LMFDB - Number field 30.2.301337616246914973122131519082482771026611328125.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1)

gp: K = bnfinit(y^30 - 5*y^29 + 8*y^28 + 19*y^27 - 63*y^26 - 52*y^25 + 265*y^24 - 211*y^23 - 393*y^22 + 858*y^21 - 86*y^20 - 2380*y^19 + 4186*y^18 + 8272*y^17 - 3336*y^16 - 13036*y^15 + 7176*y^14 + 23259*y^13 + 6583*y^12 - 9115*y^11 - 7328*y^10 + 1279*y^9 + 4440*y^8 + 2429*y^7 - 349*y^6 - 1189*y^5 - 88*y^4 + 271*y^3 + 100*y^2 - 13*y - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1)

$ x^{30} - 5 x^{29} + 8 x^{28} + 19 x^{27} - 63 x^{26} - 52 x^{25} + 265 x^{24} - 211 x^{23} - 393 x^{22} + \cdots - 1 $ x^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$30$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[2, 14]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$301337616246914973122131519082482771026611328125$ 301337616246914973122131519082482771026611328125 $\medspace = 5^{15}\cdot 439^{14}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$38.25$	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:	$5^{1/2}439^{1/2}\approx 46.850827100489916$
Ramified primes:	$5$, $439$ 5, 439	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{5}) $
$\card{ \Aut(K/\Q) }$:	$2$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{9}a^{13}+\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{4}{9}a^{6}-\frac{4}{9}a^{5}-\frac{4}{9}a^{4}-\frac{4}{9}a^{3}-\frac{4}{9}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{27}a^{20}+\frac{1}{27}a^{18}-\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{1}{27}a^{15}+\frac{1}{27}a^{14}+\frac{1}{27}a^{13}-\frac{4}{27}a^{12}+\frac{1}{27}a^{11}-\frac{4}{27}a^{10}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{5}{27}a^{7}-\frac{4}{27}a^{6}+\frac{5}{27}a^{5}+\frac{1}{3}a^{4}-\frac{13}{27}a^{3}-\frac{8}{27}a-\frac{8}{27}$, $\frac{1}{27}a^{21}+\frac{1}{27}a^{19}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}+\frac{1}{27}a^{16}+\frac{1}{27}a^{15}+\frac{1}{27}a^{14}-\frac{4}{27}a^{13}+\frac{1}{27}a^{12}-\frac{4}{27}a^{11}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{4}{27}a^{8}-\frac{4}{27}a^{7}+\frac{5}{27}a^{6}+\frac{1}{3}a^{5}-\frac{13}{27}a^{4}-\frac{8}{27}a^{2}-\frac{8}{27}a+\frac{1}{3}$, $\frac{1}{27}a^{22}-\frac{1}{27}a^{19}+\frac{1}{27}a^{18}-\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{4}{27}a^{14}-\frac{4}{27}a^{11}+\frac{4}{27}a^{10}-\frac{4}{27}a^{9}-\frac{4}{27}a^{8}+\frac{4}{27}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{5}{27}a^{3}+\frac{13}{27}a^{2}-\frac{4}{27}a-\frac{13}{27}$, $\frac{1}{27}a^{23}+\frac{1}{27}a^{19}+\frac{1}{27}a^{17}-\frac{1}{27}a^{16}-\frac{1}{27}a^{15}+\frac{4}{27}a^{14}+\frac{4}{27}a^{13}+\frac{4}{27}a^{12}-\frac{1}{27}a^{11}+\frac{4}{27}a^{10}-\frac{1}{27}a^{9}+\frac{2}{9}a^{7}-\frac{7}{27}a^{6}+\frac{11}{27}a^{5}-\frac{7}{27}a^{4}-\frac{1}{9}a^{3}+\frac{2}{27}a^{2}-\frac{4}{9}a+\frac{7}{27}$, $\frac{1}{27}a^{24}-\frac{1}{9}a^{8}+\frac{2}{27}$, $\frac{1}{27}a^{25}-\frac{1}{9}a^{9}+\frac{2}{27}a$, $\frac{1}{81}a^{26}+\frac{1}{81}a^{25}-\frac{1}{81}a^{22}-\frac{2}{81}a^{19}-\frac{1}{81}a^{18}+\frac{4}{81}a^{17}-\frac{2}{81}a^{16}-\frac{1}{27}a^{15}-\frac{7}{81}a^{14}+\frac{2}{27}a^{13}+\frac{2}{27}a^{12}-\frac{11}{81}a^{11}-\frac{10}{81}a^{10}+\frac{1}{81}a^{9}-\frac{11}{81}a^{8}+\frac{13}{27}a^{7}+\frac{8}{81}a^{6}-\frac{11}{27}a^{5}-\frac{5}{27}a^{4}-\frac{32}{81}a^{3}+\frac{1}{81}a^{2}+\frac{4}{27}a+\frac{13}{81}$, $\frac{1}{729}a^{27}+\frac{2}{729}a^{26}+\frac{10}{729}a^{25}-\frac{4}{243}a^{24}+\frac{11}{729}a^{23}-\frac{7}{729}a^{22}+\frac{1}{81}a^{21}+\frac{7}{729}a^{20}+\frac{11}{243}a^{19}-\frac{4}{243}a^{18}+\frac{29}{729}a^{17}-\frac{38}{729}a^{16}-\frac{13}{729}a^{15}+\frac{86}{729}a^{14}-\frac{10}{243}a^{13}-\frac{101}{729}a^{12}+\frac{2}{81}a^{11}-\frac{40}{243}a^{10}-\frac{115}{729}a^{9}+\frac{43}{729}a^{8}+\frac{326}{729}a^{7}+\frac{182}{729}a^{6}-\frac{80}{243}a^{5}-\frac{131}{729}a^{4}-\frac{88}{729}a^{3}-\frac{167}{729}a^{2}+\frac{220}{729}a-\frac{47}{729}$, $\frac{1}{384183}a^{28}-\frac{260}{384183}a^{27}-\frac{847}{384183}a^{26}-\frac{2965}{384183}a^{25}-\frac{112}{384183}a^{24}+\frac{1}{279}a^{23}-\frac{4088}{384183}a^{22}+\frac{1132}{384183}a^{21}+\frac{5597}{384183}a^{20}+\frac{448}{14229}a^{19}+\frac{18815}{384183}a^{18}-\frac{14665}{384183}a^{17}-\frac{20954}{384183}a^{16}+\frac{334}{42687}a^{15}+\frac{53290}{384183}a^{14}+\frac{60436}{384183}a^{13}-\frac{29842}{384183}a^{12}+\frac{18671}{128061}a^{11}+\frac{16835}{384183}a^{10}+\frac{58487}{384183}a^{9}-\frac{35114}{384183}a^{8}+\frac{12025}{42687}a^{7}+\frac{85303}{384183}a^{6}+\frac{11051}{22599}a^{5}-\frac{166322}{384183}a^{4}+\frac{7787}{384183}a^{3}+\frac{10474}{42687}a^{2}-\frac{52430}{128061}a-\frac{56923}{384183}$, $\frac{1}{69\!\cdots\!49}a^{29}-\frac{21\!\cdots\!77}{27\!\cdots\!59}a^{28}+\frac{97\!\cdots\!06}{69\!\cdots\!49}a^{27}-\frac{33\!\cdots\!46}{69\!\cdots\!49}a^{26}-\frac{63\!\cdots\!37}{69\!\cdots\!49}a^{25}+\frac{11\!\cdots\!42}{69\!\cdots\!49}a^{24}-\frac{40\!\cdots\!27}{23\!\cdots\!83}a^{23}+\frac{11\!\cdots\!38}{69\!\cdots\!49}a^{22}-\frac{50\!\cdots\!12}{69\!\cdots\!49}a^{21}-\frac{71\!\cdots\!67}{40\!\cdots\!97}a^{20}+\frac{29\!\cdots\!43}{69\!\cdots\!49}a^{19}+\frac{41\!\cdots\!32}{23\!\cdots\!83}a^{18}+\frac{16\!\cdots\!42}{69\!\cdots\!49}a^{17}-\frac{12\!\cdots\!91}{23\!\cdots\!83}a^{16}+\frac{13\!\cdots\!11}{77\!\cdots\!61}a^{15}-\frac{29\!\cdots\!74}{69\!\cdots\!49}a^{14}-\frac{11\!\cdots\!14}{69\!\cdots\!49}a^{13}+\frac{59\!\cdots\!11}{69\!\cdots\!49}a^{12}+\frac{19\!\cdots\!56}{69\!