LMFDB - Number field 46.46.48046142766286951052967407399087703375688302701387777971505263706078295125591942766227445314201.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043)

gp: K = bnfinit(y^46 - y^45 - 93*y^44 + 93*y^43 + 4043*y^42 - 4043*y^41 - 109133*y^40 + 109133*y^39 + 2049107*y^38 - 2049107*y^37 - 28412909*y^36 + 28412909*y^35 + 301468435*y^34 - 301468435*y^33 - 2502522989*y^32 + 2502522989*y^31 + 16478342035*y^30 - 16478342035*y^29 - 86750923885*y^28 + 86750923885*y^27 + 366341854099*y^26 - 366341854099*y^25 - 1240077995117*y^24 + 1240077995117*y^23 + 3349693002643*y^22 - 3349693002643*y^21 - 7159013444717*y^20 + 7159013444717*y^19 + 11947725550483*y^18 - 11947725550483*y^17 - 15279377517677*y^16 + 15279377517677*y^15 + 14582606492563*y^14 - 14582606492563*y^13 - 10009615633517*y^12 + 10009615633517*y^11 + 4689183798163*y^10 - 4689183798163*y^9 - 1389267094637*y^8 + 1389267094637*y^7 + 231653143443*y^6 - 231653143443*y^5 - 17719200877*y^4 + 17719200877*y^3 + 416969619*y^2 - 416969619*y + 22705043, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043)

$ x^{46} - x^{45} - 93 x^{44} + 93 x^{43} + 4043 x^{42} - 4043 x^{41} - 109133 x^{40} + 109133 x^{39} + \cdots + 22705043 $ x^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$46$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[46, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$480\!\cdots\!201$ 48046142766286951052967407399087703375688302701387777971505263706078295125591942766227445314201 $\medspace = 7^{23}\cdot 47^{45}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$114.37$	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:	$7^{1/2}47^{45/46}\approx 114.36599809649049$
Ramified primes:	$7$, $47$ 7, 47	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{329}) $
$\card{ \Gal(K/\Q) }$:	$46$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$329=7\cdot 47$
Dirichlet character group:	$\lbrace$$\chi_{329}(1,·)$, $\chi_{329}(258,·)$, $\chi_{329}(132,·)$, $\chi_{329}(8,·)$, $\chi_{329}(265,·)$, $\chi_{329}(139,·)$, $\chi_{329}(13,·)$, $\chi_{329}(146,·)$, $\chi_{329}(148,·)$, $\chi_{329}(279,·)$, $\chi_{329}(260,·)$, $\chi_{329}(155,·)$, $\chi_{329}(69,·)$, $\chi_{329}(160,·)$, $\chi_{329}(162,·)$, $\chi_{329}(36,·)$, $\chi_{329}(293,·)$, $\chi_{329}(167,·)$, $\chi_{329}(41,·)$, $\chi_{329}(174,·)$, $\chi_{329}(125,·)$, $\chi_{329}(50,·)$, $\chi_{329}(181,·)$, $\chi_{329}(183,·)$, $\chi_{329}(316,·)$, $\chi_{329}(62,·)$, $\chi_{329}(309,·)$, $\chi_{329}(64,·)$, $\chi_{329}(321,·)$, $\chi_{329}(267,·)$, $\chi_{329}(197,·)$, $\chi_{329}(71,·)$, $\chi_{329}(328,·)$, $\chi_{329}(76,·)$, $\chi_{329}(204,·)$, $\chi_{329}(288,·)$, $\chi_{329}(90,·)$, $\chi_{329}(223,·)$, $\chi_{329}(225,·)$, $\chi_{329}(104,·)$, $\chi_{329}(106,·)$, $\chi_{329}(239,·)$, $\chi_{329}(190,·)$, $\chi_{329}(169,·)$, $\chi_{329}(20,·)$, $\chi_{329}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2603047}a^{24}+\frac{1049835}{2603047}a^{23}-\frac{48}{2603047}a^{22}+\frac{1165483}{2603047}a^{21}+\frac{1008}{2603047}a^{20}+\frac{117763}{2603047}a^{19}-\frac{12160}{2603047}a^{18}+\frac{219330}{2603047}a^{17}+\frac{93024}{2603047}a^{16}+\frac{759101}{2603047}a^{15}-\frac{470016}{2603047}a^{14}-\frac{71742}{2603047}a^{13}-\frac{1018919}{2603047}a^{12}+\frac{1291289}{2603047}a^{11}-\frac{911321}{2603047}a^{10}+\frac{1275144}{2603047}a^{9}-\frac{264014}{2603047}a^{8}-\frac{488954}{2603047}a^{7}+\frac{1105998}{2603047}a^{6}-\frac{541713}{2603047}a^{5}-\frac{845863}{2603047}a^{4}-\frac{834491}{2603047}a^{3}-\frac{294912}{2603047}a^{2}+\frac{1259066}{2603047}a+\frac{8192}{2603047}$, $\frac{1}{2603047}a^{25}-\frac{50}{2603047}a^{23}-\frac{503377}{2603047}a^{22}+\frac{1100}{2603047}a^{21}-\frac{1278835}{2603047}a^{20}-\frac{14000}{2603047}a^{19}+\frac{870442}{2603047}a^{18}+\frac{114000}{2603047}a^{17}-\frac{577640}{2603047}a^{16}-\frac{620160}{2603047}a^{15}+\frac{380204}{2603047}a^{14}-\frac{318247}{2603047}a^{13}-\frac{617573}{2603047}a^{12}-\frac{451506}{2603047}a^{11}+\frac