LMFDB - Number field 36.0.111655439513580694005446762483626621692955058293302970913053089.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144)

gp: K = bnfinit(y^36 - 3*y^34 - y^33 + 3*y^32 + 6*y^31 + 2*y^30 + 6*y^29 - 6*y^28 - 48*y^27 - 24*y^26 + 66*y^25 + 140*y^24 + 66*y^23 - 207*y^22 - 350*y^21 - 93*y^20 + 333*y^19 + 751*y^18 + 666*y^17 - 372*y^16 - 2800*y^15 - 3312*y^14 + 2112*y^13 + 8960*y^12 + 8448*y^11 - 6144*y^10 - 24576*y^9 - 6144*y^8 + 12288*y^7 + 8192*y^6 + 49152*y^5 + 49152*y^4 - 32768*y^3 - 196608*y^2 + 262144, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144)

$ x^{36} - 3 x^{34} - x^{33} + 3 x^{32} + 6 x^{31} + 2 x^{30} + 6 x^{29} - 6 x^{28} - 48 x^{27} + \cdots + 262144 $ x^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$36$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 18]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$111655439513580694005446762483626621692955058293302970913053089$ 111655439513580694005446762483626621692955058293302970913053089 $\medspace = 3^{48}\cdot 7^{30}\cdot 199^{6}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$52.91$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$3^{4/3}7^{5/6}199^{1/2}\approx 308.91327934286153$
Ramified primes:	$3$, $7$, $199$ 3, 7, 199	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$12$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{15}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{6}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{19}-\frac{1}{8}a^{18}-\frac{1}{8}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{22}+\frac{1}{16}a^{20}-\frac{1}{16}a^{19}-\frac{1}{16}a^{18}-\frac{1}{8}a^{17}-\frac{1}{8}a^{16}-\frac{1}{8}a^{15}+\frac{1}{8}a^{14}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}+\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{3}{8}a^{7}-\frac{1}{16}a^{6}+\frac{5}{16}a^{5}+\frac{3}{16}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{23}+\frac{1}{32}a^{21}-\frac{1}{32}a^{20}-\frac{1}{32}a^{19}+\frac{1}{16}a^{18}-\frac{1}{16}a^{17}-\frac{1}{16}a^{16}+\frac{1}{16}a^{15}-\frac{1}{4}a^{14}+\frac{1}{16}a^{12}-\frac{1}{8}a^{11}-\frac{3}{16}a^{10}+\frac{1}{32}a^{9}-\frac{3}{16}a^{8}-\frac{9}{32}a^{7}+\frac{5}{32}a^{6}-\frac{13}{32}a^{5}-\frac{1}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{24}+\frac{1}{64}a^{22}-\frac{1}{64}a^{21}+\frac{7}{64}a^{20}+\frac{1}{32}a^{19}-\frac{1}{32}a^{18}-\frac{1}{32}a^{17}+\frac{1}{32}a^{16}+\frac{1}{8}a^{15}-\frac{1}{4}a^{14}+\frac{1}{32}a^{13}+\frac{3}{16}a^{12}+\frac{5}{32}a^{11}+\frac{1}{64}a^{10}+\frac{5}{32}a^{9}+\frac{7}{64}a^{8}-\frac{11}{64}a^{7}-\frac{21}{64}a^{6}-\frac{9}{32}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{256}a^{25}-\frac{1}{128}a^{24}-\frac{3}{256}a^{23}-\frac{3}{256}a^{22}+\frac{5}{256}a^{21}+\frac{3}{32}a^{20}-\frac{1}{128}a^{19}-\frac{11}{128}a^{18}-\frac{1}{128}a^{17}+\frac{3}{64}a^{16}+\frac{3}{32}a^{15}+\frac{17}{128}a^{14}+\frac{1}{32}a^{13}-\frac{11}{128}a^{12}+\frac{29}{256}a^{11}-\frac{1}{8}a^{10}-\frac{49}{256}a^{9}-\frac{1}{256}a^{8}-\frac{59}{256}a^{7}+\frac{9}{64}a^{6}-\frac{1}{16}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{36352}a^{26}-\frac{1}{2272}a^{25}+\frac{41}{36352}a^{24}-\frac{553}{36352}a^{23}-\frac{961}{36352}a^{22}+\frac{857}{18176}a^{21}+\frac{1591}{18176}a^{20}-\frac{1061}{18176}a^{19}-\frac{903}{18176}a^{18}-\frac{11}{4544}a^{17}+\frac{83}{2272}a^{16}-\frac{2087}{18176}a^{15}-\frac{85}{9088}a^{14}-\frac{3827}{18176}a^{13}+\frac{1521}{36352}a^{12}-\frac{3307}{18176}a^{11}+\frac{2175}{36352}a^{10}-\frac{8579}{36352}a^{9}+\frac{6179}{36352}a^{8}+\frac{7711}{18176}a^{7}-\frac{2125}{4544}a^{6}+\frac{657}{2272}a^{5}+\frac{61}{284}a^{4}-\frac{257}{568}a^{3}-\frac{125}{284}a^{2}+\frac{127}{284}a-\frac{39}{142}$, $\frac{1}{72704}a^{27}+\frac{69}{72704}a^{25}-\frac{465}{72704}a^{24}+\frac{699}{72704}a^{23}-\frac{441}{36352}a^{22}-\frac{1027}{36352}a^{21}-\frac{2869}{36352}a^{20}+\frac{1149}{36352}a^{19}+\frac{35}{2272}a^{18}-\frac{649}{9088}a^{17}+\frac{3425}{36352}a^{16}-\frac{2581}{18176}a^{15}-\frac{6263}{36352}a^{14}+\frac{8561}{72704}a^{13}-\frac{1079}{36352}a^{12}+\frac{13643}{72704}a^{11}+\frac{3501}{72704}a^{10}+\frac{2679}{72704}a^{9}+\frac{2473}{36352}a^{8}+\frac{4969}{18176}a^{7}+\frac{597}{2272}a^{6}-\frac{251}{2272}a^{5}+\frac{133}{1136}a^{4}-\frac{33}{71}a^{3}+\frac{257}{568}a^{2}-\frac{17}{284}a+\frac{43}{142}$, $\frac{1}{145408}a^{28}+\frac{1}{145408}a^{26}+\frac{55}{145408}a^{25}-\frac{953}{145408}a^{24}+\frac{1037}{72704}a^{23}+\frac{691}{72704}a^{22}+\frac{1051}{72704}a^{21}-\frac{255}{72704}a^{20}+\frac{1279}{18176}a^{19}-\frac{97}{2272}a^{18}+\frac{6985}{72704}a^{17}-\frac{4141}{36352}a^{16}+\frac{10693}{72704}a^{15}-\frac{5807}{145408}a^{14}-\frac{15755}{72704}a^{13}+\frac{31767}{145