LMFDB - Number field 44.0.6091524468845815638846997953624350524431889522210325630882624592675376276321.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304)

gp: K = bnfinit(y^44 - y^43 - y^42 + 3*y^41 - y^40 - 5*y^39 + 7*y^38 + 3*y^37 - 17*y^36 + 11*y^35 + 23*y^34 - 45*y^33 - y^32 + 91*y^31 - 89*y^30 - 93*y^29 + 271*y^28 - 85*y^27 - 457*y^26 + 627*y^25 + 287*y^24 - 1541*y^23 + 967*y^22 - 3082*y^21 + 1148*y^20 + 5016*y^19 - 7312*y^18 - 2720*y^17 + 17344*y^16 - 11904*y^15 - 22784*y^14 + 46592*y^13 - 1024*y^12 - 92160*y^11 + 94208*y^10 + 90112*y^9 - 278528*y^8 + 98304*y^7 + 458752*y^6 - 655360*y^5 - 262144*y^4 + 1572864*y^3 - 1048576*y^2 - 2097152*y + 4194304, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304)

$ x^{44} - x^{43} - x^{42} + 3 x^{41} - x^{40} - 5 x^{39} + 7 x^{38} + 3 x^{37} - 17 x^{36} + \cdots + 4194304 $ x^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$44$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 22]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$6091524468845815638846997953624350524431889522210325630882624592675376276321$ 6091524468845815638846997953624350524431889522210325630882624592675376276321 $\medspace = 7^{22}\cdot 23^{42}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$52.77$	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:	$7^{1/2}23^{21/22}\approx 52.76915456165355$
Ramified primes:	$7$, $23$ 7, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$44$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$161=7\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{161}(1,·)$, $\chi_{161}(132,·)$, $\chi_{161}(134,·)$, $\chi_{161}(8,·)$, $\chi_{161}(139,·)$, $\chi_{161}(13,·)$, $\chi_{161}(15,·)$, $\chi_{161}(146,·)$, $\chi_{161}(20,·)$, $\chi_{161}(22,·)$, $\chi_{161}(153,·)$, $\chi_{161}(155,·)$, $\chi_{161}(29,·)$, $\chi_{161}(160,·)$, $\chi_{161}(34,·)$, $\chi_{161}(27,·)$, $\chi_{161}(36,·)$, $\chi_{161}(6,·)$, $\chi_{161}(41,·)$, $\chi_{161}(43,·)$, $\chi_{161}(48,·)$, $\chi_{161}(50,·)$, $\chi_{161}(55,·)$, $\chi_{161}(57,·)$, $\chi_{161}(62,·)$, $\chi_{161}(64,·)$, $\chi_{161}(71,·)$, $\chi_{161}(76,·)$, $\chi_{161}(78,·)$, $\chi_{161}(141,·)$, $\chi_{161}(83,·)$, $\chi_{161}(85,·)$, $\chi_{161}(90,·)$, $\chi_{161}(97,·)$, $\chi_{161}(99,·)$, $\chi_{161}(104,·)$, $\chi_{161}(106,·)$, $\chi_{161}(111,·)$, $\chi_{161}(113,·)$, $\chi_{161}(118,·)$, $\chi_{161}(120,·)$, $\chi_{161}(148,·)$, $\chi_{161}(125,·)$, $\chi_{161}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{1934}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{393}{967}$, $\frac{1}{3868}a^{24}-\frac{1}{3868}a^{23}+\frac{1}{4}a^{22}+\frac{1}{4}a^{21}+\frac{1}{4}a^{20}+\frac{1}{4}a^{19}+\frac{1}{4}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}+\frac{1}{4}a^{11}+\frac{1}{4}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{393}{1934}a+\frac{287}{967}$, $\frac{1}{7736}a^{25}-\frac{1}{7736}a^{24}-\frac{1}{7736}a^{23}-\frac{3}{8}a^{22}+\frac{1}{8}a^{21}-\frac{3}{8}a^{20}+\frac{1}{8}a^{19}-\frac{3}{8}a^{18}+\frac{1}{8}a^{17}-\frac{3}{8}a^{16}+\frac{1}{8}a^{15}-\frac{3}{8}a^{14}+\frac{1}{8}a^{13}-\frac{3}{8}a^{12}+\frac{1}{8}a^{11}-\frac{3}{8}a^{10}+\frac{1}{8}a^{9}-\frac{3}{8}a^{8}+\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{1}{8}a^{5}-\frac{3}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1541}{3868}a^{2}+\frac{287}{1934}a-\frac{340}{967}$, $\frac{1}{15472}a^{26}-\frac{1}{15472}a^{25}-\frac{1}{15472}a^{24}+\frac{3}{15472}a^{23}-\frac{7}{16}a^{22}-\frac{3}{16}a^{21}+\frac{1}{16}a^{20}+\frac{5}{16}a^{19}-\frac{7}{16}a^{18}-\frac{3}{16}a^{17}+\frac{1}{16}a^{16}+\frac{5}{16}a^{15}-\frac{7}{16}a^{14}-\frac{3}{16}a^{13}+\frac{1}{16}a^{12}+\frac{5}{16}a^{11}-\frac{7}{16}a^{10}-\frac{3}{16}a^{9}+\frac{1}{16}a^{8}+\frac{5}{16}a^{7}-\frac{7}{16}a^{6}-\frac{3}{16}a^{5}+\frac{1}{16}a^{4}-\frac{1541}{7736}a^{3}+\frac{287}{3868}a^{2}+\frac{627}{1934}a-\frac{457}{967}$, $\frac{1}{30944}a^{27}-\frac{1}{30944}a^{26}-\frac{1}{30944}a^{25}+\frac{3}{30944}a^{24}-\frac{1}{30944}a^{23}+\frac{13}{32}a^{22}+\frac{1}{32}a^{21}+\frac{5}{32}a^{20}-\frac{7}{32}a^{19}-\frac{3}{32}a^{18}-\frac{15}{32}a^{17}-\frac{11}{32}a^{16}+\frac{9}{32}a^{15}+\frac{13}{32}a^{14}+\frac{1}{32}a^{13}+\frac{5}{32}a^{12}-\frac{7}{32}a^{11}-\frac{3}{32}a^{10}-\frac{15}{32}a^{9}-\frac{11}{32}a^{8}+\frac{9}{32}a^{7}+\frac{13}{32}a^{6}+\frac{1}{32}a^{5}-\frac{1541}{15472}a^{4}+\frac{287}{7736}a^{3}+\frac{627}{3868}a^{2}-\frac{457}{1934}a-\frac{85}{967}$, $\frac{1}{61888}a^{28}-\frac{1}{61888}a^{27}-\frac{1}{61888}a^{26}+\frac{3}{61888}a^{25}-\frac{1}{61888}a^{24}-\frac{5}{61888}a^{23}+\frac{1}{64}a^{22}-\frac{27}{64}a^{21}+\frac{25}{64}a^{20}+\frac{29}{64}a^{19}-\frac{15}{64}a^{18}+\frac{21}{64}a^{17}+\frac{9}{64}a^{16}+\frac{13}{64}a^{15}-\frac{31}{64}a^{14}+\frac{5}{64}a^{13}-\frac{7}{64}a^{12}-\frac{3}{64}a^{11}+\frac{17}{64}a^{10}-\frac{11}{64}a^{9}-\frac{23}{64}a^{8}-\frac{19}{64}a^{7}+\frac{1}{64}a^{6}-\frac{1541}{30944}