LMFDB - Number field 29.1.77103436114042117740511038546742321228145336964361.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552)

gp: K = bnfinit(y^29 - 7*y^28 + 47*y^27 - 186*y^26 + 710*y^25 - 970*y^24 + 1144*y^23 + 2298*y^22 - 8465*y^21 + 16505*y^20 - 20589*y^19 + 3590*y^18 + 30661*y^17 - 81803*y^16 + 117497*y^15 - 94780*y^14 + 10260*y^13 + 127678*y^12 - 249606*y^11 + 245472*y^10 - 99463*y^9 - 191187*y^8 + 582067*y^7 - 785770*y^6 + 915704*y^5 - 793384*y^4 + 541792*y^3 - 324448*y^2 + 144320*y - 23552, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552)

$ x^{29} - 7 x^{28} + 47 x^{27} - 186 x^{26} + 710 x^{25} - 970 x^{24} + 1144 x^{23} + 2298 x^{22} + \cdots - 23552 $ x^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$29$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[1, 14]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$77103436114042117740511038546742321228145336964361$ 77103436114042117740511038546742321228145336964361 $\medspace = 3659^{14}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$52.51$	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:	$3659^{1/2}\approx 60.489668539346454$
Ramified primes:	$3659$ 3659	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$1$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{5}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{11}-\frac{1}{16}a^{10}+\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{1}{16}a^{7}-\frac{3}{16}a^{6}+\frac{1}{8}a^{5}-\frac{3}{16}a^{4}+\frac{3}{16}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{12}+\frac{1}{16}a^{11}+\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1}{4}a^{7}-\frac{3}{16}a^{6}+\frac{3}{16}a^{5}+\frac{3}{16}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{15}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{5}+\frac{1}{16}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{16}+\frac{1}{16}a^{10}-\frac{1}{4}a^{7}-\frac{3}{32}a^{4}+\frac{1}{4}a$, $\frac{1}{32}a^{17}+\frac{1}{16}a^{11}-\frac{3}{32}a^{5}$, $\frac{1}{64}a^{18}-\frac{1}{64}a^{17}-\frac{1}{32}a^{14}-\frac{1}{32}a^{13}+\frac{1}{32}a^{12}-\frac{3}{32}a^{10}+\frac{1}{32}a^{8}-\frac{3}{32}a^{7}+\frac{13}{64}a^{6}+\frac{9}{64}a^{5}-\frac{1}{32}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{19}-\frac{1}{64}a^{17}-\frac{1}{32}a^{15}-\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{3}{32}a^{10}+\frac{3}{32}a^{9}-\frac{9}{64}a^{7}+\frac{5}{32}a^{6}-\frac{13}{64}a^{5}+\frac{7}{32}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{64}a^{20}-\frac{1}{64}a^{17}-\frac{1}{32}a^{14}-\frac{1}{16}a^{12}+\frac{1}{32}a^{11}+\frac{1}{16}a^{9}+\frac{1}{64}a^{8}-\frac{1}{4}a^{7}-\frac{3}{16}a^{6}+\frac{15}{64}a^{5}+\frac{3}{16}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{1664}a^{21}-\frac{9}{1664}a^{20}-\frac{3}{832}a^{19}-\frac{3}{1664}a^{18}+\frac{17}{1664}a^{17}-\frac{3}{416}a^{16}-\frac{1}{64}a^{15}+\frac{3}{832}a^{14}-\frac{7}{416}a^{13}-\frac{23}{832}a^{12}+\frac{45}{832}a^{11}+\frac{1}{16}a^{10}+\frac{189}{1664}a^{9}+\frac{3}{1664}a^{8}-\frac{55}{832}a^{7}+\frac{337}{1664}a^{6}-\frac{235}{1664}a^{5}-\frac{47}{416}a^{4}+\frac{111}{416}a^{3}-\frac{3}{13}a^{2}-\frac{41}{104}a-\frac{4}{13}$, $\frac{1}{3328}a^{22}-\frac{9}{3328}a^{20}+\frac{21}{3328}a^{19}+\frac{1}{208}a^{18}-\frac{41}{3328}a^{17}+\frac{11}{1664}a^{16}+\frac{1}{104}a^{15}+\frac{1}{128}a^{14}+\frac{33}{1664}a^{13}-\frac{29}{832}a^{12}+\frac{145}{1664}a^{11}+\frac{397}{3328}a^{10}-\frac{55}{832}a^{9}-\frac{161}{3328}a^{8}+\frac{569}{3328}a^{7}-\frac{35}{832}a^{6}+\frac{375}{3328}a^{5}+\frac{11}{64}a^{4}+\frac{331}{832}a^{3}+\frac{21}{104}a^{2}+\frac{41}{208}a-\frac{5}{13}$, $\frac{1}{3328}a^{23}-\frac{1}{3328}a^{21}+\frac{1}{3328}a^{20}+\frac{5}{832}a^{19}-\frac{1}{256}a^{18}+\frac{1}{1664}a^{17}+\frac{5}{416}a^{16}-\frac{3}{128}a^{15}-\frac{47}{1664}a^{14}-\frac{7}{832}a^{13}+\frac{5}{128}a^{12}-\frac{131}{3328}a^{11}-\frac{55}{832}a^{10}+\frac{207}{3328}a^{9}+\frac{333}{3328}a^{8}+\frac{35}{416}a^{7}-\frac{101}{3328}a^{6}+\frac{2}{13}a^{5}+\frac{137}{832}a^{4}+\frac{95}{208}a^{3}+\frac{57}{208}a^{2}-\frac{6}{13}a-\frac{3}{13}$, $\frac{1}{6656}a^{24}-\frac{1}{6656}a^{23}-\frac{1}{6656}a^{22}-\frac{1}{3328}a^{21}-\frac{49}{6656}a^{20}-\frac{9}{6656}a^{19}+\frac{27}{6656}a^{18}-\frac{15}{3328}a^{17}+\frac{17}{3328}a^{16}+\frac{11}{832}a^{15}+\frac{21}{3328}a^{14}+\frac{31}{3328}a^{13}+\frac{339}{6656}a^{12}-\frac{657}{6656}a^{11}-\frac{405}{6656}a^{10}+\frac{309}{3328}a^{9}-\frac{61}{512}a^{8}-\frac{1397}{6656}a^{7}-\frac{1