408}a^{12}+\frac{18733}{145408}a^{11}+\frac{18363}{145408}a^{10}+\frac{8171}{72704}a^{9}-\frac{1695}{9088}a^{8}+\frac{1361}{9088}a^{7}+\frac{2007}{4544}a^{6}+\frac{81}{568}a^{5}+\frac{203}{1136}a^{4}+\frac{333}{1136}a^{3}-\frac{169}{568}a^{2}-\frac{32}{71}a+\frac{12}{71}$, $\frac{1}{290816}a^{29}+\frac{1}{290816}a^{27}-\frac{1}{290816}a^{26}-\frac{57}{290816}a^{25}-\frac{111}{145408}a^{24}-\frac{2001}{145408}a^{23}+\frac{695}{145408}a^{22}+\frac{6281}{145408}a^{21}+\frac{3997}{36352}a^{20}-\frac{913}{18176}a^{19}+\frac{12113}{145408}a^{18}-\frac{2909}{72704}a^{17}-\frac{8315}{145408}a^{16}-\frac{26527}{290816}a^{15}+\frac{11941}{145408}a^{14}+\frac{24167}{290816}a^{13}+\frac{6261}{290816}a^{12}+\frac{61579}{290816}a^{11}+\frac{19975}{145408}a^{10}-\frac{6953}{36352}a^{9}-\frac{1907}{36352}a^{8}+\frac{21}{18176}a^{7}+\frac{715}{4544}a^{6}+\frac{3}{1136}a^{5}+\frac{751}{2272}a^{4}-\frac{263}{1136}a^{3}+\frac{15}{142}a^{2}-\frac{13}{284}a-\frac{11}{142}$, $\frac{1}{581632}a^{30}+\frac{1}{581632}a^{28}-\frac{1}{581632}a^{27}+\frac{7}{581632}a^{26}+\frac{513}{290816}a^{25}+\frac{1583}{290816}a^{24}-\frac{2233}{290816}a^{23}-\frac{5159}{290816}a^{22}+\frac{4361}{72704}a^{21}+\frac{1023}{36352}a^{20}+\frac{23729}{290816}a^{19}-\frac{3405}{145408}a^{18}+\frac{13861}{290816}a^{17}-\frac{41503}{581632}a^{16}+\frac{14693}{290816}a^{15}-\frac{138457}{581632}a^{14}+\frac{134389}{581632}a^{13}-\frac{117}{581632}a^{12}-\frac{4233}{290816}a^{11}-\frac{6593}{72704}a^{10}+\frac{13241}{72704}a^{9}+\frac{8123}{36352}a^{8}-\frac{1167}{18176}a^{7}+\frac{439}{2272}a^{6}+\frac{469}{4544}a^{5}+\frac{979}{2272}a^{4}+\frac{69}{284}a^{3}+\frac{123}{568}a^{2}+\frac{1}{4}a+\frac{57}{142}$, $\frac{1}{147734528}a^{31}-\frac{5}{18466816}a^{30}+\frac{253}{147734528}a^{29}+\frac{119}{147734528}a^{28}-\frac{725}{147734528}a^{27}-\frac{185}{73867264}a^{26}-\frac{64759}{73867264}a^{25}+\frac{260891}{73867264}a^{24}+\frac{972933}{73867264}a^{23}+\frac{69621}{2308352}a^{22}-\frac{697059}{18466816}a^{21}-\frac{129865}{1040384}a^{20}+\frac{408655}{36933632}a^{19}-\frac{5617487}{73867264}a^{18}-\frac{17970207}{147734528}a^{17}+\frac{7114493}{73867264}a^{16}-\frac{16740525}{147734528}a^{15}-\frac{31382059}{147734528}a^{14}-\frac{16997657}{147734528}a^{13}-\frac{16170975}{73867264}a^{12}+\frac{77033}{520192}a^{11}-\frac{193509}{9233408}a^{10}-\frac{64935}{9233408}a^{9}-\frac{547363}{2308352}a^{8}+\frac{85391}{1154176}a^{7}-\frac{466863}{1154176}a^{6}-\frac{73475}{577088}a^{5}-\frac{86823}{288544}a^{4}-\frac{33653}{72136}a^{3}+\frac{271}{1016}a^{2}+\frac{1656}{9017}a+\frac{1873}{9017}$, $\frac{1}{590938112}a^{32}-\frac{1}{295469056}a^{31}-\frac{251}{590938112}a^{30}+\frac{589}{590938112}a^{29}+\frac{749}{590938112}a^{28}-\frac{47}{18466816}a^{27}-\frac{669}{295469056}a^{26}+\frac{422853}{295469056}a^{25}-\frac{778921}{295469056}a^{24}+\frac{2088903}{147734528}a^{23}-\frac{1201015}{73867264}a^{22}-\frac{14285599}{295469056}a^{21}+\frac{3728313}{73867264}a^{20}+\frac{19211197}{295469056}a^{19}-\frac{47933795}{590938112}a^{18}+\frac{2570027}{36933632}a^{17}+\frac{30265207}{590938112}a^{16}+\frac{47198431}{590938112}a^{15}-\frac{82925315}{590938112}a^{14}+\frac{3177599}{147734528}a^{13}+\frac{13498447}{73867264}a^{12}-\frac{555087}{4616704}a^{11}-\frac{952667}{18466816}a^{10}+\frac{568783}{4616704}a^{9}-\frac{356907}{2308352}a^{8}+\frac{266985}{1154176}a^{7}-\frac{359803}{1154176}a^{6}+\frac{105793}{577088}a^{5}-\frac{31059}{144272}a^{4}-\frac{66351}{144272}a^{3}-\frac{2177}{9017}a^{2}+\frac{9429}{36068}a-\frac{1089}{36068}$, $\frac{1}{1181876224}a^{33}+\frac{1}{1181876224}a^{31}+\frac{7}{1181876224}a^{30}+\frac{1671}{1181876224}a^{29}-\frac{11}{590938112}a^{28}-\frac{2517}{590938112}a^{27}-\frac{749}{590938112}a^{26}+\frac{555249}{590938112}a^{25}-\frac{665221}{147734528}a^{24}-\frac{526155}{36933632}a^{23}+\frac{15099929}{590938112}a^{22}+\frac{12302483}{295469056}a^{21}-\frac{20910011}{590938112}a^{20}+\frac{143211761}{1181876224}a^{19}-\frac{59115243}{590938112}a^{18}+\frac{80479543}{1181876224}a^{17}+\frac{128360061}{1181876224}a^{16}+\frac{13209883}{1181876224}a^{15}+\frac{141457155}{590938112}a^{14}-\frac{742085}{36933632}a^{13}-\frac{2751725}{18466816}a^{12}+\frac{1966587}{9233408}a^{11}+\frac{892383}{4616704}a^{10}+\frac{839791}{4616704}a^{9}-\frac{1074365}{4616704}a^{8}+\frac{82997}{577088}a^{7}+\frac{28013}{577088}a^{6}+\frac{133}{18034}a^{5}+\frac{8939}{144272}a^{4}-\frac{34517}{72136}a^{3}-\frac{859}{72136}a^{2}+\frac{19811}{72136}a+\frac{149}{36068}$, $\frac{1}{2363752448}a^{34}+\frac{1}{2363752448}a^{32}+\frac{7}{2363752448}a^{31}+\frac{1671}{2363752448}a^{30}-\frac{11}{1181876224}a^{29}-\frac{2517}{1181876224}a^{28}-\frac{749}{1181876224}a^{27}+\frac{2545}{1181876224}a^{26}+\frac{391419}{295