\cdots\!49}a^{11}+\frac{14\!\cdots\!32}{13\!\cdots\!99}a^{10}+\frac{19\!\cdots\!31}{69\!\cdots\!49}a^{9}+\frac{31\!\cdots\!15}{77\!\cdots\!61}a^{8}+\frac{32\!\cdots\!39}{23\!\cdots\!83}a^{7}-\frac{18\!\cdots\!88}{69\!\cdots\!49}a^{6}+\frac{30\!\cdots\!14}{77\!\cdots\!61}a^{5}+\frac{15\!\cdots\!37}{69\!\cdots\!49}a^{4}-\frac{42\!\cdots\!40}{23\!\cdots\!83}a^{3}+\frac{17\!\cdots\!38}{69\!\cdots\!49}a^{2}-\frac{96\!\cdots\!06}{25\!\cdots\!87}a+\frac{16\!\cdots\!62}{69\!\cdots\!49}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 - 1/3, 1/3*a^9 - 1/3*a, 1/3*a^10 - 1/3*a^2, 1/3*a^11 - 1/3*a^3, 1/3*a^12 - 1/3*a^4, 1/3*a^13 - 1/3*a^5, 1/3*a^14 - 1/3*a^6, 1/9*a^15 + 1/9*a^14 + 1/9*a^13 + 1/9*a^12 + 1/9*a^11 + 1/9*a^10 + 1/9*a^9 + 1/9*a^8 - 4/9*a^7 - 4/9*a^6 - 4/9*a^5 - 4/9*a^4 - 4/9*a^3 - 4/9*a^2 - 4/9*a - 4/9, 1/9*a^16 + 1/9*a^8 - 2/9, 1/9*a^17 + 1/9*a^9 - 2/9*a, 1/9*a^18 + 1/9*a^10 - 2/9*a^2, 1/9*a^19 + 1/9*a^11 - 2/9*a^3, 1/27*a^20 + 1/27*a^18 - 1/27*a^17 - 1/27*a^16 + 1/27*a^15 + 1/27*a^14 + 1/27*a^13 - 4/27*a^12 + 1/27*a^11 - 4/27*a^10 - 1/9*a^9 - 1/9*a^8 + 5/27*a^7 - 4/27*a^6 + 5/27*a^5 + 1/3*a^4 - 13/27*a^3 - 8/27*a - 8/27, 1/27*a^21 + 1/27*a^19 - 1/27*a^18 - 1/27*a^17 + 1/27*a^16 + 1/27*a^15 + 1/27*a^14 - 4/27*a^13 + 1/27*a^12 - 4/27*a^11 - 1/9*a^10 - 1/9*a^9 - 4/27*a^8 - 4/27*a^7 + 5/27*a^6 + 1/3*a^5 - 13/27*a^4 - 8/27*a^2 - 8/27*a + 1/3, 1/27*a^22 - 1/27*a^19 + 1/27*a^18 - 1/27*a^17 - 1/27*a^16 + 4/27*a^14 - 4/27*a^11 + 4/27*a^10 - 4/27*a^9 - 4/27*a^8 + 4/27*a^6 + 1/3*a^5 - 1/3*a^4 + 5/27*a^3 + 13/27*a^2 - 4/27*a - 13/27, 1/27*a^23 + 1/27*a^19 + 1/27*a^17 - 1/27*a^16 - 1/27*a^15 + 4/27*a^14 + 4/27*a^13 + 4/27*a^12 - 1/27*a^11 + 4/27*a^10 - 1/27*a^9 + 2/9*a^7 - 7/27*a^6 + 11/27*a^5 - 7/27*a^4 - 1/9*a^3 + 2/27*a^2 - 4/9*a + 7/27, 1/27*a^24 - 1/9*a^8 + 2/27, 1/27*a^25 - 1/9*a^9 + 2/27*a, 1/81*a^26 + 1/81*a^25 - 1/81*a^22 - 2/81*a^19 - 1/81*a^18 + 4/81*a^17 - 2/81*a^16 - 1/27*a^15 - 7/81*a^14 + 2/27*a^13 + 2/27*a^12 - 11/81*a^11 - 10/81*a^10 + 1/81*a^9 - 11/81*a^8 + 13/27*a^7 + 8/81*a^6 - 11/27*a^5 - 5/27*a^4 - 32/81*a^3 + 1/81*a^2 + 4/27*a + 13/81, 1/729*a^27 + 2/729*a^26 + 10/729*a^25 - 4/243*a^24 + 11/729*a^23 - 7/729*a^22 + 1/81*a^21 + 7/729*a^20 + 11/243*a^19 - 4/243*a^18 + 29/729*a^17 - 38/729*a^16 - 13/729*a^15 + 86/729*a^14 - 10/243*a^13 - 101/729*a^12 + 2/81*a^11 - 40/243*a^10 - 115/729*a^9 + 43/729*a^8 + 326/729*a^7 + 182/729*a^6 - 80/243*a^5 - 131/729*a^4 - 88/729*a^3 - 167/729*a^2 + 220/729*a - 47/729, 1/384183*a^28 - 260/384183*a^27 - 847/384183*a^26 - 2965/384183*a^25 - 112/384183*a^24 + 1/279*a^23 - 4088/384183*a^22 + 1132/384183*a^21 + 5597/384183*a^20 + 448/14229*a^19 + 18815/384183*a^18 - 14665/384183*a^17 - 20954/384183*a^16 + 334/42687*a^15 + 53290/384183*a^14 + 60436/384183*a^13 - 29842/384183*a^12 + 18671/128061*a^11 + 16835/384183*a^10 + 58487/384183*a^9 - 35114/384183*a^8 + 12025/42687*a^7 + 85303/384183*a^6 + 11051/22599*a^5 - 166322/384183*a^4 + 7787/384183*a^3 + 10474/42687*a^2 - 52430/128061*a - 56923/384183, 1/6932787919812818813867113376921047991049*a^29 - 2153464308234134186111767009577/2760966913505702434833577609287553959*a^28 + 970866685316710843053001252395220706/6932787919812818813867113376921047991049*a^27 - 33093227870933222437200936937984443946/6932787919812818813867113376921047991049*a^26 - 63260015356179361793332499850241201937/6932787919812818813867113376921047991049*a^25 + 112786505585563284222044559681333556342/6932787919812818813867113376921047991049*a^24 - 40306421851876579696840883628992433827/2310929306604272937955704458973682663683*a^23 + 117398551423359694093899420270077188538/6932787919812818813867113376921047991049*a^22 - 50740576993151739325591307068645201112/6932787919812818813867113376921047991049*a^21 - 7101360271869078289308228435849127367/407811054106636400815712551583591058297*a^20 + 298686993682945136523790333949380241543/6932787919812818813867113376921047991049*a^19 + 41670478074274955750636118241830360332/2310929306604272937955704458973682663683*a^18 + 160629448965266180846727285419116589542/6932787919812818813867113376921047991049*a^17 - 128139266846633417657081684006436781891/2310929306604272937955704458973682663683*a^16 + 13204749581417629795335256186484005111/770309768868090979318568152991227554561*a^15 - 292584915461408707535096495633993477974/6932787919812818813867113376921047991049*a^14 - 1126925107580585852878934794282464997514/6932787919812818813867113376921047991049*a^13 + 59366859038605883876112624759289008911/6932787919812818813867113376921047991049*a^12 + 195349447251961042896658894096099465856/6932787919812818813867113376921047991049*a^11 + 14575708550595851788280901405743425432/135937018035545466938570850527863686099*a^10 + 198696386812724344793597817304947872731/6932787919812818813867113376921047991049*a^9 + 31383265940718125495287212902493520715/770309768868090979318568152991227554561*a^8 + 322087799195464009567506905642194300139/2310929306604272937955704458973682663683*a^7 - 1860244480352380588947016078422033983488/6932787919812818813867113376921047991049*a^6 + 304042728037087395132654496874621507314/770309768868090979318568152991227554561*a^5 + 1547545954021488284910428209144879051637/6932787919812818813867113376921047991049*a^4 - 425515906131768765781378707840133084640/2310929306604272937955704458973682663683*a^3 + 1778951606093810719888709702795166272038/6932787919812818813867113376921047991049*a^2 - 96591288373017859824994113018363798506/256769922956030326439522717663742518187*a + 1624677260981336965511385422630217276962/6932787919812818813867113376921047991049