{1047564}{2603047}a^{10}-\frac{1260188}{2603047}a^{9}+\frac{807223}{2603047}a^{8}+\frac{1260188}{2603047}a^{7}-\frac{807223}{2603047}a^{6}-\frac{80974}{2603047}a^{5}-\frac{720701}{2603047}a^{4}+\frac{1271847}{2603047}a^{3}+\frac{1185359}{2603047}a^{2}+\frac{102400}{2603047}a+\frac{218968}{2603047}$, $\frac{1}{2603047}a^{26}-\frac{72567}{2603047}a^{23}-\frac{1300}{2603047}a^{22}-\frac{271719}{2603047}a^{21}+\frac{36400}{2603047}a^{20}-\frac{1050549}{2603047}a^{19}-\frac{494000}{2603047}a^{18}-\frac{23328}{2603047}a^{17}-\frac{1175054}{2603047}a^{16}-\frac{710451}{2603047}a^{15}-\frac{391624}{2603047}a^{14}+\frac{1001421}{2603047}a^{13}+\frac{663484}{2603047}a^{12}+\frac{535839}{2603047}a^{11}+\frac{28608}{2603047}a^{10}-\frac{511752}{2603047}a^{9}+\frac{1074723}{2603047}a^{8}+\frac{775547}{2603047}a^{7}+\frac{554939}{2603047}a^{6}+\frac{827166}{2603047}a^{5}+\frac{627449}{2603047}a^{4}+\frac{1109561}{2603047}a^{3}+\frac{975082}{2603047}a^{2}+\frac{699140}{2603047}a+\frac{409600}{2603047}$, $\frac{1}{2603047}a^{27}-\frac{1404}{2603047}a^{23}-\frac{1151888}{2603047}a^{22}+\frac{41184}{2603047}a^{21}-\frac{788329}{2603047}a^{20}-\frac{589680}{2603047}a^{19}-\frac{5115}{2603047}a^{18}-\frac{84302}{2603047}a^{17}+\frac{61286}{2603047}a^{16}-\frac{389971}{2603047}a^{15}+\frac{1075190}{2603047}a^{14}+\frac{655770}{2603047}a^{13}+\frac{190801}{2603047}a^{12}+\frac{511565}{2603047}a^{11}+\frac{669323}{2603047}a^{10}-\frac{1268432}{2603047}a^{9}+\frac{497529}{2603047}a^{8}+\frac{763678}{2603047}a^{7}+\frac{35881}{2603047}a^{6}-\frac{1247075}{2603047}a^{5}-\frac{782500}{2603047}a^{4}-\frac{850954}{2603047}a^{3}-\frac{530577}{2603047}a^{2}+\frac{102322}{2603047}a+\frac{974148}{2603047}$, $\frac{1}{2603047}a^{28}-\frac{508150}{2603047}a^{23}-\frac{26208}{2603047}a^{22}+\frac{836287}{2603047}a^{21}+\frac{825552}{2603047}a^{20}-\frac{1260871}{2603047}a^{19}+\frac{1064387}{2603047}a^{18}+\frac{841060}{2603047}a^{17}+\frac{63375}{2603047}a^{16}-\frac{396276}{2603047}a^{15}-\frac{675803}{2603047}a^{14}+\frac{983866}{2603047}a^{13}-\frac{977908}{2603047}a^{12}-\frac{684680}{2603047}a^{11}-\frac{63992}{2603047}a^{10}-\frac{96631}{2603047}a^{9}-\frac{279304}{2603047}a^{8}+\frac{748873}{2603047}a^{7}+\frac{158105}{2603047}a^{6}-\frac{1257828}{2603047}a^{5}+\frac{1149873}{2603047}a^{4}-\frac{784791}{2603047}a^{3}-\frac{69653}{2603047}a^{2}+\frac{1233899}{2603047}a+\frac{1089380}{2603047}$, $\frac{1}{2603047}a^{29}-\frac{29232}{2603047}a^{23}-\frac{127490}{2603047}a^{22}+\frac{964656}{2603047}a^{21}+\frac{757117}{2603047}a^{20}+\frac{885354}{2603047}a^{19}-\frac{1232409}{2603047}a^{18}+\frac{542523}{2603047}a^{17}+\frac{1018851}{2603047}a^{16}-\frac{1228442}{2603047}a^{15}-\frac{275143}{2603047}a^{14}-\frac{1001973}{2603047}a^{13}-\frac{104901}{2603047}a^{12}+\frac{162739}{2603047}a^{11}-\frac{595387}{2603047}a^{10}+\frac{669821}{2603047}a^{9}+\frac{474106}{2603047}a^{8}-\frac{980845}{2603047}a^{7}+\frac{763337}{2603047}a^{6}-\frac{693874}{2603047}a^{5}-\frac{535413}{2603047}a^{4}+\frac{97185}{2603047}a^{3}-\frac{883111}{2603047}a^{2}+\frac{364291}{2603047}a+\frac{492647}{2603047}$, $\frac{1}{2603047}a^{30}-\frac{1274900}{2603047}a^{23}-\frac{438480}{2603047}a^{22}-\frac{1126010}{2603047}a^{21}-\frac{885354}{2603047}a^{20}-\frac{12527}{2603047}a^{19}-\frac{904205}{2603047}a^{18}+\frac{1168650}{2603047}a^{17}+\frac{468058}{2603047}a^{16}-\frac{1210386}{2603047}a^{15}+\frac{975428}{2603047}a^{14}+\frac{788837}{2603047}a^{13}-\frac{813695}{2603047}a^{12}-\frac{419886}{2603047}a^{11}+\frac{517347}{2603047}a^{10}-\frac{149526}{2603047}a^{9}-\frac{603738}{2603047}a^{8}+\frac{991086}{2603047}a^{7}-\frac{4078}{2603047}a^{6}+\frac{1048119}{2603047}a^{5}+\frac{173422}{2603047}a^{4}+\frac{1032461}{2603047}a^{3}+\frac{788371}{2603047}a^{2}+\frac{1028426}{2603047}a-\frac{11780}{2603047}$, $\frac{1}{2603047}a^{31}-\frac{503440}{2603047}a^{23}+\frac{151918}{2603047}a^{22}-\frac{500241}{2603047}a^{21}-\frac{818545}{2603047}a^{20}-\frac{797324}{2603047}a^{19}-\frac{470465