151}{6656}a^{6}+\frac{11}{208}a^{5}+\frac{301}{1664}a^{4}+\frac{189}{416}a^{3}+\frac{47}{416}a^{2}+\frac{5}{13}a-\frac{1}{13}$, $\frac{1}{306176}a^{25}+\frac{11}{306176}a^{24}-\frac{17}{306176}a^{23}-\frac{21}{153088}a^{22}-\frac{77}{306176}a^{21}-\frac{3}{13312}a^{20}+\frac{1379}{306176}a^{19}+\frac{53}{11776}a^{18}+\frac{2315}{153088}a^{17}-\frac{1}{1664}a^{16}-\frac{3175}{153088}a^{15}-\frac{3453}{153088}a^{14}-\frac{449}{23552}a^{13}-\frac{17621}{306176}a^{12}-\frac{11197}{306176}a^{11}+\frac{6281}{153088}a^{10}+\frac{1707}{306176}a^{9}-\frac{2373}{23552}a^{8}-\frac{52391}{306176}a^{7}+\frac{8635}{38272}a^{6}+\frac{8159}{38272}a^{5}+\frac{661}{9568}a^{4}-\frac{979}{2392}a^{3}-\frac{49}{104}a^{2}-\frac{53}{4784}a+\frac{1}{13}$, $\frac{1}{23269376}a^{26}+\frac{7}{126464}a^{24}+\frac{191}{23269376}a^{23}+\frac{1719}{23269376}a^{22}-\frac{301}{11634688}a^{21}+\frac{4099}{727168}a^{20}+\frac{162711}{23269376}a^{19}-\frac{2111}{612352}a^{18}-\frac{146399}{11634688}a^{17}+\frac{25943}{11634688}a^{16}-\frac{28749}{1454336}a^{15}-\frac{129419}{23269376}a^{14}+\frac{163731}{11634688}a^{13}-\frac{2635}{45448}a^{12}+\frac{578479}{23269376}a^{11}-\frac{57887}{1224704}a^{10}-\frac{533619}{11634688}a^{9}+\frac{23611}{505856}a^{8}-\frac{1740317}{23269376}a^{7}+\frac{8971}{612352}a^{6}-\frac{72991}{363584}a^{5}-\frac{25531}{223744}a^{4}+\frac{140241}{363584}a^{3}+\frac{212161}{727168}a^{2}-\frac{175367}{363584}a+\frac{73}{988}$, $\frac{1}{77\!\cdots\!08}a^{27}+\frac{1003364641}{77\!\cdots\!08}a^{26}-\frac{12732044201}{96\!\cdots\!76}a^{25}+\frac{14348545361}{33\!\cdots\!96}a^{24}-\frac{261437087097}{29\!\cdots\!08}a^{23}-\frac{7075224106163}{77\!\cdots\!08}a^{22}+\frac{3481158309595}{38\!\cdots\!04}a^{21}+\frac{17000181779923}{59\!\cdots\!16}a^{20}+\frac{465783261641501}{77\!\cdots\!08}a^{19}-\frac{19284418923381}{96\!\cdots\!76}a^{18}+\frac{70758806726225}{48\!\cdots\!88}a^{17}+\frac{581445909765791}{38\!\cdots\!04}a^{16}-\frac{13\!\cdots\!91}{77\!\cdots\!08}a^{15}-\frac{2740215813125}{77\!\cdots\!08}a^{14}+\frac{315087468740011}{38\!\cdots\!04}a^{13}+\frac{39\!\cdots\!71}{77\!\cdots\!08}a^{12}-\frac{26\!\cdots\!27}{38\!\cdots\!04}a^{11}+\frac{53\!\cdots\!25}{77\!\cdots\!08}a^{10}+\frac{30260596868299}{326961562390528}a^{9}+\frac{92\!\cdots\!81}{77\!\cdots\!08}a^{8}-\frac{97\!\cdots\!47}{77\!\cdots\!08}a^{7}+\frac{88\!\cdots\!73}{38\!\cdots\!04}a^{6}+\frac{22\!\cdots\!25}{96\!\cdots\!76}a^{5}+\frac{19\!\cdots\!69}{96\!\cdots\!76}a^{4}-\frac{619728370329145}{24\!\cdots\!44}a^{3}-\frac{63556245260825}{185487809433088}a^{2}+\frac{48655840106577}{12\!\cdots\!72}a+\frac{261103363169}{3276279242704}$, $\frac{1}{81\!\cdots\!44}a^{28}+\frac{2229146955127}{40\!\cdots\!72}a^{27}-\frac{14\!\cdots\!67}{81\!\cdots\!44}a^{26}+\frac{64\!\cdots\!99}{81\!\cdots\!44}a^{25}-\frac{60\!\cdots\!79}{81\!\cdots\!44}a^{24}-\frac{79\!\cdots\!05}{81\!\cdots\!44}a^{23}+\frac{26\!\cdots\!15}{23\!\cdots\!48}a^{22}-\frac{29\!\cdots\!99}{81\!\cdots\!44}a^{21}-\frac{66\!\cdots\!99}{10\!\cdots\!68}a^{20}+\frac{52\!\cdots\!21}{81\!\cdots\!44}a^{19}-\frac{12\!\cdots\!67}{17\!\cdots\!12}a^{18}-\frac{46\!\cdots\!27}{49\!\cdots\!84}a^{17}-\frac{40\!\cdots\!77}{81\!\cdots\!44}a^{16}-\frac{42\!\cdots\!49}{20\!\cdots\!36}a^{15}+\frac{56\!\cdots\!57}{81\!\cdots\!44}a^{14}-\frac{69\!\cdots\!47}{81\!\cdots\!44}a^{13}+\frac{18\!\cdots\!39}{35\!\cdots\!28}a^{12}+\frac{69\!\cdots\!55}{81\!\cdots\!44}a^{11}+\frac{88\!\cdots\!85}{81\!\cdots\!44}a^{10}+\frac{34\!\cdots\!13}{81\!\cdots\!44}a^{9}-\frac{36\!\cdots\!73}{40\!\cdots\!72}a^{8}-\frac{83\!\cdots\!25}{81\!\cdots\!44}a^{7}+\frac{30\!\cdots\!63}{13\!\cdots\!12}a^{6}-\frac{64\!\cdots\!91}{39\!\cdots\!68}a^{5}+\frac{15\!\cdots\!85}{10\!\cdots\!68}a^{4}+\frac{17\!\cdots\!29}{41\!\cdots\!16}a^{3}-\frac{43\!\cdots\!03}{19\!\cdots\!84}a^{2}+\frac{24\!\cdots\!95}{55\!\cdots\!52}a+\frac{53\!\cdots\!41}{34\!