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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:	$15$	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:	$\frac{51\!\cdots\!04}{69\!\cdots\!49}a^{29}-\frac{82\!\cdots\!22}{23\!\cdots\!83}a^{28}+\frac{37\!\cdots\!70}{69\!\cdots\!49}a^{27}+\frac{10\!\cdots\!65}{69\!\cdots\!49}a^{26}-\frac{30\!\cdots\!15}{69\!\cdots\!49}a^{25}-\frac{31\!\cdots\!18}{69\!\cdots\!49}a^{24}+\frac{43\!\cdots\!78}{23\!\cdots\!83}a^{23}-\frac{88\!\cdots\!90}{69\!\cdots\!49}a^{22}-\frac{21\!\cdots\!73}{69\!\cdots\!49}a^{21}+\frac{40\!\cdots\!35}{69\!\cdots\!49}a^{20}+\frac{16\!\cdots\!83}{69\!\cdots\!49}a^{19}-\frac{13\!\cdots\!58}{77\!\cdots\!61}a^{18}+\frac{11\!\cdots\!76}{40\!\cdots\!97}a^{17}+\frac{15\!\cdots\!54}{23\!\cdots\!83}a^{16}-\frac{38\!\cdots\!79}{25\!\cdots\!87}a^{15}-\frac{68\!\cdots\!30}{69\!\cdots\!49}a^{14}+\frac{26\!\cdots\!78}{69\!\cdots\!49}a^{13}+\frac{12\!\cdots\!77}{69\!\cdots\!49}a^{12}+\frac{51\!\cdots\!91}{69\!\cdots\!49}a^{11}-\frac{14\!\cdots\!93}{25\!\cdots\!87}a^{10}-\frac{43\!\cdots\!06}{69\!\cdots\!49}a^{9}+\frac{57\!\cdots\!57}{23\!\cdots\!83}a^{8}+\frac{75\!\cdots\!07}{23\!\cdots\!83}a^{7}+\frac{15\!\cdots\!83}{69\!\cdots\!49}a^{6}+\frac{17\!\cdots\!28}{23\!\cdots\!83}a^{5}-\frac{60\!\cdots\!41}{69\!\cdots\!49}a^{4}-\frac{44\!\cdots\!41}{23\!\cdots\!83}a^{3}+\frac{11\!\cdots\!02}{69\!\cdots\!49}a^{2}+\frac{14\!\cdots\!35}{15\!\cdots\!11}a-\frac{81\!\cdots\!28}{69\!\cdots\!49}$, $\frac{27\!\cdots\!57}{77\!\cdots\!61}a^{29}-\frac{15\!\cdots\!72}{77\!\cdots\!61}a^{28}+\frac{91\!\cdots\!81}{23\!\cdots\!83}a^{27}+\frac{10\!\cdots\!18}{23\!\cdots\!83}a^{26}-\frac{58\!\cdots\!84}{23\!\cdots\!83}a^{25}-\frac{32\!\cdots\!45}{85\!\cdots\!29}a^{24}+\frac{22\!\cdots\!48}{23\!\cdots\!83}a^{23}-\frac{30\!\cdots\!72}{23\!\cdots\!83}a^{22}-\frac{34\!\cdots\!68}{50\!\cdots\!37}a^{21}+\frac{81\!\cdots\!48}{23\!\cdots\!83}a^{20}-\frac{60\!\cdots\!39}{25\!\cdots\!87}a^{19}-\frac{56\!\cdots\!35}{77\!\cdots\!61}a^{18}+\frac{44\!\cdots\!64}{23\!\cdots\!83}a^{17}+\frac{42\!\cdots\!00}{23\!\cdots\!83}a^{16}-\frac{55\!\cdots\!84}{23\!\cdots\!83}a^{15}-\frac{73\!\cdots\!92}{23\!\cdots\!83}a^{14}+\frac{36\!\cdots\!26}{77\!\cdots\!61}a^{13}+\frac{12\!\cdots\!19}{23\!\cdots\!83}a^{12}-\frac{11\!\cdots\!94}{77\!\cdots\!61}a^{11}-\frac{19\!\cdots\!51}{77\!\cdots\!61}a^{10}-\frac{33\!\cdots\!83}{74\!\cdots\!93}a^{9}+\frac{26\!\cdots\!02}{23\!\cdots\!83}a^{8}+\frac{17\!\cdots\!39}{23\!\cdots\!83}a^{7}+\frac{67\!\cdots\!60}{23\!\cdots\!83}a^{6}-\frac{26\!\cdots\!74}{77\!\cdots\!61}a^{5}-\frac{52\!\cdots\!23}{23\!\cdots\!83}a^{4}+\frac{36\!\cdots\!10}{23\!\cdots\!83}a^{3}+\frac{80\!\cdots\!36}{23\!\cdots\!83}a^{2}-\frac{11\!\cdots\!81}{23\!\cdots\!83}a-\frac{22\!\cdots\!31}{23\!\cdots\!83}$, $\frac{51\!\cdots\!58}{69\!\cdots\!49}a^{29}-\frac{63\!\cdots\!63}{23\!\cdots\!83}a^{28}-\frac{33\!\cdots\!38}{69\!\cdots\!49}a^{27}+\frac{20\!\cdots\!10}{69\!\cdots\!49}a^{26}-\frac{30\!\cdots\!56}{69\!\cdots\!49}a^{25}-\frac{86\!\cdots\!28}{69\!\cdots\!49}a^{24}+\frac{59\!\cdots\!89}{23\!\cdots\!83}a^{23}+\frac{99\!\cdots\!95}{69\!\cdots\!49}a^{22}-\frac{64\!\cdots\!14}{69\!\cdots\!49}a^{21}+\frac{52\!\cdots\!46}{69\!\cdots\!49}a^{20}+\frac{86\!\cdots\!46}{69\!\cdots\!49}a^{19}-\frac{26\!\cdots\!01}{77\!\cdots\!61}a^{18}+\frac{10\!\cdots\!39}{69\!\cdots\!49}a^{17}+\frac{31\!\cdots\!76}{23\!\cdots\!83}a^{16}-\frac{20\!\cdots\!81}{77\!\cdots\!61}a^{15}-\frac{95\!\cdots\!37}{40\!\cdots\!97}a^{14}+\frac{89\!\cdots\!41}{40\!\cdots\!97}a^{13}+\frac{16\!\cdots\!60}{40\!\cdots\!97}a^{12}+\frac{62\!\cdots\!00}{69\!\cdots\!49}a^{11}-\frac{78\!\cdots\!85}{25\!\cdots\!87}a^{10}-\frac{86\!\cdots\!07}{69\!\cdots\!49}a^{9}+\frac{12\!\cdots\!79}{23\!\cdots\!83}a^{8}+\frac{20\!\cdots\!24}{23\!\cdots\!83}a^{7}+\frac{20\!\cdots\!67}{69\!\cdots\!49}a^{6}-\frac{58\!\cdots\!11}{23\!\cdots\!83}a^{5}-\frac{65\!\cdots\!26}{22\!\cdots\!79}a^{4}-\frac{10\!\cdots\!99}{23\!\cdots\!83}a^{3}+\frac{96\!\cdots\!28}{69\!\cdots\!49}a^{2}-\frac{86\!\cdots\!58}{77\!\cdots\!61}a-\frac{92\!\cdots\!34}{69\!\cdots\!49}$, $\frac{94\!\cdots\!22}{69\!\cdots\!49}a^{29}-\frac{19\!\cdots\!24}{23\!\cdots\!83}a^{28}+\frac{13\!\cdots\!13}{69\!