469056}a^{25}-\frac{211195}{73867264}a^{24}+\frac{2192665}{1181876224}a^{23}-\frac{1433837}{590938112}a^{22}-\frac{12586939}{1181876224}a^{21}-\frac{13235983}{2363752448}a^{20}+\frac{14947093}{1181876224}a^{19}+\frac{285565239}{2363752448}a^{18}+\frac{22500989}{2363752448}a^{17}-\frac{162094821}{2363752448}a^{16}+\frac{47757571}{1181876224}a^{15}-\frac{17274437}{73867264}a^{14}-\frac{9052957}{36933632}a^{13}+\frac{1959983}{18466816}a^{12}+\frac{993983}{9233408}a^{11}+\frac{1402909}{9233408}a^{10}+\frac{1957633}{9233408}a^{9}-\frac{78095}{4616704}a^{8}+\frac{7797}{288544}a^{7}+\frac{117571}{577088}a^{6}+\frac{43649}{144272}a^{5}+\frac{3379}{72136}a^{4}-\frac{27275}{144272}a^{3}+\frac{17271}{144272}a^{2}-\frac{7217}{72136}a+\frac{12}{71}$, $\frac{1}{4727504896}a^{35}+\frac{1}{4727504896}a^{33}-\frac{1}{4727504896}a^{32}-\frac{9}{4727504896}a^{31}-\frac{1663}{2363752448}a^{30}+\frac{39}{2363752448}a^{29}+\frac{5071}{2363752448}a^{28}+\frac{1569}{2363752448}a^{27}-\frac{1243}{590938112}a^{26}-\frac{277599}{295469056}a^{25}-\frac{5985983}{2363752448}a^{24}+\frac{3725035}{1181876224}a^{23}-\frac{30806043}{2363752448}a^{22}-\frac{72669471}{4727504896}a^{21}+\frac{120923733}{2363752448}a^{20}-\frac{40509337}{4727504896}a^{19}+\frac{507011}{66584576}a^{18}+\frac{244874107}{4727504896}a^{17}+\frac{76392615}{2363752448}a^{16}-\frac{133834703}{590938112}a^{15}-\frac{71377873}{590938112}a^{14}+\frac{1430081}{9233408}a^{13}-\frac{17309631}{73867264}a^{12}+\frac{3572439}{36933632}a^{11}+\frac{1935429}{9233408}a^{10}+\frac{2065747}{9233408}a^{9}+\frac{101205}{4616704}a^{8}-\frac{447053}{1154176}a^{7}+\frac{189005}{577088}a^{6}+\frac{158903}{577088}a^{5}-\frac{58447}{144272}a^{4}+\frac{66237}{288544}a^{3}+\frac{54359}{144272}a^{2}+\frac{7773}{18034}a-\frac{8891}{36068}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a, 1/2*a^9 - 1/2*a^2, 1/2*a^10 - 1/2*a^3, 1/2*a^11 - 1/2*a^4, 1/2*a^12 - 1/2*a^5, 1/2*a^13 - 1/2*a^6, 1/2*a^14 - 1/2*a^7, 1/2*a^15 - 1/2*a, 1/4*a^16 - 1/4*a^2, 1/4*a^17 - 1/4*a^3, 1/4*a^18 - 1/4*a^4, 1/4*a^19 - 1/4*a^5, 1/4*a^20 - 1/4*a^6, 1/8*a^21 - 1/8*a^19 - 1/8*a^18 - 1/8*a^17 - 1/4*a^15 - 1/4*a^14 - 1/4*a^13 - 1/4*a^10 - 1/4*a^8 + 1/8*a^7 - 1/4*a^6 + 1/8*a^5 - 3/8*a^4 - 1/8*a^3 - 1/2*a^2 - 1/2*a, 1/16*a^22 + 1/16*a^20 - 1/16*a^19 - 1/16*a^18 - 1/8*a^17 - 1/8*a^16 - 1/8*a^15 + 1/8*a^14 + 1/8*a^11 - 1/4*a^10 + 1/8*a^9 + 1/16*a^8 - 3/8*a^7 - 1/16*a^6 + 5/16*a^5 + 3/16*a^4 + 1/8*a^3 - 1/2*a, 1/32*a^23 + 1/32*a^21 - 1/32*a^20 - 1/32*a^19 + 1/16*a^18 - 1/16*a^17 - 1/16*a^16 + 1/16*a^15 - 1/4*a^14 + 1/16*a^12 - 1/8*a^11 - 3/16*a^10 + 1/32*a^9 - 3/16*a^8 - 9/32*a^7 + 5/32*a^6 - 13/32*a^5 - 1/16*a^4 + 1/4*a^3 - 1/4*a^2 - 1/2*a, 1/64*a^24 + 1/64*a^22 - 1/64*a^21 + 7/64*a^20 + 1/32*a^19 - 1/32*a^18 - 1/32*a^17 + 1/32*a^16 + 1/8*a^15 - 1/4*a^14 + 1/32*a^13 + 3/16*a^12 + 5/32*a^11 + 1/64*a^10 + 5/32*a^9 + 7/64*a^8 - 11/64*a^7 - 21/64*a^6 - 9/32*a^5 + 3/8*a^4 - 1/8*a^3 - 1/2*a^2, 1/256*a^25 - 1/128*a^24 - 3/256*a^23 - 3/256*a^22 + 5/256*a^21 + 3/32*a^20 - 1/128*a^19 - 11/128*a^18 - 1/128*a^17 + 3/64*a^16 + 3/32*a^15 + 17/128*a^14 + 1/32*a^13 - 11/128*a^12 + 29/256*a^11 - 1/8*a^10 - 49/256*a^9 - 1/256*a^8 - 59/256*a^7 + 9/64*a^6 - 1/16*a^5 - 1/8*a^3 - 1/2*a - 1/2, 1/36352*a^26 - 1/2272*a^25 + 41/36352*a^24 - 553/36352*a^23 - 961/36352*a^22 + 857/18176*a^21 + 1591/18176*a^20 - 1061/18176*a^19 - 903/18176*a^18 - 11/4544*a^17 + 83/2272*a^16 - 2087/18176*a^15 - 85/9088*a^14 - 3827/18176*a^13 + 1521/36352*a^12 - 3307/18176*a^11 + 2175/36352*a^10 - 8579/36352*a^9 + 6179/36352*a^8 + 7711/18176*a^7 - 2125/4544*a^6 + 657/2272*a^5 + 61/284*a^4 - 257/568*a^3 - 125/284*a^2 + 127/284*a - 39/142, 1/72704*a^27 + 69/72704*a^25 - 465/72704*a^24 + 699/72704*a^23 - 441/36352*a^22 - 1027/36352*a^21 - 2869/36352*a^20 + 1149/36352*a^19 + 35/2272*a^18 - 649/9088*a^17 + 3425/36352*a^16 - 2581/18176*a^15 - 6263/36352*a^14 + 8561/72704*a^13 - 1079/36352*a^12 + 13643/72704*a^11 + 3501/72704*a^10 + 2679/72704*a^9 + 2473/36352*a^8 + 4969/18176*a^7 + 597/2272*a^6 - 251/2272*a^5 + 133/1136*a^4 - 33/71*a^3 + 257/568*a^2 - 17/284*a + 43/142, 1/145408*a^28 + 1/145408*a^26 + 55/145408*a^25 - 953/145408*a^24 + 1037/72704*a^23 + 691/72704*a^22 + 1051/72704*a^21 - 255/72704*a^20 + 1279/18176*a^19 - 97/2272*a^18 + 6985/72704*a^17 - 4141/36352*a^16 + 10693/72704*a^15 - 5807/145408*a^14 - 15755/72704*a^13 + 31767/145408*a^12 + 18733/145408*a^11 + 18363/145408*a^10 + 8171/72704*a^9 - 1695/9088*a^8 + 1361/9088*a^7 + 2007/4544*a^6 + 81/568*a^5 + 203/1136*a^4 + 333/1136*a^3 - 169/568*a^2 - 32/71*a + 12/71, 1/290816*a^29 + 1/290816*a^27 - 1/290816*a^26 - 57/290816*a^25 - 111/145408*a^24 - 2001/145408*a^23 + 695/145408*a^22 + 6281/145408*a^21 + 3997/36352*a^20 - 913/18176*a^19 + 12113/145408*a^18 - 2909/72704*a^17 - 8315/145408*a^16 - 26527/290816*a^15 + 11941/145408*a^14 + 24167/290816*a^13 + 6261/290816*a^12 + 61579/290816*a^11 + 19975/145408*a^10 - 6953/36352*a^9 - 1907/36352*a^8 + 21/18176*a^7 + 715/4544*a^6 + 3/1136*a^5 + 751/2272*a^4 - 263/1136*a^3 + 15/142*a^2 - 13/284*a - 11/142, 1/581632*a^30 + 1/581632*a^28 - 1/581632*a^27 + 7/581632*a^26 + 513/290816*a^25 + 1583/290816*a^24 - 2233/290816*a^23 - 5159/290816*a^22 + 4361/72704*a^21 + 1023/36352*a^20 + 23729/290816*a^19 - 3405/145408*a^18 + 13861/290816*a^17 - 41503/581632*a^16 + 14693/290816*a^15 - 138457/581632*a^14 + 134389/581632*a^13 - 117/581632*a^12 - 4233/290816*a^11 - 6593/72704*a^10 + 13241/72704*a^9 + 8123/36352*a^8 - 1167/18176*a^7 + 439/2272*a^6 + 469/4544*a^5 + 979/2272*a^4 + 69/284*a^3 + 123/568*a^2 + 1/4*a + 57/142, 1/147734528*a^31 - 5/18466816*a^30 + 253/147734528*a^29 + 119/147734528*a^28 - 725/147734528*a^27 - 185/73867264*a^26 - 64759/73867264*a^25 + 260891/73867264*a^24 + 972933/73867264*a^23 + 69621/2308352*a^22 - 697059/18466816*a^21 - 129865/1040384*a^20 + 408655/36933632*a^19 - 5617487/73867264*a^18 - 17970207/147734528*a^17 + 7114493/73867264*a^16 - 16740525/147734528*a^15 - 31382059/147734528*a^14 - 16997657/147734528*a^13 - 16170975/73867264*a^12 + 77033/520192*a^11 - 193509/9233408*a^10 - 64935/9233408*a^9 - 547363/2308352*a^8 + 85391/1154176*a^7 - 466863/1154176*a^6 - 73475/577088*a^5 - 86823/288544*a^4 - 33653/72136*a^3 + 271/1016*a^2 + 1656/9017*a + 1873/9017, 1/590938112*a^32 - 1/295469056*a^31 - 251/590938112*a^30 + 589/590938112*a^29 + 749/590938112*a^28 - 47/18466816*a^27 - 669/295469056*a^26 + 422853/295469056*a^25 - 778921/295469056*a^24 + 2088903/147734528*a^23 - 1201015/73867264*a^22 - 14285599/295469056*a^21 + 3728313/73867264*a^20 + 19211197/295469056*a^19 - 47933795/590938112*a^18 + 2570027/36933632*a^17 + 30265207/590938112*a^16 + 47198431/590938112*a^15 - 82925315/590938112*a^14 + 3177599/147734528*a^13 + 13498447/73867264*a^12 - 555087/4616704*a^11 - 952667/18466816*a^10 + 568783/4616704*a^9 - 356907/2308352*a^8 + 266985/1154176*a^7 - 359803/1154176*a^6 + 105793/577088*a^5 - 31059/144272*a^4 - 66351/144272*a^3 - 2177/9017*a^2 + 9429/36068*a - 1089/36068, 1/1181876224*a^33 + 1/1181876224*a^31 + 7/1181876224*a^30 + 1671/1181876224*a^29 - 11/590938112*a^28 - 2517/590938112*a^27 - 749/590938112*a^26 + 555249/590938112*a^25 - 665221/147734528*a^24 - 526155/36933632*a^23 + 15099929/590938112*a^22 + 12302483/295469056*a^21 - 20910011/590938112*a^20 + 143211761/1181876224*a^19 - 59115243/590938112*a^18 + 80479543/1181876224*a^17 + 128360061/1181876224*a^16 + 13209883/1181876224*a^15 + 141457155/590938112*a^14 - 742085/36933632*a^13 - 2751725/18466816*a^12 + 1966587/9233408*a^11 + 892383/4616704*a^10 + 839791/4616704*a^9 - 1074365/4616704*a^8 + 82997/577088*a^7 + 28013/577088*a^6 + 133/18034*a^5 + 8939/144272*a^4 - 34517/72136*a^3 - 859/72136*a^2 + 19811/72136*a + 149/36068, 1/2363752448*a^34 + 1/2363752448*a^32 + 7/2363752448*a^31 + 1671/2363752448*a^30 - 11/1181876224*a^29 - 2517/1181876224*a^28 - 749/1181876224*a^27 + 2545/1181876224*a^26 + 391419/295469056*a^25 - 211195/73867264*a^24 + 2192665/1181876224*a^23 - 1433837/590938112*a^22 - 12586939/1181876224*a^21 - 13235983/2363752448*a^20 + 14947093/1181876224*a^19 + 285565239/2363752448*a^18 + 22500989/2363752448*a^17 - 162094821/2363752448*a^16 + 47757571/1181876224*a^15 - 17274437/73867264*a^14 - 9052957/36933632*a^13 + 1959983/18466816*a^12 + 993983/9233408*a^11 + 1402909/9233408*a^10 + 1957633/9233408*a^9 - 78095/4616704*a^8 + 7797/288544*a^7 + 117571/577088*a^6 + 43649/144272*a^5 + 3379/72136*a^4 - 27275/144272*a^3 + 17271/144272*a^2 - 7217/72136*a + 12/71, 1/4727504896*a^35 + 1/4727504896*a^33 - 1/4727504896*a^32 - 9/4727504896*a^31 - 1663/2363752448*a^30 + 39/2363752448*a^29 + 5071/2363752448*a^28 + 1569/2363752448*a^27 - 1243/590938112*a^26 - 277599/295469056*a^25 - 5985983/2363752448*a^24 + 3725035/1181876224*a^23 - 30806043/2363752448*a^22 - 72669471/4727504896*a^21 + 120923733/2363752448*a^20 - 40509337/4727504896*a^19 + 507011/66584576*a^18 + 244874107/4727504896*a^17 + 76392615/2363752448*a^16 - 133834703/590938112*a^15 - 71377873/590938112*a^14 + 1430081/9233408*a^13 - 17309631/73867264*a^12 + 3572439/36933632*a^11 + 1935429/9233408*a^10 + 2065747/9233408*a^9 + 101205/4616704*a^8 - 447053/1154176*a^7 + 189005/577088*a^6 + 158903/577088*a^5 - 58447/144272*a^4 + 66237/288544*a^3 + 54359/144272*a^2 + 7773/18034*a - 8891/36068

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

