LMFDB - Number field 29.1.3587518960220469771354937124331329329937375710441.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776)

gp: K = bnfinit(y^29 - y^28 - 7*y^27 - 50*y^26 + 119*y^25 + 495*y^24 + 59*y^23 - 530*y^22 + 637*y^21 + 2591*y^20 + 2977*y^19 + 1178*y^18 + 2213*y^17 + 5385*y^16 + 5965*y^15 + 6650*y^14 + 10883*y^13 + 16281*y^12 + 23955*y^11 + 21146*y^10 - 8059*y^9 - 7103*y^8 + 28957*y^7 + 37734*y^6 + 35524*y^5 + 2552*y^4 - 22800*y^3 - 19616*y^2 - 17024*y - 11776, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^29 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^29 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776)

$ x^{29} - x^{28} - 7 x^{27} - 50 x^{26} + 119 x^{25} + 495 x^{24} + 59 x^{23} - 530 x^{22} + 637 x^{21} + \cdots - 11776 $ x^29 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776

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:	$3587518960220469771354937124331329329937375710441$ 3587518960220469771354937124331329329937375710441 $\medspace = 2939^{14}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$47.24$	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:	$2939^{1/2}\approx 54.2125446737192$
Ramified primes:	$2939$ 2939	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{5}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{10}-\frac{1}{8}a^{7}+\frac{1}{8}a^{4}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{11}-\frac{1}{8}a^{8}+\frac{1}{8}a^{5}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}+\frac{1}{16}a^{11}+\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{3}{16}a^{7}-\frac{3}{16}a^{6}+\frac{3}{16}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{16}+\frac{1}{16}a^{10}-\frac{1}{4}a^{7}-\frac{3}{32}a^{4}+\frac{1}{4}a$, $\frac{1}{32}a^{17}+\frac{1}{16}a^{11}-\frac{3}{32}a^{5}$, $\frac{1}{32}a^{18}-\frac{1}{16}a^{12}-\frac{1}{8}a^{9}+\frac{1}{32}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{64}a^{19}-\frac{1}{64}a^{18}-\frac{1}{64}a^{17}-\frac{1}{32}a^{15}+\frac{1}{32}a^{14}+\frac{1}{16}a^{11}+\frac{1}{32}a^{10}-\frac{3}{32}a^{9}-\frac{1}{32}a^{8}-\frac{9}{64}a^{7}-\frac{7}{64}a^{6}+\frac{5}{64}a^{5}-\frac{5}{32}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{20}-\frac{1}{64}a^{17}+\frac{1}{32}a^{14}+\frac{3}{32}a^{11}+\frac{5}{64}a^{8}-\frac{5}{64}a^{5}-\frac{1}{8}a^{2}$, $\frac{1}{64}a^{21}-\frac{1}{64}a^{18}-\frac{1}{32}a^{15}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}+\frac{1}{32}a^{12}+\frac{1}{16}a^{11}+\frac{1}{16}a^{10}+\frac{1}{64}a^{9}+\frac{1}{16}a^{8}-\frac{3}{16}a^{7}+\frac{15}{64}a^{6}+\frac{3}{16}a^{5}+\frac{3}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{22}-\frac{1}{128}a^{20}-\frac{1}{128}a^{18}-\frac{1}{64}a^{17}+\frac{1}{64}a^{15}+\frac{1}{32}a^{14}-\frac{3}{64}a^{13}+\frac{1}{32}a^{12}+\frac{3}{64}a^{11}-\frac{1}{128}a^{10}+\frac{7}{64}a^{9}+\frac{5}{128}a^{8}+\frac{15}{64}a^{7}-\frac{27}{128}a^{6}+\frac{5}{32}a^{5}-\frac{3}{16}a^{4}-\frac{7}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{23}-\frac{1}{128}a^{21}-\frac{1}{128}a^{19}-\frac{1}{64}a^{18}-\frac{1}{64}a^{16}-\frac{1}{32}a^{15}+\frac{1}{64}a^{14}-\frac{1}{32}a^{13}-\frac{1}{64}a^{12}-\frac{9}{128}a^{11}+\frac{7}{64}a^{10}-\frac{3}{128}a^{9}-\frac{5}{64}a^{8}-\frac{19}{128}a^{7}-\frac{1}{32}a^{6}+\frac{1}{8}a^{5}-\frac{5}{32}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{128}a^{24}+\frac{1}{128}a^{18}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}+\frac{3}{128}a^{12}+\frac{1}{16}a^{11}+\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{3}{16}a^{7}+\frac{27}{128}a^{6}+\frac{3}{16}a^{5}+\frac{3}{16}a^{4}-\frac{3}{16}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{359168}a^{25}-\frac{787}{359168}a^{24}-\frac{191}{179584}a^{23}+\frac{873}{359168}a^{22}+\frac{509}{89792}a^{21}-\frac{2261}{359168}a^{20}-\frac{1617}{359168}a^{19}-\frac{363}{179584}a^{18}+\frac{987}{179584}a^{17}+\frac{1321}{89792}a^{16}-\frac{1077}{179584}a^{15}-\frac{2167}{44896}a^{14}-\frac{20347}{359168}a^{13}+\frac{9179}{359168}a^{12}-\frac{10761}{89792}a^{11}-\frac{37317}{359168}a^{10}+\frac{17671}{179584}a^{9}+\frac{1251}{15616}a^{8}-\frac{