}{2603047}a^{18}-\frac{229776}{2603047}a^{17}+\frac{265894}{2603047}a^{16}-\frac{194661}{2603047}a^{15}-\frac{1190163}{2603047}a^{14}+\frac{1175991}{2603047}a^{13}-\frac{884200}{2603047}a^{12}-\frac{975139}{2603047}a^{11}+\frac{705554}{2603047}a^{10}-\frac{461048}{2603047}a^{9}-\frac{862132}{2603047}a^{8}-\frac{175306}{2603047}a^{7}+\frac{1178030}{2603047}a^{6}+\frac{287574}{2603047}a^{5}+\frac{604921}{2603047}a^{4}-\frac{448159}{2603047}a^{3}-\frac{774741}{2603047}a^{2}+\frac{1283835}{2603047}a+\frac{556236}{2603047}$, $\frac{1}{2603047}a^{32}+\frac{1215344}{2603047}a^{23}-\frac{1237938}{2603047}a^{22}-\frac{278248}{2603047}a^{21}-\frac{923969}{2603047}a^{20}-\frac{864217}{2603047}a^{19}+\frac{306368}{2603047}a^{18}+\frac{1110401}{2603047}a^{17}+\frac{389322}{2603047}a^{16}-\frac{521934}{2603047}a^{15}+\frac{1102392}{2603047}a^{14}+\frac{1203492}{2603047}a^{13}+\frac{1297509}{2603047}a^{12}-\frac{321113}{2603047}a^{11}-\frac{1062397}{2603047}a^{10}-\frac{611818}{2603047}a^{9}-\frac{1200599}{2603047}a^{8}-\frac{684175}{2603047}a^{7}-\frac{847841}{2603047}a^{6}-\frac{756656}{2603047}a^{5}+\frac{1154039}{2603047}a^{4}-\frac{756263}{2603047}a^{3}+\frac{778294}{2603047}a^{2}-\frac{628647}{2603047}a+\frac{954032}{2603047}$, $\frac{1}{2603047}a^{33}+\frac{214389}{2603047}a^{23}+\frac{791230}{2603047}a^{22}-\frac{51789}{2603047}a^{21}+\frac{104168}{2603047}a^{20}+\frac{1084097}{2603047}a^{19}-\frac{407425}{2603047}a^{18}-\frac{1188257}{2603047}a^{17}-\frac{1144886}{2603047}a^{16}-\frac{1031706}{2603047}a^{15}-\frac{1129060}{2603047}a^{14}+\frac{844445}{2603047}a^{13}-\frac{365099}{2603047}a^{12}+\frac{17205}{2603047}a^{11}+\frac{32623}{2603047}a^{10}-\frac{160403}{2603047}a^{9}-\frac{44861}{2603047}a^{8}-\frac{534248}{2603047}a^{7}+\frac{429033}{2603047}a^{6}+\frac{944977}{2603047}a^{5}+\frac{223040}{2603047}a^{4}+\frac{442152}{2603047}a^{3}+\frac{153557}{2603047}a^{2}+\frac{1221231}{2603047}a+\frac{556727}{2603047}$, $\frac{1}{2603047}a^{34}+\frac{174270}{2603047}a^{23}-\frac{173305}{2603047}a^{22}-\frac{149189}{2603047}a^{21}+\frac{1032886}{2603047}a^{20}-\frac{546379}{2603047}a^{19}+\frac{131936}{2603047}a^{18}+\frac{959799}{2603047}a^{17}+\frac{192072}{2603047}a^{16}+\frac{1068138}{2603047}a^{15}+\frac{552252}{2603047}a^{14}-\frac{1074184}{2603047}a^{13}-\frac{58497}{2603047}a^{12}+\frac{1129746}{2603047}a^{11}+\frac{138787}{2603047}a^{10}+\frac{310157}{2603047}a^{9}+\frac{509230}{2603047}a^{8}-\frac{517598}{2603047}a^{7}+\frac{1294032}{2603047}a^{6}-\frac{13555}{2603047}a^{5}+\frac{292557}{2603047}a^{4}+\frac{1027293}{2603047}a^{3}-\frac{901631}{2603047}a^{2}-\frac{1179188}{2603047}a+\frac{782037}{2603047}$, $\frac{1}{2603047}a^{35}+\frac{239640}{2603047}a^{23}+\frac{406630}{2603047}a^{22}+\frac{258745}{2603047}a^{21}+\frac{796657}{2603047}a^{20}-\frac{3526}{2603047}a^{19}+\frac{1202741}{2603047}a^{18}+\frac{695120}{2603047}a^{17}-\frac{1050673}{2603047}a^{16}-\frac{1130478}{2603047}a^{15}+\frac{1137234}{2603047}a^{14}-\frac{14898}{2603047}a^{13}+\frac{1292771}{2603047}a^{12}+\frac{617907}{2603047}a^{11}-\frac{882737}{2603047}a^{10}+\frac{683693}{2603047}a^{9}+\frac{346457}{2603047}a^{8}+\frac{564067}{2603047}a^{7}+\frac{330100}{2603047}a^{6}-\frac{88482}{2603047}a^{5}-\frac{979307}{2603047}a^{4}-\frac{1184857}{2603047}a^{3}+\frac{1178131}{2603047}a^{2}-\frac{612059}{2603047}a-\frac{1150084}{2603047}$, $\frac{1}{2603047}a^{36}-\frac{163267}{2603047}a^{23}-\frac{1253770}{2603047}a^{22}+\frac{981449}{2603047}a^{21}+\frac{522725}{2603047}a^{20}+\frac{109948}{2603047}a^{19}-\frac{695120}{2603047}a^{18}-\frac{566849}{2603047}a^{17}-\frac{907330}{2603047}a^{16}-\frac{1092905}{2603047}a^{15}+\frac{775652}{2603047}a^{14}+\frac{420216}{2603047}a^{13}+\frac{749326}{2603047}a^{12}-\frac{357431}{2603047}a^{11}-\frac{789073}{2603047}a^{10}-\frac{871326}{2603047}a^{9}-\frac{781355}{2603047}a^{8}-\frac{290998}{2603047}a^{7}+\frac{796338}{2603047}a^{6}+\frac{1170123}{2603047}a^{5}-\frac{448474}{2603047}a^{4}-\frac{484404}{2603047}a^{3}-\frac{626