\cdots\!72}$ 1, a, a^2, a^3, 1/2*a^4 - 1/2*a, 1/2*a^5 - 1/2*a^2, 1/2*a^6 - 1/2*a^3, 1/2*a^7 - 1/2*a, 1/4*a^8 - 1/4*a^2, 1/4*a^9 - 1/4*a^3, 1/4*a^10 - 1/4*a^4, 1/4*a^11 - 1/4*a^5, 1/8*a^12 - 1/8*a^9 - 1/8*a^6 + 1/8*a^3, 1/16*a^13 - 1/16*a^12 - 1/8*a^11 - 1/16*a^10 + 1/16*a^9 - 1/8*a^8 - 1/16*a^7 - 3/16*a^6 + 1/8*a^5 - 3/16*a^4 + 3/16*a^3 + 1/8*a^2 + 1/4*a, 1/16*a^14 - 1/16*a^12 + 1/16*a^11 + 1/16*a^9 + 1/16*a^8 - 1/4*a^7 - 3/16*a^6 + 3/16*a^5 + 3/16*a^3 - 3/8*a^2 + 1/4*a, 1/16*a^15 - 1/8*a^11 - 1/8*a^9 - 1/8*a^8 - 1/4*a^7 + 1/8*a^5 + 1/16*a^3 + 1/8*a^2 + 1/4*a, 1/32*a^16 + 1/16*a^10 - 1/4*a^7 - 3/32*a^4 + 1/4*a, 1/32*a^17 + 1/16*a^11 - 3/32*a^5, 1/64*a^18 - 1/64*a^17 - 1/32*a^14 - 1/32*a^13 + 1/32*a^12 - 3/32*a^10 + 1/32*a^8 - 3/32*a^7 + 13/64*a^6 + 9/64*a^5 - 1/32*a^4 - 1/4*a^3 - 1/8*a^2 + 1/4*a, 1/64*a^19 - 1/64*a^17 - 1/32*a^15 - 1/32*a^12 - 1/32*a^11 - 3/32*a^10 + 3/32*a^9 - 9/64*a^7 + 5/32*a^6 - 13/64*a^5 + 7/32*a^4 - 3/16*a^3 + 1/4*a^2, 1/64*a^20 - 1/64*a^17 - 1/32*a^14 - 1/16*a^12 + 1/32*a^11 + 1/16*a^9 + 1/64*a^8 - 1/4*a^7 - 3/16*a^6 + 15/64*a^5 + 3/16*a^3 - 1/4*a^2 + 1/4*a, 1/1664*a^21 - 9/1664*a^20 - 3/832*a^19 - 3/1664*a^18 + 17/1664*a^17 - 3/416*a^16 - 1/64*a^15 + 3/832*a^14 - 7/416*a^13 - 23/832*a^12 + 45/832*a^11 + 1/16*a^10 + 189/1664*a^9 + 3/1664*a^8 - 55/832*a^7 + 337/1664*a^6 - 235/1664*a^5 - 47/416*a^4 + 111/416*a^3 - 3/13*a^2 - 41/104*a - 4/13, 1/3328*a^22 - 9/3328*a^20 + 21/3328*a^19 + 1/208*a^18 - 41/3328*a^17 + 11/1664*a^16 + 1/104*a^15 + 1/128*a^14 + 33/1664*a^13 - 29/832*a^12 + 145/1664*a^11 + 397/3328*a^10 - 55/832*a^9 - 161/3328*a^8 + 569/3328*a^7 - 35/832*a^6 + 375/3328*a^5 + 11/64*a^4 + 331/832*a^3 + 21/104*a^2 + 41/208*a - 5/13, 1/3328*a^23 - 1/3328*a^21 + 1/3328*a^20 + 5/832*a^19 - 1/256*a^18 + 1/1664*a^17 + 5/416*a^16 - 3/128*a^15 - 47/1664*a^14 - 7/832*a^13 + 5/128*a^12 - 131/3328*a^11 - 55/832*a^10 + 207/3328*a^9 + 333/3328*a^8 + 35/416*a^7 - 101/3328*a^6 + 2/13*a^5 + 137/832*a^4 + 95/208*a^3 + 57/208*a^2 - 6/13*a - 3/13, 1/6656*a^24 - 1/6656*a^23 - 1/6656*a^22 - 1/3328*a^21 - 49/6656*a^20 - 9/6656*a^19 + 27/6656*a^18 - 15/3328*a^17 + 17/3328*a^16 + 11/832*a^15 + 21/3328*a^14 + 31/3328*a^13 + 339/6656*a^12 - 657/6656*a^11 - 405/6656*a^10 + 309/3328*a^9 - 61/512*a^8 - 1397/6656*a^7 - 1151/6656*a^6 + 11/208*a^5 + 301/1664*a^4 + 189/416*a^3 + 47/416*a^2 + 5/13*a - 1/13, 1/306176*a^25 + 11/306176*a^24 - 17/306176*a^23 - 21/153088*a^22 - 77/306176*a^21 - 3/13312*a^20 + 1379/306176*a^19 + 53/11776*a^18 + 2315/153088*a^17 - 1/1664*a^16 - 3175/153088*a^15 - 3453/153088*a^14 - 449/23552*a^13 - 17621/306176*a^12 - 11197/306176*a^11 + 6281/153088*a^10 + 1707/306176*a^9 - 2373/23552*a^8 - 52391/306176*a^7 + 8635/38272*a^6 + 8159/38272*a^5 + 661/9568*a^4 - 979/2392*a^3 - 49/104*a^2 - 53/4784*a + 1/13, 1/23269376*a^26 + 7/126464*a^24 + 191/23269376*a^23 + 1719/23269376*a^22 - 301/11634688*a^21 + 4099/727168*a^20 + 162711/23269376*a^19 - 2111/612352*a^18 - 146399/11634688*a^17 + 25943/11634688*a^16 - 28749/1454336*a^15 - 129419/23269376*a^14 + 163731/11634688*a^13 - 2635/45448*a^12 + 578479/23269376*a^11 - 57887/1224704*a^10 - 533619/11634688*a^9 + 23611/505856*a^8 - 1740317/23269376*a^7 + 8971/612352*a^6 - 72991/363584*a^5 - 25531/223744*a^4 + 140241/363584*a^3 + 212161/727168*a^2 - 175367/363584*a + 73/988, 1/77162928724164608*a^27 + 1003364641/77162928724164608*a^26 - 12732044201/9645366090520576*a^25 + 14348545361/3354909944528896*a^24 - 261437087097/2967804950929408*a^23 - 7075224106163/77162928724164608*a^22 + 3481158309595/38581464362082304*a^21 + 17000181779923/5935609901858816*a^20 + 465783261641501/77162928724164608*a^19 - 19284418923381/9645366090520576*a^18 + 70758806726225/4822683045260288*a^17 + 581445909765791/38581464362082304*a^16 - 1319854945552891/77162928724164608*a^15 - 2740215813125/77162928724164608*a^14 + 315087468740011/38581464362082304*a^13 + 3991031730294671/77162928724164608*a^12 - 2630765209500527/38581464362082304*a^11 + 5384823552449725/77162928724164608*a^10 + 30260596868299/326961562390528*a^9 + 9225470237912381/77162928724164608*a^8 - 9732207301119547/77162928724164608*a^7 + 8862989339089673/38581464362082304*a^6 + 2217476553913125/9645366090520576*a^5 + 1910522689713569/9645366090520576*a^4 - 619728370329145/2411341522630144*a^3 - 63556245260825/185487809433088*a^2 + 48655840106577/1205670761315072*a + 261103363169/3276279242704, 1/8181594473904530216746692050944*a^28 + 2229146955127/4090797236952265108373346025472*a^27 - 145414354073906263853067/8181594473904530216746692050944*a^26 + 6425519837908305834795999/8181594473904530216746692050944*a^25 - 607770813895353108907552079/8181594473904530216746692050944*a^24 - 79206506749197482353896805/8181594473904530216746692050944*a^23 + 2613239436210306433452815/23177321455820198914296578048*a^22 - 292001098392336423017606699/8181594473904530216746692050944*a^21 - 6695885046011675839903995999/1022699309238066277093336506368*a^20 + 52977417459091256155096281121/8181594473904530216746692050944*a^19 - 12368970819467476880421467/1736331594631691472144883712*a^18 - 466605655230128073752885427/49286713698220061546666819584*a^17 - 