5165}{359168}a^{7}-\frac{26109}{179584}a^{6}-\frac{22227}{89792}a^{5}-\frac{4781}{22448}a^{4}+\frac{237}{976}a^{3}+\frac{1453}{11224}a^{2}-\frac{1185}{5612}a-\frac{1}{61}$, $\frac{1}{359168}a^{26}+\frac{375}{359168}a^{24}+\frac{481}{359168}a^{23}-\frac{1189}{359168}a^{22}-\frac{2155}{359168}a^{21}-\frac{505}{89792}a^{20}-\frac{2187}{359168}a^{19}-\frac{117}{7808}a^{18}-\frac{657}{179584}a^{17}-\frac{1069}{179584}a^{16}-\frac{141}{7808}a^{15}+\frac{1441}{359168}a^{14}+\frac{757}{179584}a^{13}-\frac{22173}{359168}a^{12}-\frac{36107}{359168}a^{11}+\frac{6399}{359168}a^{10}+\frac{38273}{359168}a^{9}-\frac{5423}{179584}a^{8}-\frac{11}{5888}a^{7}+\frac{2603}{89792}a^{6}+\frac{20115}{89792}a^{5}-\frac{173}{1403}a^{4}-\frac{435}{5612}a^{3}+\frac{1905}{11224}a^{2}+\frac{1711}{5612}a+\frac{6}{61}$, $\frac{1}{412420951496704}a^{27}+\frac{2076281}{6444077367136}a^{26}-\frac{2646223}{12888154734272}a^{25}-\frac{346326983169}{206210475748352}a^{24}+\frac{1463033329189}{412420951496704}a^{23}+\frac{259282335059}{206210475748352}a^{22}+\frac{183644285661}{103105237874176}a^{21}-\frac{140899861469}{103105237874176}a^{20}-\frac{1624817459419}{412420951496704}a^{19}-\frac{177118778683}{51552618937088}a^{18}-\frac{1076908980525}{103105237874176}a^{17}-\frac{42339298785}{206210475748352}a^{16}-\frac{4626194107113}{412420951496704}a^{15}+\frac{5153611907897}{206210475748352}a^{14}+\frac{342950034625}{6444077367136}a^{13}-\frac{1673674349115}{51552618937088}a^{12}+\frac{22648002286811}{412420951496704}a^{11}-\frac{7994206665053}{103105237874176}a^{10}-\frac{11549227505821}{103105237874176}a^{9}+\frac{6836322410973}{206210475748352}a^{8}-\frac{2216804329687}{9591184918528}a^{7}+\frac{247012967465}{3380499602432}a^{6}-\frac{258768458957}{1690249801216}a^{5}+\frac{11353559665101}{51552618937088}a^{4}-\frac{10291156989907}{25776309468544}a^{3}+\frac{5715030153655}{12888154734272}a^{2}-\frac{840033193837}{3222038683568}a-\frac{13727415149}{35022159604}$, $\frac{1}{11\!\cdots\!76}a^{28}-\frac{33\!\cdots\!83}{27\!\cdots\!44}a^{27}-\frac{96\!\cdots\!11}{22\!\cdots\!56}a^{26}-\frac{10\!\cdots\!17}{55\!\cdots\!88}a^{25}+\frac{22\!\cdots\!85}{11\!\cdots\!76}a^{24}-\frac{18\!\cdots\!15}{55\!\cdots\!88}a^{23}+\frac{93\!\cdots\!53}{27\!\cdots\!44}a^{22}-\frac{11\!\cdots\!53}{27\!\cdots\!44}a^{21}-\frac{75\!\cdots\!91}{11\!\cdots\!76}a^{20}-\frac{18\!\cdots\!31}{27\!\cdots\!44}a^{19}+\frac{20\!\cdots\!63}{27\!\cdots\!44}a^{18}-\frac{60\!\cdots\!61}{55\!\cdots\!88}a^{17}+\frac{75\!\cdots\!35}{11\!\cdots\!76}a^{16}+\frac{59\!\cdots\!35}{55\!\cdots\!88}a^{15}-\frac{48\!\cdots\!49}{13\!\cdots\!72}a^{14}+\frac{16\!\cdots\!49}{34\!\cdots\!68}a^{13}+\frac{10\!\cdots\!15}{18\!\cdots\!16}a^{12}-\frac{10\!\cdots\!31}{13\!\cdots\!72}a^{11}-\frac{45\!\cdots\!71}{27\!\cdots\!44}a^{10}-\frac{63\!\cdots\!55}{55\!\cdots\!88}a^{9}+\frac{41\!\cdots\!25}{35\!\cdots\!96}a^{8}+\frac{28\!\cdots\!63}{55\!\cdots\!88}a^{7}-\frac{70\!\cdots\!81}{12\!\cdots\!28}a^{6}+\frac{11\!\cdots\!47}{13\!\cdots\!72}a^{5}+\frac{11\!\cdots\!87}{69\!\cdots\!36}a^{4}-\frac{48\!\cdots\!81}{15\!\cdots\!16}a^{3}+\frac{10\!\cdots\!39}{21\!\cdots\!48}a^{2}+\frac{20\!\cdots\!43}{21\!\cdots\!48}a-\frac{99\!\cdots\!37}{23\!\cdots\!69}$ 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, 1/8*a^13 - 1/8*a^10 - 1/8*a^7 + 1/8*a^4, 1/8*a^14 - 1/8*a^11 - 1/8*a^8 + 1/8*a^5, 1/16*a^15 - 1/16*a^14 - 1/16*a^13 - 1/16*a^12 + 1/16*a^11 + 1/16*a^10 - 1/16*a^9 + 1/16*a^8 - 3/16*a^7 - 3/16*a^6 + 3/16*a^5 + 3/16*a^4 + 1/4*a^3 - 1/4*a^2, 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, 1/32*a^18 - 1/16*a^12 - 1/8*a^9 + 1/32*a^6 + 1/8*a^3, 1/64*a^19 - 1/64*a^18 - 1/64*a^17 - 1/32*a^15 + 1/32*a^14 + 1/16*a^11 + 1/32*a^10 - 3/32*a^9 - 1/32*a^8 - 9/64*a^7 - 7/64*a^6 + 5/64*a^5 - 5/32*a^4 + 1/4*a^3 - 1/8*a^2 + 1/4*a, 1/64*a^20 - 1/64*a^17 + 1/32*a^14 + 3/32*a^11 + 5/64*a^8 - 5/64*a^5 - 1/8*a^2, 1/64*a^21 - 1/64*a^18 - 1/32*a^15 - 1/16*a^14 - 1/16*a^13 + 1/32*a^12 + 1/16*a^11 + 1/16*a^10 + 1/64*a^9 + 1/16*a^8 - 3/16*a^7 + 15/64*a^6 + 3/16*a^5 + 3/16*a^4 - 1/4*a^3 - 1/4*a^2, 1/128*a^22 - 1/128*a^20 - 1/128*a^18 - 1/64*a^17 + 1/64*a^15 + 