$C_{7}\times C_{147}$, which has order $1029$ (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:	$17$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( \frac{9567}{2363752448} a^{35} + \frac{3189}{2363752448} a^{34} - \frac{9567}{2363752448} a^{33} - \frac{9567}{1181876224} a^{32} - \frac{3189}{1181876224} a^{31} - \frac{64165}{2363752448} a^{30} + \frac{9567}{1181876224} a^{29} + \frac{9567}{147734528} a^{28} + \frac{9567}{295469056} a^{27} - \frac{105237}{1181876224} a^{26} - \frac{111615}{590938112} a^{25} - \frac{105237}{1181876224} a^{24} + \frac{447855}{1181876224} a^{23} + \frac{558075}{1181876224} a^{22} + \frac{296577}{2363752448} a^{21} - \frac{1061937}{2363752448} a^{20} - \frac{2394939}{2363752448} a^{19} - \frac{1061937}{1181876224} a^{18} + \frac{296577}{590938112} a^{17} + \frac{16076631}{2363752448} a^{16} + \frac{660123}{147734528} a^{15} - \frac{105237}{36933632} a^{14} - \frac{111615}{9233408} a^{13} - \frac{105237}{9233408} a^{12} + \frac{9567}{1154176} a^{11} + \frac{9567}{288544} a^{10} + \frac{255325}{9233408} a^{9} - \frac{9567}{577088} a^{8} - \frac{3189}{288544} a^{7} - \frac{9567}{144272} a^{6} - \frac{9567}{144272} a^{5} + \frac{3189}{72136} a^{4} + \frac{9567}{36068} a^{3} - \frac{1665}{144272} a^{2} - \frac{3189}{9017} a \) (9567)/(2363752448)a^(35) + (3189)/(2363752448)a^(34) - (9567)/(2363752448)a^(33) - (9567)/(1181876224)a^(32) - (3189)/(1181876224)a^(31) - (64165)/(2363752448)a^(30) + (9567)/(1181876224)a^(29) + (9567)/(147734528)a^(28) + (9567)/(295469056)a^(27) - (105237)/(1181876224)a^(26) - (111615)/(590938112)a^(25) - (105237)/(1181876224)a^(24) + (447855)/(1181876224)a^(23) + (558075)/(1181876224)a^(22) + (296577)/(2363752448)a^(21) - (1061937)/(2363752448)a^(20) - (2394939)/(2363752448)a^(19) - (1061937)/(1181876224)a^(18) + (296577)/(590938112)a^(17) + (16076631)/(2363752448)a^(16) + (660123)/(147734528)a^(15) - (105237)/(36933632)a^(14) - (111615)/(9233408)a^(13) - (105237)/(9233408)a^(12) + (9567)/(1154176)a^(11) + (9567)/(288544)a^(10) + (255325)/(9233408)a^(9) - (9567)/(577088)a^(8) - (3189)/(288544)a^(7) - (9567)/(144272)a^(6) - (9567)/(144272)a^(5) + (3189)/(72136)a^(4) + (9567)/(36068)a^(3) - (1665)/(144272)a^(2) - (3189)/(9017)*a (order $14$)	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:	$\frac{75435}{4727504896}a^{35}-\frac{5337}{295469056}a^{34}-\frac{226305}{4727504896}a^{33}-\frac{75435}{4727504896}a^{32}+\frac{226305}{4727504896}a^{31}+\frac{226305}{2363752448}a^{30}+\frac{75435}{2363752448}a^{29}+\frac{226305}{2363752448}a^{28}-\frac{865897}{2363752448}a^{27}-\frac{226305}{295469056}a^{26}-\frac{226305}{590938112}a^{25}+\frac{2489355}{2363752448}a^{24}+\frac{2640225}{1181876224}a^{23}+\frac{2489355}{2363752448}a^{22}-\frac{15615045}{4727504896}a^{21}-\frac{11911389}{2363752448}a^{20}-\frac{7015455}{4727504896}a^{19}+\frac{25119855}{4727504896}a^{18}+\frac{56651685}{4727504896}a^{17}+\frac{25119855}{2363752448}a^{16}-\frac{7015455}{1181876224}a^{15}-\frac{13201125}{295469056}a^{14}-\frac{1168599}{36933632}a^{13}+\frac{2489355}{73867264}a^{12}+\frac{2640225}{18466816}a^{11}+\frac{2489355}{18466816}a^{10}-\frac{226305}{2308352}a^{9}-\frac{226305}{577088}a^{8}-\frac{226305}{2308352}a^{7}-\frac{46597}{577088}a^{6}+\frac{75435}{577088}a^{5}+\frac{226305}{288544}a^{4}+\frac{226305}{288544}a^{3}-\frac{75435}{144272}a^{2}-\frac{226305}{72136}a+1$, $\frac{56301}{4727504896}a^{35}-\frac{39165}{1181876224}a^{34}-\frac{50595}{4727504896}a^{33}+\frac{157503}{4727504896}a^{32}+\frac{4281}{66584576}a^{31}+\frac{193135}{2363752448}a^{30}-\frac{138369}{2363752448}a^{29}+\frac{8343}{2363752448}a^{28}-\frac{1137103}{2363752448}a^{27}-\frac{20763}{73867264}a^{26}+\frac{137325}{295469056}a^{25}+\frac{3478509}{2363752448}a^{24}+\frac{1121685}{1181876224}a^{23}-\frac{3169095}{2363752448}a^{22}-\frac{20490939}{4727504896}a^{21}-\frac{4133337}{2363752448}a^{20}+\frac{14656131}{4727504896}a^{19}+\frac{35402373}{4727504896}a^{18}+\frac{32670699}{4727504896}a^{17}-\frac{15322971}{2363752448}a^{16}-\frac{1443789}{73867264}a^{15}-\frac{21701073}{590938112}a^{14}+\frac{2789319}{147734528}a^{13}+\frac{1604771}{18466816}a^{12}+\frac{3903621}{36933632}a^{11}-\frac{394083}{18466816}a^{10}-\frac{1115615}{4616704}a^{9}-\frac{336141}{1154176}a^{8}+\frac{577887}{2308352}a^{7}+\frac{80677}{1154176}a^{6}+\frac{92649}{144272}a^{5}+\frac{11319}{18034}a^{4}-\frac{44901}{288544}a^{3}-\frac{171105}{144272}a^{2}-\frac{42087}{18034}a+\frac{97335}{36068}$, $\frac{9039}{590938112}a^{34}+\frac{3013}{590938112}a^{33}-\frac{9039}{590938112}a^{32}-\frac{9039}{295469056}a^{31}-\frac{3013}{295469056}a^{30}-\frac{56821}{590938112}a^{29}+\frac{9039}{295469056}a^{28}+\frac{9039}{36933632}a^{27}+\frac{9039}{73867264}a^{26}-\frac{99429}{295469056}a^{25}-\frac{105455}{147734528}a^{24}-\frac{99429}{295469056}a^{23}+\frac{125407}{295469056}a^{22}+\frac{527275}{295469056}a^{21}+\frac{280209}{590938112}a^{20}-\frac{1003329}{590938112}a^{19}-\frac{2262763}{590938112}a^{18}-\frac