a^{5}+\frac{287}{15472}a^{4}+\frac{627}{7736}a^{3}-\frac{457}{3868}a^{2}-\frac{85}{1934}a+\frac{271}{967}$, $\frac{1}{123776}a^{29}-\frac{1}{123776}a^{28}-\frac{1}{123776}a^{27}+\frac{3}{123776}a^{26}-\frac{1}{123776}a^{25}-\frac{5}{123776}a^{24}+\frac{7}{123776}a^{23}-\frac{27}{128}a^{22}+\frac{25}{128}a^{21}+\frac{29}{128}a^{20}+\frac{49}{128}a^{19}+\frac{21}{128}a^{18}+\frac{9}{128}a^{17}-\frac{51}{128}a^{16}+\frac{33}{128}a^{15}-\frac{59}{128}a^{14}-\frac{7}{128}a^{13}-\frac{3}{128}a^{12}+\frac{17}{128}a^{11}-\frac{11}{128}a^{10}-\frac{23}{128}a^{9}+\frac{45}{128}a^{8}+\frac{1}{128}a^{7}-\frac{1541}{61888}a^{6}+\frac{287}{30944}a^{5}+\frac{627}{15472}a^{4}-\frac{457}{7736}a^{3}-\frac{85}{3868}a^{2}+\frac{271}{1934}a-\frac{93}{967}$, $\frac{1}{247552}a^{30}-\frac{1}{247552}a^{29}-\frac{1}{247552}a^{28}+\frac{3}{247552}a^{27}-\frac{1}{247552}a^{26}-\frac{5}{247552}a^{25}+\frac{7}{247552}a^{24}+\frac{3}{247552}a^{23}-\frac{103}{256}a^{22}-\frac{99}{256}a^{21}+\frac{49}{256}a^{20}-\frac{107}{256}a^{19}+\frac{9}{256}a^{18}-\frac{51}{256}a^{17}+\frac{33}{256}a^{16}+\frac{69}{256}a^{15}+\frac{121}{256}a^{14}-\frac{3}{256}a^{13}+\frac{17}{256}a^{12}-\frac{11}{256}a^{11}-\frac{23}{256}a^{10}+\frac{45}{256}a^{9}+\frac{1}{256}a^{8}-\frac{1541}{123776}a^{7}+\frac{287}{61888}a^{6}+\frac{627}{30944}a^{5}-\frac{457}{15472}a^{4}-\frac{85}{7736}a^{3}+\frac{271}{3868}a^{2}-\frac{93}{1934}a-\frac{89}{967}$, $\frac{1}{495104}a^{31}-\frac{1}{495104}a^{30}-\frac{1}{495104}a^{29}+\frac{3}{495104}a^{28}-\frac{1}{495104}a^{27}-\frac{5}{495104}a^{26}+\frac{7}{495104}a^{25}+\frac{3}{495104}a^{24}-\frac{17}{495104}a^{23}-\frac{99}{512}a^{22}-\frac{207}{512}a^{21}-\frac{107}{512}a^{20}+\frac{9}{512}a^{19}+\frac{205}{512}a^{18}-\frac{223}{512}a^{17}-\frac{187}{512}a^{16}+\frac{121}{512}a^{15}+\frac{253}{512}a^{14}+\frac{17}{512}a^{13}-\frac{11}{512}a^{12}-\frac{23}{512}a^{11}+\frac{45}{512}a^{10}+\frac{1}{512}a^{9}-\frac{1541}{247552}a^{8}+\frac{287}{123776}a^{7}+\frac{627}{61888}a^{6}-\frac{457}{30944}a^{5}-\frac{85}{15472}a^{4}+\frac{271}{7736}a^{3}-\frac{93}{3868}a^{2}-\frac{89}{1934}a+\frac{91}{967}$, $\frac{1}{990208}a^{32}-\frac{1}{990208}a^{31}-\frac{1}{990208}a^{30}+\frac{3}{990208}a^{29}-\frac{1}{990208}a^{28}-\frac{5}{990208}a^{27}+\frac{7}{990208}a^{26}+\frac{3}{990208}a^{25}-\frac{17}{990208}a^{24}+\frac{11}{990208}a^{23}+\frac{305}{1024}a^{22}-\frac{107}{1024}a^{21}-\frac{503}{1024}a^{20}-\frac{307}{1024}a^{19}+\frac{289}{1024}a^{18}+\frac{325}{1024}a^{17}+\frac{121}{1024}a^{16}+\frac{253}{1024}a^{15}-\frac{495}{1024}a^{14}-\frac{11}{1024}a^{13}-\frac{23}{1024}a^{12}+\frac{45}{1024}a^{11}+\frac{1}{1024}a^{10}-\frac{1541}{495104}a^{9}+\frac{287}{247552}a^{8}+\frac{627}{123776}a^{7}-\frac{457}{61888}a^{6}-\frac{85}{30944}a^{5}+\frac{271}{15472}a^{4}-\frac{93}{7736}a^{3}-\frac{89}{3868}a^{2}+\frac{91}{1934}a-\frac{1}{967}$, $\frac{1}{1980416}a^{33}-\frac{1}{1980416}a^{32}-\frac{1}{1980416}a^{31}+\frac{3}{1980416}a^{30}-\frac{1}{1980416}a^{29}-\frac{5}{1980416}a^{28}+\frac{7}{1980416}a^{27}+\frac{3}{1980416}a^{26}-\frac{17}{1980416}a^{25}+\frac{11}{1980416}a^{24}+\frac{23}{1980416}a^{23}+\frac{917}{2048}a^{22}+\frac{521}{2048}a^{21}-\frac{307}{2048}a^{20}-\frac{735}{2048}a^{19}-\frac{699}{2048}a^{18}+\frac{121}{2048}a^{17}-\frac{771}{2048}a^{16}+\frac{529}{2048}a^{15}+\frac{1013}{2048}a^{14}-\frac{23}{2048}a^{13}+\frac{45}{2048}a^{12}+\frac{1}{2048}a^{11}-\frac{1541}{990208}a^{10}+\frac{287}{495104}a^{9}+\frac{627}{247552}a^{8}-\frac{457}{123776}a^{7}-\frac{85}{61888}a^{6}+\frac{271}{30944}a^{5}-\frac{93}{15472}a^{4}-\frac{89}{7736}a^{3}+\frac{91}{3868}a^{2}-\frac{1}{1934}a-\frac{45}{967}$, $\frac{1}{3960832}a^{34}-\frac{1}{3960832}a^{33}-\frac{1}{3960832}a^{32}+\frac{3}{3960832}a^{31}-\frac{1}{3960832}a^{30}-\frac{5}{3960832}a^{29}+\frac{7}{3960832}a^{28}+\frac{3}{3960832}a^{27}-\frac{17}{3960832}a^{26}+\frac{11}{3960832}a^{25}+\frac{23}{3960832}a^{24}-\frac{45}{3960832}a^{23}+\frac{521}{4096}a^{22}+\frac{1741}{4096}a^{21}+\frac{1313}{4096}a^{20}-\frac{699}{4096}a^{19}-\frac{1927}{4096}a^{18}-\frac{771}{4096}a^{17}+\frac{529}{4096}a^{16}+\frac{1013}{4096}a^{15}+\frac{2025}{4096}a^{14}+\frac{45}{4096}a^{13}+\frac{1}{4096}a^{12}-\frac{1541}{1980416}a^{11}+\frac{287}{990208}a^{10}+\frac{627}{495104}a^{9}-\frac{457}{247552}a^{8}-\frac{85}{123776}a^{7}+\frac{271}{61888}a^{6}-\frac{93}{30944}a^{5}-\frac{89}{15472}a^{4}+\frac{91}{7736}a^{3}-\frac{1}{3868}a^{2}-\frac{45}{1934}a+\frac{23}{967}$, $\frac{1}{7921664}a^{35}-\frac{1}{7921664}a^{34}-\frac{1}{7921664}a^{33}+\frac{3}{7921664}a^{32}-\frac{1}{7921664}a^{31}-\frac{5}{7921664}a^{30}+\frac{7}{7921664}a^{29}+\frac{3}{7921664}a^{28}-\frac{17}{7921664}a^{27}+\frac{11}{7921664}a^{26}+\frac{23}{7921664}a^{25}-\frac{45}{7921664}a^{24}-\frac{1}{7921664}a^{23}-\frac{2355}{8192}a^{22}+\frac{1313}{8192}a^{21}+\frac{3397}{8192}a^{20}+\frac{2169}{8192}a^{19}-\frac{771}{8192}a^{18}-\frac{3567}{8192}a^{17}-\frac{3083}{8192}