1/32*a^14 - 3/64*a^13 + 1/32*a^12 + 3/64*a^11 - 1/128*a^10 + 7/64*a^9 + 5/128*a^8 + 15/64*a^7 - 27/128*a^6 + 5/32*a^5 - 3/16*a^4 - 7/16*a^3 + 1/4*a^2 - 1/2*a, 1/128*a^23 - 1/128*a^21 - 1/128*a^19 - 1/64*a^18 - 1/64*a^16 - 1/32*a^15 + 1/64*a^14 - 1/32*a^13 - 1/64*a^12 - 9/128*a^11 + 7/64*a^10 - 3/128*a^9 - 5/64*a^8 - 19/128*a^7 - 1/32*a^6 + 1/8*a^5 - 5/32*a^4 - 3/8*a^3 - 1/2*a^2 - 1/4*a, 1/128*a^24 + 1/128*a^18 - 1/16*a^14 - 1/16*a^13 + 3/128*a^12 + 1/16*a^11 + 1/16*a^10 - 1/16*a^9 + 1/16*a^8 - 3/16*a^7 + 27/128*a^6 + 3/16*a^5 + 3/16*a^4 - 3/16*a^3 - 1/4*a^2, 1/359168*a^25 - 787/359168*a^24 - 191/179584*a^23 + 873/359168*a^22 + 509/89792*a^21 - 2261/359168*a^20 - 1617/359168*a^19 - 363/179584*a^18 + 987/179584*a^17 + 1321/89792*a^16 - 1077/179584*a^15 - 2167/44896*a^14 - 20347/359168*a^13 + 9179/359168*a^12 - 10761/89792*a^11 - 37317/359168*a^10 + 17671/179584*a^9 + 1251/15616*a^8 - 5165/359168*a^7 - 26109/179584*a^6 - 22227/89792*a^5 - 4781/22448*a^4 + 237/976*a^3 + 1453/11224*a^2 - 1185/5612*a - 1/61, 1/359168*a^26 + 375/359168*a^24 + 481/359168*a^23 - 1189/359168*a^22 - 2155/359168*a^21 - 505/89792*a^20 - 2187/359168*a^19 - 117/7808*a^18 - 657/179584*a^17 - 1069/179584*a^16 - 141/7808*a^15 + 1441/359168*a^14 + 757/179584*a^13 - 22173/359168*a^12 - 36107/359168*a^11 + 6399/359168*a^10 + 38273/359168*a^9 - 5423/179584*a^8 - 11/5888*a^7 + 2603/89792*a^6 + 20115/89792*a^5 - 173/1403*a^4 - 435/5612*a^3 + 1905/11224*a^2 + 1711/5612*a + 6/61, 1/412420951496704*a^27 + 2076281/6444077367136*a^26 - 2646223/12888154734272*a^25 - 346326983169/206210475748352*a^24 + 1463033329189/412420951496704*a^23 + 259282335059/206210475748352*a^22 + 183644285661/103105237874176*a^21 - 140899861469/103105237874176*a^20 - 1624817459419/412420951496704*a^19 - 177118778683/51552618937088*a^18 - 1076908980525/103105237874176*a^17 - 42339298785/206210475748352*a^16 - 4626194107113/412420951496704*a^15 + 5153611907897/206210475748352*a^14 + 342950034625/6444077367136*a^13 - 1673674349115/51552618937088*a^12 + 22648002286811/412420951496704*a^11 - 7994206665053/103105237874176*a^10 - 11549227505821/103105237874176*a^9 + 6836322410973/206210475748352*a^8 - 2216804329687/9591184918528*a^7 + 247012967465/3380499602432*a^6 - 258768458957/1690249801216*a^5 + 11353559665101/51552618937088*a^4 - 10291156989907/25776309468544*a^3 + 5715030153655/12888154734272*a^2 - 840033193837/3222038683568*a - 13727415149/35022159604, 1/11144994447581623263273115852158976*a^28 - 3373261618969304683/2786248611895405815818278963039744*a^27 - 9698632740070619444923511/22469746870124240450147410992256*a^26 - 1011864625493885596774257317/5572497223790811631636557926079488*a^25 + 22315771071283690805350464218485/11144994447581623263273115852158976*a^24 - 18907146491876009542174930594115/5572497223790811631636557926079488*a^23 + 933642467039186620626294259653/2786248611895405815818278963039744*a^22 - 11954190034509046406686079473953/2786248611895405815818278963039744*a^21 - 75646364394429446864741583146691/11144994447581623263273115852158976*a^20 - 18913258852341881381305278494931/2786248611895405815818278963039744*a^19 + 20774650484348326809470729322363/2786248611895405815818278963039744*a^18 - 60872197403698336222939067402161/5572497223790811631636557926079488*a^17 + 75491199933600334474894458113935/11144994447581623263273115852158976*a^16 + 59594925431835565758543417451735/5572497223790811631636557926079488*a^15 - 48263358125029974659590372767549/1393124305947702907909139481519872*a^14 + 16417434821056948806130642447049/348281076486925726977284870379968*a^13 + 10353868119635271432051224023615/182704827009534807594641243478016*a^12 - 106963196964385034849908513899531/1393124305947702907909139481519872*a^11 - 45992402833673676534214517218371/2786248611895405815818278963039744*a^10 - 632863925969831305882886428133555/5572497223790811631636557926079488*a^9 + 41901518404871696774350747688925/359515949921987847202358575876096*a^8 + 28404532215726691699596279435863/5572497223790811631636557926079488*a^7 - 7050716361711239067226647422881/121141243995452426774707781001728*a^6 + 111066485706359068863687522171247/1393124305947702907909139481519872*a^5 + 116288622698331561098965606056687/696562152973851453954569740759936*a^4 - 4894178256617425299005025812081/15142655499431553346838472625216*a^3 + 10878495738948014427700965607439/21767567280432857936080304398748*a^2 + 2071809838747545712476605612643/21767567280432857936080304398748*a - 99662773610393508429807022037/236603992178618021044351134769