40539081002108544773506791477/8181594473904530216746692050944*a^16 - 4226709481768994556402631649/2045398618476132554186673012736*a^15 + 56631461642532933816965678757/8181594473904530216746692050944*a^14 - 69615242631981448898246996147/8181594473904530216746692050944*a^13 + 18963397449751163845858211139/355721498865414357249856176128*a^12 + 696758728371073847747404517255/8181594473904530216746692050944*a^11 + 885346189391873733043599644685/8181594473904530216746692050944*a^10 + 340004323456127957223831928113/8181594473904530216746692050944*a^9 - 361752595188922427263976917873/4090797236952265108373346025472*a^8 - 83007820237493759139775112925/8181594473904530216746692050944*a^7 + 306284202111884180994157463/131961201192008551883011162112*a^6 - 6494110117431717178283561291/39334588816848702965128327168*a^5 + 154542471808164440895629349385/1022699309238066277093336506368*a^4 + 1785136913558617700880009629/4123787537250267246344098816*a^3 - 4358423265633799904483926203/19667294408424351482564163584*a^2 + 2461497756275785857135869995/5558148419772099332029002752*a + 53933202961865624049220541/347384276235756208251812672

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

Trivial group, which has order $1$ (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:	$14$	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:	$\frac{68\!\cdots\!35}{81\!\cdots\!44}a^{28}-\frac{30\!\cdots\!89}{40\!\cdots\!72}a^{27}+\frac{40\!\cdots\!59}{81\!\cdots\!44}a^{26}-\frac{78\!\cdots\!29}{35\!\cdots\!28}a^{25}+\frac{51\!\cdots\!55}{62\!\cdots\!88}a^{24}-\frac{13\!\cdots\!07}{81\!\cdots\!44}a^{23}+\frac{25\!\cdots\!45}{17\!\cdots\!96}a^{22}+\frac{10\!\cdots\!37}{43\!\cdots\!76}a^{21}-\frac{14\!\cdots\!83}{10\!\cdots\!44}a^{20}+\frac{20\!\cdots\!51}{81\!\cdots\!44}a^{19}-\frac{15\!\cdots\!91}{53\!\cdots\!72}a^{18}+\frac{58\!\cdots\!05}{40\!\cdots\!72}a^{17}+\frac{52\!\cdots\!69}{81\!\cdots\!44}a^{16}-\frac{17\!\cdots\!61}{12\!\cdots\!96}a^{15}+\frac{13\!\cdots\!03}{81\!\cdots\!44}a^{14}-\frac{53\!\cdots\!61}{62\!\cdots\!88}a^{13}-\frac{83\!\cdots\!89}{81\!\cdots\!44}a^{12}+\frac{13\!\cdots\!03}{43\!\cdots\!76}a^{11}-\frac{31\!\cdots\!53}{81\!\cdots\!44}a^{10}+\frac{19\!\cdots\!59}{81\!\cdots\!44}a^{9}+\frac{43\!\cdots\!35}{31\!\cdots\!44}a^{8}-\frac{44\!\cdots\!07}{81\!\cdots\!44}a^{7}+\frac{34\!\cdots\!67}{40\!\cdots\!72}a^{6}-\frac{58\!\cdots\!99}{62\!\cdots\!52}a^{5}+\frac{21\!\cdots\!03}{28\!\cdots\!72}a^{4}-\frac{27\!\cdots\!63}{55\!\cdots\!52}a^{3}+\frac{64\!\cdots\!59}{25\!\cdots\!92}a^{2}-\frac{10\!\cdots\!57}{12\!\cdots\!96}a+\frac{90\!\cdots\!47}{34\!\cdots\!72}$, $\frac{34\!\cdots\!75}{82\!\cdots\!24}a^{28}-\frac{11\!\cdots\!53}{41\!\cdots\!12}a^{27}+\frac{15\!\cdots\!79}{82\!\cdots\!24}a^{26}-\frac{61\!\cdots\!71}{82\!\cdots\!24}a^{25}+\frac{23\!\cdots\!27}{82\!\cdots\!24}a^{24}-\frac{29\!\cdots\!91}{82\!\cdots\!24}a^{23}+\frac{10\!\cdots\!33}{23\!\cdots\!08}a^{22}+\frac{83\!\cdots\!75}{82\!\cdots\!24}a^{21}-\frac{68\!\cdots\!09}{20\!\cdots\!56}a^{20}+\frac{53\!\cdots\!19}{82\!\cdots\!24}a^{19}-\frac{80\!\cdots\!33}{10\!\cdots\!28}a^{18}+\frac{20\!\cdots\!61}{41\!\cdots\!12}a^{17}+\frac{10\!\cdots\!49}{82\!\cdots\!24}a^{16}-\frac{83\!\cdots\!09}{25\!\cdots\!32}a^{15}+\frac{36\!\cdots\!23}{82\!\cdots\!24}a^{14}-\frac{28\!\cdots\!49}{82\!\cdots\!24}a^{13}+\frac{13\!\cdots\!19}{82\!\cdots\!24}a^{12}+\frac{43\!\cdots\!65}{82\!\cdots\!24}a^{11}-\frac{80\!\cdots\!17}{82\!\cdots\!24}a^{10}+\frac{76\!\cdots\!63}{82\!\cdots\!24}a^{9}-\frac{14\!\cdots\!53}{41\!\cdots\!12}a^{8}-\frac{67\!\cdots\!31}{82\!\cdots\!24}a^{7}+\frac{96\!\cdots\!07}{41\!\cdots\!12}a^{6}-\frac{15\!\cdots\!91}{51\!\cdots\!64}a^{5}+\frac{36\!\cdots\!67}{10\!\cdots\!28}a^{4}-\frac{37\!\cdots\!41}{12\!\cdots\!16}a^{3}+\frac{48\!\cdots\!51}{25\!\cdots\!32}a^{2}-\frac{14\!\cdots\!77}{12\!\cdots\!16}a+\frac{10\!\cdots\!39}{34\!\cdots\!12}$, $\frac{46\!\cdots\!19}{62\!\cdots\!88}a^{28}-\frac{19\!\cdots\!21}{17\!\cdots\!64}a^{27}+\frac{47\!\cdots\!87}{81\!\cdots\!44}a^{26}-\frac{24\!\cdots\!23}{81\!\cdots\!44}a^{25}+\frac{74\!\cdots\!95}{81\!\cdots\!44}a^{24}-\frac{19\!\cdots\!43}{81\!\cdots\!44}a^{23}-\frac{63\!\cdots\!27}{23\!\cdots\!48}a^{22}+\frac{16\!\cdots\!91}{81\!\cdots\!44}a^{21}-\frac{24\!\cdots\!63}{10\!\cdots\!68}a^{20}+\frac{71\!\cdots\!83}{81\!\cdots\!44}a^{19}-\frac{58\!\cdots\!97}{10\!\cdots\!68}a^{18}-\frac{11\!\cdots\!39}{40\!\cdots\!72}a^{17}+\frac{85\!\cdots\!13}{81\!\cdots\!44}a^{16}-\frac{91\!\cdots\!83}{20\!\cdots\!36}a^{15}+\frac{11\!\cdots\!65}{13\!