{1003329}{295469056}a^{17}+\frac{280209}{147734528}a^{16}+\frac{15586759}{590938112}a^{15}+\frac{623691}{36933632}a^{14}-\frac{99429}{9233408}a^{13}-\frac{105455}{2308352}a^{12}-\frac{99429}{2308352}a^{11}+\frac{9039}{288544}a^{10}+\frac{9039}{72136}a^{9}+\frac{88609}{1154176}a^{8}-\frac{9039}{144272}a^{7}-\frac{3013}{72136}a^{6}-\frac{9039}{36068}a^{5}-\frac{9039}{36068}a^{4}+\frac{3013}{18034}a^{3}+\frac{9039}{9017}a^{2}-\frac{54321}{36068}a-\frac{12052}{9017}$, $\frac{3729}{1181876224}a^{35}+\frac{11187}{590938112}a^{34}-\frac{11187}{1181876224}a^{33}-\frac{41019}{1181876224}a^{32}-\frac{86579}{1181876224}a^{31}+\frac{55935}{590938112}a^{29}+\frac{78309}{590938112}a^{28}+\frac{145431}{590938112}a^{27}-\frac{63393}{295469056}a^{26}-\frac{123057}{147734528}a^{25}-\frac{649711}{590938112}a^{24}+\frac{70851}{147734528}a^{23}+\frac{1331253}{590938112}a^{22}+\frac{2673693}{1181876224}a^{21}-\frac{220011}{147734528}a^{20}-\frac{6208785}{1181876224}a^{19}-\frac{7014249}{1181876224}a^{18}+\frac{5669789}{1181876224}a^{17}+\frac{4128003}{295469056}a^{16}+\frac{1465497}{73867264}a^{15}+\frac{227469}{73867264}a^{14}-\frac{861399}{18466816}a^{13}-\frac{302049}{4616704}a^{12}-\frac{11187}{4616704}a^{11}+\frac{175539}{1154176}a^{10}+\frac{55935}{288544}a^{9}+\frac{18645}{577088}a^{8}-\frac{78309}{288544}a^{7}-\frac{33561}{144272}a^{6}-\frac{26103}{72136}a^{5}+\frac{11187}{72136}a^{4}+\frac{19419}{36068}a^{3}+\frac{18645}{18034}a^{2}-\frac{22374}{9017}$, $\frac{18515}{2363752448}a^{35}-\frac{14749}{295469056}a^{34}+\frac{120249}{2363752448}a^{33}+\frac{210233}{2363752448}a^{32}+\frac{259307}{2363752448}a^{31}-\frac{57649}{590938112}a^{30}-\frac{106197}{295469056}a^{29}-\frac{74931}{1181876224}a^{28}-\frac{450379}{1181876224}a^{27}+\frac{507473}{590938112}a^{26}+\frac{1290697}{590938112}a^{25}+\frac{1931639}{1181876224}a^{24}-\frac{1294783}{590938112}a^{23}-\frac{7421579}{1181876224}a^{22}-\frac{12171117}{2363752448}a^{21}+\frac{8550233}{1181876224}a^{20}+\frac{34225571}{2363752448}a^{19}+\frac{387153}{33292288}a^{18}-\frac{31703885}{2363752448}a^{17}-\frac{22981325}{590938112}a^{16}-\frac{44018589}{1181876224}a^{15}+\frac{100195}{18466816}a^{14}+\frac{8762477}{73867264}a^{13}+\frac{10294305}{73867264}a^{12}-\frac{521943}{9233408}a^{11}-\frac{3543187}{9233408}a^{10}-\frac{2170087}{4616704}a^{9}+\frac{236987}{2308352}a^{8}+\frac{2651731}{2308352}a^{7}+\frac{132485}{288544}a^{6}+\frac{6473}{8128}a^{5}-\frac{37387}{36068}a^{4}-\frac{286917}{144272}a^{3}-\frac{22997}{9017}a^{2}-\frac{8087}{72136}a+\frac{189987}{18034}$, $\frac{14319}{2363752448}a^{35}+\frac{24877}{590938112}a^{34}-\frac{747}{33292288}a^{33}-\frac{198291}{2363752448}a^{32}-\frac{223207}{2363752448}a^{31}+\frac{20391}{1181876224}a^{30}+\frac{255567}{1181876224}a^{29}+\frac{314293}{1181876224}a^{28}+\frac{599223}{1181876224}a^{27}-\frac{189599}{295469056}a^{26}-\frac{554091}{295469056}a^{25}-\frac{2036873}{1181876224}a^{24}+\frac{768423}{590938112}a^{23}+\frac{6063463}{1181876224}a^{22}+\frac{11163927}{2363752448}a^{21}-\frac{4786263}{1181876224}a^{20}-\frac{25089179}{2363752448}a^{19}-\frac{28198281}{2363752448}a^{18}+\frac{13136181}{2363752448}a^{17}+\frac{36806813}{1181876224}a^{16}+\frac{24772869}{590938112}a^{15}+\frac{1114797}{295469056}a^{14}-\frac{7804853}{73867264}a^{13}-\frac{2548793}{18466816}a^{12}+\frac{138387}{18466816}a^{11}+\frac{5204323}{18466816}a^{10}+\frac{1942581}{4616704}a^{9}+\frac{30813}{1154176}a^{8}-\frac{382263}{577088}a^{7}-\frac{278133}{577088}a^{6}-\frac{232617}{288544}a^{5}+\frac{24877}{72136}a^{4}+\frac{112005}{72136}a^{3}+\frac{163581}{72136}a^{2}-\frac{6797}{36068}a-\frac{88271}{18034}$, $\frac{187365}{4727504896}a^{35}-\frac{42957}{1181876224}a^{34}-\frac{55651}{4727504896}a^{33}+\frac{172671}{4727504896}a^{32}+\frac{334287}{4727504896}a^{31}+\frac{198191}{2363752448}a^{30}-\frac{174753}{2363752448}a^{29}+\frac{914179}{2363752448}a^{28}-\frac{1258447}{2363752448}a^{27}-\frac{22659}{73867264}a^{26}+\frac{158181}{295469056}a^{25}+\frac{3832429}{2363752448}a^{24}+\frac{1205109}{1181876224}a^{23}-\frac{3683463}{2363752448}a^{22}-\frac{29975027}{4727504896}a^{21}-\frac{4368441}{2363752448}a^{20}+\frac{16339779}{4727504896}a^{19}+\frac{39199429}{4727504896}a^{18}+\frac{36037995}{4727504896}a^{17}-\frac{16263387}{2363752448}a^{16}-\frac{989225}{36933632}a^{15}-\frac{8275871}{147734528}a^{14}+\frac{3123015}{147734528}a^{13}+\frac{1781731}{18466816}a^{12}+\frac{4237317}{36933632}a^{11}-\frac{515427}{18466816}a^{10}-\frac{1236959}{4616704}a^{9}-\frac{1390587}{4616704}a^{8}+\frac{923173}{1154176}a^{7}+\frac{90789}{1154176}a^{6}+\frac{100233}{144272}a^{5}+\frac{12267}{18034}a^{4}-\frac{55013}{288544}a^{3}-\frac{201441}{144272}a^{2}-\frac{84939}{36068}a+\frac{86774}{9017}$, $\frac{9787}{1181