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:	$\frac{97\!\cdots\!59}{48\!\cdots\!12}a^{28}-\frac{19\!\cdots\!97}{55\!\cdots\!88}a^{27}-\frac{38\!\cdots\!39}{34\!\cdots\!68}a^{26}-\frac{51\!\cdots\!89}{55\!\cdots\!88}a^{25}+\frac{33\!\cdots\!37}{11\!\cdots\!76}a^{24}+\frac{21\!\cdots\!61}{27\!\cdots\!44}a^{23}-\frac{54\!\cdots\!47}{13\!\cdots\!72}a^{22}-\frac{24\!\cdots\!61}{59\!\cdots\!52}a^{21}+\frac{18\!\cdots\!65}{11\!\cdots\!76}a^{20}+\frac{19\!\cdots\!19}{55\!\cdots\!88}a^{19}+\frac{10\!\cdots\!23}{27\!\cdots\!44}a^{18}+\frac{46\!\cdots\!71}{55\!\cdots\!88}a^{17}+\frac{55\!\cdots\!91}{11\!\cdots\!76}a^{16}+\frac{19\!\cdots\!91}{27\!\cdots\!44}a^{15}+\frac{17\!\cdots\!35}{27\!\cdots\!44}a^{14}+\frac{21\!\cdots\!99}{22\!\cdots\!52}a^{13}+\frac{12\!\cdots\!99}{11\!\cdots\!76}a^{12}+\frac{12\!\cdots\!43}{55\!\cdots\!88}a^{11}+\frac{75\!\cdots\!01}{27\!\cdots\!44}a^{10}+\frac{13\!\cdots\!77}{55\!\cdots\!88}a^{9}-\frac{40\!\cdots\!73}{11\!\cdots\!76}a^{8}+\frac{58\!\cdots\!23}{27\!\cdots\!44}a^{7}+\frac{26\!\cdots\!95}{13\!\cdots\!72}a^{6}+\frac{99\!\cdots\!75}{22\!\cdots\!56}a^{5}+\frac{22\!\cdots\!89}{87\!\cdots\!92}a^{4}-\frac{35\!\cdots\!29}{17\!\cdots\!84}a^{3}-\frac{39\!\cdots\!85}{17\!\cdots\!84}a^{2}-\frac{55\!\cdots\!59}{71\!\cdots\!36}a-\frac{99\!\cdots\!77}{47\!\cdots\!38}$, $\frac{60\!\cdots\!45}{55\!\cdots\!88}a^{28}-\frac{30\!\cdots\!59}{13\!\cdots\!72}a^{27}-\frac{70\!\cdots\!39}{13\!\cdots\!72}a^{26}-\frac{12\!\cdots\!53}{27\!\cdots\!44}a^{25}+\frac{20\!\cdots\!95}{11\!\cdots\!04}a^{24}+\frac{37\!\cdots\!55}{11\!\cdots\!56}a^{23}-\frac{26\!\cdots\!77}{69\!\cdots\!36}a^{22}+\frac{54\!\cdots\!41}{69\!\cdots\!36}a^{21}+\frac{95\!\cdots\!37}{55\!\cdots\!88}a^{20}+\frac{16\!\cdots\!59}{87\!\cdots\!92}a^{19}+\frac{23\!\cdots\!41}{13\!\cdots\!72}a^{18}+\frac{57\!\cdots\!19}{27\!\cdots\!44}a^{17}+\frac{56\!\cdots\!43}{91\!\cdots\!08}a^{16}+\frac{21\!\cdots\!47}{27\!\cdots\!44}a^{15}+\frac{10\!\cdots\!63}{13\!\cdots\!72}a^{14}+\frac{92\!\cdots\!33}{69\!\cdots\!36}a^{13}+\frac{92\!\cdots\!19}{55\!\cdots\!88}a^{12}+\frac{24\!\cdots\!37}{13\!\cdots\!72}a^{11}+\frac{76\!\cdots\!07}{30\!\cdots\!32}a^{10}+\frac{53\!\cdots\!11}{27\!\cdots\!44}a^{9}-\frac{10\!\cdots\!03}{11\!\cdots\!04}a^{8}+\frac{22\!\cdots\!51}{59\!\cdots\!52}a^{7}+\frac{62\!\cdots\!33}{13\!\cdots\!72}a^{6}+\frac{32\!\cdots\!65}{69\!\cdots\!36}a^{5}+\frac{83\!\cdots\!11}{34\!\cdots\!68}a^{4}-\frac{32\!\cdots\!19}{17\!\cdots\!84}a^{3}-\frac{11\!\cdots\!91}{43\!\cdots\!96}a^{2}-\frac{35\!\cdots\!59}{21\!\cdots\!48}a-\frac{34\!\cdots\!43}{23\!\cdots\!69}$, $\frac{10\!\cdots\!67}{11\!\cdots\!76}a^{28}+\frac{87\!\cdots\!57}{11\!\cdots\!76}a^{27}-\frac{41\!\cdots\!51}{69\!\cdots\!36}a^{26}-\frac{32\!\cdots\!11}{55\!\cdots\!88}a^{25}+\frac{82\!\cdots\!85}{11\!\cdots\!76}a^{24}+\frac{59\!\cdots\!47}{11\!\cdots\!76}a^{23}+\frac{57\!\cdots\!21}{55\!\cdots\!88}a^{22}+\frac{12\!\cdots\!53}{13\!\cdots\!72}a^{21}+\frac{36\!\cdots\!45}{35\!\cdots\!96}a^{20}+\frac{14\!\cdots\!71}{48\!\cdots\!12}a^{19}+\frac{19\!\cdots\!71}{27\!\cdots\!44}a^{18}+\frac{56\!\cdots\!39}{55\!\cdots\!88}a^{17}+\frac{14\!\cdots\!31}{11\!\cdots\!76}a^{16}+\frac{17\!\cdots\!29}{11\!\cdots\!76}a^{15}+\frac{10\!\cdots\!81}{55\!\cdots\!88}a^{14}+\frac{10\!\cdots\!23}{60\!\cdots\!96}a^{13}+\frac{20\!\cdots\!49}{11\!\cdots\!76}a^{12}+\frac{24\!\cdots\!35}{11\!\cdots\!76}a^{11}+\frac{49\!\cdots\!31}{13\!\cdots\!72}a^{10}+\frac{29\!\cdots\!89}{55\!\cdots\!88}a^{9}+\frac{49\!\cdots\!91}{11\!\cdots\!76}a^{8}+\frac{83\!\cdots\!41}{11\!\cdots\!76}a^{7}-\frac{13\!\cdots\!13}{55\!\cdots\!88}a^{6}-\frac{11\!\cdots\!03}{27\!\cdots\!44}a^{5}-\frac{42\!\cdots\!97}{13\!\cdots\!72}a^{4}-\frac{95\!\cdots\!01}{69\!\cdots\!36}a^{3}-\frac{17\!