\cdots\!16}a^{14}+\frac{77\!\cdots\!27}{81\!\cdots\!44}a^{13}-\frac{10\!\cdots\!57}{81\!\cdots\!44}a^{12}+\frac{17\!\cdots\!45}{81\!\cdots\!44}a^{11}-\frac{61\!\cdots\!65}{81\!\cdots\!44}a^{10}+\frac{50\!\cdots\!15}{81\!\cdots\!44}a^{9}+\frac{21\!\cdots\!73}{40\!\cdots\!72}a^{8}-\frac{10\!\cdots\!91}{81\!\cdots\!44}a^{7}+\frac{28\!\cdots\!71}{40\!\cdots\!72}a^{6}-\frac{60\!\cdots\!87}{16\!\cdots\!64}a^{5}-\frac{40\!\cdots\!97}{10\!\cdots\!68}a^{4}-\frac{12\!\cdots\!93}{29\!\cdots\!72}a^{3}+\frac{15\!\cdots\!89}{82\!\cdots\!32}a^{2}+\frac{12\!\cdots\!07}{12\!\cdots\!96}a+\frac{58\!\cdots\!03}{34\!\cdots\!72}$, $\frac{29\!\cdots\!03}{35\!\cdots\!28}a^{28}-\frac{16\!\cdots\!61}{31\!\cdots\!44}a^{27}+\frac{27\!\cdots\!37}{81\!\cdots\!44}a^{26}-\frac{75\!\cdots\!57}{62\!\cdots\!88}a^{25}+\frac{35\!\cdots\!93}{81\!\cdots\!44}a^{24}-\frac{15\!\cdots\!45}{81\!\cdots\!44}a^{23}-\frac{75\!\cdots\!81}{23\!\cdots\!48}a^{22}+\frac{28\!\cdots\!01}{81\!\cdots\!44}a^{21}-\frac{31\!\cdots\!17}{51\!\cdots\!84}a^{20}+\frac{43\!\cdots\!33}{81\!\cdots\!44}a^{19}+\frac{24\!\cdots\!69}{10\!\cdots\!68}a^{18}-\frac{99\!\cdots\!85}{40\!\cdots\!72}a^{17}+\frac{33\!\cdots\!83}{81\!\cdots\!44}a^{16}-\frac{77\!\cdots\!63}{20\!\cdots\!36}a^{15}-\frac{65\!\cdots\!41}{35\!\cdots\!28}a^{14}+\frac{61\!\cdots\!73}{81\!\cdots\!44}a^{13}-\frac{10\!\cdots\!51}{81\!\cdots\!44}a^{12}+\frac{10\!\cdots\!95}{81\!\cdots\!44}a^{11}-\frac{32\!\cdots\!95}{81\!\cdots\!44}a^{10}-\frac{46\!\cdots\!61}{33\!\cdots\!52}a^{9}+\frac{11\!\cdots\!83}{40\!\cdots\!72}a^{8}-\frac{13\!\cdots\!51}{43\!\cdots\!76}a^{7}+\frac{93\!\cdots\!97}{40\!\cdots\!72}a^{6}+\frac{41\!\cdots\!97}{51\!\cdots\!84}a^{5}-\frac{30\!\cdots\!83}{10\!\cdots\!68}a^{4}+\frac{52\!\cdots\!35}{12\!\cdots\!96}a^{3}-\frac{52\!\cdots\!57}{13\!\cdots\!68}a^{2}+\frac{25\!\cdots\!09}{12\!\cdots\!96}a-\frac{12\!\cdots\!35}{34\!\cdots\!72}$, $\frac{14\!\cdots\!81}{40\!\cdots\!72}a^{28}-\frac{59\!\cdots\!49}{20\!\cdots\!36}a^{27}+\frac{77\!\cdots\!65}{40\!\cdots\!72}a^{26}-\frac{32\!\cdots\!49}{40\!\cdots\!72}a^{25}+\frac{11\!\cdots\!61}{40\!\cdots\!72}a^{24}-\frac{19\!\cdots\!01}{40\!\cdots\!72}a^{23}+\frac{29\!\cdots\!19}{11\!\cdots\!24}a^{22}+\frac{50\!\cdots\!05}{40\!\cdots\!72}a^{21}-\frac{55\!\cdots\!35}{12\!\cdots\!96}a^{20}+\frac{12\!\cdots\!43}{17\!\cdots\!64}a^{19}-\frac{30\!\cdots\!61}{51\!\cdots\!84}a^{18}-\frac{64\!\cdots\!25}{20\!\cdots\!36}a^{17}+\frac{62\!\cdots\!91}{31\!\cdots\!44}a^{16}-\frac{35\!\cdots\!03}{10\!\cdots\!68}a^{15}+\frac{15\!\cdots\!65}{40\!\cdots\!72}a^{14}-\frac{66\!\cdots\!31}{40\!\cdots\!72}a^{13}-\frac{83\!\cdots\!75}{31\!\cdots\!44}a^{12}+\frac{21\!\cdots\!59}{31\!\cdots\!44}a^{11}-\frac{28\!\cdots\!87}{31\!\cdots\!44}a^{10}+\frac{23\!\cdots\!53}{40\!\cdots\!72}a^{9}+\frac{61\!\cdots\!71}{20\!\cdots\!36}a^{8}-\frac{54\!\cdots\!17}{40\!\cdots\!72}a^{7}+\frac{46\!\cdots\!09}{20\!\cdots\!36}a^{6}-\frac{67\!\cdots\!95}{25\!\cdots\!92}a^{5}+\frac{11\!\cdots\!41}{51\!\cdots\!84}a^{4}-\frac{10\!\cdots\!09}{63\!\cdots\!48}a^{3}+\frac{12\!\cdots\!33}{12\!\cdots\!96}a^{2}-\frac{17\!\cdots\!19}{49\!\cdots\!96}a+\frac{26\!\cdots\!97}{48\!\cdots\!44}$, $\frac{29\!\cdots\!31}{81\!\cdots\!44}a^{28}-\frac{10\!\cdots\!25}{40\!\cdots\!72}a^{27}+\frac{13\!\cdots\!67}{81\!\cdots\!44}a^{26}-\frac{49\!\cdots\!55}{81\!\cdots\!44}a^{25}+\frac{18\!\cdots\!35}{81\!\cdots\!44}a^{24}-\frac{17\!\cdots\!31}{81\!\cdots\!44}a^{23}-\frac{22\!\cdots\!11}{23\!\cdots\!48}a^{22}+\frac{10\!\cdots\!19}{81\!\cdots\!44}a^{21}-\frac{60\!\cdots\!79}{20\!\cdots\!36}a^{20}+\frac{28\!\cdots\!67}{81\!\cdots\!44}a^{19}-\frac{17\!\cdots\!09}{10\!\cdots\!68}a^{18}-\frac{22\!\cdots\!47}{40\!\cdots\!72}a^{17}+\frac{11\!\cdots\!25}{81\!\cdots\!44}a^{16}-\frac{10\!\cdots\!07}{53\!\cdots\!72}a^{15}+\frac{12\!\cdots\!47}{81\!\cdots\!44}a^{14}+\frac{25\!\cdots\!11}{81\!\cdots\!44}a^{13}-\frac{11\!\cdots\!83}{43\!\cdots\!76}a^{12}+\frac{33\!\cdots\!09}{81\!\cdots\!44}a^{11}-\frac{40\!\cdots\!03}{98\!\cdots\!68}a^{10}+\frac{55\!\cdots\!83}{81\!\cdots\!44}a^{9}+\frac{18\!\cdots\!23}{40\!\cdots\!72}a^{8}-\frac{69\!\cdots\!87}{81\!\cdots\!44}a^{7}+\frac{51\!\cdots\!15}{40\!\cdots\!72}a^{6}-\frac{57\!\cdots\!07}{51\!\cdots\!84}a^{5}+\frac{81\!\cdots\!35}{10\!\cdots\!68}a^{4}-\frac{67\!\cdots\!77}{12\!\cdots\!96}a^{3}+\frac{55\!\cdots\!35}{25\!\cdots\!92}a^{2}-\frac{64\!\cdots\!73}{12\!\cdots\!96}a-\frac{36\!\cdots\!37}{34\!\cdots\!72}$, $\frac{93\!\cdots\!79}{81\!\cdots\!44}a^{28}-\frac{37\!\cdots\!13}{40\!\cdots\!72}a^{27}+\frac{47\!