429}{2603047}a^{2}+\frac{657540}{2603047}a-\frac{433442}{2603047}$, $\frac{1}{2603047}a^{37}-\frac{678634}{2603047}a^{23}+\frac{953774}{2603047}a^{22}+\frac{96939}{2603047}a^{21}+\frac{691123}{2603047}a^{20}+\frac{11459}{2603047}a^{19}+\frac{231292}{2603047}a^{18}+\frac{929248}{2603047}a^{17}+\frac{480305}{2603047}a^{16}+\frac{644855}{2603047}a^{15}+\frac{143504}{2603047}a^{14}-\frac{1243335}{2603047}a^{13}-\frac{678128}{2603047}a^{12}+\frac{712513}{2603047}a^{11}+\frac{649487}{2603047}a^{10}-\frac{941920}{2603047}a^{9}-\frac{1209463}{2603047}a^{8}+\frac{989016}{2603047}a^{7}+\frac{775199}{2603047}a^{6}-\frac{576926}{2603047}a^{5}+\frac{56713}{2603047}a^{4}+\frac{614501}{2603047}a^{3}-\frac{179605}{2603047}a^{2}+\frac{873590}{2603047}a-\frac{482894}{2603047}$, $\frac{1}{2603047}a^{38}+\frac{715264}{2603047}a^{23}-\frac{1240929}{2603047}a^{22}+\frac{1250395}{2603047}a^{21}-\frac{526830}{2603047}a^{20}-\frac{541960}{2603047}a^{19}+\frac{398798}{2603047}a^{18}+\frac{445018}{2603047}a^{17}+\frac{798227}{2603047}a^{16}+\frac{1081097}{2603047}a^{15}-\frac{511240}{2603047}a^{14}+\frac{152532}{2603047}a^{13}+\frac{1040947}{2603047}a^{12}+\frac{99210}{2603047}a^{11}+\frac{976249}{2603047}a^{10}+\frac{522200}{2603047}a^{9}-\frac{162850}{2603047}a^{8}+\frac{779641}{2603047}a^{7}-\frac{508268}{2603047}a^{6}+\frac{921434}{2603047}a^{5}+\frac{956940}{2603047}a^{4}-\frac{445673}{2603047}a^{3}-\frac{1168023}{2603047}a^{2}-\frac{458706}{2603047}a-\frac{738664}{2603047}$, $\frac{1}{2603047}a^{39}+\frac{957909}{2603047}a^{23}-\frac{859591}{2603047}a^{22}-\frac{154545}{2603047}a^{21}-\frac{484053}{2603047}a^{20}+\frac{762239}{2603047}a^{19}+\frac{1275231}{2603047}a^{18}-\frac{221344}{2603047}a^{17}+\frac{847128}{2603047}a^{16}+\frac{1032638}{2603047}a^{15}-\frac{446341}{2603047}a^{14}-\frac{957723}{2603047}a^{13}+\frac{285860}{2603047}a^{12}-\frac{1025554}{2603047}a^{11}-\frac{1182467}{2603047}a^{10}+\frac{1259182}{2603047}a^{9}-\frac{158325}{2603047}a^{8}+\frac{908950}{2603047}a^{7}-\frac{633503}{2603047}a^{6}+\frac{12128}{2603047}a^{5}-\frac{894863}{2603047}a^{4}-\frac{1077546}{2603047}a^{3}-\frac{1038630}{2603047}a^{2}+\frac{436314}{2603047}a+\frac{16109}{2603047}$, $\frac{1}{2603047}a^{40}+\frac{908139}{2603047}a^{23}-\frac{1029759}{2603047}a^{22}-\frac{1105176}{2603047}a^{21}+\frac{920404}{2603047}a^{20}+\frac{682456}{2603047}a^{19}-\frac{683229}{2603047}a^{18}-\frac{204378}{2603047}a^{17}+\frac{10726}{2603047}a^{16}+\frac{641112}{2603047}a^{15}+\frac{780560}{2603047}a^{14}-\frac{450509}{2603047}a^{13}-\frac{39162}{2603047}a^{12}+\frac{763715}{2603047}a^{11}+\frac{798957}{2603047}a^{10}-\frac{76612}{2603047}a^{9}+\frac{661344}{2603047}a^{8}-\frac{1252168}{2603047}a^{7}-\frac{90960}{2603047}a^{6}+\frac{1252945}{2603047}a^{5}+\frac{454090}{2603047}a^{4}+\frac{903553}{2603047}a^{3}+\frac{1016600}{2603047}a^{2}-\frac{870375}{2603047}a+\frac{996177}{2603047}$, $\frac{1}{2603047}a^{41}+\frac{63490}{2603047}a^{23}+\frac{836744}{2603047}a^{22}+\frac{88843}{2603047}a^{21}-\frac{1052159}{2603047}a^{20}+\frac{329709}{2603047}a^{19}+\frac{640488}{2603047}a^{18}+\frac{437249}{2603047}a^{17}+\frac{1206114}{2603047}a^{16}-\frac{902422}{2603047}a^{15}-\frac{428204}{2603047}a^{14}+\frac{5613}{2603047}a^{13}+\frac{110084}{2603047}a^{12}+\frac{968239}{2603047}a^{11}+\frac{1110968}{2603047}a^{10}-\frac{228970}{2603047}a^{9}-\frac{1295298}{2603047}a^{8}-\frac{63802}{2603047}a^{7}+\frac{35408}{2603047}a^{6}-\frac{1299380}{2603047}a^{5}+\frac{309763}{2603047}a^{4}-\frac{646449}{2603047}a^{3}+\frac{521704}{2603047}a^{2}+\frac{674082}{2603047}a+\frac{33638}{2603047}$, $\frac{1}{2603047}a^{42}+\frac{434076}{2603047}a^{23}+\frac{533316}{2603047}a^{22}-\frac{750760}{2603047}a^{21}-\frac{1195083}{2603047}a^{20}-\frac{181398}{2603047}a^{19}-\frac{629310}{2603047}a^{18}-\frac{357183}{2603047}a^{17}-\frac{682539}{2603047}a^{16}-\frac{335489}{2603047}a^{15}-\frac{9355}{2603047}a^{14}-\frac{322586}{2603047}a^{13}+\frac{1211505}{2603047}a^{12}+\frac{137623}{2603047}a^{11}-\frac