LMFDB - Number field 44.44.191158492466532603992882058813386761401107535956984758058028173625115327713733756218489.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597)

gp: K = bnfinit(y^44 - y^43 - 86*y^42 + 81*y^41 + 3445*y^40 - 3040*y^39 - 85351*y^38 + 70151*y^37 + 1464534*y^36 - 1113779*y^35 - 18468767*y^34 + 12899872*y^33 + 177261061*y^32 - 112761701*y^31 - 1323060098*y^30 + 759251593*y^29 + 7780382701*y^28 - 3984124736*y^27 - 36301909279*y^26 + 16381285599*y^25 + 134698547590*y^24 - 52792119595*y^23 - 396744216375*y^22 + 132783613203*y^21 + 922279857882*y^20 - 258361552805*y^19 - 1675313579545*y^18 + 383501034280*y^17 + 2343182319195*y^16 - 425622641659*y^15 - 2471777086526*y^14 + 343273729047*y^13 + 1911152162483*y^12 - 192962821056*y^11 - 1040820630209*y^10 + 70437336577*y^9 + 377125555258*y^8 - 13999395253*y^7 - 83413545257*y^6 + 289196448*y^5 + 9759539347*y^4 + 438024589*y^3 - 453637806*y^2 - 49033617*y - 368597, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^44 - x^43 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^44 - x^43 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597)

$ x^{44} - x^{43} - 86 x^{42} + 81 x^{41} + 3445 x^{40} - 3040 x^{39} - 85351 x^{38} + 70151 x^{37} + \cdots - 368597 $ x^44 - x^43 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597

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:	$[44, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$191\!\cdots\!489$ 191158492466532603992882058813386761401107535956984758058028173625115327713733756218489 $\medspace = 3^{22}\cdot 7^{22}\cdot 23^{42}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$91.40$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$3^{1/2}7^{1/2}23^{21/22}\approx 91.39885677323893$
Ramified primes:	$3$, $7$, $23$ 3, 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:	$483=3\cdot 7\cdot 23$
Dirichlet character group:	$\lbrace$$\chi_{483}(1,·)$, $\chi_{483}(260,·)$, $\chi_{483}(134,·)$, $\chi_{483}(400,·)$, $\chi_{483}(146,·)$, $\chi_{483}(281,·)$, $\chi_{483}(155,·)$, $\chi_{483}(412,·)$, $\chi_{483}(286,·)$, $\chi_{483}(160,·)$, $\chi_{483}(34,·)$, $\chi_{483}(167,·)$, $\chi_{483}(41,·)$, $\chi_{483}(428,·)$, $\chi_{483}(176,·)$, $\chi_{483}(433,·)$, $\chi_{483}(181,·)$, $\chi_{483}(440,·)$, $\chi_{483}(188,·)$, $\chi_{483}(190,·)$, $\chi_{483}(64,·)$, $\chi_{483}(454,·)$, $\chi_{483}(113,·)$, $\chi_{483}(76,·)$, $\chi_{483}(461,·)$, $\chi_{483}(463,·)$, $\chi_{483}(209,·)$, $\chi_{483}(335,·)$, $\chi_{483}(211,·)$, $\chi_{483}(85,·)$, $\chi_{483}(470,·)$, $\chi_{483}(344,·)$, $\chi_{483}(218,·)$, $\chi_{483}(475,·)$, $\chi_{483}(97,·)$, $\chi_{483}(358,·)$, $\chi_{483}(104,·)$, $\chi_{483}(365,·)$, $\chi_{483}(232,·)$, $\chi_{483}(244,·)$, $\chi_{483}(62,·)$, $\chi_{483}(169,·)$, $\chi_{483}(377,·)$, $\chi_{483}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{967}a^{23}-\frac{46}{967}a^{21}-\frac{47}{967}a^{19}+\frac{149}{967}a^{17}-\frac{354}{967}a^{15}-\frac{282}{967}a^{13}+\frac{180}{967}a^{11}+\frac{199}{967}a^{9}+\frac{148}{967}a^{7}-\frac{421}{967}a^{5}-\frac{168}{967}a^{3}+\frac{279}{967}a-\frac{181}{967}$, $\frac{1}{967}a^{24}-\frac{46}{967}a^{22}-\frac{47}{967}a^{20}+\frac{149}{967}a^{18}-\frac{354}{967}a^{16}-\frac{282}{967}a^{14}+\frac{180}{967}a^{12}+\frac{199}{967}a^{10}+\frac{148}{967}a^{8}-\frac{421}{967}a^{6}-\frac{168}{967}a^{4}+\frac{279}{967}a^{2}-\frac{181}{967}a$, $\frac{1}{967}a^{25}-\frac{229}{967}a^{21}-\frac{79}{967}a^{19}-\frac{269}{967}a^{17}-\frac{127}{967}a^{15}-\frac{221}{967}a^{13}-\frac{224}{967}a^{11}-\frac{368}{967}a^{9}-\frac{382}{967}a^{7}-\frac{194}{967}a^{5}+\frac{287}{967}a^{3}-\frac{181}{967}a^{2}+\frac{263}{967}a+\frac{377}{967}$, $\frac{1}{967}a^{26}-\frac{229}{967}a^{22}-\frac{79}{967}a^{20}-\frac{269}{967}a^{18}-\frac{127}{967}a^{16}-\frac{221}{967}a^{14}-\frac{224}{967}a^{12}-\frac{368}{967}a^{10}-\frac{382}{967}a^{8}-\frac{194}{967}a^{6}+\frac{287}{967}a^{4}-\frac{181}{967}a^{3}+\frac{263}{967}a^{2}+\frac{377}{967}a$, $\frac{1}{967}a^{27}+\frac{24}{967}a^{21}-\frac{395