876224}a^{35}-\frac{34355}{2363752448}a^{34}-\frac{68505}{1181876224}a^{33}-\frac{116909}{2363752448}a^{32}+\frac{26277}{2363752448}a^{31}+\frac{214779}{2363752448}a^{30}+\frac{116909}{1181876224}a^{29}+\frac{91167}{1181876224}a^{28}-\frac{126211}{1181876224}a^{27}-\frac{937551}{1181876224}a^{26}-\frac{15837}{18466816}a^{25}+\frac{128301}{590938112}a^{24}+\frac{2440865}{1181876224}a^{23}+\frac{729273}{295469056}a^{22}-\frac{119403}{147734528}a^{21}-\frac{9294263}{2363752448}a^{20}-\frac{2565309}{590938112}a^{19}+\frac{3500757}{2363752448}a^{18}+\frac{25504259}{2363752448}a^{17}+\frac{37402479}{2363752448}a^{16}+\frac{7163421}{1181876224}a^{15}-\frac{16719185}{590938112}a^{14}-\frac{5085873}{73867264}a^{13}-\frac{1795949}{73867264}a^{12}+\frac{3811045}{36933632}a^{11}+\frac{1781517}{9233408}a^{10}+\frac{600909}{9233408}a^{9}-\frac{332223}{1154176}a^{8}-\frac{224031}{577088}a^{7}-\frac{50537}{144272}a^{6}-\frac{217745}{577088}a^{5}+\frac{149889}{288544}a^{4}+\frac{156057}{144272}a^{3}+\frac{58187}{144272}a^{2}-\frac{149889}{72136}a-\frac{133403}{36068}$, $\frac{9717}{590938112}a^{35}-\frac{7097}{590938112}a^{34}-\frac{29397}{1181876224}a^{33}-\frac{1487}{590938112}a^{32}+\frac{50517}{1181876224}a^{31}+\frac{115331}{1181876224}a^{30}-\frac{19795}{1181876224}a^{29}+\frac{23003}{147734528}a^{28}-\frac{135901}{590938112}a^{27}-\frac{462557}{590938112}a^{26}-\frac{179993}{590938112}a^{25}+\frac{32577}{36933632}a^{24}+\frac{514851}{295469056}a^{23}+\frac{2905}{4653056}a^{22}-\frac{1182849}{590938112}a^{21}-\frac{1219689}{295469056}a^{20}-\frac{4241233}{1181876224}a^{19}+\frac{868269}{295469056}a^{18}+\frac{8958307}{1181876224}a^{17}+\frac{12383285}{1181876224}a^{16}-\frac{23367}{1181876224}a^{15}-\frac{11001219}{295469056}a^{14}-\frac{2880509}{73867264}a^{13}+\frac{3751195}{73867264}a^{12}+\frac{4309603}{36933632}a^{11}+\frac{475903}{4616704}a^{10}-\frac{104711}{4616704}a^{9}-\frac{345903}{1154176}a^{8}-\frac{56493}{577088}a^{7}-\frac{114061}{577088}a^{6}+\frac{226567}{577088}a^{5}+\frac{194273}{288544}a^{4}+\frac{29811}{36068}a^{3}+\frac{1005}{72136}a^{2}-\frac{228597}{72136}a+\frac{12631}{18034}$, $\frac{123481}{4727504896}a^{35}-\frac{43303}{2363752448}a^{34}-\frac{467139}{4727504896}a^{33}-\frac{232663}{4727504896}a^{32}+\frac{521121}{4727504896}a^{31}+\frac{230071}{1181876224}a^{30}+\frac{27999}{2363752448}a^{29}+\frac{328457}{2363752448}a^{28}-\frac{383765}{2363752448}a^{27}-\frac{1382367}{1181876224}a^{26}-\frac{760147}{590938112}a^{25}+\frac{1793657}{2363752448}a^{24}+\frac{14983}{4161536}a^{23}+\frac{7803277}{2363752448}a^{22}-\frac{16276867}{4727504896}a^{21}-\frac{9307243}{1181876224}a^{20}-\frac{18885049}{4727504896}a^{19}+\frac{15507971}{4727504896}a^{18}+\frac{59379017}{4727504896}a^{17}+\frac{13133605}{590938112}a^{16}+\frac{5129433}{295469056}a^{15}-\frac{1818783}{36933632}a^{14}-\frac{17206295}{147734528}a^{13}-\frac{135879}{2308352}a^{12}+\frac{378023}{2308352}a^{11}+\frac{6119091}{18466816}a^{10}+\frac{692031}{4616704}a^{9}-\frac{874041}{2308352}a^{8}-\frac{519001}{2308352}a^{7}-\frac{719505}{1154176}a^{6}-\frac{154485}{144272}a^{5}+\frac{112689}{288544}a^{4}+\frac{872765}{288544}a^{3}+\frac{23647}{9017}a^{2}-\frac{71815}{18034}a-\frac{64599}{18034}$, $\frac{19221}{590938112}a^{35}-\frac{62717}{1181876224}a^{34}-\frac{61169}{1181876224}a^{33}+\frac{19375}{1181876224}a^{32}+\frac{17059}{295469056}a^{31}+\frac{90125}{590938112}a^{30}-\frac{101629}{1181876224}a^{29}+\frac{143091}{590938112}a^{28}-\frac{216489}{295469056}a^{27}-\frac{142711}{147734528}a^{26}+\frac{61071}{590938112}a^{25}+\frac{154779}{73867264}a^{24}+\frac{1954307}{590938112}a^{23}-\frac{705159}{590938112}a^{22}-\frac{461881}{73867264}a^{21}-\frac{5338167}{1181876224}a^{20}-\frac{1232363}{1181876224}a^{19}+\frac{14508247}{1181876224}a^{18}+\frac{1288971}{73867264}a^{17}+\frac{1471347}{147734528}a^{16}-\frac{28358885}{1181876224}a^{15}-\frac{22967439}{295469056}a^{14}-\frac{1150235}{73867264}a^{13}+\frac{149347}{2308352}a^{12}+\frac{9836441}{36933632}a^{11}+\frac{1350489}{18466816}a^{10}-\frac{200491}{577088}a^{9}-\frac{1206867}{2308352}a^{8}+\frac{357307}{1154176}a^{7}-\frac{158173}{577088}a^{6}+\frac{113471}{288544}a^{5}+\frac{489435}{288544}a^{4}+\frac{22621}{144272}a^{3}-\frac{92351}{36068}a^{2}-\frac{310139}{72136}a+\frac{71565}{18034}$, $\frac{174061}{4727504896}a^{35}+\frac{25359}{590938112}a^{34}-\frac{442111}{4727504896}a^{33}-\frac{999229}{4727504896}a^{32}-\frac{417673}{4727504896}a^{31}+\frac{574723}{2363752448}a^{30}+\frac{752597}{2363752448}a^{29}+\frac{1316363}{2363752448}a^{28}+\frac{347393}{2363752448}a^{27}-\frac{166887}{73867264}a^{26}-\frac{2180475}{590938112}a^{25}-\frac{2359803}{2363752448}a^{24}+\frac{7099811}{1181876224}a^{23}+\frac{24064229}{2363752448}a^{22}+