{987396}{2603047}a^{10}-\frac{220064}{2603047}a^{9}+\frac{1165425}{2603047}a^{8}-\frac{213654}{2603047}a^{7}+\frac{1286519}{2603047}a^{6}-\frac{391878}{2603047}a^{5}-\frac{267236}{2603047}a^{4}-\frac{63344}{2603047}a^{3}+\frac{919891}{2603047}a^{2}-\frac{1096379}{2603047}a+\frac{499320}{2603047}$, $\frac{1}{2603047}a^{43}-\frac{14995}{2603047}a^{23}-\frac{739488}{2603047}a^{22}+\frac{599800}{2603047}a^{21}-\frac{418110}{2603047}a^{20}-\frac{84312}{2603047}a^{19}-\frac{972339}{2603047}a^{18}-\frac{127594}{2603047}a^{17}+\frac{1246798}{2603047}a^{16}-\frac{830536}{2603047}a^{15}+\frac{724864}{2603047}a^{14}-\frac{162411}{2603047}a^{13}-\frac{500397}{2603047}a^{12}+\frac{765244}{2603047}a^{11}-\frac{95211}{2603047}a^{10}+\frac{1069514}{2603047}a^{9}+\frac{180188}{2603047}a^{8}-\frac{160216}{2603047}a^{7}+\frac{187625}{2603047}a^{6}+\frac{697254}{2603047}a^{5}+\frac{1175753}{2603047}a^{4}+\frac{1223828}{2603047}a^{3}+\frac{479567}{2603047}a^{2}+\frac{708330}{2603047}a-\frac{188390}{2603047}$, $\frac{1}{2603047}a^{44}+\frac{911128}{2603047}a^{23}-\frac{119960}{2603047}a^{22}-\frac{858083}{2603047}a^{21}-\frac{587634}{2603047}a^{20}+\frac{17980}{2603047}a^{19}-\frac{253504}{2603047}a^{18}-\frac{151260}{2603047}a^{17}-\frac{1168848}{2603047}a^{16}+\frac{319828}{2603047}a^{15}+\frac{998945}{2603047}a^{14}-\frac{1213276}{2603047}a^{13}-\frac{642318}{2603047}a^{12}-\frac{1283289}{2603047}a^{11}-\frac{795178}{2603047}a^{10}-\frac{1018794}{2603047}a^{9}+\frac{184341}{2603047}a^{8}+\frac{1105794}{2603047}a^{7}+\frac{1124827}{2603047}a^{6}-\frac{304042}{2603047}a^{5}-\frac{446873}{2603047}a^{4}+\frac{133951}{2603047}a^{3}+\frac{1079743}{2603047}a^{2}-\frac{393611}{2603047}a+\frac{495831}{2603047}$, $\frac{1}{2603047}a^{45}-\frac{311891}{2603047}a^{23}+\frac{1227309}{2603047}a^{22}+\frac{432051}{2603047}a^{21}+\frac{476547}{2603047}a^{20}+\frac{177172}{2603047}a^{19}+\frac{597188}{2603047}a^{18}-\frac{351851}{2603047}a^{17}-\frac{1240924}{2603047}a^{16}+\frac{220058}{2603047}a^{15}+\frac{644520}{2603047}a^{14}+\frac{389441}{2603047}a^{13}+\frac{650028}{2603047}a^{12}+\frac{29984}{2603047}a^{11}-\frac{1282954}{2603047}a^{10}-\frac{1250581}{2603047}a^{9}-\frac{1125778}{2603047}a^{8}-\frac{276923}{2603047}a^{7}+\frac{1123136}{2603047}a^{6}+\frac{487627}{2603047}a^{5}+\frac{266031}{2603047}a^{4}-\frac{8733}{2603047}a^{3}+\frac{57503}{2603047}a^{2}+\frac{831424}{2603047}a-\frac{1024827}{2603047}$ 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, a^23, 1/2603047*a^24 + 1049835/2603047*a^23 - 48/2603047*a^22 + 1165483/2603047*a^21 + 1008/2603047*a^20 + 117763/2603047*a^19 - 12160/2603047*a^18 + 219330/2603047*a^17 + 93024/2603047*a^16 + 759101/2603047*a^15 - 470016/2603047*a^14 - 71742/2603047*a^13 - 1018919/2603047*a^12 + 1291289/2603047*a^11 - 911321/2603047*a^10 + 1275144/2603047*a^9 - 264014/2603047*a^8 - 488954/2603047*a^7 + 1105998/2603047*a^6 - 541713/2603047*a^5 - 845863/2603047*a^4 - 834491/2603047*a^3 - 294912/2603047*a^2 + 1259066/2603047*a + 8192/2603047, 1/2603047*a^25 - 50/2603047*a^23 - 503377/2603047*a^22 + 1100/2603047*a^21 - 1278835/2603047*a^20 - 14000/2603047*a^19 + 870442/2603047*a^18 + 114000/2603047*a^17 - 577640/2603047*a^16 - 620160/2603047*a^15 + 380204/2603047*a^14 - 318247/2603047*a^13 - 617573/2603047*a^12 - 451506/2603047*a^11 + 1047564/2603047*a^10 - 1260188/2603047*a^9 + 807223/2603047*a^8 + 1260188/2603047*a^7 - 807223/2603047*a^6 - 80974/2603047*a^5 - 720701/2603047*a^4 + 1271847/2603047*a^3 + 1185359/2603047*a^2 + 102400/2603047*a + 218968/2603047, 1/2603047*a^26 - 72567/2603047*a^23 - 1300/2603047*a^22 - 271719/2603047*a^21 + 36400/2603047*a^20 - 1050549/2603047*a^19 - 494000/2603047*a^18 - 23328/2603047*a^17 - 1175054/2603047*a^16 - 710451/2603047*a^15 - 391624/2603047*a^14 + 1001421/2603047*a^13 + 663484/2603047*a^12 + 535839/2603047*a^11 + 28608/2603047*a^10 - 511752/2603047*a^9 + 1074723/2603047*a^8 + 775547/2603047*a^7 + 554939/2603047*a^6 + 827166/2603047*a^5 + 627449/2603047*a^4 + 1109561/2603047*a^3 + 