}{967}a^{19}+\frac{149}{967}a^{17}-\frac{59}{967}a^{15}-\frac{13}{967}a^{13}+\frac{238}{967}a^{11}-\frac{260}{967}a^{9}-\frac{147}{967}a^{7}-\frac{389}{967}a^{5}-\frac{181}{967}a^{4}+\frac{471}{967}a^{3}+\frac{377}{967}a^{2}+\frac{69}{967}a+\frac{132}{967}$, $\frac{1}{967}a^{28}+\frac{24}{967}a^{22}-\frac{395}{967}a^{20}+\frac{149}{967}a^{18}-\frac{59}{967}a^{16}-\frac{13}{967}a^{14}+\frac{238}{967}a^{12}-\frac{260}{967}a^{10}-\frac{147}{967}a^{8}-\frac{389}{967}a^{6}-\frac{181}{967}a^{5}+\frac{471}{967}a^{4}+\frac{377}{967}a^{3}+\frac{69}{967}a^{2}+\frac{132}{967}a$, $\frac{1}{967}a^{29}-\frac{258}{967}a^{21}+\frac{310}{967}a^{19}+\frac{233}{967}a^{17}-\frac{220}{967}a^{15}+\frac{237}{967}a^{13}+\frac{255}{967}a^{11}-\frac{88}{967}a^{9}-\frac{73}{967}a^{7}-\frac{181}{967}a^{6}-\frac{62}{967}a^{5}+\frac{377}{967}a^{4}+\frac{233}{967}a^{3}+\frac{132}{967}a^{2}+\frac{73}{967}a+\frac{476}{967}$, $\frac{1}{967}a^{30}-\frac{258}{967}a^{22}+\frac{310}{967}a^{20}+\frac{233}{967}a^{18}-\frac{220}{967}a^{16}+\frac{237}{967}a^{14}+\frac{255}{967}a^{12}-\frac{88}{967}a^{10}-\frac{73}{967}a^{8}-\frac{181}{967}a^{7}-\frac{62}{967}a^{6}+\frac{377}{967}a^{5}+\frac{233}{967}a^{4}+\frac{132}{967}a^{3}+\frac{73}{967}a^{2}+\frac{476}{967}a$, $\frac{1}{967}a^{31}+\frac{46}{967}a^{21}-\frac{289}{967}a^{19}-\frac{458}{967}a^{17}-\frac{197}{967}a^{15}+\frac{24}{967}a^{13}-\frac{64}{967}a^{11}+\frac{18}{967}a^{9}-\frac{181}{967}a^{8}+\frac{409}{967}a^{7}+\frac{377}{967}a^{6}-\frac{81}{967}a^{5}+\frac{132}{967}a^{4}+\frac{244}{967}a^{3}+\frac{476}{967}a^{2}+\frac{424}{967}a-\frac{282}{967}$, $\frac{1}{967}a^{32}+\frac{46}{967}a^{22}-\frac{289}{967}a^{20}-\frac{458}{967}a^{18}-\frac{197}{967}a^{16}+\frac{24}{967}a^{14}-\frac{64}{967}a^{12}+\frac{18}{967}a^{10}-\frac{181}{967}a^{9}+\frac{409}{967}a^{8}+\frac{377}{967}a^{7}-\frac{81}{967}a^{6}+\frac{132}{967}a^{5}+\frac{244}{967}a^{4}+\frac{476}{967}a^{3}+\frac{424}{967}a^{2}-\frac{282}{967}a$, $\frac{1}{967}a^{33}-\frac{107}{967}a^{21}-\frac{230}{967}a^{19}-\frac{282}{967}a^{17}-\frac{131}{967}a^{15}+\frac{337}{967}a^{13}+\frac{441}{967}a^{11}-\frac{181}{967}a^{10}-\frac{42}{967}a^{9}+\frac{377}{967}a^{8}-\frac{120}{967}a^{7}+\frac{132}{967}a^{6}+\frac{270}{967}a^{5}+\frac{476}{967}a^{4}+\frac{416}{967}a^{3}-\frac{282}{967}a^{2}-\frac{263}{967}a-\frac{377}{967}$, $\frac{1}{18005734367}a^{34}-\frac{591175}{18005734367}a^{33}+\frac{8219207}{18005734367}a^{32}-\frac{4614865}{18005734367}a^{31}-\frac{7492431}{18005734367}a^{30}+\frac{8807653}{18005734367}a^{29}+\frac{1722207}{18005734367}a^{28}+\frac{6507841}{18005734367}a^{27}-\frac{2886529}{18005734367}a^{26}+\frac{5395227}{18005734367}a^{25}-\frac{7252309}{18005734367}a^{24}+\frac{6524189}{18005734367}a^{23}-\frac{8116723272}{18005734367}a^{22}-\frac{7617802541}{18005734367}a^{21}+\frac{2967351794}{18005734367}a^{20}+\frac{6954293200}{18005734367}a^{19}+\frac{8986155566}{18005734367}a^{18}-\frac{1318943358}{18005734367}a^{17}-\frac{5376394790}{18005734367}a^{16}-\frac{6956511157}{18005734367}a^{15}+\frac{526187471}{18005734367}a^{14}-\frac{7729633096}{18005734367}a^{13}-\frac{8236571390}{18005734367}a^{12}-\frac{6473597624}{18005734367}a^{11}-\frac{8830373379}{18005734367}a^{10}-\frac{1822289900}{18005734367}a^{9}+\frac{4660704427}{18005734367}a^{8}-\frac{3095868076}{18005734367}a^{7}+\frac{6437991265}{18005734367}a^{6}-\frac{3128984668}{18005734367}a^{5}-\frac{5591973051}{18005734367}a^{4}+\frac{5300710589}{18005734367}a^{3}-\frac{8347844180}{18005734367}a^{2}-\frac{7999597184}{18005734367}a+\frac{5880106311}{18005734367}$, $\frac{1}{18005734367}a^{35}+\frac{2891151}{18005734367}a^{33}+\frac{6392008}{18005734367}a^{32}-\frac{5698688}{18005734367}a^{31}+\frac{7084706}{18005734367}a^{30}+\frac{6077847}{18005734367}a^{29}-\frac{1739413}{18005734367}a^{28}+\frac{1326428}{18005734367}a^{27}-\frac{8685904}{18005734367}a^{26}+\frac{5979523}{18005734367}a^{25}-\frac{1487832}{18005734367}a^{24}+\frac{6541902}{18005734367}a^{23}+\frac{4188803716}{18005734367}a^{22}+\frac{4295754719}{18005734