\cdots\!49}a^{27}+\frac{80\!\cdots\!35}{69\!\cdots\!49}a^{26}-\frac{77\!\cdots\!62}{69\!\cdots\!49}a^{25}+\frac{22\!\cdots\!51}{69\!\cdots\!49}a^{24}+\frac{95\!\cdots\!92}{23\!\cdots\!83}a^{23}-\frac{28\!\cdots\!45}{40\!\cdots\!97}a^{22}-\frac{67\!\cdots\!30}{69\!\cdots\!49}a^{21}+\frac{11\!\cdots\!85}{69\!\cdots\!49}a^{20}-\frac{10\!\cdots\!36}{69\!\cdots\!49}a^{19}-\frac{20\!\cdots\!76}{77\!\cdots\!61}a^{18}+\frac{62\!\cdots\!54}{69\!\cdots\!49}a^{17}+\frac{91\!\cdots\!85}{23\!\cdots\!83}a^{16}-\frac{11\!\cdots\!29}{77\!\cdots\!61}a^{15}-\frac{76\!\cdots\!94}{69\!\cdots\!49}a^{14}+\frac{17\!\cdots\!65}{69\!\cdots\!49}a^{13}+\frac{39\!\cdots\!64}{22\!\cdots\!79}a^{12}-\frac{14\!\cdots\!19}{69\!\cdots\!49}a^{11}-\frac{44\!\cdots\!60}{25\!\cdots\!87}a^{10}-\frac{13\!\cdots\!87}{69\!\cdots\!49}a^{9}+\frac{22\!\cdots\!37}{23\!\cdots\!83}a^{8}+\frac{11\!\cdots\!45}{23\!\cdots\!83}a^{7}-\frac{14\!\cdots\!93}{69\!\cdots\!49}a^{6}-\frac{63\!\cdots\!19}{23\!\cdots\!83}a^{5}-\frac{57\!\cdots\!91}{40\!\cdots\!97}a^{4}+\frac{84\!\cdots\!90}{74\!\cdots\!93}a^{3}+\frac{18\!\cdots\!49}{69\!\cdots\!49}a^{2}-\frac{29\!\cdots\!43}{77\!\cdots\!61}a-\frac{69\!\cdots\!05}{69\!\cdots\!49}$, $\frac{26\!\cdots\!73}{23\!\cdots\!83}a^{29}-\frac{16\!\cdots\!07}{25\!\cdots\!87}a^{28}+\frac{10\!\cdots\!55}{77\!\cdots\!61}a^{27}+\frac{24\!\cdots\!12}{23\!\cdots\!83}a^{26}-\frac{18\!\cdots\!98}{23\!\cdots\!83}a^{25}+\frac{80\!\cdots\!66}{23\!\cdots\!83}a^{24}+\frac{71\!\cdots\!52}{23\!\cdots\!83}a^{23}-\frac{11\!\cdots\!01}{23\!\cdots\!83}a^{22}-\frac{24\!\cdots\!00}{23\!\cdots\!83}a^{21}+\frac{83\!\cdots\!11}{77\!\cdots\!61}a^{20}-\frac{20\!\cdots\!82}{23\!\cdots\!83}a^{19}-\frac{16\!\cdots\!39}{77\!\cdots\!61}a^{18}+\frac{54\!\cdots\!51}{85\!\cdots\!29}a^{17}+\frac{10\!\cdots\!97}{23\!\cdots\!83}a^{16}-\frac{18\!\cdots\!50}{23\!\cdots\!83}a^{15}-\frac{22\!\cdots\!73}{23\!\cdots\!83}a^{14}+\frac{38\!\cdots\!49}{23\!\cdots\!83}a^{13}+\frac{12\!\cdots\!92}{77\!\cdots\!61}a^{12}-\frac{15\!\cdots\!10}{23\!\cdots\!83}a^{11}-\frac{84\!\cdots\!31}{77\!\cdots\!61}a^{10}-\frac{88\!\cdots\!22}{77\!\cdots\!61}a^{9}+\frac{89\!\cdots\!24}{13\!\cdots\!99}a^{8}+\frac{63\!\cdots\!94}{23\!\cdots\!83}a^{7}-\frac{26\!\cdots\!61}{23\!\cdots\!83}a^{6}-\frac{16\!\cdots\!05}{77\!\cdots\!61}a^{5}-\frac{11\!\cdots\!31}{25\!\cdots\!87}a^{4}+\frac{19\!\cdots\!52}{23\!\cdots\!83}a^{3}+\frac{41\!\cdots\!01}{13\!\cdots\!99}a^{2}-\frac{92\!\cdots\!38}{23\!\cdots\!83}a-\frac{40\!\cdots\!33}{28\!\cdots\!43}$, $\frac{22\!\cdots\!41}{69\!\cdots\!49}a^{29}-\frac{60\!\cdots\!70}{23\!\cdots\!83}a^{28}+\frac{91\!\cdots\!95}{69\!\cdots\!49}a^{27}-\frac{16\!\cdots\!72}{69\!\cdots\!49}a^{26}-\frac{25\!\cdots\!19}{69\!\cdots\!49}a^{25}+\frac{10\!\cdots\!19}{69\!\cdots\!49}a^{24}+\frac{51\!\cdots\!41}{77\!\cdots\!61}a^{23}-\frac{43\!\cdots\!06}{69\!\cdots\!49}a^{22}+\frac{52\!\cdots\!35}{69\!\cdots\!49}a^{21}+\frac{22\!\cdots\!92}{40\!\cdots\!97}a^{20}-\frac{14\!\cdots\!47}{69\!\cdots\!49}a^{19}+\frac{90\!\cdots\!60}{77\!\cdots\!61}a^{18}+\frac{34\!\cdots\!72}{69\!\cdots\!49}a^{17}-\frac{26\!\cdots\!07}{23\!\cdots\!83}a^{16}-\frac{33\!\cdots\!63}{23\!\cdots\!83}a^{15}+\frac{68\!\cdots\!61}{69\!\cdots\!49}a^{14}+\frac{48\!\cdots\!36}{22\!\cdots\!79}a^{13}-\frac{16\!\cdots\!54}{69\!\cdots\!49}a^{12}-\frac{26\!\cdots\!95}{69\!\cdots\!49}a^{11}-\frac{28\!\cdots\!03}{45\!\cdots\!33}a^{10}+\frac{42\!\cdots\!03}{69\!\cdots\!49}a^{9}+\frac{55\!\cdots\!01}{82\!\cdots\!77}a^{8}-\frac{34\!\cdots\!43}{23\!\cdots\!83}a^{7}-\frac{31\!\cdots\!20}{69\!\cdots\!49}a^{6}-\frac{69\!\cdots\!69}{23\!\cdots\!83}a^{5}+\frac{21\!\cdots\!43}{69\!\cdots\!49}a^{4}+\frac{21\!\cdots\!84}{23\!\cdots\!83}a^{3}-\frac{33\!\cdots\!74}{69\!\cdots\!49}a^{2}+\frac{69\!\cdots\!18}{23\!\cdots\!83}a-\frac{82\!\cdots\!50}{69\!\cdots\!49}$, $\frac{15\!\cdots\!11}{69\!\cdots\!49}a^{29}-\frac{30\!\cdots\!44}{23\!\cdots\!83}a^{28}+\frac{21\!\cdots\!51}{69\!\cdots\!49}a^{27}+\frac{11\!\cdots\!46}{69\!\cdots\!49}a^{26}-\frac{11\!\cdots\!36}{69\!\cdots\!49}a^{25}+\frac{32\!\cdots\!11}{69\!\cdots\!49}a^{24}+\frac{14\!\cdots\!31}{23\!\cdots\!83}a^{23}-\frac{75\!\cdots\!30}{69\!\cdots\!49}a^{22}-\frac{33\!\cdots\!63}{69\!\cdots\!49}a^{21}+\frac{16\!