a^{16}+\frac{2025}{8192}a^{15}-\frac{4051}{8192}a^{14}+\frac{1}{8192}a^{13}-\frac{1541}{3960832}a^{12}+\frac{287}{1980416}a^{11}+\frac{627}{990208}a^{10}-\frac{457}{495104}a^{9}-\frac{85}{247552}a^{8}+\frac{271}{123776}a^{7}-\frac{93}{61888}a^{6}-\frac{89}{30944}a^{5}+\frac{91}{15472}a^{4}-\frac{1}{7736}a^{3}-\frac{45}{3868}a^{2}+\frac{23}{1934}a+\frac{11}{967}$, $\frac{1}{15843328}a^{36}-\frac{1}{15843328}a^{35}-\frac{1}{15843328}a^{34}+\frac{3}{15843328}a^{33}-\frac{1}{15843328}a^{32}-\frac{5}{15843328}a^{31}+\frac{7}{15843328}a^{30}+\frac{3}{15843328}a^{29}-\frac{17}{15843328}a^{28}+\frac{11}{15843328}a^{27}+\frac{23}{15843328}a^{26}-\frac{45}{15843328}a^{25}-\frac{1}{15843328}a^{24}+\frac{91}{15843328}a^{23}-\frac{6879}{16384}a^{22}-\frac{4795}{16384}a^{21}+\frac{2169}{16384}a^{20}+\frac{7421}{16384}a^{19}+\frac{4625}{16384}a^{18}-\frac{3083}{16384}a^{17}-\frac{6167}{16384}a^{16}-\frac{4051}{16384}a^{15}+\frac{1}{16384}a^{14}-\frac{1541}{7921664}a^{13}+\frac{287}{3960832}a^{12}+\frac{627}{1980416}a^{11}-\frac{457}{990208}a^{10}-\frac{85}{495104}a^{9}+\frac{271}{247552}a^{8}-\frac{93}{123776}a^{7}-\frac{89}{61888}a^{6}+\frac{91}{30944}a^{5}-\frac{1}{15472}a^{4}-\frac{45}{7736}a^{3}+\frac{23}{3868}a^{2}+\frac{11}{1934}a-\frac{17}{967}$, $\frac{1}{31686656}a^{37}-\frac{1}{31686656}a^{36}-\frac{1}{31686656}a^{35}+\frac{3}{31686656}a^{34}-\frac{1}{31686656}a^{33}-\frac{5}{31686656}a^{32}+\frac{7}{31686656}a^{31}+\frac{3}{31686656}a^{30}-\frac{17}{31686656}a^{29}+\frac{11}{31686656}a^{28}+\frac{23}{31686656}a^{27}-\frac{45}{31686656}a^{26}-\frac{1}{31686656}a^{25}+\frac{91}{31686656}a^{24}-\frac{89}{31686656}a^{23}+\frac{11589}{32768}a^{22}+\frac{2169}{32768}a^{21}+\frac{7421}{32768}a^{20}-\frac{11759}{32768}a^{19}-\frac{3083}{32768}a^{18}-\frac{6167}{32768}a^{17}+\frac{12333}{32768}a^{16}+\frac{1}{32768}a^{15}-\frac{1541}{15843328}a^{14}+\frac{287}{7921664}a^{13}+\frac{627}{3960832}a^{12}-\frac{457}{1980416}a^{11}-\frac{85}{990208}a^{10}+\frac{271}{495104}a^{9}-\frac{93}{247552}a^{8}-\frac{89}{123776}a^{7}+\frac{91}{61888}a^{6}-\frac{1}{30944}a^{5}-\frac{45}{15472}a^{4}+\frac{23}{7736}a^{3}+\frac{11}{3868}a^{2}-\frac{17}{1934}a+\frac{3}{967}$, $\frac{1}{63373312}a^{38}-\frac{1}{63373312}a^{37}-\frac{1}{63373312}a^{36}+\frac{3}{63373312}a^{35}-\frac{1}{63373312}a^{34}-\frac{5}{63373312}a^{33}+\frac{7}{63373312}a^{32}+\frac{3}{63373312}a^{31}-\frac{17}{63373312}a^{30}+\frac{11}{63373312}a^{29}+\frac{23}{63373312}a^{28}-\frac{45}{63373312}a^{27}-\frac{1}{63373312}a^{26}+\frac{91}{63373312}a^{25}-\frac{89}{63373312}a^{24}-\frac{93}{63373312}a^{23}+\frac{2169}{65536}a^{22}-\frac{25347}{65536}a^{21}+\frac{21009}{65536}a^{20}+\frac{29685}{65536}a^{19}-\frac{6167}{65536}a^{18}+\frac{12333}{65536}a^{17}+\frac{1}{65536}a^{16}-\frac{1541}{31686656}a^{15}+\frac{287}{15843328}a^{14}+\frac{627}{7921664}a^{13}-\frac{457}{3960832}a^{12}-\frac{85}{1980416}a^{11}+\frac{271}{990208}a^{10}-\frac{93}{495104}a^{9}-\frac{89}{247552}a^{8}+\frac{91}{123776}a^{7}-\frac{1}{61888}a^{6}-\frac{45}{30944}a^{5}+\frac{23}{15472}a^{4}+\frac{11}{7736}a^{3}-\frac{17}{3868}a^{2}+\frac{3}{1934}a+\frac{7}{967}$, $\frac{1}{126746624}a^{39}-\frac{1}{126746624}a^{38}-\frac{1}{126746624}a^{37}+\frac{3}{126746624}a^{36}-\frac{1}{126746624}a^{35}-\frac{5}{126746624}a^{34}+\frac{7}{126746624}a^{33}+\frac{3}{126746624}a^{32}-\frac{17}{126746624}a^{31}+\frac{11}{126746624}a^{30}+\frac{23}{126746624}a^{29}-\frac{45}{126746624}a^{28}-\frac{1}{126746624}a^{27}+\frac{91}{126746624}a^{26}-\frac{89}{126746624}a^{25}-\frac{93}{126746624}a^{24}+\frac{271}{126746624}a^{23}-\frac{25347}{131072}a^{22}+\frac{21009}{131072}a^{21}+\frac{29685}{131072}a^{20}+\frac{59369}{131072}a^{19}+\frac{12333}{131072}a^{18}+\frac{1}{131072}a^{17}-\frac{1541}{63373312}a^{16}+\frac{287}{31686656}a^{15}+\frac{627}{15843328}a^{14}-\frac{457}{7921664}a^{13}-\frac{85}{3960832}a^{12}+\frac{271}{1980416}a^{11}-\frac{93}{990208}a^{10}-\frac{89}{495104}a^{9}+\frac{91}{247552}a^{8}-\frac{1}{123776}a^{7}-\frac{45}{61888}a^{6}+\frac{23}{30944}a^{5}+\frac{11}{15472}a^{4}-\frac{17}{7736}a^{3}+\frac{3}{3868}a^{2}+\frac{7}{1934}a-\frac{5}{967}$, $\frac{1}{253493248}a^{40}-\frac{1}{253493248}a^{39}-\frac{1}{253493248}a^{38}+\frac{3}{253493248}a^{37}-\frac{1}{253493248}a^{36}-\frac{5}{253493248}a^{35}+\frac{7}{253493248}a^{34}+\frac{3}{253493248}a^{33}-\frac{17}{253493248}a^{32}+\frac{11}{253493248}a^{31}+\frac{23}{253493248}a^{30}-\frac{45}{253493248}a^{29}-\frac{1}{253493248}a^{28}+\frac{91}{253493248}a^{27}-\frac{89}{253493248}a^{26}-\frac{93}{253493248}a^{25}+\frac{271}{253493248}a^{24}-\frac{85}{253493248}a^{23}-\frac{110063}{262144}a^{22}-\frac{101387}{262144}a^{21}+\frac{59369}{262144}a^{20}-\frac{118739}{262144}a^{19}+\frac{1}{262144}a^{18}-\frac{1541}{126746624}a^{17}+\frac{287}{63373312}a^{16}+\frac{627}{31686656}a^{15}-\frac{457}{15843328}a^{14}-\frac{85}{7921664}a^{13}+\frac{271}{3960832}a^{12}-\frac{93}{1980416}a^{11}-\frac{89}{990208}a^{10}+\frac{91}{495104}a^{9}-\frac{1}{247552}a^{8}-\frac{45}{123776}a^{7}+\frac{23}{61888}a^{6}+\frac{11}{30944}a^{5}-\frac{17}{15472}a^{4}+\frac{3}{7736}a^{3}+\frac{7}{3868}a^{2}-\frac{5}{1934}a-\frac{1}{967}$, $\frac{1}{506986496}a^{41}-\frac{1}{506986496}a^{40}-\frac{1}{506986496}a^{39}+\frac{3}{506986496}a^{38}-\frac{1}{506986496}a^{37}-\frac{5}{506986496}a^{36}+\frac{7}{506986496}a^{35}+\frac{3}{506986496}a^{34}-\frac{17}{506986496}a^{33}+\frac{11}{506986496}a^{32}+\frac{23}{506986496}a^{31}-\frac{45}{506986496}a^{30}-\frac{1}{506986496}a^{29}+\frac{91}{506986496}a^{28}-\frac{89}{506986496}a^{27}-\frac{93}{506986496}a^{26}+\frac{271}{506986496}a^{25}-\frac{85}{506986496}a^{24}-\frac{457}{506986496}a^{23}-\frac{101387}{524288}a^{22}-\frac{202775}{524288}a^{21}-\frac{118739}{524288}a^{20}+\frac{1}{524288}a^{19}-\frac{1541}{253493248}a^{18}+\frac{287}{126746624}a^{17}+\frac{627}{63373312}a^{16}-\frac{457}{31686656}a^{15}-\frac{85}{15843328}a^{14}+\frac{271}{7921664}a^{13}-\frac{93}{3960832}a^{12}-\frac{89}{1980416}a^{11}+\frac{91}{990208}a^{10}-\frac{1}{495104}a^{9}-\frac{45}{247552}a^{8}+\frac{23}{123776}a^{7}+\frac{11}{61888}a^{6}-\frac{17}{30944}a^{5}+\frac{3}{15472}a^{4}+\frac{7}{7736}a^{3}-\frac{5}{3868}a^{2}-\frac{1}{1934}a+\frac{3}{967}$, $\frac{1}{1013972992}a^{42}-\frac{1}{1013972992}a^{41}-\frac{1}{1013972992}a^{40}+\frac{3}{1013972992}a^{39}-\frac{1}{1013972992}a^{38}-\frac{5}{1013972992}a^{37}+\frac{7}{1013972992}a^{36}+\frac{3}{1013972992}a^{35}-\frac{17}{1013972992}a^{34}+\frac{11}{1013972992}a^{33}+\frac{23}{1013972992}a^{32}-\frac{45}{1013972992}a^{31}-\frac{1}{1013972992}a^{30}+\frac{91}{1013972992}a^{29}-\frac{89}{1013972992}a^{28}-\frac{93}{1013972992}a^{27}+\frac{271}{1013972992}a^{26}-\frac{85}{1013972992}a^{25}-\frac{457}{1013972992}a^{24}+\frac{627}{1013972992}a^{23}-\frac{202775}{1048576}a^{22}+\frac{405549}{1048576}a^{21}+\frac{1}{1048576}a^{20}-\frac{1541}{506986496}a^{19}+\frac{287}{253493248}a^{18}+\frac{627}{126746624}a^{17}-\frac{457}{63373312}a^{16}-\frac{85}{31686656}a^{15}+\frac{271}{15843328}a^{14}-\frac{93}{7921664}a^{13}-\frac{89}{3960832}a^{12}+\frac{91}{1980416}a^{11}-\frac{1}{990208}a^{10}-\frac{45}{495104}a^{9}+\frac{23}{247552}a^{8}+\frac{11}{123776}a^{7}-\frac{17}{61888}a^{6}+\frac{3}{30944}a^{5}+\frac{7}{15472}a^{4}-\frac{5}{7736}a^{3}-\frac{1}{3868}a^{2}+\frac{3}{1934}a-\frac{1}{967}$, $\frac{1}{2027945984}a^{43}-\frac{1}{2027945984}a^{42}-\frac{1}{2027945984}a^{41}+\frac{3}{2027945984}a^{40}-\frac{1}{2027945984}a^{39}-\frac{5}{2027945984}a^{38}+\frac{7}{2027945984}a^{37}+\frac{3}{2027945984}a^{36}-\frac{17}{2027945984}a^{35}+\frac{11}{2027945984}a^{34}+\frac{23}{2027945984}a^{33}-\frac{45}{2027945984}a^{32}-\frac{1}{2027945984}a^{31}+\frac{91}{2027945984}a^{30}-\frac{89}{2027945984}a^{29}-\frac{93}{2027945984}a^{28}+\frac{271}{2027945984}a^{27}-\frac{85}{2027945984}a^{26}-\frac{457}{2027945984}a^{25}+\frac{627}{2027945984}a^{24}+\frac{287}{2027945984}a^{23}+\frac{405549}{2097152}a^{22}+\frac{1}{2097152}a^{21}-\frac{1541}{1013972992}a^{20}+\frac{287}{506986496}a^{19}+\frac{627}{253493248}a^{18}-\frac{457}{126746624}a^{17}-\frac{85}{63373312}a^{16}+\frac{271}{31686656}a^{15}-\frac{93}{15843328}a^{14}-\frac{89}{7921664}a^{13}+\frac{91}{3960832}a^{12}-\frac{1}{1980416}a^{11}-\frac{45}{990208}a^{10}+\frac{23}{495104}a^{9}+\frac{11}{247552}a^{8}-\frac{17}{123776}a^{7}+\frac{3}{61888}a^{6}+\frac{7}{30944}a^{5}-\frac{5}{15472}a^{4}-\frac{1}{7736}a^{3}+\frac{3}{3868}a^{2}-\frac{1}{1934}a-\frac{1}{967}$ 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, 1/1934*a^23 - 1/2*a^22 - 1/2*a^21 - 1/2*a^20 - 1/2*a^19 - 1/2*a^18 - 1/2*a^17 - 1/2*a^16 - 1/2*a^15 - 1/2*a^14 - 1/2*a^13 - 1/2*a^12 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a + 393/967, 1/3868*a^24 - 1/3868*a^23 + 1/4*a^22 + 1/4*a^21 + 1/4*a^20 + 1/4*a^19 + 1/4*a^18 + 1/4*a^17 + 1/4*a^16 + 1/4*a^15 + 1/4*a^14 + 1/4*a^13 + 1/4*a^12 + 1/4*a^11 + 1/4*a^10 + 1/4*a^9 + 1/4*a^8 + 1/4*a^7 + 1/4*a^6 + 1/4*a^5 + 1/4*a^4 + 1/4*a^3 + 1/4*a^2 + 393/1934*a + 287/967, 1/7736*a^25 - 1/7736*a^24 - 1/7736*a^23 - 3/8*a^22 + 1/8*a^21 - 3/8*a^20 + 1/8*a^19 - 3/8*a^18 + 1/8*a^17 - 3/8*a^16 + 1/8*a^15 - 3/8*a^14 + 1/8*a^13 - 3/8*a^12 + 1/8*a^11 - 3/8*a^10 + 1/8*a^9 - 3/8*a^8 + 1/8*a^7 - 3/8*a^6 + 1/8*a^5 - 3/8*a^4 + 1/8*a^3 - 1541/3868*a^2 + 287/1934*a - 340/967, 1/15472*a^26 - 1/15472*a^25 - 1/15472*a^24 + 3/15472*a^23 - 7/16*a^22 - 3/16*a^21 + 1/16*a^20 + 5/16*a^19 - 7/16*a^18 - 3/16*a^17 + 1/16*a^16 + 5/16*a^15 - 7/16*a^14 - 3/16*a^13 + 1/16*a^12 + 5/16*a^11 - 7/16*a^10 - 3/16*a^9 + 1/16*a^8 + 5/16*a^7 - 7/16*a^6 - 3/16*a^5 + 1/16*a^4 - 1541/7736*a^3 + 287/3868*a^2 + 627/1934*a - 457/967, 1/30944*a^27 - 1/30944*a^26 - 1/30944*a^25 + 3/30944*a^24 - 1/30944*a^23 + 13/32*a^22 + 1/32*a^21 + 5/32*a^20 - 7/32*a^19 - 3/32*a^18 - 15/32*a^17 - 11/32*a^16 + 9/32*a^15 + 13/32*a^14 + 1/32*a^13 + 5/32*a^12 - 7/32*a^11 - 3/32*a^10 - 15/32*a^9 - 11/32*a^8 + 9/32*a^7 + 13/32*a^6 + 1/32*a^5 - 1541/15472*a^4 + 287/7736*a^3 + 627/3868*a^2 - 457/1934*a - 85/967, 1/61888*a^28 - 1/61888*a^27 - 1/61888*a^26 + 3/61888*a^25 - 1/61888*a^24 - 5/61888*a^23 + 1/64*a^22 - 27/64*a^21 + 25/64*a^20 + 29/64*a^19 - 15/64*a^18 + 21/64*a^17 + 9/64*a^16 + 13/64*a^15 - 31/64*a^14 + 5/64*a^13 - 7/64*a^12 - 3/64*a^11 + 17/64*a^10 - 11/64*a^9 - 23/64*a^8 - 19/64*a^7 + 1/64*a^6 - 1541/30944*a^5 + 287/15472*a^4 + 627/7736*a^3 - 457/3868*a^2 - 85/1934*a + 271/967, 1/123776*a^29 - 1/123776*a^28 - 1/123776*a^27 + 3/123776*a^26 - 1/123776*a^25 - 5/123776*a^24 + 7/123776*a^23 - 27/128*a^22 + 25/128*a^21 + 29/128*a^20 + 49/128*a^19 + 21/128*a^18 + 9/128*a^17 - 51/128*a^16 + 33/128*a^15 - 59/128*a^14 - 7/128*a^13 - 3/128*a^12 + 17/128*a^11 - 11/128*a^10 - 23/128*a^9 + 45/128*a^8 + 1/128*a^7 - 1541/61888*a^6 + 287/30944*a^5 + 627/15472*a^4 - 457/7736*a^3 - 85/3868*a^2 + 271/1934*a - 93/967, 1/247552*a^30 - 1/247552*a^29 - 1/247552*a^28 + 3/247552*a^27 - 1/247552*a^26 - 5/247552*a^25 + 7/247552*a^24 + 3/247552*a^23 - 103/256*a^22 - 99/256*a^21 + 49/256*a^20 - 107/256*a^19 + 9/256*a^18 - 51/256*a^17 + 33/256*a^16 + 69/256*a^15 + 121/256*a^14 - 3/256*a^13 + 17/256*a^12 - 11/256*a^11 - 23/256*a^10 + 45/256*a^9 + 1/256*a^8 - 1541/123776*a^7 + 287/61888*a^6 + 627/30944*a^5 - 457/15472*a^4 - 85/7736*a^3 + 271/3868*a^2 - 93/1934*a - 89/967, 1/495104*a^31 - 1/495104*a^30 - 1/495104*a^29 + 3/495104*a^28 - 1/495104*a^27 - 5/495104*a^26 + 7/495104*a^25 + 3/495104*a^24 - 17/495104*a^23 - 99/512*a^22 - 207/512*a^21 - 107/512*a^20 + 9/512*a^19 + 205/512*a^18 - 223/512*a^17 - 187/512*a^16 + 121/512*a^15 + 253/512*a^14 + 17/512*a^13 - 11/512*a^12 - 23/512*a^11 + 45/512*a^10 + 1/512*a^9 - 1541/247552*a^8 + 287/123776*a^7 + 627/61888*a^6 - 457/30944*a^5 - 85/15472*a^4 + 271/7736*a^3 - 93/3868*a^2 - 89/1934*a + 91/967, 1/990208*a^32 - 1/990208*a^31 - 1/990208*a^30 + 3/990208*a^29 - 1/990208*a^28 - 5/990208*a^27 + 7/990208*a^26 + 3/990208*a^25 - 17/990208*a^24 + 11/990208*a^23 + 305/1024*a^22 - 107/1024*a^21 - 503/1024*a^20 - 307/1024*a^19 + 289/1024*a^18 + 325/1024*a^17 + 121/1024*a^16 + 253/1024*a^15 - 495/1024*a^14 - 11/1024*a^13 - 23/1024*a^12 + 45/1024*a^11 + 1/1024*a^10 - 1541/495104*a^9 + 287/247552*a^8 + 627/123776*a^7 - 457/61888*a^6 - 85/30944*a^5 + 271/15472*a^4 - 93/7736*a^3 - 89/3868*a^2 + 91/1934*a - 1/967, 1/1980416*a^33 - 1/1980416*a^32 - 1/1980416*a^31 + 3/1980416*a^30 - 1/1980416*a^29 - 5/1980416*a^28 + 7/1980416*a^27 + 3/1980416*a^26 - 17/1980416*a^25 + 11/1980416*a^24 + 23/1980416*a^23 + 917/2048*a^22 + 521/2048*a^21 - 307/2048*a^20 - 735/2048*a^19 - 699/2048*a^18 + 121/2048*a^17 - 771/2048*a^16 + 529/2048*a^15 + 1013/2048*a^14 - 23/2048*a^13 + 45/2048*a^12 + 1/2048*a^11 - 1541/990208*a^10 + 287/495104*a^9 + 627/247552*a^8 - 457/123776*a^7 - 85/61888*a^6 + 271/30944*a^5 - 93/15472*a^4 - 89/7736*a^3 + 91/3868*a^2 - 1/1934*a - 45/967, 1/3960832*a^34 - 1/3960832*a^33 - 1/3960832*a^32 + 3/3960832*a^31 - 1/3960832*a^30 - 5/3960832*a^29 + 7/3960832*a^28 + 3/3960832*a^27 - 17/3960832*a^26 + 11/3960832*a^25 + 23/3960832*a^24 - 45/3960832*a^23 + 521/4096*a^22 + 1741/4096*a^21 + 1313/4096*a^20 - 699/4096*a^19 - 1927/4096*a^18 - 771/4096*a^17 + 529/4096*a^16 + 1013/4096*a^15 + 2025/4096*a^14 + 45/4096*a^13 + 1/4096*a^12 - 1541/1980416*a^11 + 287/990208*a^10 + 627/495104*a^9 - 457/247552*a^8 - 85/123776*a^7 + 271/61888*a^6 - 93/30944*a^5 - 89/15472*a^4 + 91/7736*a^3 - 1/3868*a^2 - 45/1934*a + 23/967, 1/7921664*a^35 - 1/7921664*a^34 - 1/7921664*a^33 + 3/7921664*a^32 - 1/7921664*a^31 - 5/7921664*a^30 + 7/7921664*a^29 + 3/7921664*a^28 - 17/7921664*a^27 + 11/7921664*a^26 + 23/7921664*a^25 - 45/7921664*a^24 - 1/7921664*a^23 - 2355/8192*a^22 + 1313/8192*a^21 + 3397/8192*a^20 + 2169/8192*a^19 - 771/8192*a^18 - 3567/8192*a^17 - 3083/8192*a^16 + 2025/8192*a^15 - 4051/8192*a^14 + 1/8192*a^13 - 1541/3960832*a^12 + 287/1980416*a^11 + 627/990208*a^10 - 457/495104*a^9 - 85/247552*a^8 + 271/123776*a^7 - 93/61888*a^6 - 89/30944*a^5 + 91/15472*a^4 - 1/7736*a^3 - 45/3868*a^2 + 23/1934*a + 11/967, 1/15843328*a^36 - 1/15843328*a^35 - 1/15843328*a^34 + 3/15843328*a^33 - 1/15843328*a^32 - 5/15843328*a^31 + 7/15843328*a^30 + 3/15843328*a^29 - 17/15843328*a^28 + 11/15843328*a^27 + 23/15843328*a^26 - 45/15843328*a^25 - 1/15843328*a^24 + 91/15843328*a^23 - 6879/16384*a^22 - 4795/16384*a^21 + 2169/16384*a^20 + 7421/16384*a^19 + 4625/16384*a^18 - 3083/16384*a^17 - 6167/16384*a^16 - 4051/16384*a^15 + 1/16384*a^14 - 1541/7921664*a^13 + 287/3960832*a^12 + 627/1980416*a^11 - 457/990208*a^10 - 85/495104*a^9 + 271/247552*a^8 - 93/123776*a^7 - 89/61888*a^6 + 91/30944*a^5 - 1/15472*a^4 - 45/7736*a^3 + 23/3868*a^2 + 11/1934*a - 17/967, 1/31686656*a^37 - 1/31686656*a^36 - 1/31686656*a^35 + 3/31686656*a^34 - 1/31686656*a^33 - 5/31686656*a^32 + 7/31686656*a^31 + 3/31686656*a^30 - 17/31686656*a^29 + 11/31686656*a^28 + 23/31686656*a^27 - 45/31686656*a^26 - 1/31686656*a^25 + 