975082/2603047*a^2 + 699140/2603047*a + 409600/2603047, 1/2603047*a^27 - 1404/2603047*a^23 - 1151888/2603047*a^22 + 41184/2603047*a^21 - 788329/2603047*a^20 - 589680/2603047*a^19 - 5115/2603047*a^18 - 84302/2603047*a^17 + 61286/2603047*a^16 - 389971/2603047*a^15 + 1075190/2603047*a^14 + 655770/2603047*a^13 + 190801/2603047*a^12 + 511565/2603047*a^11 + 669323/2603047*a^10 - 1268432/2603047*a^9 + 497529/2603047*a^8 + 763678/2603047*a^7 + 35881/2603047*a^6 - 1247075/2603047*a^5 - 782500/2603047*a^4 - 850954/2603047*a^3 - 530577/2603047*a^2 + 102322/2603047*a + 974148/2603047, 1/2603047*a^28 - 508150/2603047*a^23 - 26208/2603047*a^22 + 836287/2603047*a^21 + 825552/2603047*a^20 - 1260871/2603047*a^19 + 1064387/2603047*a^18 + 841060/2603047*a^17 + 63375/2603047*a^16 - 396276/2603047*a^15 - 675803/2603047*a^14 + 983866/2603047*a^13 - 977908/2603047*a^12 - 684680/2603047*a^11 - 63992/2603047*a^10 - 96631/2603047*a^9 - 279304/2603047*a^8 + 748873/2603047*a^7 + 158105/2603047*a^6 - 1257828/2603047*a^5 + 1149873/2603047*a^4 - 784791/2603047*a^3 - 69653/2603047*a^2 + 1233899/2603047*a + 1089380/2603047, 1/2603047*a^29 - 29232/2603047*a^23 - 127490/2603047*a^22 + 964656/2603047*a^21 + 757117/2603047*a^20 + 885354/2603047*a^19 - 1232409/2603047*a^18 + 542523/2603047*a^17 + 1018851/2603047*a^16 - 1228442/2603047*a^15 - 275143/2603047*a^14 - 1001973/2603047*a^13 - 104901/2603047*a^12 + 162739/2603047*a^11 - 595387/2603047*a^10 + 669821/2603047*a^9 + 474106/2603047*a^8 - 980845/2603047*a^7 + 763337/2603047*a^6 - 693874/2603047*a^5 - 535413/2603047*a^4 + 97185/2603047*a^3 - 883111/2603047*a^2 + 364291/2603047*a + 492647/2603047, 1/2603047*a^30 - 1274900/2603047*a^23 - 438480/2603047*a^22 - 1126010/2603047*a^21 - 885354/2603047*a^20 - 12527/2603047*a^19 - 904205/2603047*a^18 + 1168650/2603047*a^17 + 468058/2603047*a^16 - 1210386/2603047*a^15 + 975428/2603047*a^14 + 788837/2603047*a^13 - 813695/2603047*a^12 - 419886/2603047*a^11 + 517347/2603047*a^10 - 149526/2603047*a^9 - 603738/2603047*a^8 + 991086/2603047*a^7 - 4078/2603047*a^6 + 1048119/2603047*a^5 + 173422/2603047*a^4 + 1032461/2603047*a^3 + 788371/2603047*a^2 + 1028426/2603047*a - 11780/2603047, 1/2603047*a^31 - 503440/2603047*a^23 + 151918/2603047*a^22 - 500241/2603047*a^21 - 818545/2603047*a^20 - 797324/2603047*a^19 - 470465/2603047*a^18 - 229776/2603047*a^17 + 265894/2603047*a^16 - 194661/2603047*a^15 - 1190163/2603047*a^14 + 1175991/2603047*a^13 - 884200/2603047*a^12 - 975139/2603047*a^11 + 705554/2603047*a^10 - 461048/2603047*a^9 - 862132/2603047*a^8 - 175306/2603047*a^7 + 1178030/2603047*a^6 + 287574/2603047*a^5 + 604921/2603047*a^4 - 448159/2603047*a^3 - 774741/2603047*a^2 + 1283835/2603047*a + 556236/2603047, 1/2603047*a^32 + 1215344/2603047*a^23 - 1237938/2603047*a^22 - 278248/2603047*a^21 - 923969/2603047*a^20 - 864217/2603047*a^19 + 306368/2603047*a^18 + 1110401/2603047*a^17 + 389322/2603047*a^16 - 521934/2603047*a^15 + 1102392/2603047*a^14 + 1203492/2603047*a^13 + 1297509/2603047*a^12 - 321113/2603047*a^11 - 1062397/2603047*a^10 - 611818/2603047*a^9 - 1200599/2603047*a^8 - 684175/2603047*a^7 - 847841/2603047*a^6 - 756656/2603047*a^5 + 1154039/2603047*a^4 - 756263/2603047*a^3 + 778294/2603047*a^2 - 628647/2603047*a + 954032/2603047, 1/2603047*a^33 + 214389/2603047*a^23 + 791230/2603047*a^22 - 51789/2603047*a^21 + 104168/2603047*a^20 + 1084097/2603047*a^19 - 407425/2603047*a^18 - 1188257/2603047*a^17 - 1144886/2603047*a^16 - 1031706/2603047*a^15 - 1129060/2603047*a^14 + 844445/2603047*a^13 - 365099/2603047*a^12 + 17205/2603047*a^11 + 32623/2603047*a^10 - 160403/2603047*a^9 - 44861/2603047*a^8 - 534248/2603047*a^7 + 429033/2603047*a^6 + 944977/2603047*a^5 + 223040/2603047*a^4 + 442152/2603047*a^3 + 153557/2603047*a^2 + 1221231/2603047*a + 556727/2603047, 1/2603047*a^34 + 174270/2603047*a^23 - 173305/2603047*a^22 - 149189/2603047*a^21 + 1032886/2603047*a^20 - 546379/2603047*a^19 + 131936/2603047*a^18 + 959799/2603047*a^17 + 192072/2603047*a^16 + 1068138/2603047*a^15 + 552252/2603047*a^14 - 1074184/2603047*a^13 - 58497/2603047*a^12 + 1129746/2603047*a^11 + 138787/2603047*a^10 + 310157/2603047*a^9 + 509230/2603047*a^8 - 517598/2603047*a^7 + 1294032/2603047*a^6 - 13555/2603047*a^5 + 292557/2603047*a^4 + 1027293/2603047*a^3 - 901631/2603047*a^2 - 1179188/2603047*a + 782037/2603047, 1/2603047*a^35 + 239640/2603047*a^23 + 406630/2603047*a^22 + 258745/2603047*a^21 + 796657/2603047*a^20 - 3526/2603047*a^19 + 1202741/2603047*a^18 + 695120/2603047*a^17 - 1050673/2603047*a^16 - 1130478/2603047*a^15 + 1137234/2603047*a^14 - 14898/2603047*a^13 + 1292771/2603047*a^12 + 617907/2603047*a^11 - 882737/2603047*a^10 + 683693/2603047*a^9 + 346457/2603047*a^8 + 564067/2603047*a^7 + 330100/2603047*a^6 - 88482/2603047*a^5 - 979307/2603047*a^4 - 1184857/2603047*a^3 + 1178131/2603047*a^2 - 612059/2603047*a - 1150084/2603047, 1/2603047*a^36 - 163267/2603047*a^23 - 1253770/2603047*a^22 + 981449/2603047*a^21 + 522725/2603047*a^20 + 109948/2603047*a^19 - 695120/2603047*a^18 - 566849/2603047*a^17 - 907330/2603047*a^16 - 1092905/2603047*a^15 + 775652/2603047*a^14 + 420216/2603047*a^13 + 749326/2603047*a^12 - 357431/2603047*a^11 - 789073/2603047*a^10 - 871326/2603047*a^9 - 781355/2603047*a^8 - 290998/2603047*a^7 + 796338/2603047*a^6 + 1170123/2603047*a^5 - 448474/2603047*a^4 - 484404/2603047*a^3 - 626429/2603047*a^2 + 657540/2603047*a - 433442/2603047, 1/2603047*a^37 - 678634/2603047*a^23 + 953774/2603047*a^22 + 96939/2603047*a^21 + 691123/2603047*a^20 + 11459/2603047*a^19 + 231292/2603047*a^18 + 929248/2603047*a^17 + 480305/2603047*a^16 + 644855/2603047*a^15 + 143504/2603047*a^14 - 1243335/2603047*a^13 - 678128/2603047*a^12 + 712513/2603047*a^11 + 649487/2603047*a^10 - 941920/2603047*a^9 - 1209463/2603047*a^8 + 989016/2603047*a^7 + 775199/2603047*a^6 - 576926/2603047*a^5 + 56713/2603047*a^4 + 614501/2603047*a^3 - 179605/2603047*a^2 + 873590/2603047*a - 482894/2603047, 1/2603047*a^38 + 715264/2603047*a^23 - 1240929/2603047*a^22 + 1250395/2603047*a^21 - 526830/2603047*a^20 - 541960/2603047*a^19 + 398798/2603047*a^18 + 445018/2603047*a^17 + 798227/2603047*a^16 + 1081097/2603047*a^15 - 511240/2603047*a^14 + 152532/2603047*a^13 + 1040947/2603047*a^12 + 99210/2603047*a^11 + 976249/2603047*a^10 + 522200/2603047*a^9 - 162850/2603047*a^8 + 779641/2603047*a^7 - 508268/2603047*a^6 + 921434/2603047*a^5 + 956940/2603047*a^4 - 445673/2603047*a^3 - 1168023/2603047*a^2 - 458706/2603047*a - 738664/2603047, 1/2603047*a^39 + 957909/2603047*a^23 - 859591/2603047*a^22 - 154545/2603047*a^21 - 484053/2603047*a^20 + 762239/2603047*a^19 + 1275231/2603047*a^18 - 221344/2603047*a^17 + 847128/2603047*a^16 + 1032638/2603047*a^15 - 446341/2603047*a^14 - 957723/2603047*a^13 + 285860/2603047*a^12 - 1025554/2603047*a^11 - 1182467/2603047*a^10 + 1259182/2603047*a^9 - 158325/2603047*a^8 + 908950/2603047*a^7 - 633503/2603047*a^6 + 12128/2603047*a^5 - 894863/2603047*a^4 - 1077546/2603047*a^3 - 1038630/2603047*a^2 + 436314/2603047*a + 16109/2603047, 1/2603047*a^40 + 908139/2603047*a^23 - 1029759/2603047*a^22 - 1105176/2603047*a^21 + 920404/2603047*a^20 + 682456/2603047*a^19 - 683229/2603047*a^18 - 204378/2603047*a^17 + 10726/2603047*a^16 + 641112/2603047*a^15 + 780560/2603047*a^14 - 450509/2603047*a^13 - 39162/2603047*a^12 + 763715/2603047*a^11 + 798957/2603047*a^10 - 76612/2603047*a^9 + 661344/2603047*a^8 - 1252168/2603047*a^7 - 90960/2603047*a^6 + 1252945/2603047*a^5 + 454090/2603047*a^4 + 903553/2603047*a^3 + 1016600/2603047*a^2 - 870375/2603047*a + 996177/2603047, 1/2603047*a^41 + 63490/2603047*a^23 + 836744/2603047*a^22 + 88843/2603047*a^21 - 1052159/2603047*a^20 + 329709/2603047*a^19 + 640488/2603047*a^18 + 437249/2603047*a^17 + 1206114/2603047*a^16 - 902422/2603047*a^15 - 428204/2603047*a^14 + 5613/2603047*a^13 + 110084/2603047*a^12 + 968239/2603047*a^11 + 1110968/2603047*a^10 - 228970/2603047*a^9 - 1295298/2603047*a^8 - 63802/2603047*a^7 + 35408/2603047*a^6 - 1299380/2603047*a^5 + 309763/2603047*a^4 - 646449/2603047*a^3 + 521704/2603047*a^2 + 674082/2603047*a + 33638/2603047, 