\cdots\!85}{69\!\cdots\!49}a^{20}-\frac{16\!\cdots\!04}{69\!\cdots\!49}a^{19}-\frac{58\!\cdots\!13}{15\!\cdots\!11}a^{18}+\frac{96\!\cdots\!64}{69\!\cdots\!49}a^{17}+\frac{46\!\cdots\!11}{77\!\cdots\!61}a^{16}-\frac{47\!\cdots\!75}{23\!\cdots\!83}a^{15}-\frac{11\!\cdots\!24}{69\!\cdots\!49}a^{14}+\frac{26\!\cdots\!52}{69\!\cdots\!49}a^{13}+\frac{17\!\cdots\!69}{69\!\cdots\!49}a^{12}-\frac{16\!\cdots\!96}{69\!\cdots\!49}a^{11}-\frac{17\!\cdots\!38}{77\!\cdots\!61}a^{10}-\frac{31\!\cdots\!10}{69\!\cdots\!49}a^{9}+\frac{21\!\cdots\!15}{23\!\cdots\!83}a^{8}+\frac{47\!\cdots\!26}{77\!\cdots\!61}a^{7}+\frac{15\!\cdots\!29}{69\!\cdots\!49}a^{6}-\frac{77\!\cdots\!56}{23\!\cdots\!83}a^{5}-\frac{13\!\cdots\!87}{69\!\cdots\!49}a^{4}+\frac{60\!\cdots\!49}{85\!\cdots\!29}a^{3}+\frac{12\!\cdots\!81}{69\!\cdots\!49}a^{2}-\frac{51\!\cdots\!38}{23\!\cdots\!83}a-\frac{26\!\cdots\!12}{69\!\cdots\!49}$, $\frac{75\!\cdots\!54}{69\!\cdots\!49}a^{29}-\frac{12\!\cdots\!54}{23\!\cdots\!83}a^{28}+\frac{54\!\cdots\!89}{69\!\cdots\!49}a^{27}+\frac{15\!\cdots\!37}{69\!\cdots\!49}a^{26}-\frac{45\!\cdots\!10}{69\!\cdots\!49}a^{25}-\frac{45\!\cdots\!26}{69\!\cdots\!49}a^{24}+\frac{64\!\cdots\!18}{23\!\cdots\!83}a^{23}-\frac{76\!\cdots\!53}{40\!\cdots\!97}a^{22}-\frac{31\!\cdots\!97}{69\!\cdots\!49}a^{21}+\frac{60\!\cdots\!27}{69\!\cdots\!49}a^{20}+\frac{76\!\cdots\!58}{22\!\cdots\!79}a^{19}-\frac{19\!\cdots\!28}{77\!\cdots\!61}a^{18}+\frac{28\!\cdots\!74}{69\!\cdots\!49}a^{17}+\frac{22\!\cdots\!37}{23\!\cdots\!83}a^{16}-\frac{18\!\cdots\!06}{85\!\cdots\!29}a^{15}-\frac{10\!\cdots\!26}{69\!\cdots\!49}a^{14}+\frac{39\!\cdots\!41}{69\!\cdots\!49}a^{13}+\frac{18\!\cdots\!59}{69\!\cdots\!49}a^{12}+\frac{76\!\cdots\!49}{69\!\cdots\!49}a^{11}-\frac{21\!\cdots\!22}{25\!\cdots\!87}a^{10}-\frac{63\!\cdots\!45}{69\!\cdots\!49}a^{9}+\frac{84\!\cdots\!87}{23\!\cdots\!83}a^{8}+\frac{11\!\cdots\!89}{23\!\cdots\!83}a^{7}+\frac{23\!\cdots\!80}{69\!\cdots\!49}a^{6}+\frac{26\!\cdots\!05}{23\!\cdots\!83}a^{5}-\frac{52\!\cdots\!63}{40\!\cdots\!97}a^{4}-\frac{65\!\cdots\!94}{23\!\cdots\!83}a^{3}+\frac{17\!\cdots\!65}{69\!\cdots\!49}a^{2}-\frac{13\!\cdots\!25}{85\!\cdots\!29}a+\frac{51\!\cdots\!04}{69\!\cdots\!49}$, $\frac{35\!\cdots\!94}{69\!\cdots\!49}a^{29}-\frac{60\!\cdots\!65}{23\!\cdots\!83}a^{28}+\frac{29\!\cdots\!86}{69\!\cdots\!49}a^{27}+\frac{68\!\cdots\!65}{69\!\cdots\!49}a^{26}-\frac{23\!\cdots\!99}{69\!\cdots\!49}a^{25}-\frac{17\!\cdots\!49}{69\!\cdots\!49}a^{24}+\frac{32\!\cdots\!03}{23\!\cdots\!83}a^{23}-\frac{80\!\cdots\!92}{69\!\cdots\!49}a^{22}-\frac{14\!\cdots\!87}{69\!\cdots\!49}a^{21}+\frac{32\!\cdots\!34}{69\!\cdots\!49}a^{20}-\frac{33\!\cdots\!78}{69\!\cdots\!49}a^{19}-\frac{29\!\cdots\!45}{23\!\cdots\!83}a^{18}+\frac{15\!\cdots\!13}{69\!\cdots\!49}a^{17}+\frac{32\!\cdots\!48}{77\!\cdots\!61}a^{16}-\frac{17\!\cdots\!02}{77\!\cdots\!61}a^{15}-\frac{48\!\cdots\!77}{69\!\cdots\!49}a^{14}+\frac{31\!\cdots\!57}{69\!\cdots\!49}a^{13}+\frac{87\!\cdots\!95}{69\!\cdots\!49}a^{12}+\frac{11\!\cdots\!92}{69\!\cdots\!49}a^{11}-\frac{14\!\cdots\!82}{23\!\cdots\!83}a^{10}-\frac{21\!\cdots\!94}{69\!\cdots\!49}a^{9}+\frac{47\!\cdots\!76}{23\!\cdots\!83}a^{8}+\frac{53\!\cdots\!76}{23\!\cdots\!83}a^{7}+\frac{44\!\cdots\!19}{69\!\cdots\!49}a^{6}-\frac{70\!\cdots\!84}{23\!\cdots\!83}a^{5}-\frac{31\!\cdots\!77}{69\!\cdots\!49}a^{4}-\frac{45\!\cdots\!78}{45\!\cdots\!33}a^{3}+\frac{83\!\cdots\!04}{69\!\cdots\!49}a^{2}+\frac{37\!\cdots\!75}{77\!\cdots\!61}a-\frac{23\!\cdots\!09}{69\!\cdots\!49}$, $\frac{33\!\cdots\!28}{69\!\cdots\!49}a^{29}-\frac{53\!\cdots\!11}{23\!\cdots\!83}a^{28}+\frac{23\!\cdots\!65}{69\!\cdots\!49}a^{27}+\frac{21\!\cdots\!78}{22\!\cdots\!79}a^{26}-\frac{19\!\cdots\!88}{69\!\cdots\!49}a^{25}-\frac{20\!\cdots\!74}{69\!\cdots\!49}a^{24}+\frac{94\!\cdots\!08}{77\!\cdots\!61}a^{23}-\frac{55\!\cdots\!81}{69\!\cdots\!49}a^{22}-\frac{14\!\cdots\!42}{69\!\cdots\!49}a^{21}+\frac{26\!\cdots\!77}{69\!\cdots\!49}a^{20}+\frac{21\!\cdots\!62}{69\!\cdots\!49}a^{19}-\frac{26\!\cdots\!24}{23\!\cdots\!83}a^{18}+\frac{12\!\cdots\!20}{69\!\cdots\!49}a^{17}+\frac{58\!\cdots\!12}{13\!\cdots\!99}a^{16}-\frac{22\!\cdots\!75}{23\!\cdots\!83}a^{15}-\frac{44\!\cdots\!17}{69\!