91/31686656*a^24 - 89/31686656*a^23 + 11589/32768*a^22 + 2169/32768*a^21 + 7421/32768*a^20 - 11759/32768*a^19 - 3083/32768*a^18 - 6167/32768*a^17 + 12333/32768*a^16 + 1/32768*a^15 - 1541/15843328*a^14 + 287/7921664*a^13 + 627/3960832*a^12 - 457/1980416*a^11 - 85/990208*a^10 + 271/495104*a^9 - 93/247552*a^8 - 89/123776*a^7 + 91/61888*a^6 - 1/30944*a^5 - 45/15472*a^4 + 23/7736*a^3 + 11/3868*a^2 - 17/1934*a + 3/967, 1/63373312*a^38 - 1/63373312*a^37 - 1/63373312*a^36 + 3/63373312*a^35 - 1/63373312*a^34 - 5/63373312*a^33 + 7/63373312*a^32 + 3/63373312*a^31 - 17/63373312*a^30 + 11/63373312*a^29 + 23/63373312*a^28 - 45/63373312*a^27 - 1/63373312*a^26 + 91/63373312*a^25 - 89/63373312*a^24 - 93/63373312*a^23 + 2169/65536*a^22 - 25347/65536*a^21 + 21009/65536*a^20 + 29685/65536*a^19 - 6167/65536*a^18 + 12333/65536*a^17 + 1/65536*a^16 - 1541/31686656*a^15 + 287/15843328*a^14 + 627/7921664*a^13 - 457/3960832*a^12 - 85/1980416*a^11 + 271/990208*a^10 - 93/495104*a^9 - 89/247552*a^8 + 91/123776*a^7 - 1/61888*a^6 - 45/30944*a^5 + 23/15472*a^4 + 11/7736*a^3 - 17/3868*a^2 + 3/1934*a + 7/967, 1/126746624*a^39 - 1/126746624*a^38 - 1/126746624*a^37 + 3/126746624*a^36 - 1/126746624*a^35 - 5/126746624*a^34 + 7/126746624*a^33 + 3/126746624*a^32 - 17/126746624*a^31 + 11/126746624*a^30 + 23/126746624*a^29 - 45/126746624*a^28 - 1/126746624*a^27 + 91/126746624*a^26 - 89/126746624*a^25 - 93/126746624*a^24 + 271/126746624*a^23 - 25347/131072*a^22 + 21009/131072*a^21 + 29685/131072*a^20 + 59369/131072*a^19 + 12333/131072*a^18 + 1/131072*a^17 - 1541/63373312*a^16 + 287/31686656*a^15 + 627/15843328*a^14 - 457/7921664*a^13 - 85/3960832*a^12 + 271/1980416*a^11 - 93/990208*a^10 - 89/495104*a^9 + 91/247552*a^8 - 1/123776*a^7 - 45/61888*a^6 + 23/30944*a^5 + 11/15472*a^4 - 17/7736*a^3 + 3/3868*a^2 + 7/1934*a - 5/967, 1/253493248*a^40 - 1/253493248*a^39 - 1/253493248*a^38 + 3/253493248*a^37 - 1/253493248*a^36 - 5/253493248*a^35 + 7/253493248*a^34 + 3/253493248*a^33 - 17/253493248*a^32 + 11/253493248*a^31 + 23/253493248*a^30 - 45/253493248*a^29 - 1/253493248*a^28 + 91/253493248*a^27 - 89/253493248*a^26 - 93/253493248*a^25 + 271/253493248*a^24 - 85/253493248*a^23 - 110063/262144*a^22 - 101387/262144*a^21 + 59369/262144*a^20 - 118739/262144*a^19 + 1/262144*a^18 - 1541/126746624*a^17 + 287/63373312*a^16 + 627/31686656*a^15 - 457/15843328*a^14 - 85/7921664*a^13 + 271/3960832*a^12 - 93/1980416*a^11 - 89/990208*a^10 + 91/495104*a^9 - 1/247552*a^8 - 45/123776*a^7 + 23/61888*a^6 + 11/30944*a^5 - 17/15472*a^4 + 3/7736*a^3 + 7/3868*a^2 - 5/1934*a - 1/967, 1/506986496*a^41 - 1/506986496*a^40 - 1/506986496*a^39 + 3/506986496*a^38 - 1/506986496*a^37 - 5/506986496*a^36 + 7/506986496*a^35 + 3/506986496*a^34 - 17/506986496*a^33 + 11/506986496*a^32 + 23/506986496*a^31 - 45/506986496*a^30 - 1/506986496*a^29 + 91/506986496*a^28 - 89/506986496*a^27 - 93/506986496*a^26 + 271/506986496*a^25 - 85/506986496*a^24 - 457/506986496*a^23 - 101387/524288*a^22 - 202775/524288*a^21 - 118739/524288*a^20 + 1/524288*a^19 - 1541/253493248*a^18 + 287/126746624*a^17 + 627/63373312*a^16 - 457/31686656*a^15 - 85/15843328*a^14 + 271/7921664*a^13 - 93/3960832*a^12 - 89/1980416*a^11 + 91/990208*a^10 - 1/495104*a^9 - 45/247552*a^8 + 23/123776*a^7 + 11/61888*a^6 - 17/30944*a^5 + 3/15472*a^4 + 7/7736*a^3 - 5/3868*a^2 - 1/1934*a + 3/967, 1/1013972992*a^42 - 1/1013972992*a^41 - 1/1013972992*a^40 + 3/1013972992*a^39 - 1/1013972992*a^38 - 5/1013972992*a^37 + 7/1013972992*a^36 + 3/1013972992*a^35 - 17/1013972992*a^34 + 11/1013972992*a^33 + 23/1013972992*a^32 - 45/1013972992*a^31 - 1/1013972992*a^30 + 91/1013972992*a^29 - 89/1013972992*a^28 - 93/1013972992*a^27 + 271/1013972992*a^26 - 85/1013972992*a^25 - 457/1013972992*a^24 + 627/1013972992*a^23 - 202775/1048576*a^22 + 405549/1048576*a^21 + 1/1048576*a^20 - 1541/506986496*a^19 + 287/253493248*a^18 + 627/126746624*a^17 - 457/63373312*a^16 - 85/31686656*a^15 + 271/15843328*a^14 - 93/7921664*a^13 - 89/3960832*a^12 + 91/1980416*a^11 - 1/990208*a^10 - 45/495104*a^9 + 23/247552*a^8 + 11/123776*a^7 - 17/61888*a^6 + 3/30944*a^5 + 7/15472*a^4 - 5/7736*a^3 - 1/3868*a^2 + 3/1934*a - 1/967, 1/2027945984*a^43 - 1/2027945984*a^42 - 1/2027945984*a^41 + 3/2027945984*a^40 - 1/2027945984*a^39 - 5/2027945984*a^38 + 7/2027945984*a^37 + 3/2027945984*a^36 - 17/2027945984*a^35 + 11/2027945984*a^34 + 23/2027945984*a^33 - 45/2027945984*a^32 - 1/2027945984*a^31 + 91/2027945984*a^30 - 89/2027945984*a^29 - 93/2027945984*a^28 + 271/2027945984*a^27 - 85/2027945984*a^26 - 457/2027945984*a^25 + 627/2027945984*a^24 + 287/2027945984*a^23 + 405549/2097152*a^22 + 1/2097152*a^21 - 1541/1013972992*a^20 + 287/506986496*a^19 + 627/253493248*a^18 - 457/126746624*a^17 - 85/63373312*a^16 + 271/31686656*a^15 - 93/15843328*a^14 - 89/7921664*a^13 + 91/3960832*a^12 - 1/1980416*a^11 - 45/990208*a^10 + 23/495104*a^9 + 11/247552*a^8 - 17/123776*a^7 + 3/61888*a^6 + 7/30944*a^5 - 5/15472*a^4 - 1/7736*a^3 + 3/3868*a^2 - 1/1934*a - 1/967