\cdots\!49}{34\!\cdots\!68}a^{2}+\frac{31\!\cdots\!81}{20\!\cdots\!44}a+\frac{29\!\cdots\!19}{94\!\cdots\!76}$, $\frac{13\!\cdots\!51}{27\!\cdots\!44}a^{28}-\frac{51\!\cdots\!23}{55\!\cdots\!88}a^{27}-\frac{19\!\cdots\!13}{69\!\cdots\!36}a^{26}-\frac{10\!\cdots\!41}{48\!\cdots\!68}a^{25}+\frac{37\!\cdots\!49}{48\!\cdots\!68}a^{24}+\frac{10\!\cdots\!37}{55\!\cdots\!88}a^{23}-\frac{37\!\cdots\!53}{27\!\cdots\!44}a^{22}-\frac{22\!\cdots\!99}{13\!\cdots\!72}a^{21}+\frac{43\!\cdots\!03}{89\!\cdots\!24}a^{20}+\frac{49\!\cdots\!81}{55\!\cdots\!88}a^{19}+\frac{47\!\cdots\!97}{69\!\cdots\!36}a^{18}-\frac{14\!\cdots\!73}{34\!\cdots\!68}a^{17}+\frac{78\!\cdots\!35}{69\!\cdots\!36}a^{16}+\frac{99\!\cdots\!07}{55\!\cdots\!88}a^{15}+\frac{37\!\cdots\!81}{27\!\cdots\!44}a^{14}+\frac{14\!\cdots\!71}{69\!\cdots\!36}a^{13}+\frac{10\!\cdots\!77}{27\!\cdots\!44}a^{12}+\frac{27\!\cdots\!23}{55\!\cdots\!88}a^{11}+\frac{10\!\cdots\!05}{13\!\cdots\!72}a^{10}+\frac{22\!\cdots\!53}{57\!\cdots\!88}a^{9}-\frac{10\!\cdots\!83}{13\!\cdots\!72}a^{8}+\frac{17\!\cdots\!95}{55\!\cdots\!88}a^{7}+\frac{35\!\cdots\!65}{27\!\cdots\!44}a^{6}+\frac{10\!\cdots\!79}{13\!\cdots\!72}a^{5}+\frac{76\!\cdots\!09}{69\!\cdots\!36}a^{4}-\frac{28\!\cdots\!35}{34\!\cdots\!68}a^{3}-\frac{92\!\cdots\!45}{17\!\cdots\!84}a^{2}-\frac{37\!\cdots\!49}{10\!\cdots\!72}a-\frac{28\!\cdots\!97}{47\!\cdots\!38}$, $\frac{22\!\cdots\!37}{48\!\cdots\!12}a^{28}-\frac{21\!\cdots\!73}{11\!\cdots\!76}a^{27}-\frac{32\!\cdots\!77}{13\!\cdots\!72}a^{26}-\frac{66\!\cdots\!23}{55\!\cdots\!88}a^{25}+\frac{14\!\cdots\!37}{11\!\cdots\!76}a^{24}+\frac{86\!\cdots\!01}{11\!\cdots\!76}a^{23}-\frac{44\!\cdots\!49}{55\!\cdots\!88}a^{22}-\frac{77\!\cdots\!39}{13\!\cdots\!72}a^{21}+\frac{15\!\cdots\!99}{11\!\cdots\!76}a^{20}+\frac{10\!\cdots\!91}{11\!\cdots\!76}a^{19}-\frac{84\!\cdots\!89}{27\!\cdots\!44}a^{18}-\frac{12\!\cdots\!23}{24\!\cdots\!56}a^{17}-\frac{35\!\cdots\!37}{10\!\cdots\!96}a^{16}+\frac{23\!\cdots\!35}{11\!\cdots\!76}a^{15}-\frac{22\!\cdots\!01}{55\!\cdots\!88}a^{14}-\frac{98\!\cdots\!07}{13\!\cdots\!72}a^{13}-\frac{41\!\cdots\!19}{11\!\cdots\!76}a^{12}-\frac{55\!\cdots\!11}{11\!\cdots\!76}a^{11}-\frac{82\!\cdots\!85}{69\!\cdots\!36}a^{10}-\frac{14\!\cdots\!35}{55\!\cdots\!88}a^{9}-\frac{39\!\cdots\!81}{11\!\cdots\!76}a^{8}+\frac{19\!\cdots\!23}{11\!\cdots\!76}a^{7}+\frac{27\!\cdots\!21}{55\!\cdots\!88}a^{6}-\frac{49\!\cdots\!65}{27\!\cdots\!44}a^{5}-\frac{97\!\cdots\!95}{13\!\cdots\!72}a^{4}-\frac{14\!\cdots\!81}{22\!\cdots\!56}a^{3}+\frac{83\!\cdots\!81}{34\!\cdots\!68}a^{2}+\frac{11\!\cdots\!91}{20\!\cdots\!44}a+\frac{32\!\cdots\!17}{94\!\cdots\!76}$, $\frac{35910741216911}{10\!\cdots\!72}a^{28}-\frac{16181745095931}{21\!\cdots\!44}a^{27}-\frac{1783560844403}{56\!\cdots\!44}a^{26}-\frac{15561159155975}{87\!\cdots\!76}a^{25}+\frac{16\!\cdots\!69}{53\!\cdots\!36}a^{24}+\frac{47\!\cdots\!37}{21\!\cdots\!44}a^{23}+\frac{74\!\cdots\!03}{10\!\cdots\!72}a^{22}-\frac{19\!\cdots\!51}{53\!\cdots\!36}a^{21}+\frac{16\!\cdots\!97}{10\!\cdots\!72}a^{20}+\frac{28\!\cdots\!33}{21\!\cdots\!44}a^{19}+\frac{18\!\cdots\!77}{13\!\cdots\!84}a^{18}-\frac{18081515664787}{30\!\cdots\!88}a^{17}+\frac{498541702141519}{26\!\cdots\!68}a^{16}+\frac{56\!\cdots\!63}{21\!\cdots\!44}a^{15}+\frac{29\!\cdots\!41}{10\!\cdots\!72}a^{14}+\frac{24\!\cdots\!87}{13\!\cdots\!84}a^{13}+\frac{32\!\cdots\!29}{10\!\cdots\!72}a^{12}+\frac{13\!\cdots\!47}{21\!\cdots\!44}a^{11}+\frac{43\!\cdots\!59}{53\!\cdots\!36}a^{10}+\frac{41\!\cdots\!53}{66\!\cdots\!92}a^{9}-\frac{32\!\cdots\!41}{53\!\cdots\!36}a^{8}-\frac{20\!\cdots\!45}{21\!\cdots\!44}a^{7}+\frac{15\!\cdots\!45}{10\!\cdots\!72}a^{6}+\frac{10\!\cdots\!47}{53\!\cdots\!36}a^{5}+\frac{19\!\cdots\!77}{26\!\cdots\!68}a^{4}-\frac{88\!\cdots\!99}{13\!\cdots\!84}a^{3}-\frac{12\!\cdots\!21}{66\!\cdots\!92}a^{2}-\frac{82\!\cdots\!45}{16\!\cdots\!48}a-\frac{98969079991049}{180706463928244}$, $\frac{38\!\cdots\!35}{27\!\cdots\!44}a^{28}-\frac{41\!\cdots\!31}{69\!\cdots\!36}a^{27}-\frac{60\!\cdots\!41}{69\!\cdots\!36}a^{26}-\frac{50\!\cdots\!83}{13\!\cdots\!72}a^{25}+\frac{11\!\cdots\!67}{27\!