\cdots\!49}a^{14}+\frac{17\!\cdots\!03}{69\!\cdots\!49}a^{13}+\frac{80\!\cdots\!81}{69\!\cdots\!49}a^{12}+\frac{33\!\cdots\!71}{69\!\cdots\!49}a^{11}-\frac{88\!\cdots\!06}{23\!\cdots\!83}a^{10}-\frac{26\!\cdots\!58}{69\!\cdots\!49}a^{9}-\frac{46\!\cdots\!72}{23\!\cdots\!83}a^{8}+\frac{45\!\cdots\!73}{23\!\cdots\!83}a^{7}+\frac{10\!\cdots\!05}{69\!\cdots\!49}a^{6}+\frac{24\!\cdots\!76}{23\!\cdots\!83}a^{5}-\frac{37\!\cdots\!35}{69\!\cdots\!49}a^{4}-\frac{40\!\cdots\!81}{23\!\cdots\!83}a^{3}+\frac{98\!\cdots\!77}{69\!\cdots\!49}a^{2}+\frac{12\!\cdots\!00}{23\!\cdots\!83}a+\frac{33\!\cdots\!05}{69\!\cdots\!49}$, $\frac{21\!\cdots\!91}{69\!\cdots\!49}a^{29}-\frac{35\!\cdots\!25}{23\!\cdots\!83}a^{28}+\frac{17\!\cdots\!67}{69\!\cdots\!49}a^{27}+\frac{38\!\cdots\!40}{69\!\cdots\!49}a^{26}-\frac{13\!\cdots\!41}{69\!\cdots\!49}a^{25}-\frac{10\!\cdots\!19}{69\!\cdots\!49}a^{24}+\frac{20\!\cdots\!04}{25\!\cdots\!87}a^{23}-\frac{47\!\cdots\!57}{69\!\cdots\!49}a^{22}-\frac{46\!\cdots\!22}{40\!\cdots\!97}a^{21}+\frac{18\!\cdots\!57}{69\!\cdots\!49}a^{20}-\frac{28\!\cdots\!60}{69\!\cdots\!49}a^{19}-\frac{16\!\cdots\!82}{23\!\cdots\!83}a^{18}+\frac{91\!\cdots\!38}{69\!\cdots\!49}a^{17}+\frac{56\!\cdots\!39}{23\!\cdots\!83}a^{16}-\frac{26\!\cdots\!66}{23\!\cdots\!83}a^{15}-\frac{26\!\cdots\!23}{69\!\cdots\!49}a^{14}+\frac{16\!\cdots\!20}{69\!\cdots\!49}a^{13}+\frac{48\!\cdots\!50}{69\!\cdots\!49}a^{12}+\frac{10\!\cdots\!38}{69\!\cdots\!49}a^{11}-\frac{66\!\cdots\!39}{23\!\cdots\!83}a^{10}-\frac{14\!\cdots\!62}{69\!\cdots\!49}a^{9}+\frac{13\!\cdots\!55}{23\!\cdots\!83}a^{8}+\frac{31\!\cdots\!48}{23\!\cdots\!83}a^{7}+\frac{46\!\cdots\!19}{69\!\cdots\!49}a^{6}-\frac{35\!\cdots\!80}{23\!\cdots\!83}a^{5}-\frac{24\!\cdots\!24}{69\!\cdots\!49}a^{4}+\frac{19\!\cdots\!84}{23\!\cdots\!83}a^{3}+\frac{61\!\cdots\!69}{69\!\cdots\!49}a^{2}+\frac{65\!\cdots\!73}{23\!\cdots\!83}a-\frac{33\!\cdots\!32}{69\!\cdots\!49}$, $\frac{24\!\cdots\!30}{69\!\cdots\!49}a^{29}-\frac{13\!\cdots\!94}{77\!\cdots\!61}a^{28}+\frac{19\!\cdots\!86}{69\!\cdots\!49}a^{27}+\frac{46\!\cdots\!41}{69\!\cdots\!49}a^{26}-\frac{15\!\cdots\!39}{69\!\cdots\!49}a^{25}-\frac{12\!\cdots\!02}{69\!\cdots\!49}a^{24}+\frac{21\!\cdots\!97}{23\!\cdots\!83}a^{23}-\frac{53\!\cdots\!76}{69\!\cdots\!49}a^{22}-\frac{95\!\cdots\!58}{69\!\cdots\!49}a^{21}+\frac{21\!\cdots\!99}{69\!\cdots\!49}a^{20}-\frac{24\!\cdots\!87}{69\!\cdots\!49}a^{19}-\frac{19\!\cdots\!23}{23\!\cdots\!83}a^{18}+\frac{10\!\cdots\!01}{69\!\cdots\!49}a^{17}+\frac{68\!\cdots\!45}{23\!\cdots\!83}a^{16}-\frac{88\!\cdots\!06}{77\!\cdots\!61}a^{15}-\frac{31\!\cdots\!67}{69\!\cdots\!49}a^{14}+\frac{17\!\cdots\!20}{69\!\cdots\!49}a^{13}+\frac{56\!\cdots\!78}{69\!\cdots\!49}a^{12}+\frac{16\!\cdots\!78}{69\!\cdots\!49}a^{11}-\frac{68\!\cdots\!28}{23\!\cdots\!83}a^{10}-\frac{16\!\cdots\!29}{69\!\cdots\!49}a^{9}+\frac{20\!\cdots\!33}{45\!\cdots\!33}a^{8}+\frac{34\!\cdots\!85}{23\!\cdots\!83}a^{7}+\frac{58\!\cdots\!18}{69\!\cdots\!49}a^{6}-\frac{79\!\cdots\!38}{77\!\cdots\!61}a^{5}-\frac{26\!\cdots\!84}{69\!\cdots\!49}a^{4}-\frac{25\!\cdots\!59}{23\!\cdots\!83}a^{3}+\frac{38\!\cdots\!93}{40\!\cdots\!97}a^{2}+\frac{24\!\cdots\!50}{77\!\cdots\!61}a-\frac{34\!\cdots\!73}{69\!\cdots\!49}$, $\frac{13\!\cdots\!33}{40\!\cdots\!97}a^{29}-\frac{12\!\cdots\!52}{77\!\cdots\!61}a^{28}+\frac{16\!\cdots\!13}{69\!\cdots\!49}a^{27}+\frac{48\!\cdots\!49}{69\!\cdots\!49}a^{26}-\frac{13\!\cdots\!98}{69\!\cdots\!49}a^{25}-\frac{15\!\cdots\!94}{69\!\cdots\!49}a^{24}+\frac{11\!\cdots\!72}{13\!\cdots\!99}a^{23}-\frac{37\!\cdots\!07}{69\!\cdots\!49}a^{22}-\frac{10\!\cdots\!77}{69\!\cdots\!49}a^{21}+\frac{18\!\cdots\!78}{69\!\cdots\!49}a^{20}+\frac{16\!\cdots\!86}{69\!\cdots\!49}a^{19}-\frac{18\!\cdots\!29}{23\!\cdots\!83}a^{18}+\frac{87\!\cdots\!81}{69\!\cdots\!49}a^{17}+\frac{22\!\cdots\!14}{74\!\cdots\!93}a^{16}-\frac{39\!\cdots\!71}{77\!\cdots\!61}a^{15}-\frac{31\!\cdots\!73}{69\!\cdots\!49}a^{14}+\frac{10\!\cdots\!02}{69\!\cdots\!49}a^{13}+\frac{57\!\cdots\!24}{69\!\cdots\!49}a^{12}+\frac{87\!\cdots\!10}{22\!\cdots\!79}a^{11}-\frac{52\!\cdots\!47}{23\!\cdots\!83}a^{10}-\frac{20\!\cdots\!71}{69\!\cdots\!49}a^{9}-\frac{10\!\cdots\!61}{77\!\cdots\!61}a^{8}+\frac{33\!