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:	$21$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( \frac{287}{2027945984} a^{43} + \frac{287}{2027945984} a^{42} - \frac{861}{2027945984} a^{41} + \frac{287}{2027945984} a^{40} + \frac{1435}{2027945984} a^{39} - \frac{2009}{2027945984} a^{38} - \frac{861}{2027945984} a^{37} + \frac{4879}{2027945984} a^{36} - \frac{3157}{2027945984} a^{35} - \frac{6601}{2027945984} a^{34} + \frac{12915}{2027945984} a^{33} + \frac{287}{2027945984} a^{32} - \frac{26117}{2027945984} a^{31} + \frac{25543}{2027945984} a^{30} + \frac{26691}{2027945984} a^{29} - \frac{77777}{2027945984} a^{28} + \frac{24395}{2027945984} a^{27} + \frac{131159}{2027945984} a^{26} - \frac{179949}{2027945984} a^{25} - \frac{82369}{2027945984} a^{24} + \frac{442267}{2027945984} a^{23} - \frac{287}{2097152} a^{22} + \frac{1541}{2097152} a^{21} - \frac{82369}{506986496} a^{20} - \frac{179949}{253493248} a^{19} + \frac{131159}{126746624} a^{18} + \frac{24395}{63373312} a^{17} - \frac{77777}{31686656} a^{16} + \frac{26691}{15843328} a^{15} + \frac{25543}{7921664} a^{14} - \frac{26117}{3960832} a^{13} + \frac{287}{1980416} a^{12} + \frac{12915}{990208} a^{11} - \frac{6601}{495104} a^{10} - \frac{3157}{247552} a^{9} + \frac{4879}{123776} a^{8} - \frac{861}{61888} a^{7} - \frac{2009}{30944} a^{6} + \frac{1435}{15472} a^{5} + \frac{287}{7736} a^{4} - \frac{861}{3868} a^{3} + \frac{287}{1934} a^{2} + \frac{287}{967} a - \frac{574}{967} \) (287)/(2027945984)a^(43) + (287)/(2027945984)a^(42) - (861)/(2027945984)a^(41) + (287)/(2027945984)a^(40) + (1435)/(2027945984)a^(39) - (2009)/(2027945984)a^(38) - (861)/(2027945984)a^(37) + (4879)/(2027945984)a^(36) - (3157)/(2027945984)a^(35) - (6601)/(2027945984)a^(34) + (12915)/(2027945984)a^(33) + (287)/(2027945984)a^(32) - (26117)/(2027945984)a^(31) + (25543)/(2027945984)a^(30) + (26691)/(2027945984)a^(29) - (77777)/(2027945984)a^(28) + (24395)/(2027945984)a^(27) + (131159)/(2027945984)a^(26) - (179949)/(2027945984)a^(25) - (82369)/(2027945984)a^(24) + (442267)/(2027945984)a^(23) - (287)/(2097152)a^(22) + (1541)/(2097152)a^(21) - (82369)/(506986496)a^(20) - (179949)/(253493248)a^(19) + (131159)/(126746624)a^(18) + (24395)/(63373312)a^(17) - (77777)/(31686656)a^(16) + (26691)/(15843328)a^(15) + (25543)/(7921664)a^(14) - (26117)/(3960832)a^(13) + (287)/(1980416)a^(12) + (12915)/(990208)a^(11) - (6601)/(495104)a^(10) - (3157)/(247552)a^(9) + (4879)/(123776)a^(8) - (861)/(61888)a^(7) - (2009)/(30944)a^(6) + (1435)/(15472)a^(5) + (287)/(7736)a^(4) - (861)/(3868)a^(3) + (287)/(1934)a^(2) + (287)/(967)*a - (574)/(967) (order $46$)	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^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304)