\cdots\!44}a^{24}+\frac{48\!\cdots\!87}{13\!\cdots\!72}a^{23}-\frac{17\!\cdots\!19}{69\!\cdots\!36}a^{22}-\frac{92\!\cdots\!27}{30\!\cdots\!32}a^{21}+\frac{10\!\cdots\!63}{96\!\cdots\!36}a^{20}+\frac{90\!\cdots\!97}{69\!\cdots\!36}a^{19}-\frac{10\!\cdots\!85}{15\!\cdots\!16}a^{18}-\frac{24\!\cdots\!15}{13\!\cdots\!72}a^{17}-\frac{20\!\cdots\!25}{12\!\cdots\!28}a^{16}-\frac{18\!\cdots\!83}{13\!\cdots\!72}a^{15}-\frac{13\!\cdots\!81}{69\!\cdots\!36}a^{14}-\frac{64\!\cdots\!97}{43\!\cdots\!96}a^{13}-\frac{99\!\cdots\!71}{27\!\cdots\!44}a^{12}-\frac{16\!\cdots\!95}{17\!\cdots\!84}a^{11}-\frac{85\!\cdots\!43}{69\!\cdots\!36}a^{10}-\frac{83\!\cdots\!45}{13\!\cdots\!72}a^{9}-\frac{35\!\cdots\!03}{27\!\cdots\!44}a^{8}-\frac{14\!\cdots\!45}{17\!\cdots\!68}a^{7}+\frac{49\!\cdots\!83}{34\!\cdots\!68}a^{6}+\frac{21\!\cdots\!05}{17\!\cdots\!84}a^{5}+\frac{18\!\cdots\!49}{14\!\cdots\!16}a^{4}-\frac{14\!\cdots\!75}{21\!\cdots\!48}a^{3}-\frac{57\!\cdots\!31}{43\!\cdots\!96}a^{2}-\frac{52\!\cdots\!07}{21\!\cdots\!48}a-\frac{13\!\cdots\!35}{23\!\cdots\!69}$, $\frac{83\!\cdots\!03}{15\!\cdots\!24}a^{28}-\frac{91\!\cdots\!35}{11\!\cdots\!76}a^{27}-\frac{70\!\cdots\!49}{13\!\cdots\!72}a^{26}-\frac{76\!\cdots\!91}{34\!\cdots\!68}a^{25}+\frac{76\!\cdots\!39}{91\!\cdots\!08}a^{24}+\frac{33\!\cdots\!57}{11\!\cdots\!76}a^{23}-\frac{20\!\cdots\!81}{55\!\cdots\!88}a^{22}-\frac{15\!\cdots\!01}{27\!\cdots\!44}a^{21}+\frac{33\!\cdots\!99}{27\!\cdots\!44}a^{20}+\frac{96\!\cdots\!21}{11\!\cdots\!76}a^{19}-\frac{93\!\cdots\!79}{13\!\cdots\!72}a^{18}-\frac{20\!\cdots\!77}{27\!\cdots\!44}a^{17}-\frac{24\!\cdots\!41}{55\!\cdots\!88}a^{16}+\frac{43\!\cdots\!31}{11\!\cdots\!76}a^{15}-\frac{11\!\cdots\!03}{55\!\cdots\!88}a^{14}+\frac{14\!\cdots\!55}{69\!\cdots\!36}a^{13}+\frac{26\!\cdots\!23}{43\!\cdots\!96}a^{12}-\frac{26\!\cdots\!77}{11\!\cdots\!76}a^{11}+\frac{34\!\cdots\!93}{27\!\cdots\!44}a^{10}-\frac{17\!\cdots\!27}{27\!\cdots\!44}a^{9}-\frac{12\!\cdots\!95}{55\!\cdots\!88}a^{8}+\frac{41\!\cdots\!05}{35\!\cdots\!96}a^{7}+\frac{82\!\cdots\!21}{55\!\cdots\!88}a^{6}+\frac{12\!\cdots\!33}{12\!\cdots\!28}a^{5}-\frac{21\!\cdots\!75}{13\!\cdots\!72}a^{4}-\frac{16\!\cdots\!07}{69\!\cdots\!36}a^{3}-\frac{69\!\cdots\!85}{34\!\cdots\!68}a^{2}-\frac{74\!\cdots\!45}{87\!\cdots\!92}a-\frac{10\!\cdots\!37}{94\!\cdots\!76}$, $\frac{98\!\cdots\!95}{48\!\cdots\!12}a^{28}-\frac{34\!\cdots\!15}{55\!\cdots\!88}a^{27}-\frac{90\!\cdots\!79}{13\!\cdots\!72}a^{26}-\frac{48\!\cdots\!53}{55\!\cdots\!88}a^{25}+\frac{47\!\cdots\!49}{11\!\cdots\!76}a^{24}+\frac{72\!\cdots\!45}{17\!\cdots\!84}a^{23}-\frac{12\!\cdots\!13}{13\!\cdots\!72}a^{22}-\frac{44\!\cdots\!03}{12\!\cdots\!28}a^{21}-\frac{31\!\cdots\!83}{11\!\cdots\!76}a^{20}-\frac{25\!\cdots\!95}{55\!\cdots\!88}a^{19}+\frac{10\!\cdots\!95}{27\!\cdots\!44}a^{18}-\frac{29\!\cdots\!01}{55\!\cdots\!88}a^{17}-\frac{32\!\cdots\!65}{11\!\cdots\!76}a^{16}-\frac{44\!\cdots\!49}{13\!\cdots\!72}a^{15}-\frac{65\!\cdots\!09}{27\!\cdots\!44}a^{14}-\frac{62\!\cdots\!55}{13\!\cdots\!72}a^{13}-\frac{19\!\cdots\!19}{48\!\cdots\!12}a^{12}-\frac{21\!\cdots\!63}{55\!\cdots\!88}a^{11}-\frac{13\!\cdots\!09}{27\!\cdots\!44}a^{10}-\frac{15\!\cdots\!87}{55\!\cdots\!88}a^{9}-\frac{12\!\cdots\!93}{11\!\cdots\!76}a^{8}-\frac{70\!\cdots\!25}{34\!\cdots\!68}a^{7}-\frac{77\!\cdots\!65}{69\!\cdots\!36}a^{6}+\frac{31\!\cdots\!21}{34\!\cdots\!68}a^{5}+\frac{53\!\cdots\!51}{34\!\cdots\!68}a^{4}+\frac{52\!\cdots\!09}{30\!\cdots\!48}a^{3}+\frac{10\!\cdots\!73}{17\!\cdots\!84}a^{2}-\frac{17\!\cdots\!87}{18\!\cdots\!52}a+\frac{74\!\cdots\!33}{15\!\cdots\!98}$, $\frac{22\!\cdots\!87}{27\!\cdots\!44}a^{28}-\frac{88\!\cdots\!75}{55\!\cdots\!88}a^{27}-\frac{12\!\cdots\!61}{22\!\cdots\!56}a^{26}-\frac{45\!\cdots\!95}{13\!\cdots\!72}a^{25}+\frac{19\!\cdots\!57}{13\!\cdots\!72}a^{24}+\frac{18\!\cdots\!57}{55\!\cdots\!88}a^{23}-\frac{12\!\cdots\!33}{27\!\cdots\!44}a^{22}-\frac{98\!\cdots\!15}{13\!\cdots\!72}a^{21}+\frac{31\!\cdots\!09}{27\!\cdots\!44}a^{20}+\frac{11\!\cdots\!17}{55\!\cdots\!88}a^{19}-\frac{59\!\cdots\!23}{21\!\cdots\!48}a^{18}-\frac{11\!\cdots\!33}{34\!\cdots\!68}a^{17}+\frac{12\!