\cdots\!91}{81\!\cdots\!44}a^{26}-\frac{19\!\cdots\!83}{81\!\cdots\!44}a^{25}+\frac{68\!\cdots\!39}{81\!\cdots\!44}a^{24}-\frac{98\!\cdots\!27}{81\!\cdots\!44}a^{23}-\frac{48\!\cdots\!63}{23\!\cdots\!48}a^{22}+\frac{31\!\cdots\!99}{81\!\cdots\!44}a^{21}-\frac{18\!\cdots\!71}{15\!\cdots\!72}a^{20}+\frac{33\!\cdots\!45}{26\!\cdots\!24}a^{19}-\frac{52\!\cdots\!53}{10\!\cdots\!68}a^{18}-\frac{63\!\cdots\!51}{40\!\cdots\!72}a^{17}+\frac{11\!\cdots\!47}{26\!\cdots\!24}a^{16}-\frac{30\!\cdots\!91}{80\!\cdots\!84}a^{15}+\frac{60\!\cdots\!21}{35\!\cdots\!28}a^{14}+\frac{15\!\cdots\!15}{81\!\cdots\!44}a^{13}-\frac{28\!\cdots\!37}{81\!\cdots\!44}a^{12}-\frac{15\!\cdots\!73}{43\!\cdots\!76}a^{11}+\frac{11\!\cdots\!43}{81\!\cdots\!44}a^{10}-\frac{71\!\cdots\!43}{26\!\cdots\!24}a^{9}+\frac{14\!\cdots\!83}{40\!\cdots\!72}a^{8}-\frac{14\!\cdots\!27}{81\!\cdots\!44}a^{7}-\frac{11\!\cdots\!39}{13\!\cdots\!28}a^{6}+\frac{18\!\cdots\!57}{51\!\cdots\!84}a^{5}-\frac{86\!\cdots\!49}{10\!\cdots\!68}a^{4}+\frac{89\!\cdots\!67}{12\!\cdots\!96}a^{3}-\frac{25\!\cdots\!67}{43\!\cdots\!88}a^{2}+\frac{67\!\cdots\!23}{12\!\cdots\!96}a-\frac{79\!\cdots\!57}{34\!\cdots\!72}$, $\frac{61\!\cdots\!83}{40\!\cdots\!72}a^{28}-\frac{20\!\cdots\!27}{34\!\cdots\!04}a^{27}+\frac{14\!\cdots\!15}{40\!\cdots\!72}a^{26}-\frac{64\!\cdots\!33}{13\!\cdots\!12}a^{25}+\frac{45\!\cdots\!95}{40\!\cdots\!72}a^{24}+\frac{14\!\cdots\!37}{69\!\cdots\!48}a^{23}-\frac{34\!\cdots\!11}{89\!\cdots\!48}a^{22}+\frac{90\!\cdots\!13}{13\!\cdots\!12}a^{21}+\frac{25\!\cdots\!67}{10\!\cdots\!68}a^{20}-\frac{11\!\cdots\!97}{40\!\cdots\!72}a^{19}+\frac{27\!\cdots\!29}{51\!\cdots\!84}a^{18}-\frac{13\!\cdots\!63}{20\!\cdots\!36}a^{17}+\frac{14\!\cdots\!45}{40\!\cdots\!72}a^{16}+\frac{68\!\cdots\!19}{55\!\cdots\!52}a^{15}-\frac{10\!\cdots\!85}{40\!\cdots\!72}a^{14}+\frac{12\!\cdots\!23}{40\!\cdots\!72}a^{13}-\frac{63\!\cdots\!45}{40\!\cdots\!72}a^{12}-\frac{65\!\cdots\!79}{40\!\cdots\!72}a^{11}+\frac{20\!\cdots\!03}{40\!\cdots\!72}a^{10}-\frac{29\!\cdots\!17}{40\!\cdots\!72}a^{9}+\frac{32\!\cdots\!97}{65\!\cdots\!56}a^{8}+\frac{98\!\cdots\!83}{69\!\cdots\!08}a^{7}-\frac{15\!\cdots\!89}{20\!\cdots\!36}a^{6}+\frac{21\!\cdots\!75}{13\!\cdots\!68}a^{5}-\frac{87\!\cdots\!09}{51\!\cdots\!84}a^{4}+\frac{91\!\cdots\!11}{63\!\cdots\!48}a^{3}-\frac{13\!\cdots\!77}{12\!\cdots\!96}a^{2}+\frac{33\!\cdots\!03}{63\!\cdots\!48}a-\frac{24\!\cdots\!53}{17\!\cdots\!36}$, $\frac{32\!\cdots\!95}{81\!\cdots\!44}a^{28}-\frac{78\!\cdots\!37}{31\!\cdots\!44}a^{27}+\frac{99\!\cdots\!75}{62\!\cdots\!88}a^{26}-\frac{45\!\cdots\!95}{81\!\cdots\!44}a^{25}+\frac{16\!\cdots\!99}{81\!\cdots\!44}a^{24}-\frac{63\!\cdots\!75}{81\!\cdots\!44}a^{23}-\frac{27\!\cdots\!33}{10\!\cdots\!76}a^{22}+\frac{94\!\cdots\!35}{62\!\cdots\!88}a^{21}-\frac{51\!\cdots\!89}{20\!\cdots\!36}a^{20}+\frac{36\!\cdots\!21}{26\!\cdots\!24}a^{19}+\frac{20\!\cdots\!91}{10\!\cdots\!68}a^{18}-\frac{37\!\cdots\!35}{40\!\cdots\!72}a^{17}+\frac{36\!\cdots\!67}{26\!\cdots\!24}a^{16}-\frac{24\!\cdots\!07}{25\!\cdots\!92}a^{15}-\frac{28\!\cdots\!01}{81\!\cdots\!44}a^{14}+\frac{19\!\cdots\!23}{81\!\cdots\!44}a^{13}-\frac{30\!\cdots\!01}{81\!\cdots\!44}a^{12}+\frac{24\!\cdots\!89}{81\!\cdots\!44}a^{11}-\frac{37\!\cdots\!21}{62\!\cdots\!88}a^{10}-\frac{47\!\cdots\!09}{11\!\cdots\!88}a^{9}+\frac{32\!\cdots\!67}{40\!\cdots\!72}a^{8}-\frac{61\!\cdots\!99}{81\!\cdots\!44}a^{7}+\frac{26\!\cdots\!19}{40\!\cdots\!72}a^{6}-\frac{73\!\cdots\!45}{22\!\cdots\!08}a^{5}-\frac{58\!\cdots\!89}{10\!\cdots\!68}a^{4}+\frac{84\!\cdots\!95}{12\!\cdots\!96}a^{3}-\frac{16\!\cdots\!13}{25\!\cdots\!92}a^{2}+\frac{68\!\cdots\!87}{12\!\cdots\!96}a-\frac{62\!\cdots\!53}{34\!\cdots\!72}$, $\frac{24\!\cdots\!43}{81\!\cdots\!44}a^{28}-\frac{11\!\cdots\!57}{40\!\cdots\!72}a^{27}+\frac{15\!\cdots\!75}{81\!\cdots\!44}a^{26}-\frac{70\!\cdots\!83}{81\!\cdots\!44}a^{25}+\frac{27\!\cdots\!59}{81\!\cdots\!44}a^{24}-\frac{47\!\cdots\!89}{64\!\cdots\!72}a^{23}+\frac{22\!\cdots\!61}{23\!\cdots\!48}a^{22}+\frac{32\!\cdots\!35}{81\!\cdots\!44}a^{21}-\frac{37\!\cdots\!71}{88\!\cdots\!32}a^{20}+\frac{95\!\cdots\!91}{81\!\cdots\!44}a^{19}-\frac{69\!\cdots\!19}{44\!\cdots\!16}a^{18}+\frac{21\!\cdots\!23}{21\!\cdots\!88}a^{17}+\frac{10\!\cdots\!29}{81\!\cdots\!44}a^{16}-\frac{14\!\cdots\!75}{25\!\cdots\!92}a^{15}+\frac{36\!\cdots\!17}{43\!\cdots\!76}a^{14}-\frac{54\!\cdots\!13}{62\!\cdots\!88}a^{13}+\frac{27\!\cdots\!23}{81\!\cdots\!44}a^{12}+\frac{36\!\cdots\!95}{43\!\cdots\!76}a^{11}-\frac{14\!