1/2603047*a^42 + 434076/2603047*a^23 + 533316/2603047*a^22 - 750760/2603047*a^21 - 1195083/2603047*a^20 - 181398/2603047*a^19 - 629310/2603047*a^18 - 357183/2603047*a^17 - 682539/2603047*a^16 - 335489/2603047*a^15 - 9355/2603047*a^14 - 322586/2603047*a^13 + 1211505/2603047*a^12 + 137623/2603047*a^11 - 987396/2603047*a^10 - 220064/2603047*a^9 + 1165425/2603047*a^8 - 213654/2603047*a^7 + 1286519/2603047*a^6 - 391878/2603047*a^5 - 267236/2603047*a^4 - 63344/2603047*a^3 + 919891/2603047*a^2 - 1096379/2603047*a + 499320/2603047, 1/2603047*a^43 - 14995/2603047*a^23 - 739488/2603047*a^22 + 599800/2603047*a^21 - 418110/2603047*a^20 - 84312/2603047*a^19 - 972339/2603047*a^18 - 127594/2603047*a^17 + 1246798/2603047*a^16 - 830536/2603047*a^15 + 724864/2603047*a^14 - 162411/2603047*a^13 - 500397/2603047*a^12 + 765244/2603047*a^11 - 95211/2603047*a^10 + 1069514/2603047*a^9 + 180188/2603047*a^8 - 160216/2603047*a^7 + 187625/2603047*a^6 + 697254/2603047*a^5 + 1175753/2603047*a^4 + 1223828/2603047*a^3 + 479567/2603047*a^2 + 708330/2603047*a - 188390/2603047, 1/2603047*a^44 + 911128/2603047*a^23 - 119960/2603047*a^22 - 858083/2603047*a^21 - 587634/2603047*a^20 + 17980/2603047*a^19 - 253504/2603047*a^18 - 151260/2603047*a^17 - 1168848/2603047*a^16 + 319828/2603047*a^15 + 998945/2603047*a^14 - 1213276/2603047*a^13 - 642318/2603047*a^12 - 1283289/2603047*a^11 - 795178/2603047*a^10 - 1018794/2603047*a^9 + 184341/2603047*a^8 + 1105794/2603047*a^7 + 1124827/2603047*a^6 - 304042/2603047*a^5 - 446873/2603047*a^4 + 133951/2603047*a^3 + 1079743/2603047*a^2 - 393611/2603047*a + 495831/2603047, 1/2603047*a^45 - 311891/2603047*a^23 + 1227309/2603047*a^22 + 432051/2603047*a^21 + 476547/2603047*a^20 + 177172/2603047*a^19 + 597188/2603047*a^18 - 351851/2603047*a^17 - 1240924/2603047*a^16 + 220058/2603047*a^15 + 644520/2603047*a^14 + 389441/2603047*a^13 + 650028/2603047*a^12 + 29984/2603047*a^11 - 1282954/2603047*a^10 - 1250581/2603047*a^9 - 1125778/2603047*a^8 - 276923/2603047*a^7 + 1123136/2603047*a^6 + 487627/2603047*a^5 + 266031/2603047*a^4 - 8733/2603047*a^3 + 57503/2603047*a^2 + 831424/2603047*a - 1024827/2603047

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:	$45$	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^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043)

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^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043, 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^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043);

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^46 - x^45 - 93*x^44 + 93*x^43 + 4043*x^42 - 4043*x^41 - 109133*x^40 + 109133*x^39 + 2049107*x^38 - 2049107*x^37 - 28412909*x^36 + 28412909*x^35 + 301468435*x^34 - 301468435*x^33 - 2502522989*x^32 + 2502522989*x^31 + 16478342035*x^30 - 16478342035*x^29 - 86750923885*x^28 + 86750923885*x^27 + 366341854099*x^26 - 366341854099*x^25 - 1240077995117*x^24 + 1240077995117*x^23 + 3349693002643*x^22 - 3349693002643*x^21 - 7159013444717*x^20 + 7159013444717*x^19 + 11947725550483*x^18 - 11947725550483*x^17 - 15279377517677*x^16 + 15279377517677*x^15 + 14582606492563*x^14 - 14582606492563*x^13 - 10009615633517*x^12 + 10009615633517*x^11 + 4689183798163*x^10 - 4689183798163*x^9 - 1389267094637*x^8 + 1389267094637*x^7 + 231653143443*x^6 - 231653143443*x^5 - 17719200877*x^4 + 17719200877*x^3 + 416969619*x^2 - 416969619*x + 22705043);

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

$C_{46}$ (as 46T1):

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 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$ is not computed

Intermediate fields

$\Q(\sqrt{329}) $, $\Q(\zeta_{47})^+$

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	$23^{2}$	$46$	$23^{2}$	R	$46$	$23^{2}$	$46$	$23^{2}$	$46$	$46$	$23^{2}$	$23^{2}$	$23^{2}$	$46$	R	$23^{2}$	$46$

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 $46$	$2$	$23$	$23$
$47$ 47		Deg $46$	$46$	$1$	$45$

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