\cdots\!09}{74\!\cdots\!44}a^{16}+\frac{28\!\cdots\!75}{55\!\cdots\!88}a^{15}+\frac{60\!\cdots\!81}{27\!\cdots\!44}a^{14}-\frac{90\!\cdots\!21}{34\!\cdots\!68}a^{13}-\frac{12\!\cdots\!47}{45\!\cdots\!04}a^{12}+\frac{22\!\cdots\!51}{55\!\cdots\!88}a^{11}+\frac{19\!\cdots\!93}{13\!\cdots\!72}a^{10}+\frac{30\!\cdots\!73}{34\!\cdots\!68}a^{9}-\frac{50\!\cdots\!93}{19\!\cdots\!44}a^{8}-\frac{86\!\cdots\!49}{55\!\cdots\!88}a^{7}+\frac{82\!\cdots\!29}{27\!\cdots\!44}a^{6}+\frac{28\!\cdots\!59}{13\!\cdots\!72}a^{5}+\frac{90\!\cdots\!41}{69\!\cdots\!36}a^{4}-\frac{89\!\cdots\!73}{44\!\cdots\!92}a^{3}-\frac{65\!\cdots\!99}{40\!\cdots\!88}a^{2}-\frac{19\!\cdots\!35}{43\!\cdots\!96}a-\frac{68\!\cdots\!87}{47\!\cdots\!38}$, $\frac{11\!\cdots\!83}{11\!\cdots\!76}a^{28}-\frac{42\!\cdots\!95}{24\!\cdots\!56}a^{27}-\frac{10\!\cdots\!45}{15\!\cdots\!16}a^{26}-\frac{25\!\cdots\!43}{55\!\cdots\!88}a^{25}+\frac{17\!\cdots\!95}{11\!\cdots\!76}a^{24}+\frac{26\!\cdots\!69}{60\!\cdots\!64}a^{23}-\frac{10\!\cdots\!27}{34\!\cdots\!68}a^{22}-\frac{18\!\cdots\!49}{27\!\cdots\!44}a^{21}+\frac{94\!\cdots\!59}{11\!\cdots\!76}a^{20}+\frac{23\!\cdots\!01}{11\!\cdots\!04}a^{19}+\frac{33\!\cdots\!61}{27\!\cdots\!44}a^{18}-\frac{62\!\cdots\!47}{55\!\cdots\!88}a^{17}+\frac{51\!\cdots\!81}{11\!\cdots\!76}a^{16}+\frac{33\!\cdots\!49}{13\!\cdots\!72}a^{15}+\frac{11\!\cdots\!91}{27\!\cdots\!44}a^{14}+\frac{90\!\cdots\!57}{13\!\cdots\!72}a^{13}+\frac{47\!\cdots\!49}{11\!\cdots\!76}a^{12}+\frac{43\!\cdots\!75}{55\!\cdots\!88}a^{11}+\frac{35\!\cdots\!07}{27\!\cdots\!44}a^{10}+\frac{19\!\cdots\!03}{55\!\cdots\!88}a^{9}-\frac{30\!\cdots\!43}{11\!\cdots\!76}a^{8}-\frac{10\!\cdots\!13}{13\!\cdots\!72}a^{7}+\frac{18\!\cdots\!79}{69\!\cdots\!36}a^{6}+\frac{78\!\cdots\!03}{44\!\cdots\!92}a^{5}+\frac{77\!\cdots\!75}{34\!\cdots\!68}a^{4}-\frac{98\!\cdots\!83}{87\!\cdots\!92}a^{3}-\frac{30\!\cdots\!77}{17\!\cdots\!84}a^{2}+\frac{11\!\cdots\!49}{43\!\cdots\!96}a-\frac{20\!\cdots\!35}{15\!\cdots\!98}$, $\frac{19\!\cdots\!13}{27\!\cdots\!44}a^{28}-\frac{56\!\cdots\!39}{59\!\cdots\!52}a^{27}-\frac{65\!\cdots\!33}{13\!\cdots\!72}a^{26}-\frac{57\!\cdots\!61}{17\!\cdots\!84}a^{25}+\frac{26\!\cdots\!95}{27\!\cdots\!44}a^{24}+\frac{28\!\cdots\!37}{89\!\cdots\!24}a^{23}-\frac{13\!\cdots\!07}{13\!\cdots\!72}a^{22}-\frac{51\!\cdots\!41}{13\!\cdots\!72}a^{21}+\frac{48\!\cdots\!57}{59\!\cdots\!52}a^{20}+\frac{44\!\cdots\!87}{27\!\cdots\!44}a^{19}+\frac{22\!\cdots\!97}{43\!\cdots\!96}a^{18}+\frac{24\!\cdots\!93}{13\!\cdots\!72}a^{17}+\frac{61\!\cdots\!65}{27\!\cdots\!44}a^{16}+\frac{57\!\cdots\!95}{27\!\cdots\!44}a^{15}+\frac{74\!\cdots\!15}{69\!\cdots\!36}a^{14}+\frac{15\!\cdots\!95}{13\!\cdots\!72}a^{13}+\frac{38\!\cdots\!51}{27\!\cdots\!44}a^{12}+\frac{11\!\cdots\!27}{27\!\cdots\!44}a^{11}+\frac{30\!\cdots\!57}{34\!\cdots\!68}a^{10}+\frac{16\!\cdots\!83}{34\!\cdots\!68}a^{9}-\frac{36\!\cdots\!37}{27\!\cdots\!44}a^{8}-\frac{10\!\cdots\!93}{27\!\cdots\!44}a^{7}+\frac{92\!\cdots\!53}{13\!\cdots\!72}a^{6}-\frac{27\!\cdots\!21}{69\!\cdots\!36}a^{5}-\frac{18\!\cdots\!97}{34\!\cdots\!68}a^{4}-\frac{29\!\cdots\!15}{17\!\cdots\!84}a^{3}+\frac{18\!\cdots\!61}{87\!\cdots\!92}a^{2}+\frac{22\!\cdots\!84}{12\!\cdots\!09}a-\frac{16\!\cdots\!27}{23\!\cdots\!69}$, $\frac{12\!\cdots\!31}{11\!\cdots\!76}a^{28}+\frac{42\!\cdots\!67}{48\!\cdots\!12}a^{27}-\frac{10\!\cdots\!39}{13\!\cdots\!72}a^{26}-\frac{39\!\cdots\!95}{55\!\cdots\!88}a^{25}+\frac{19\!\cdots\!09}{11\!\cdots\!76}a^{24}+\frac{24\!\cdots\!53}{35\!\cdots\!96}a^{23}+\frac{68\!\cdots\!85}{55\!\cdots\!88}a^{22}+\frac{15\!\cdots\!21}{17\!\cdots\!84}a^{21}+\frac{11\!\cdots\!51}{11\!\cdots\!76}a^{20}+\frac{38\!\cdots\!45}{11\!\cdots\!76}a^{19}+\frac{22\!\cdots\!55}{27\!\cdots\!44}a^{18}+\frac{62\!\cdots\!43}{55\!\cdots\!88}a^{17}+\frac{14\!\cdots\!39}{11\!\cdots\!76}a^{16}+\frac{17\!\cdots\!81}{11\!\cdots\!76}a^{15}+\frac{15\!\cdots\!17}{91\!\cdots\!08}a^{14}+\frac{21\!\cdots\!81}{13\!\cdots\!72}a^{13}+\frac{15\!\cdots\!29}{11\!\cdots\!76}a^{12}+\frac{19\!\cdots\!83}{11\!\cdots\!76}a^{11}+\frac{46\!\cdots\!77}{13\!\cdots\!72}a^{10}+\frac{29\!\cdots\!29}{55\!