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^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304, 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^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304);

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^44 - x^43 - x^42 + 3*x^41 - x^40 - 5*x^39 + 7*x^38 + 3*x^37 - 17*x^36 + 11*x^35 + 23*x^34 - 45*x^33 - x^32 + 91*x^31 - 89*x^30 - 93*x^29 + 271*x^28 - 85*x^27 - 457*x^26 + 627*x^25 + 287*x^24 - 1541*x^23 + 967*x^22 - 3082*x^21 + 1148*x^20 + 5016*x^19 - 7312*x^18 - 2720*x^17 + 17344*x^16 - 11904*x^15 - 22784*x^14 + 46592*x^13 - 1024*x^12 - 92160*x^11 + 94208*x^10 + 90112*x^9 - 278528*x^8 + 98304*x^7 + 458752*x^6 - 655360*x^5 - 262144*x^4 + 1572864*x^3 - 1048576*x^2 - 2097152*x + 4194304);

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

$C_2\times C_{22}$ (as 44T2):

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)

An abelian group of order 44

The 44 conjugacy class representatives for $C_2\times C_{22}$

Character table for $C_2\times C_{22}$

Intermediate fields

$\Q(\sqrt{161}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{-23}) $, $\Q(\sqrt{-7}, \sqrt{-23})$, $\Q(\zeta_{23})^+$, 22.22.78048218870425324004237696277333187889.1, 22.0.3393400820453274956705986794666660343.1, $\Q(\zeta_{23})$

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	${\href{/padicField/2.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	R	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	R	${\href{/padicField/29.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/padicField/47.2.0.1}{2} }^{22}$	$22^{2}$	$22^{2}$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$7$ 7		Deg $44$	$2$	$22$	$22$
$23$ 23	23.22.21.17	$x^{22} + 23$	$22$	$1$	$21$	22T1	$[\ ]_{22}$
$23$ 23	23.22.21.17	$x^{22} + 23$	$22$	$1$	$21$	22T1	$[\ ]_{22}$

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