\cdots\!68}{23\!\cdots\!83}a^{7}+\frac{73\!\cdots\!07}{69\!\cdots\!49}a^{6}+\frac{12\!\cdots\!89}{15\!\cdots\!11}a^{5}-\frac{24\!\cdots\!11}{69\!\cdots\!49}a^{4}-\frac{19\!\cdots\!29}{23\!\cdots\!83}a^{3}+\frac{45\!\cdots\!63}{69\!\cdots\!49}a^{2}+\frac{10\!\cdots\!28}{28\!\cdots\!43}a+\frac{12\!\cdots\!57}{69\!\cdots\!49}$, $\frac{42\!\cdots\!48}{69\!\cdots\!49}a^{29}-\frac{68\!\cdots\!72}{23\!\cdots\!83}a^{28}+\frac{30\!\cdots\!82}{69\!\cdots\!49}a^{27}+\frac{87\!\cdots\!04}{69\!\cdots\!49}a^{26}-\frac{25\!\cdots\!31}{69\!\cdots\!49}a^{25}-\frac{27\!\cdots\!40}{69\!\cdots\!49}a^{24}+\frac{35\!\cdots\!57}{23\!\cdots\!83}a^{23}-\frac{69\!\cdots\!05}{69\!\cdots\!49}a^{22}-\frac{18\!\cdots\!24}{69\!\cdots\!49}a^{21}+\frac{33\!\cdots\!57}{69\!\cdots\!49}a^{20}+\frac{87\!\cdots\!94}{22\!\cdots\!79}a^{19}-\frac{37\!\cdots\!85}{25\!\cdots\!87}a^{18}+\frac{15\!\cdots\!78}{69\!\cdots\!49}a^{17}+\frac{14\!\cdots\!15}{25\!\cdots\!87}a^{16}-\frac{21\!\cdots\!58}{23\!\cdots\!83}a^{15}-\frac{33\!\cdots\!21}{40\!\cdots\!97}a^{14}+\frac{11\!\cdots\!26}{40\!\cdots\!97}a^{13}+\frac{60\!\cdots\!93}{40\!\cdots\!97}a^{12}+\frac{48\!\cdots\!74}{69\!\cdots\!49}a^{11}-\frac{30\!\cdots\!13}{77\!\cdots\!61}a^{10}-\frac{35\!\cdots\!87}{69\!\cdots\!49}a^{9}-\frac{59\!\cdots\!26}{23\!\cdots\!83}a^{8}+\frac{65\!\cdots\!79}{25\!\cdots\!87}a^{7}+\frac{13\!\cdots\!80}{69\!\cdots\!49}a^{6}+\frac{44\!\cdots\!28}{23\!\cdots\!83}a^{5}-\frac{44\!\cdots\!63}{69\!\cdots\!49}a^{4}-\frac{12\!\cdots\!40}{77\!\cdots\!61}a^{3}+\frac{87\!\cdots\!77}{69\!\cdots\!49}a^{2}+\frac{17\!\cdots\!23}{23\!\cdots\!83}a+\frac{27\!\cdots\!67}{69\!\cdots\!49}$, $\frac{12\!\cdots\!41}{69\!\cdots\!49}a^{29}-\frac{17\!\cdots\!24}{23\!\cdots\!83}a^{28}+\frac{60\!\cdots\!88}{69\!\cdots\!49}a^{27}+\frac{28\!\cdots\!41}{69\!\cdots\!49}a^{26}-\frac{59\!\cdots\!24}{69\!\cdots\!49}a^{25}-\frac{11\!\cdots\!04}{69\!\cdots\!49}a^{24}+\frac{86\!\cdots\!63}{23\!\cdots\!83}a^{23}-\frac{63\!\cdots\!13}{69\!\cdots\!49}a^{22}-\frac{57\!\cdots\!24}{69\!\cdots\!49}a^{21}+\frac{67\!\cdots\!89}{69\!\cdots\!49}a^{20}+\frac{45\!\cdots\!45}{69\!\cdots\!49}a^{19}-\frac{33\!\cdots\!14}{85\!\cdots\!29}a^{18}+\frac{32\!\cdots\!59}{69\!\cdots\!49}a^{17}+\frac{43\!\cdots\!67}{23\!\cdots\!83}a^{16}+\frac{16\!\cdots\!92}{25\!\cdots\!87}a^{15}-\frac{14\!\cdots\!61}{69\!\cdots\!49}a^{14}-\frac{20\!\cdots\!73}{69\!\cdots\!49}a^{13}+\frac{29\!\cdots\!05}{69\!\cdots\!49}a^{12}+\frac{17\!\cdots\!49}{40\!\cdots\!97}a^{11}+\frac{50\!\cdots\!36}{77\!\cdots\!61}a^{10}-\frac{56\!\cdots\!01}{40\!\cdots\!97}a^{9}-\frac{18\!\cdots\!55}{23\!\cdots\!83}a^{8}+\frac{25\!\cdots\!50}{74\!\cdots\!93}a^{7}+\frac{49\!\cdots\!03}{69\!\cdots\!49}a^{6}+\frac{87\!\cdots\!14}{23\!\cdots\!83}a^{5}-\frac{20\!\cdots\!06}{69\!\cdots\!49}a^{4}-\frac{17\!\cdots\!27}{23\!\cdots\!83}a^{3}-\frac{79\!\cdots\!89}{69\!\cdots\!49}a^{2}+\frac{38\!\cdots\!29}{25\!\cdots\!87}a+\frac{66\!\cdots\!96}{69\!\cdots\!49}$ 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560443989926525948040706453143201378194801/407811054106636400815712551583591058297a^9 - 1855872390892387362801477365565363229456055/2310929306604272937955704458973682663683a^8 + 25188135621500014548720733587294644735650/74546106664653965740506595450763956893a^7 + 4997238648272843713971832045413433566639703/6932787919812818813867113376921047991049a^6 + 873170133065418493305273749030393303188114/2310929306604272937955704458973682663683a^5 - 205908151760458679969420020817013640161206/6932787919812818813867113376921047991049a^4 - 173418871289949091792677857840263925077427/2310929306604272937955704458973682663683a^3 - 7959308220024946533469500146377847989189/6932787919812818813867113376921047991049a^2 + 3803101195172415407956697560680521847229/256769922956030326439522717663742518187*a + 6621418923616858568599235260109761030096/6932787919812818813867113376921047991049 (assuming GRH)	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:	$ 748323688575.9082 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 748323688575.9082 \cdot 2}{2\cdot\sqrt{301337616246914973122131519082482771026611328125}}\cr\approx \mathstrut & 0.814968771317785 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1)

