}+\frac{29}{64}a^{15}-\frac{667}{4096}a^{14}+\frac{1857}{4096}a^{13}+\frac{1917}{4096}a^{12}+\frac{493}{2048}a^{11}+\frac{469}{1024}a^{10}+\frac{15}{512}a^{9}-\frac{19}{64}a^{8}-\frac{7}{16}a^{7}-\frac{1}{16}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{232973139968}a^{40}+\frac{287065}{232973139968}a^{39}-\frac{103341}{29121642496}a^{38}+\frac{192249}{232973139968}a^{37}+\frac{1838903}{232973139968}a^{36}+\frac{425457}{232973139968}a^{35}-\frac{148579}{58243284992}a^{34}+\frac{410835}{29121642496}a^{33}-\frac{8629243}{58243284992}a^{32}+\frac{25752031}{58243284992}a^{31}+\frac{3669509}{7280410624}a^{30}-\frac{229095311}{232973139968}a^{29}-\frac{23625231}{232973139968}a^{28}+\frac{1503769635}{232973139968}a^{27}-\frac{384862707}{58243284992}a^{26}-\frac{1785088033}{58243284992}a^{25}-\frac{8779215185}{232973139968}a^{24}-\frac{11778264041}{232973139968}a^{23}-\frac{6435252649}{29121642496}a^{22}+\frac{3702629989}{232973139968}a^{21}-\frac{8920138249}{232973139968}a^{20}-\frac{49593138983}{232973139968}a^{19}-\frac{19361964231}{58243284992}a^{18}-\frac{30382771}{7280410624}a^{17}+\frac{276835557}{29121642496}a^{16}+\frac{766175957}{3640205312}a^{15}+\frac{3127092445}{7280410624}a^{14}+\frac{228242695}{910051328}a^{13}-\frac{128435369}{455025664}a^{12}+\frac{33628559}{455025664}a^{11}-\frac{37289043}{113756416}a^{10}-\frac{107104027}{227512832}a^{9}-\frac{8817353}{113756416}a^{8}+\frac{6903775}{28439104}a^{7}-\frac{3089785}{14219552}a^{6}+\frac{2550683}{14219552}a^{5}-\frac{3479141}{7109776}a^{4}+\frac{1327151}{3554888}a^{3}+\frac{433295}{888722}a^{2}-\frac{149359}{888722}a+\frac{217820}{444361}$, $\frac{1}{55\!\cdots\!92}a^{41}+\frac{64\!\cdots\!05}{55\!\cdots\!92}a^{40}+\frac{86\!\cdots\!77}{13\!\cdots\!48}a^{39}+\frac{19\!\cdots\!77}{55\!\cdots\!92}a^{38}-\frac{11\!\cdots\!09}{55\!\cdots\!92}a^{37}+\frac{82\!\cdots\!25}{55\!\cdots\!92}a^{36}+\frac{51\!\cdots\!81}{34\!\cdots\!12}a^{35}+\frac{64\!\cdots\!91}{13\!\cdots\!48}a^{34}-\frac{96\!\cdots\!01}{13\!\cdots\!48}a^{33}-\frac{32\!\cdots\!89}{13\!\cdots\!48}a^{32}+\frac{16\!\cdots\!43}{34\!\cdots\!12}a^{31}+\frac{31\!\cdots\!53}{55\!\cdots\!92}a^{30}+\frac{14\!\cdots\!97}{55\!\cdots\!92}a^{29}+\frac{21\!\cdots\!39}{55\!\cdots\!92}a^{28}-\frac{11\!\cdots\!91}{68\!\cdots\!24}a^{27}-\frac{79\!\cdots\!57}{68\!\cdots\!24}a^{26}+\frac{15\!\cdots\!31}{55\!\cdots\!92}a^{25}+\frac{11\!\cdots\!39}{55\!\cdots\!92}a^{24}-\frac{16\!\cdots\!17}{13\!\cdots\!48}a^{23}-\frac{61\!\cdots\!59}{55\!\cdots\!92}a^{22}-\frac{11\!\cdots\!25}{55\!\cdots\!92}a^{21}+\frac{12\!\cdots\!93}{55\!\cdots\!92}a^{20}+\frac{48\!\cdots\!47}{17\!\cdots\!56}a^{19}-\frac{72\!\cdots\!43}{13\!\cdots\!48}a^{18}-\frac{16\!\cdots\!17}{53\!\cdots\!08}a^{17}+\frac{21\!\cdots\!07}{43\!\cdots\!64}a^{16}+\frac{31\!\cdots\!67}{17\!\cdots\!56}a^{15}-\frac{41\!\cdots\!65}{21\!\cdots\!32}a^{14}-\frac{19\!\cdots\!11}{43\!\cdots\!64}a^{13}-\frac{13\!\cdots\!73}{26\!\cdots\!04}a^{12}-\frac{16\!\cdots\!57}{53\!\cdots\!08}a^{11}-\frac{23\!\cdots\!27}{53\!\cdots\!08}a^{10}-\frac{11\!\cdots\!29}{26\!\cdots\!04}a^{9}-\frac{28\!\cdots\!73}{67\!\cdots\!76}a^{8}-\frac{12\!\cdots\!85}{67\!\cdots\!76}a^{7}-\frac{12\!\cdots\!95}{16\!\cdots\!44}a^{6}-\frac{61\!\cdots\!21}{16\!\cdots\!44}a^{5}+\frac{55\!\cdots\!17}{21\!\cdots\!68}a^{4}-\frac{52\!\cdots\!71}{42\!\cdots\!36}a^{3}-\frac{92\!\cdots\!35}{21\!\cdots\!68}a^{2}-\frac{78\!\cdots\!67}{52\!\cdots\!17}a+\frac{23\!\cdots\!72}{52\!\cdots\!17}$ 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:  $20$
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^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)
 
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^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);
 
[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^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);
 
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^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);
 
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

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

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 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$

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.

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

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

# 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
\(7\) Copy content Toggle raw display 7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(43\) Copy content Toggle raw display 43.21.20.1$x^{21} + 43$$21$$1$$20$$C_{21}$$[\ ]_{21}$
43.21.20.1$x^{21} + 43$$21$$1$$20$$C_{21}$$[\ ]_{21}$