367}a^{21}-\frac{3643735828}{18005734367}a^{20}-\frac{6229789542}{18005734367}a^{19}+\frac{5986636351}{18005734367}a^{18}+\frac{4562480385}{18005734367}a^{17}+\frac{984673024}{18005734367}a^{16}+\frac{7974093305}{18005734367}a^{15}+\frac{3838596361}{18005734367}a^{14}+\frac{2090331415}{18005734367}a^{13}+\frac{411576391}{18005734367}a^{12}+\frac{8836545999}{18005734367}a^{11}-\frac{6234614076}{18005734367}a^{10}+\frac{7693860991}{18005734367}a^{9}+\frac{1431546227}{18005734367}a^{8}-\frac{3071692058}{18005734367}a^{7}+\frac{6807196682}{18005734367}a^{6}-\frac{8559645400}{18005734367}a^{5}+\frac{8677627270}{18005734367}a^{4}+\frac{2949437404}{18005734367}a^{3}+\frac{5566756153}{18005734367}a^{2}+\frac{244208981}{18005734367}a+\frac{1380731697}{18005734367}$, $\frac{1}{18005734367}a^{36}-\frac{2905759}{18005734367}a^{33}-\frac{4061152}{18005734367}a^{32}-\frac{7762028}{18005734367}a^{31}+\frac{483181}{18005734367}a^{30}+\frac{4181745}{18005734367}a^{29}-\frac{5695223}{18005734367}a^{28}-\frac{5196425}{18005734367}a^{27}+\frac{9298212}{18005734367}a^{26}+\frac{4256606}{18005734367}a^{25}+\frac{7560898}{18005734367}a^{24}-\frac{3718440}{18005734367}a^{23}+\frac{320156472}{18005734367}a^{22}-\frac{5478291523}{18005734367}a^{21}+\frac{7168819784}{18005734367}a^{20}-\frac{1708076999}{18005734367}a^{19}+\frac{8798946849}{18005734367}a^{18}+\frac{429821056}{18005734367}a^{17}+\frac{3131789313}{18005734367}a^{16}+\frac{8764057688}{18005734367}a^{15}+\frac{8223972137}{18005734367}a^{14}-\frac{5117701182}{18005734367}a^{13}+\frac{1608094027}{18005734367}a^{12}-\frac{432570598}{18005734367}a^{11}+\frac{6073271067}{18005734367}a^{10}+\frac{3909560052}{18005734367}a^{9}+\frac{8961572383}{18005734367}a^{8}-\frac{7474250380}{18005734367}a^{7}+\frac{2076432229}{18005734367}a^{6}-\frac{5795486410}{18005734367}a^{5}+\frac{5936703895}{18005734367}a^{4}-\frac{1192062182}{18005734367}a^{3}-\frac{6637331363}{18005734367}a^{2}-\frac{4825927838}{18005734367}a-\frac{8732205726}{18005734367}$, $\frac{1}{18005734367}a^{37}+\frac{9125479}{18005734367}a^{33}-\frac{6279756}{18005734367}a^{32}+\frac{6509615}{18005734367}a^{31}-\frac{8734360}{18005734367}a^{30}+\frac{3609934}{18005734367}a^{29}+\frac{3933531}{18005734367}a^{28}-\frac{2396245}{18005734367}a^{27}+\frac{9277550}{18005734367}a^{26}-\frac{4018357}{18005734367}a^{25}+\frac{2476382}{18005734367}a^{24}-\frac{4866021}{18005734367}a^{23}-\frac{8756667875}{18005734367}a^{22}+\frac{2613994159}{18005734367}a^{21}-\frac{1224129587}{18005734367}a^{20}-\frac{2313735423}{18005734367}a^{19}-\frac{4266003562}{18005734367}a^{18}+\frac{5286048246}{18005734367}a^{17}+\frac{7251021131}{18005734367}a^{16}-\frac{1225971537}{18005734367}a^{15}-\frac{3694278137}{18005734367}a^{14}+\frac{1068500791}{18005734367}a^{13}-\frac{7734120736}{18005734367}a^{12}+\frac{837459922}{18005734367}a^{11}-\frac{3880875919}{18005734367}a^{10}+\frac{6321693380}{18005734367}a^{9}+\frac{599308239}{18005734367}a^{8}+\frac{5824256901}{18005734367}a^{7}+\frac{3595148396}{18005734367}a^{6}+\frac{1482750779}{18005734367}a^{5}+\frac{7873587205}{18005734367}a^{4}+\frac{7201261728}{18005734367}a^{3}+\frac{7519856446}{18005734367}a^{2}-\frac{3015523811}{18005734367}a-\frac{5936071860}{18005734367}$, $\frac{1}{18005734367}a^{38}-\frac{7586857}{18005734367}a^{33}+\frac{4864371}{18005734367}a^{32}+\frac{1814300}{18005734367}a^{31}+\frac{6562056}{18005734367}a^{30}-\frac{6801359}{18005734367}a^{29}+\frac{4801230}{18005734367}a^{28}+\frac{244905}{18005734367}a^{27}+\frac{130590}{18005734367}a^{26}-\frac{964432}{18005734367}a^{25}-\frac{4928657}{18005734367}a^{24}+\frac{6133871}{18005734367}a^{23}+\frac{2551964881}{18005734367}a^{22}+\frac{756213003}{18005734367}a^{21}-\frac{8965561854}{18005734367}a^{20}+\frac{7397365223}{18005734367}a^{19}+\frac{1908001736}{18005734367}a^{18}-\frac{7007048352}{18005734367}a^{17}+\frac{6322353875}{18005734367}a^{16}+\frac{8442571972}{18005734367}a^{15}-\frac{1991824458}{18005734367}a^{14}+\frac{7844400165}{18005734367}a^{13}-\frac{7702268240}{18005734367}a^{12}-\frac{2312474263}{18005734367}a^{11}+\frac{8868333767}{18005734367}a^{10}-\frac{6687850982}{18005734367}a^{9}-\frac{7855350190}{18005734367}a^{8}-\frac{4503376574}{18005734367}a^{7}-\frac{8665280315}{18005734367}a^{6}-\frac{6424255271}{18005734367}a^{5}-\frac{3377377461}{18005734367}a^{4}-\frac{7277758159}{18005734367}a^{3}-\frac{6432875800}{18005734367}a^{2}-\frac{5736567751}{18005734367}a+\frac{3250209210}{18005734367}$, $\frac{1}{18005734367}a^{39}+\frac{4213472}{18005734367}a^{33}-\frac{4580442}{18005734367}a^{32}-\frac{5518909}{18005734367}a^{31}+\frac{914888}{18005734367}a^{30}+\frac{188146}{18005734367}a^{29}+\frac{8272986}{18005734367}a^{28}+\frac{8016888}{18005734367}a^{27}+\frac{187340}{18005734367}a^{26}+\frac{4324381}{18005734367}a^{25}+\frac{3281033}{18005734367}a^{24}-\frac{8957184}{18005734367}a^{23}-\frac{6744250151}{18005734367}a^{22}+\frac{7965373416}{18005734367}a^{21}+\frac{546580484}{18005734367}a^{20}-\frac{2922409241}{18005734367}a^{19}-\frac{3513942863}{18005734367}a^{18}-\frac{5095278699}{18005734367}a^{17}+\frac{4174270369}{18005734367}a^{16}-\frac{368972889}{18005734367}a^{15}+\frac{7145370504}{18005734367}a^{14}+\frac{7020692573}{18005734367}a^{13}-\frac{8468200194}{18005734367}a^{12}+\frac{4384435815}{18005734367}a^{11}-\frac{2277353109}{18005734367}a^{10}-\frac{4807282016}{18005734367}a^{9}+\frac{3921419956}{18005734367}a^{8}+\frac{7824816414}{18005734367}a^{7}+\frac{5111926715}{18005734367}a^{6}-\frac{8825424399}{18005734367}a^{5}-\frac{3799691411}{18005734367}a^{4}+\frac{3649891214}{18005734367}a^{3}-\frac{5608901741}{18005734367}a^{2}-\frac{3068694642}{18005734367}a+\frac{7934826520}{18005734367}$, $\frac{1}{18005734367}a^{40}-\frac{4039416}{18005734367}a^{33}-\frac{8779130}{18005734367}a^{32}-\frac{5023107}{18005734367}a^{31}-\frac{5861847}{18005734367}a^{30}-\frac{4247391}{18005734367}a^{29}-\frac{3044307}{18005734367}a^{28}+\frac{5091415}{18005734367}a^{27}+\frac{7794291}{18005734367}a^{26}+\frac{5355548}{18005734367}a^{25}+\frac{3530976}{18005734367}a^{24}-\frac{4550071}{18005734367}a^{23}+\frac{7997192356}{18005734367}a^{22}+\frac{3172132204}{18005734367}a^{21}+\frac{1518571401}{18005734367}a^{20}-\frac{5809972218}{18005734367}a^{19}-\frac{2000573561}{18005734367}a^{18}-\frac{7895674988}{18005734367}a^{17}+\frac{8571208629}{18005734367}a^{16}+\frac{5277511649}{18005734367}a^{15}+\frac{7715304425}{18005734367}a^{14}+\frac{3170224994}{18005734367}a^{13}-\frac{1147425478}{18005734367}a^{12}-\frac{5710452540}{18005734367}a^{11}-\frac{7969985851}{18005734367}a^{10}+\frac{645830128}{18005734367}a^{9}-\frac{7711633309}{18005734367}a^{8}+\frac{1259481103}{18005734367}a^{7}-\frac{391016636}{18005734367}a^{6}+\frac{5394939914}{18005734367}a^{5}+\frac{4744793178}{18005734367}a^{4}+\frac{8124331432}{18005734367}a^{3}+\frac{5293649129}{18005734367}a^{2}-\frac{909048877}{18005734367}a-\frac{2955114152}{18005734367}$, $\frac{1}{18005734367}a^{41}-\frac{6995082}{18005734367}a^{33}+\frac{4606553}{18005734367}a^{32}+\frac{6787850}{18005734367}a^{31}+\frac{1355301}{18005734367}a^{30}+\frac{4394033}{18005734367}a^{29}+\frac{3686716}{18005734367}a^{28}-\frac{3195447}{18005734367}a^{27}-\frac{9306321}{18005734367}a^{26}+\frac{2422782}{18005734367}a^{25}+\frac{8811082}{18005734367}a^{24}+\frac{7276411}{18005734367}a^{23}+\frac{7778147674}{18005734367}a^{22}-\frac{5926347193}{18005734367}a^{21}+\frac{4292606763}{18005734367}a^{20}+\frac{2068227716}{18005734367}a^{19}-\frac{57725774}{18005734367}a^{18}+\frac{6845099410}{18005734367}a^{17}+\frac{1407754133}{18005734367}a^{16}+\frac{7638905808}{18005734367}a^{15}+\frac{1356015547}{18005734367}a^{14}-\frac{1644701161}{18005734367}a^{13}+\frac{4885873290}{18005734367}a^{12}+\frac{5371344226}{18005734367}a^{11}+\frac{6325121466}{18005734367}a^{10}+\frac{8935538223}{18005734367}a^{9}+\frac{7587603073}{18005734367}a^{8}+\frac{4117245630}{18005734367}a^{7}+\frac{3465280453}{18005734367}a^{6}+\frac{8064681332}{18005734367}a^{5}+\frac{3316668054}{18005734367}a^{4}+\frac{6748597825}{18005734367