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^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1, 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^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1);

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^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1);

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

$D_{30}$ (as 30T14):

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 solvable group of order 60

The 18 conjugacy class representatives for $D_{30}$

Character table for $D_{30}$

Intermediate fields

$\Q(\sqrt{5}) $, 3.1.439.1, 5.1.192721.1, 6.2.24090125.1, 10.2.116066824503125.1, 15.1.3142328914862177479.1

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]

Sibling fields

Degree 30 sibling:	deg 30
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$30$	${\href{/padicField/3.2.0.1}{2} }^{15}$	R	$30$	$15^{2}$	${\href{/padicField/13.6.0.1}{6} }^{5}$	${\href{/padicField/17.2.0.1}{2} }^{15}$	${\href{/padicField/19.5.0.1}{5} }^{6}$	${\href{/padicField/23.2.0.1}{2} }^{15}$	$15^{2}$	${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{15}$	${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.2.0.1}{2} }^{15}$	${\href{/padicField/47.2.0.1}{2} }^{15}$	$30$	${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
$5$ 5	5.10.5.1	$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
	5.10.5.1	$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
	5.10.5.1	$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$	$2$	$5$	$5$	$C_{10}$	$[\ ]_{2}^{5}$
$439$ 439	$\Q_{439}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{439}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.2195.2t1.a.a	$1$	$ 5 \cdot 439 $	$\Q(\sqrt{-2195}) $	$C_2$ (as 2T1)	$1$	$-1$
	1.439.2t1.a.a	$1$	$ 439 $	$\Q(\sqrt{-439}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.2t1.a.a	$1$	$ 5 $	$\Q(\sqrt{5}) $	$C_2$ (as 2T1)	$1$	$1$
*	2.10975.6t3.a.a	$2$	$ 5^{2} \cdot 439 $	6.0.10575564875.1	$D_{6}$ (as 6T3)	$1$	$0$
*	2.439.3t2.a.a	$2$	$ 439 $	3.1.439.1	$S_3$ (as 3T2)	$1$	$0$
*	2.439.5t2.a.a	$2$	$ 439 $	5.1.192721.1	$D_{5}$ (as 5T2)	$1$	$0$
*	2.439.5t2.a.b	$2$	$ 439 $	5.1.192721.1	$D_{5}$ (as 5T2)	$1$	$0$
*	2.10975.10t3.a.b	$2$	$ 5^{2} \cdot 439 $	10.0.50953335956871875.1	$D_{10}$ (as 10T3)	$1$	$0$
*	2.10975.10t3.a.a	$2$	$ 5^{2} \cdot 439 $	10.0.50953335956871875.1	$D_{10}$ (as 10T3)	$1$	$0$
*	2.439.15t2.a.a	$2$	$ 439 $	15.1.3142328914862177479.1	$D_{15}$ (as 15T2)	$1$	$0$
*	2.10975.30t14.a.c	$2$	$ 5^{2} \cdot 439 $	30.2.301337616246914973122131519082482771026611328125.1	$D_{30}$ (as 30T14)	$1$	$0$
*	2.439.15t2.a.c	$2$	$ 439 $	15.1.3142328914862177479.1	$D_{15}$ (as 15T2)	$1$	$0$
*	2.10975.30t14.a.a	$2$	$ 5^{2} \cdot 439 $	30.2.301337616246914973122131519082482771026611328125.1	$D_{30}$ (as 30T14)	$1$	$0$
*	2.439.15t2.a.b	$2$	$ 439 $	15.1.3142328914862177479.1	$D_{15}$ (as 15T2)	$1$	$0$
*	2.10975.30t14.a.b	$2$	$ 5^{2} \cdot 439 $	30.2.301337616246914973122131519082482771026611328125.1	$D_{30}$ (as 30T14)	$1$	$0$
*	2.439.15t2.a.d	$2$	$ 439 $	15.1.3142328914862177479.1	$D_{15}$ (as 15T2)	$1$	$0$
*	2.10975.30t14.a.d	$2$	$ 5^{2} \cdot 439 $	30.2.301337616246914973122131519082482771026611328125.1	$D_{30}$ (as 30T14)	$1$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

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