\cdots\!85}{81\!\cdots\!44}a^{10}+\frac{16\!\cdots\!63}{81\!\cdots\!44}a^{9}-\frac{57\!\cdots\!49}{40\!\cdots\!72}a^{8}-\frac{94\!\cdots\!23}{81\!\cdots\!44}a^{7}+\frac{15\!\cdots\!91}{40\!\cdots\!72}a^{6}-\frac{27\!\cdots\!47}{51\!\cdots\!84}a^{5}+\frac{72\!\cdots\!63}{10\!\cdots\!68}a^{4}-\frac{73\!\cdots\!01}{12\!\cdots\!96}a^{3}+\frac{10\!\cdots\!51}{25\!\cdots\!92}a^{2}-\frac{33\!\cdots\!61}{12\!\cdots\!96}a+\frac{34\!\cdots\!59}{34\!\cdots\!72}$, $\frac{57\!\cdots\!85}{81\!\cdots\!44}a^{28}-\frac{27\!\cdots\!73}{31\!\cdots\!44}a^{27}+\frac{46\!\cdots\!41}{81\!\cdots\!44}a^{26}-\frac{23\!\cdots\!57}{81\!\cdots\!44}a^{25}+\frac{86\!\cdots\!25}{81\!\cdots\!44}a^{24}-\frac{22\!\cdots\!13}{81\!\cdots\!44}a^{23}+\frac{51\!\cdots\!91}{23\!\cdots\!48}a^{22}+\frac{10\!\cdots\!57}{81\!\cdots\!44}a^{21}-\frac{90\!\cdots\!83}{51\!\cdots\!84}a^{20}+\frac{29\!\cdots\!29}{81\!\cdots\!44}a^{19}-\frac{44\!\cdots\!19}{10\!\cdots\!68}a^{18}+\frac{87\!\cdots\!59}{40\!\cdots\!72}a^{17}+\frac{56\!\cdots\!39}{81\!\cdots\!44}a^{16}-\frac{34\!\cdots\!11}{20\!\cdots\!36}a^{15}+\frac{20\!\cdots\!89}{81\!\cdots\!44}a^{14}-\frac{17\!\cdots\!59}{81\!\cdots\!44}a^{13}+\frac{15\!\cdots\!05}{62\!\cdots\!88}a^{12}+\frac{22\!\cdots\!59}{81\!\cdots\!44}a^{11}-\frac{33\!\cdots\!39}{62\!\cdots\!88}a^{10}+\frac{45\!\cdots\!33}{81\!\cdots\!44}a^{9}-\frac{71\!\cdots\!45}{40\!\cdots\!72}a^{8}-\frac{14\!\cdots\!43}{35\!\cdots\!28}a^{7}+\frac{44\!\cdots\!21}{40\!\cdots\!72}a^{6}-\frac{97\!\cdots\!43}{55\!\cdots\!36}a^{5}+\frac{17\!\cdots\!53}{10\!\cdots\!68}a^{4}-\frac{19\!\cdots\!01}{12\!\cdots\!96}a^{3}+\frac{80\!\cdots\!71}{82\!\cdots\!32}a^{2}-\frac{63\!\cdots\!51}{12\!\cdots\!96}a+\frac{60\!\cdots\!25}{34\!\cdots\!72}$, $\frac{11\!\cdots\!45}{81\!\cdots\!44}a^{28}-\frac{58\!\cdots\!39}{40\!\cdots\!72}a^{27}+\frac{87\!\cdots\!53}{81\!\cdots\!44}a^{26}-\frac{42\!\cdots\!93}{81\!\cdots\!44}a^{25}+\frac{17\!\cdots\!21}{81\!\cdots\!44}a^{24}-\frac{44\!\cdots\!77}{81\!\cdots\!44}a^{23}+\frac{21\!\cdots\!23}{23\!\cdots\!48}a^{22}+\frac{25\!\cdots\!93}{62\!\cdots\!88}a^{21}-\frac{12\!\cdots\!71}{47\!\cdots\!52}a^{20}+\frac{80\!\cdots\!65}{81\!\cdots\!44}a^{19}-\frac{11\!\cdots\!59}{10\!\cdots\!68}a^{18}+\frac{32\!\cdots\!11}{40\!\cdots\!72}a^{17}+\frac{90\!\cdots\!75}{81\!\cdots\!44}a^{16}-\frac{45\!\cdots\!75}{10\!\cdots\!68}a^{15}+\frac{49\!\cdots\!25}{81\!\cdots\!44}a^{14}-\frac{22\!\cdots\!53}{35\!\cdots\!28}a^{13}+\frac{69\!\cdots\!41}{81\!\cdots\!44}a^{12}+\frac{51\!\cdots\!83}{81\!\cdots\!44}a^{11}-\frac{11\!\cdots\!79}{81\!\cdots\!44}a^{10}+\frac{51\!\cdots\!55}{35\!\cdots\!28}a^{9}-\frac{37\!\cdots\!31}{40\!\cdots\!72}a^{8}-\frac{81\!\cdots\!17}{62\!\cdots\!88}a^{7}+\frac{10\!\cdots\!65}{40\!\cdots\!72}a^{6}-\frac{23\!\cdots\!41}{51\!\cdots\!84}a^{5}+\frac{46\!\cdots\!29}{10\!\cdots\!68}a^{4}-\frac{34\!\cdots\!11}{98\!\cdots\!92}a^{3}+\frac{65\!\cdots\!45}{25\!\cdots\!92}a^{2}-\frac{62\!\cdots\!69}{55\!\cdots\!52}a+\frac{57\!\cdots\!25}{34\!\cdots\!72}$, $\frac{10\!\cdots\!57}{81\!\cdots\!44}a^{28}-\frac{31\!\cdots\!59}{40\!\cdots\!72}a^{27}+\frac{42\!\cdots\!93}{81\!\cdots\!44}a^{26}-\frac{15\!\cdots\!89}{81\!\cdots\!44}a^{25}+\frac{19\!\cdots\!83}{26\!\cdots\!24}a^{24}-\frac{81\!\cdots\!11}{13\!\cdots\!16}a^{23}+\frac{16\!\cdots\!43}{23\!\cdots\!48}a^{22}+\frac{29\!\cdots\!85}{81\!\cdots\!44}a^{21}-\frac{16\!\cdots\!45}{20\!\cdots\!36}a^{20}+\frac{10\!\cdots\!73}{81\!\cdots\!44}a^{19}-\frac{13\!\cdots\!87}{10\!\cdots\!68}a^{18}-\frac{61\!\cdots\!27}{69\!\cdots\!08}a^{17}+\frac{27\!\cdots\!39}{81\!\cdots\!44}a^{16}-\frac{73\!\cdots\!65}{10\!\cdots\!68}a^{15}+\frac{51\!\cdots\!87}{64\!\cdots\!72}a^{14}-\frac{30\!\cdots\!31}{81\!\cdots\!44}a^{13}-\frac{29\!\cdots\!51}{81\!\cdots\!44}a^{12}+\frac{11\!\cdots\!39}{81\!\cdots\!44}a^{11}-\frac{15\!\cdots\!35}{81\!\cdots\!44}a^{10}+\frac{97\!\cdots\!25}{81\!\cdots\!44}a^{9}+\frac{26\!\cdots\!27}{13\!\cdots\!28}a^{8}-\frac{10\!\cdots\!87}{43\!\cdots\!76}a^{7}+\frac{20\!\cdots\!17}{40\!\cdots\!72}a^{6}-\frac{19\!\cdots\!49}{39\!\cdots\!68}a^{5}+\frac{14\!\cdots\!53}{23\!\cdots\!64}a^{4}-\frac{48\!\cdots\!99}{12\!\cdots\!96}a^{3}+\frac{46\!\cdots\!13}{19\!\cdots\!84}a^{2}-\frac{16\!\cdots\!79}{12\!\cdots\!96}a+\frac{13\!\cdots\!53}{34\!\cdots\!72}$, $\frac{32\!\cdots\!57}{20\!\cdots\!36}a^{28}-\frac{35\!\cdots\!27}{26\!\cdots\!36}a^{27}+\frac{18\!\cdots\!55}{20\!\cdots\!36}a^{26}-\frac{82\!\cdots\!25}{20\!\cdots\!36}a^{25}+\frac{31\!\cdots\!55}{20\!\cdots\!36}a^{24}-\frac{49\!\cdots\!59}{16\!