\cdots\!88}a^{9}+\frac{43\!\cdots\!95}{11\!\cdots\!76}a^{8}-\frac{10\!\cdots\!35}{11\!\cdots\!76}a^{7}-\frac{24\!\cdots\!73}{55\!\cdots\!88}a^{6}-\frac{15\!\cdots\!59}{27\!\cdots\!44}a^{5}-\frac{47\!\cdots\!97}{13\!\cdots\!72}a^{4}-\frac{64\!\cdots\!29}{69\!\cdots\!36}a^{3}-\frac{54\!\cdots\!21}{34\!\cdots\!68}a^{2}+\frac{48\!\cdots\!63}{87\!\cdots\!92}a+\frac{50\!\cdots\!11}{94\!\cdots\!76}$, $\frac{82\!\cdots\!87}{13\!\cdots\!72}a^{28}-\frac{11\!\cdots\!03}{69\!\cdots\!36}a^{27}-\frac{10\!\cdots\!93}{17\!\cdots\!84}a^{26}-\frac{90\!\cdots\!21}{34\!\cdots\!68}a^{25}+\frac{19\!\cdots\!49}{13\!\cdots\!72}a^{24}+\frac{54\!\cdots\!17}{15\!\cdots\!16}a^{23}-\frac{75\!\cdots\!15}{17\!\cdots\!84}a^{22}-\frac{71\!\cdots\!43}{34\!\cdots\!68}a^{21}-\frac{38\!\cdots\!49}{13\!\cdots\!72}a^{20}-\frac{20\!\cdots\!99}{87\!\cdots\!92}a^{19}-\frac{29\!\cdots\!29}{69\!\cdots\!36}a^{18}-\frac{81\!\cdots\!69}{69\!\cdots\!36}a^{17}-\frac{31\!\cdots\!87}{13\!\cdots\!72}a^{16}-\frac{12\!\cdots\!29}{34\!\cdots\!68}a^{15}-\frac{15\!\cdots\!27}{34\!\cdots\!68}a^{14}-\frac{12\!\cdots\!67}{22\!\cdots\!56}a^{13}-\frac{82\!\cdots\!05}{13\!\cdots\!72}a^{12}-\frac{40\!\cdots\!75}{69\!\cdots\!36}a^{11}-\frac{20\!\cdots\!75}{34\!\cdots\!68}a^{10}-\frac{51\!\cdots\!21}{69\!\cdots\!36}a^{9}-\frac{19\!\cdots\!39}{13\!\cdots\!72}a^{8}-\frac{19\!\cdots\!45}{11\!\cdots\!76}a^{7}-\frac{88\!\cdots\!85}{69\!\cdots\!36}a^{6}-\frac{12\!\cdots\!89}{34\!\cdots\!68}a^{5}+\frac{15\!\cdots\!11}{17\!\cdots\!84}a^{4}+\frac{10\!\cdots\!03}{87\!\cdots\!92}a^{3}+\frac{35\!\cdots\!33}{43\!\cdots\!96}a^{2}+\frac{94\!\cdots\!55}{21\!\cdots\!48}a+\frac{14\!\cdots\!44}{81\!\cdots\!61}$ 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14423241658195128757986027944/8158758350986828311874177061 (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:	$ 191758984500558.16 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 191758984500558.16 \cdot 1}{2\cdot\sqrt{3587518960220469771354937124331329329937375710441}}\cr\approx \mathstrut & 15.1313185970062 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776)

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 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776, 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 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776);

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 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776);

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	${\href{/padicField/2.2.0.1}{2} }^{14}{,}\,{\href{/padicField/2.1.0.1}{1} }$	$29$	$29$	$29$	$29$	$29$	$29$	$29$	${\href{/padicField/23.2.0.1}{2} }^{14}{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }$	$29$	${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.2.0.1}{2} }^{14}{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.2.0.1}{2} }^{14}{,}\,{\href{/padicField/47.1.0.1}{1} }$	$29$	${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$2939$ 2939	$\Q_{2939}$	$x$	$1$	$1$	$0$	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	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.2939.2t1.a.a	$1$	$ 2939 $	$\Q(\sqrt{-2939}) $	$C_2$ (as 2T1)	$1$	$-1$
*	2.2939.29t2.a.k	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.g	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.n	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.c	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.j	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.l	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.b	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.d	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.e	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.f	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.i	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.h	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.a	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (as 29T2)	$1$	$0$
*	2.2939.29t2.a.m	$2$	$ 2939 $	29.1.3587518960220469771354937124331329329937375710441.1	$D_{29}$ (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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