}a^{3}-\frac{4982589257}{18005734367}a^{2}-\frac{4000887394}{18005734367}a-\frac{3544487360}{18005734367}$, $\frac{1}{18005734367}a^{42}-\frac{8415310}{18005734367}a^{33}-\frac{7784700}{18005734367}a^{32}+\frac{609845}{18005734367}a^{31}+\frac{3203185}{18005734367}a^{30}-\frac{1736724}{18005734367}a^{29}+\frac{3866743}{18005734367}a^{28}-\frac{95370}{18005734367}a^{27}-\frac{725809}{18005734367}a^{26}+\frac{7611062}{18005734367}a^{25}+\frac{6693960}{18005734367}a^{24}-\frac{5559399}{18005734367}a^{23}-\frac{8171795}{18005734367}a^{22}-\frac{960080681}{18005734367}a^{21}+\frac{4508740552}{18005734367}a^{20}+\frac{1545375374}{18005734367}a^{19}+\frac{8479026244}{18005734367}a^{18}-\frac{8061593078}{18005734367}a^{17}-\frac{6750371537}{18005734367}a^{16}+\frac{2869758516}{18005734367}a^{15}-\frac{1033755832}{18005734367}a^{14}-\frac{4934748138}{18005734367}a^{13}+\frac{7889459308}{18005734367}a^{12}-\frac{1577976098}{18005734367}a^{11}-\frac{4200062249}{18005734367}a^{10}+\frac{8138237792}{18005734367}a^{9}+\frac{6372420551}{18005734367}a^{8}+\frac{3813257769}{18005734367}a^{7}+\frac{6204503434}{18005734367}a^{6}-\frac{5075895478}{18005734367}a^{5}+\frac{3377582715}{18005734367}a^{4}+\frac{5270018394}{18005734367}a^{3}+\frac{457218847}{18005734367}a^{2}+\frac{8686783840}{18005734367}a+\frac{3126775714}{18005734367}$, $\frac{1}{18005734367}a^{43}-\frac{1990971}{18005734367}a^{33}-\frac{391816}{18005734367}a^{32}+\frac{3758901}{18005734367}a^{31}-\frac{2475566}{18005734367}a^{30}+\frac{9018397}{18005734367}a^{29}+\frac{2586857}{18005734367}a^{28}+\frac{5601314}{18005734367}a^{27}-\frac{8292976}{18005734367}a^{26}+\frac{3171583}{18005734367}a^{25}-\frac{6683343}{18005734367}a^{24}+\frac{5598220}{18005734367}a^{23}+\frac{3903195380}{18005734367}a^{22}-\frac{643848132}{18005734367}a^{21}+\frac{2418825435}{18005734367}a^{20}-\frac{2777454018}{18005734367}a^{19}+\frac{4620699613}{18005734367}a^{18}+\frac{2209981651}{18005734367}a^{17}-\frac{4324821486}{18005734367}a^{16}+\frac{5382131895}{18005734367}a^{15}-\frac{8981316434}{18005734367}a^{14}-\frac{7238956882}{18005734367}a^{13}+\frac{2416364515}{18005734367}a^{12}+\frac{3169087318}{18005734367}a^{11}+\frac{6027935631}{18005734367}a^{10}+\frac{4960877003}{18005734367}a^{9}-\frac{4844876804}{18005734367}a^{8}-\frac{1326830891}{18005734367}a^{7}+\frac{3174503826}{18005734367}a^{6}-\frac{8752341093}{18005734367}a^{5}-\frac{8365286509}{18005734367}a^{4}+\frac{8741680307}{18005734367}a^{3}+\frac{2057324183}{18005734367}a^{2}+\frac{8312953902}{18005734367}a+\frac{445165358}{18005734367}$ 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, 1/967*a^23 - 46/967*a^21 - 47/967*a^19 + 149/967*a^17 - 354/967*a^15 - 282/967*a^13 + 180/967*a^11 + 199/967*a^9 + 148/967*a^7 - 421/967*a^5 - 168/967*a^3 + 279/967*a - 181/967, 1/967*a^24 - 46/967*a^22 - 47/967*a^20 + 149/967*a^18 - 354/967*a^16 - 282/967*a^14 + 180/967*a^12 + 199/967*a^10 + 148/967*a^8 - 421/967*a^6 - 168/967*a^4 + 279/967*a^2 - 181/967*a, 1/967*a^25 - 229/967*a^21 - 79/967*a^19 - 269/967*a^17 - 127/967*a^15 - 221/967*a^13 - 224/967*a^11 - 368/967*a^9 - 382/967*a^7 - 194/967*a^5 + 287/967*a^3 - 181/967*a^2 + 263/967*a + 377/967, 1/967*a^26 - 229/967*a^22 - 79/967*a^20 - 269/967*a^18 - 127/967*a^16 - 221/967*a^14 - 224/967*a^12 - 368/967*a^10 - 382/967*a^8 - 194/967*a^6 + 287/967*a^4 - 181/967*a^3 + 263/967*a^2 + 377/967*a, 1/967*a^27 + 24/967*a^21 - 395/967*a^19 + 149/967*a^17 - 59/967*a^15 - 13/967*a^13 + 238/967*a^11 - 260/967*a^9 - 147/967*a^7 - 389/967*a^5 - 181/967*a^4 + 471/967*a^3 + 377/967*a^2 + 69/967*a + 132/967, 1/967*a^28 + 24/967*a^22 - 395/967*a^20 + 149/967*a^18 - 59/967*a^16 - 13/967*a^14 + 238/967*a^12 - 260/967*a^10 - 147/967*a^8 - 389/967*a^6 - 181/967*a^5 + 471/967*a^4 + 377/967*a^3 + 69/967*a^2 + 132/967*a, 1/967*a^29 - 258/967*a^21 + 310/967*a^19 + 233/967*a^17 - 220/967*a^15 + 237/967*a^13 + 255/967*a^11 - 88/967*a^9 - 73/967*a^7 - 181/967*a^6 - 62/967*a^5 + 377/967*a^4 + 233/967*a^3 + 132/967*a^2 + 73/967*a + 476/967, 1/967*a^30 - 258/967*a^22 + 310/967*a^20 + 233/967*a^18 - 220/967*a^16 + 237/967*a^14 + 255/967*a^12 - 88/967*a^10 - 73/967*a^8 - 181/967*a^7 - 62/967*a^6 + 377/967*a^5 + 233/967*a^4 + 132/967*a^3 + 73/967*a^2 + 476/967*a, 1/967*a^31 + 46/967*a^21 - 289/967*a^19 - 458/967*a^17 - 197/967*a^15 + 24/967*a^13 - 64/967*a^11 + 18/967*a^9 - 181/967*a^8 + 409/967*a^7 + 377/967*a^6 - 81/967*a^5 + 132/967*a^4 + 244/967*a^3 + 476/967*a^2 + 424/967*a - 282/967, 1/967*a^32 + 46/967*a^22 - 289/967*a^20 - 458/967*a^18 - 197/967*a^16 + 24/967*a^14 - 64/967*a^12 + 18/967*a^10 - 181/967*a^9 + 409/967*a^8 + 377/967*a^7 - 81/967*a^6 + 132/967*a^5 + 244/967*a^4 + 476/967*a^3 + 424/967*a^2 - 282/967*a, 1/967*a^33 - 107/967*a^21 - 230/967*a^19 - 282/967*a^17 - 131/967*a^15 + 337/967*a^13 + 441/967*a^11 - 181/967*a^10 - 42/967*a^9 + 377/967*a^8 - 120/967*a^7 + 132/967*a^6 + 270/967*a^5 + 476/967*a^4 + 416/967*a^3 - 282/967*a^2 - 263/967*a - 377/967, 1/18005734367*a^34 - 591175/18005734367*a^33 + 8219207/18005734367*a^32 - 4614865/18005734367*a^31 - 7492431/18005734367*a^30 + 8807653/18005734367*a^29 + 1722207/18005734367*a^28 + 6507841/18005734367*a^27 - 2886529/18005734367*a^26 + 5395227/18005734367*a^25 - 7252309/18005734367*a^24 + 6524189/18005734367*a^23 - 8116723272/18005734367*a^22 - 7617802541/18005734367*a^21 + 2967351794/18005734367*a^20 + 6954293200/18005734367*a^19 + 8986155566/18005734367*a^18 - 1318943358/18005734367*a^17 - 5376394790/18005734367*a^16 - 6956511157/18005734367*a^15 + 526187471/18005734367*a^14 - 7729633096/18005734367*a^13 - 8236571390/18005734367*a^12 - 6473597624/18005734367*a^11 - 8830373379/18005734367*a^10 - 1822289900/18005734367*a^9 + 4660704427/18005734367*a^8 - 3095868076/18005734367*a^7 + 6437991265/18005734367*a^6 - 3128984668/18005734367*a^5 - 5591973051/18005734367*a^4 + 5300710589/18005734367*a^3 - 8347844180/18005734367*a^2 - 7999597184/18005734367*a + 5880106311/18005734367, 1/18005734367*a^35 + 2891151/18005734367*a^33 + 6392008/18005734367*a^32 - 5698688/18005734367*a^31 + 7084706/18005734367*a^30 + 6077847/18005734367*a^29 - 1739413/18005734367*a^28 + 1326428/18005734367*a^27 - 8685904/18005734367*a^26 + 5979523/18005734367*a^25 - 1487832/18005734367*a^24 + 6541902/18005734367*a^23 + 4188803716/18005734367*a^22 + 4295754719/18005734367*a^21 - 3643735828/18005734367*a^20 - 6229789542/18005734367*a^19 + 5986636351/18005734367*a^18 + 4562480385/18005734367*a^17 + 984673024/18005734367*a^16 + 7974093305/18005734367*a^15 + 3838596361/18005734367*a^14 + 2090331415/18005734367*a^13 + 411576391/18005734367*a^12 + 8836545999/18005734367*a^11 - 6234614076/18005734367*a^10 + 7693860991/18005734367*a^9 + 1431546227/18005734367*a^8 - 3071692058/18005734367*a^7 + 6807196682/18005734367*a^6 - 8559645400/18005734367*a^5 + 8677627270/18005734367*a^4 + 2949437404/18005734367*a^3 + 5566756153/18005734367*a^2 + 244208981/18005734367*a + 1380731697/18005734367, 1/18005734367*a^36 - 2905759/18005734367*a^33 - 4061152/18005734367*a^32 - 7762028/18005734367*a^31 + 483181/18005734367*a^30 + 4181745/18005734367*a^29 - 5695223/18005734367*a^28 - 5196425/18005734367*a^27 + 9298212/18005734367*a^26 + 4256606/18005734367*a^25 + 7560898/18005734367*a^24 - 3718440/18005734367*a^23 + 320156472/18005734367*a^22 - 5478291523/18005734367*a^21 + 7168819784/18005734367*a^20 - 1708076999/18005734367*a^19 + 8798946849/18005734367*a^18 + 429821056/18005734367*a^17 + 3131789313/18005734367*a^16 + 8764057688/18005734367*a^15 + 8223972137/18005734367*a^14 - 5117701182/18005734367*a^13 + 1608094027/18005734367*a^12 - 432570598/18005734367*a^11 + 6073271067/18005734367*a^10 + 3909560052/18005734367*a^9 + 8961572383/18005734367*a^8 - 7474250380/18005734367*a^7 + 2076432229/18005734367*a^6 - 5795486410/18005734367*a^5 + 5936703895/18005734367*a^4 - 1192062182/18005734367*a^3 - 6637331363/18005734367*a^2 - 4825927838/18005734367*a - 8732205726/18005734367, 1/18005734367*a^37 + 9125479/18005734367*a^33 - 6279756/18005734367*a^32 + 6509615/18005734367*a^31 - 8734360/18005734367*a^30 + 3609934/18005734367*a^29 + 3933531/18005734367*a^28 - 2396245/18005734367*a^27 + 9277550/18005734367*a^26 - 4018357/18005734367*a^25 + 2476382/18005734367*a^24 - 4866021/18005734367*a^23 - 8756667875/18005734367*a^22 + 2613994159/18005734367*a^21 - 1224129587/18005734367*a^20 - 2313735423/18005734367*a^19 - 4266003562/18005734367*a^18 + 5286048246/18005734367*a^17 + 7251021131/18005734367*a^16 - 1225971537/18005734367*a^15 - 3694278137/18005734367*a^14 + 1068500791/18005734367*a^13 - 7734120736/18005734367*a^12 + 837459922/18005734367*a^11 - 3880875919/18005734367*a^10 + 6321693380/18005734367*a^9 + 599308239/18005734367*a^8 + 5824256901/18005734367*a^7 + 3595148396/18005734367*a^6 + 1482750779/18005734367*a^5 + 7873587205/18005734367*a^4 + 7201261728/18005734367*a^3 + 7519856446/18005734367*a^2 - 3015523811/18005734367*a - 5936071860/18005734367, 1/18005734367*a^38 - 7586857/18005734367*a^33 + 4864371/18005734367*a^32 + 1814300/18005734367*a^31 + 6562056/18005734367*a^30 - 6801359/18005734367*a^29 + 4801230/18005734367*a^28 + 244905/18005734367*a^27 + 130590/18005734367*a^26 - 964432/18005734367*a^25 - 4928657/18005734367*a^24 + 6133871/18005734367*a^23 + 2551964881/18005734367*a^22 + 756213003/18005734367*a^21 - 8965561854/18005734367*a^20 + 7397365223/18005734367*a^19 + 1908001736/18005734367*a^18 - 7007048352/18005734367*a^17 + 6322353875/18005734367*a^16 + 8442571972/18005734367*a^15 - 1991824458/18005734367*a^14 + 7844400165/18005734367*a^13 - 7702268240/18005734367*a^12 - 2312474263/18005734367*a^11 + 8868333767/18005734367*a^10 - 6687850982/18005734367*a^9 - 7855350190/18005734367*a^8 - 4503376574/18005734367*a^7 - 8665280315/18005734367*a^6 - 6424255271/18005734367*a^5 - 3377377461/18005734367*a^4 - 7277758159/18005734367*a^3 - 6432875800/18005734367*a^2 - 5736567751/18005734367*a + 3250209210/18005734367, 1/18005734367*a^39 + 4213472/18005734367*a^33 - 4580442/18005734367*a^32 - 5518909/18005734367*a^31 + 914888/18005734367*a^30 + 188146/18005734367*a^29 + 8272986/18005734367*a^28 + 8016888/18005734367*a^27 + 187340/18005734367*a^26 + 4324381/18005734367*a^25 + 3281033/18005734367*a^24 - 8957184/18005734367*a^23 - 6744250151/18005734367*a^22 + 7965373416/18005734367*a^21 + 546580484/18005734367*a^20 - 2922409241/18005734367*a^19 - 3513942863/18005734367*a^18 - 5095278699/18005734367*a^17 + 4174270369/18005734367*a^16 - 368972889/18005734367*a^15 + 7145370504/18005734367*a^14 + 7020692573/18005734367*a^13 - 8468200194/18005734367*a^12 + 4384435815/18005734367*a^11 - 2277353109/18005734367*a^10 - 4807282016/18005734367*a^9 + 3921419956/18005734367*a^8 + 7824816414/18005734367*a^7 + 5111926715/18005734367*a^6 - 8825424399/18005734367*a^5 - 3799691411/18005734367*a^4 + 3649891214/18005734367*a^3 - 5608901741/18005734367*a^2 - 3068694642/18005734367*a + 7934826520/18005734367, 1/18005734367*a^40 - 4039416/18005734367*a^33 - 8779130/18005734367*a^32 - 5023107/18005734367*a^31 - 5861847/18005734367*a^30 - 4247391/18005734367*a^29 - 3044307/18005734367*a^28 + 5091415/18005734367*a^27 + 7794291/18005734367*a^26 + 5355548/18005734367*a^25 + 3530976/18005734367*a^24 - 4550071/18005734367*a^23 + 7997192356/18005734367*a^22 + 3172132204/18005734367*a^21 + 1518571401/18005734367*a^20 - 5809972218/18005734367*a^19 - 2000573561/18005734367*a^18 - 7895674988/18005734367*a^17 + 8571208629/18005734367*a^16 + 5277511649/18005734367*a^15 + 7715304425/18005734367*a^14 + 3170224994/18005734367*a^13 - 1147425478/18005734367*a^12 - 5710452540/18005734367*a^11 - 7969985851/18005734367*a^10 + 645830128/18005734367*a^9 - 7711633309/18005734367*a^8 + 1259481103/18005734367*a^7 - 391016636/18005734367*a^6 + 5394939914/18005734367*a^5 + 4744793178/18005734367*a^4 + 8124331432/18005734367*a^3 + 5293649129/18005734367*a^2 - 909048877/18005734367*a - 2955114152/18005734367, 1/18005734367*a^41 - 6995082/18005734367*a^33 + 4606553/18005734367*a^32 + 6787850/18005734367*a^31 + 1355301/18005734367*a^30 + 4394033/18005734367*a^29 + 3686716/18005734367*a^28 - 3195447/18005734367*a^27 - 9306321/18005734367*a^26 + 2422782/18005734367*a^25 + 8811082/18005734367*a^24 + 7276411/18005734367*a^23 + 7778147674/18005734367*a^22 - 5926347193/18005734367*a^21 + 4292606763/18005734367*a^20 + 2068227716/18005734367*a^19 - 57725774/18005734367*a^18 + 6845099410/18005734367*a^17 + 1407754133/18005734367*a^16 + 7638905808/18005734367*a^15 + 1356015547/18005734367*a^14 - 1644701161/18005734367*a^13 + 4885873290/18005734367*a^12 + 5371344226/18005734367*a^11 + 6325121466/18005734367*a^10 + 8935538223/18005734367*a^9 + 7587603073/18005734367*a^8 + 4117245630/18005734367*a^7 + 3465280453/18005734367*a^6 + 8064681332/18005734367*a^5 + 3316668054/18005734367*a^4 + 6748597825/18005734367*a^3 - 4982589257/18005734367*a^2 - 4000887394/18005734367*a - 3544487360/18005734367, 1/18005734367*a^42 - 8415310/18005734367*a^33 - 7784700/18005734367*a^32 + 609845/18005734367*a^31 + 3203185/18005734367*a^30 - 1736724/18005734367*a^29 + 3866743/18005734367*a^28 - 95370/18005734367*a^27 - 725809/18005734367*a^26 + 7611062/18005734367*a^25 + 6693960/18005734367*a^24 - 5559399/18005734367*a^23 - 8171795/18005734367*a^22 - 960080681/18005734367*a^21 + 4508740552/18005734367*a^20 + 1545375374/18005734367*a^19 + 8479026244/18005734367*a^18 - 8061593078/18005734367*a^17 - 6750371537/18005734367*a^16 + 2869758516/18005734367*a^15 - 1033755832/18005734367*a^14 - 4934748138/18005734367*a^13 + 7889459308/18005734367*a^12 - 1577976098/18005734367*a^11 - 4200062249/18005734367*a^10 + 8138237792/18005734367*a^9 + 6372420551/18005734367*a^8 + 3813257769/18005734367*a^7 + 6204503434/18005734367*a^6 - 5075895478/18005734367*a^5 + 3377582715/18005734367*a^4 + 5270018394/18005734367*a^3 + 457218847/18005734367*a^2 + 8686783840/18005734367*a + 3126775714/18005734367, 1/18005734367*a^43 - 1990971/18005734367*a^33 - 391816/18005734367*a^32 + 3758901/18005734367*a^31 - 2475566/18005734367*a^30 + 9018397/18005734367*a^29 + 2586857/18005734367*a^28 + 5601314/18005734367*a^27 - 8292976/18005734367*a^26 + 3171583/18005734367*a^25 - 6683343/18005734367*a^24 + 5598220/18005734367*a^23 + 3903195380/18005734367*a^22 - 643848132/18005734367*a^21 + 2418825435/18005734367*a^20 - 2777454018/18005734367*a^19 + 4620699613/18005734367*a^18 + 2209981651/18005734367*a^17 - 4324821486/18005734367*a^16 + 5382131895/18005734367*a^15 - 8981316434/18005734367*a^14 - 7238956882/18005734367*a^13 + 2416364515/18005734367*a^12 + 3169087318/18005734367*a^11 + 6027935631/18005734367*a^10 + 4960877003/18005734367*a^9 - 4844876804/18005734367*a^8 - 1326830891/18005734367*a^7 + 3174503826/18005734367*a^6 - 8752341093/18005734367*a^5 - 8365286509/18005734367*a^4 + 8741680307/18005734367*a^3 + 2057324183/18005734367*a^2 + 8312953902/18005734367*a + 445165358/18005734367

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:	$43$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597)

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 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597, 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 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597);

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 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597);

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 $C_2\times C_{22}$ is not computed

Intermediate fields

$\Q(\sqrt{161}) $, $\Q(\sqrt{21}) $, $\Q(\sqrt{69}) $, $\Q(\sqrt{21}, \sqrt{69})$, $\Q(\zeta_{23})^+$, 22.22.78048218870425324004237696277333187889.1, 22.22.601130775140836298755595442714814879781421.1, $\Q(\zeta_{69})^+$

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	$22^{2}$	R	${\href{/padicField/5.11.0.1}{11} }^{4}$	R	$22^{2}$	$22^{2}$	${\href{/padicField/17.11.0.1}{11} }^{4}$	$22^{2}$	R	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/padicField/47.2.0.1}{2} }^{22}$	$22^{2}$	$22^{2}$

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

$p$	Polynomial	$e$	$f$	$c$
$3$ 3	Deg $44$	$2$	$22$	$22$
$7$ 7	Deg $44$	$2$	$22$	$22$
$23$ 23	Deg $44$	$22$	$2$	$42$

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