\cdots\!68}a^{23}+\frac{21\!\cdots\!77}{57\!\cdots\!12}a^{22}+\frac{91\!\cdots\!81}{20\!\cdots\!36}a^{21}-\frac{23\!\cdots\!07}{10\!\cdots\!68}a^{20}+\frac{10\!\cdots\!51}{20\!\cdots\!36}a^{19}-\frac{15\!\cdots\!05}{25\!\cdots\!92}a^{18}+\frac{15\!\cdots\!97}{53\!\cdots\!72}a^{17}+\frac{21\!\cdots\!19}{20\!\cdots\!36}a^{16}-\frac{26\!\cdots\!99}{10\!\cdots\!68}a^{15}+\frac{66\!\cdots\!59}{20\!\cdots\!36}a^{14}-\frac{27\!\cdots\!59}{15\!\cdots\!72}a^{13}-\frac{15\!\cdots\!37}{20\!\cdots\!36}a^{12}+\frac{10\!\cdots\!35}{20\!\cdots\!36}a^{11}-\frac{14\!\cdots\!57}{20\!\cdots\!36}a^{10}+\frac{10\!\cdots\!93}{20\!\cdots\!36}a^{9}-\frac{33\!\cdots\!27}{51\!\cdots\!84}a^{8}-\frac{18\!\cdots\!99}{20\!\cdots\!36}a^{7}+\frac{13\!\cdots\!19}{78\!\cdots\!36}a^{6}-\frac{62\!\cdots\!63}{31\!\cdots\!24}a^{5}+\frac{49\!\cdots\!51}{25\!\cdots\!92}a^{4}-\frac{10\!\cdots\!55}{79\!\cdots\!56}a^{3}+\frac{46\!\cdots\!23}{63\!\cdots\!48}a^{2}-\frac{10\!\cdots\!33}{31\!\cdots\!24}a+\frac{45\!\cdots\!67}{86\!\cdots\!68}$ 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10529754658974305624110404358035/2045398618476132554186673012736a^11 - 14933810414860368595838657814557/2045398618476132554186673012736a^10 + 10775996158343113200837149666393/2045398618476132554186673012736a^9 - 339807475606874276215332170327/511349654619033138546668253184a^8 - 18266693263818748190871104204599/2045398618476132554186673012736a^7 + 1320236708910228503150863979619/78669177633697405930256654336a^6 - 628788877911731254524206856863/31959353413689571159166765824a^5 + 4902634781979104791016787179351/255674827309516569273334126592a^4 - 100820160774699182662283974155/7989838353422392789791691456a^3 + 464125668558466533765843060123/63918706827379142318333531648a^2 - 102975937526099855751345482133/31959353413689571159166765824a + 45599181193517932078525467/86846069058939052062953168 (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:	$ 1075559501191057.0 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 1075559501191057.0 \cdot 1}{2\cdot\sqrt{77103436114042117740511038546742321228145336964361}}\cr\approx \mathstrut & 18.3069388830337 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552)

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^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552, 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^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552);

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^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552);

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

$D_{29}$ (as 29T2):

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 58

The 16 conjugacy class representatives for $D_{29}$

Character table for $D_{29}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.2.0.1}{2} }^{14}{,}\,{\href{/padicField/2.1.0.1}{1} }$	$29$	$29$	$29$	$29$	${\href{/padicField/13.2.0.1}{2} }^{14}{,}\,{\href{/padicField/13.1.0.1}{1} }$	$29$	${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.2.0.1}{2} }^{14}{,}\,{\href{/padicField/23.1.0.1}{1} }$	$29$	${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }$	$29$	$29$	${\href{/padicField/43.2.0.1}{2} }^{14}{,}\,{\href{/padicField/43.1.0.1}{1} }$	$29$	${\href{/padicField/53.2.0.1}{2} }^{14}{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$3659$ 3659	$\Q_{3659}$	$x$	$1$	$1$	$0$	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	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.3659.2t1.a.a	$1$	$ 3659 $	$\Q(\sqrt{-3659}) $	$C_2$ (as 2T1)	$1$	$-1$
*	2.3659.29t2.a.m	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.k	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.a	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.i	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.c	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.b	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.g	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.f	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.j	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.e	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.l	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.d	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.h	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.3659.29t2.a.n	$2$	$ 3659 $	29.1.77103436114042117740511038546742321228145336964361.1	$D_{29}$ (as 29T2)	$1$	$0$

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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