\frac{6090589}{4727504896}a^{21}-\frac{39809199}{2363752448}a^{20}-\frac{112017521}{4727504896}a^{19}-\frac{25140263}{4727504896}a^{18}+\frac{162712227}{4727504896}a^{17}+\frac{151586005}{2363752448}a^{16}+\frac{59392175}{1181876224}a^{15}-\frac{355553}{4653056}a^{14}-\frac{2144165}{9233408}a^{13}-\frac{2524339}{18466816}a^{12}+\frac{9323279}{36933632}a^{11}+\frac{6639857}{9233408}a^{10}+\frac{1097397}{2308352}a^{9}-\frac{3069845}{4616704}a^{8}-\frac{2898471}{2308352}a^{7}-\frac{519531}{577088}a^{6}-\frac{34619}{144272}a^{5}+\frac{9338}{9017}a^{4}+\frac{1052833}{288544}a^{3}+\frac{400763}{144272}a^{2}-\frac{512057}{72136}a-\frac{307219}{36068}$, $\frac{214397}{2363752448}a^{35}-\frac{195125}{2363752448}a^{34}-\frac{432947}{2363752448}a^{33}-\frac{149861}{1181876224}a^{32}+\frac{36767}{295469056}a^{31}+\frac{823359}{2363752448}a^{30}+\frac{93259}{590938112}a^{29}+\frac{124703}{147734528}a^{28}-\frac{875701}{590938112}a^{27}-\frac{4263113}{1181876224}a^{26}-\frac{654867}{295469056}a^{25}+\frac{4557045}{1181876224}a^{24}+\frac{11757457}{1181876224}a^{23}+\frac{53533}{9306112}a^{22}-\frac{28898725}{2363752448}a^{21}-\frac{51016931}{2363752448}a^{20}-\frac{27161543}{2363752448}a^{19}+\frac{22620307}{1181876224}a^{18}+\frac{71960983}{1181876224}a^{17}+\frac{155246215}{2363752448}a^{16}-\frac{11098735}{1181876224}a^{15}-\frac{30986017}{147734528}a^{14}-\frac{31252379}{147734528}a^{13}+\frac{9426729}{73867264}a^{12}+\frac{22707945}{36933632}a^{11}+\frac{12305449}{18466816}a^{10}-\frac{2861243}{9233408}a^{9}-\frac{3712831}{2308352}a^{8}-\frac{406301}{1154176}a^{7}-\frac{1190361}{1154176}a^{6}+\frac{407687}{577088}a^{5}+\frac{906733}{288544}a^{4}+\frac{509315}{144272}a^{3}-\frac{433469}{144272}a^{2}-\frac{1066225}{72136}a+\frac{33210}{9017}$, $\frac{2589}{295469056}a^{35}+\frac{47187}{2363752448}a^{34}-\frac{87525}{1181876224}a^{33}-\frac{428585}{2363752448}a^{32}-\frac{276925}{2363752448}a^{31}+\frac{9491}{2363752448}a^{30}+\frac{250295}{590938112}a^{29}+\frac{407157}{1181876224}a^{28}+\frac{597555}{1181876224}a^{27}-\frac{1620767}{1181876224}a^{26}-\frac{1921745}{590938112}a^{25}-\frac{602815}{295469056}a^{24}+\frac{3582275}{1181876224}a^{23}+\frac{1304501}{147734528}a^{22}+\frac{6959955}{1181876224}a^{21}-\frac{16804353}{2363752448}a^{20}-\frac{11998873}{590938112}a^{19}-\frac{30489915}{2363752448}a^{18}+\frac{38949425}{2363752448}a^{17}+\frac{157456603}{2363752448}a^{16}+\frac{7804413}{147734528}a^{15}-\frac{1694203}{73867264}a^{14}-\frac{15554827}{73867264}a^{13}-\frac{1785199}{9233408}a^{12}+\frac{789963}{9233408}a^{11}+\frac{696901}{1154176}a^{10}+\frac{1386707}{2308352}a^{9}-\frac{917855}{4616704}a^{8}-\frac{11903}{9017}a^{7}-\frac{720033}{577088}a^{6}-\frac{78177}{72136}a^{5}+\frac{33007}{72136}a^{4}+\frac{437791}{144272}a^{3}+\frac{187337}{144272}a^{2}-\frac{87573}{36068}a-\frac{102578}{9017}$, $\frac{57311}{2363752448}a^{35}+\frac{131209}{2363752448}a^{34}-\frac{181747}{2363752448}a^{33}-\frac{188825}{1181876224}a^{32}-\frac{80001}{1181876224}a^{31}+\frac{309875}{2363752448}a^{30}+\frac{571801}{1181876224}a^{29}+\frac{99115}{295469056}a^{28}+\frac{128075}{295469056}a^{27}-\frac{1813813}{1181876224}a^{26}-\frac{1810405}{590938112}a^{25}-\frac{1791453}{1181876224}a^{24}+\frac{5985555}{1181876224}a^{23}+\frac{12253663}{1181876224}a^{22}+\frac{9190985}{2363752448}a^{21}-\frac{29669797}{2363752448}a^{20}-\frac{50150167}{2363752448}a^{19}-\frac{12130243}{1181876224}a^{18}+\frac{11562679}{590938112}a^{17}+\frac{144859767}{2363752448}a^{16}+\frac{26247997}{590938112}a^{15}-\frac{544579}{18466816}a^{14}-\frac{8261679}{36933632}a^{13}-\frac{3589323}{18466816}a^{12}+\frac{5484585}{36933632}a^{11}+\frac{11671321}{18466816}a^{10}+\frac{4316549}{9233408}a^{9}-\frac{269123}{577088}a^{8}-\frac{1576977}{1154176}a^{7}-\frac{534855}{577088}a^{6}-\frac{61617}{72136}a^{5}+\frac{341591}{288544}a^{4}+\frac{507071}{144272}a^{3}+\frac{324751}{144272}a^{2}-\frac{103945}{36068}a-\frac{113057}{9017}$, $\frac{147399}{4727504896}a^{35}-\frac{6777}{295469056}a^{34}-\frac{214561}{4727504896}a^{33}+\frac{4207}{66584576}a^{32}-\frac{70487}{4727504896}a^{31}+\frac{55219}{2363752448}a^{30}+\frac{144261}{2363752448}a^{29}+\frac{472289}{2363752448}a^{28}-\frac{213793}{2363752448}a^{27}-\frac{352247}{590938112}a^{26}-\frac{40521}{147734528}a^{25}+\frac{566855}{2363752448}a^{24}+\frac{505189}{1181876224}a^{23}+\frac{2750715}{2363752448}a^{22}-\frac{2147417}{4727504896}a^{21}-\frac{4934013}{2363752448}a^{20}-\frac{10055351}{4727504896}a^{19}-\frac{14675405}{4727504896}a^{18}+\frac{3376181}{4727504896}a^{17}+\frac{17204229}{2363752448}a^{16}+\frac{416629}{18466816}a^{15}-\frac{7041705}{590938112}a^{14}-\frac{4283327}{73867264}a^{13}-\frac{282141}{73867264}a^{12}+\frac{3196319}{36933632}a^{11}+\frac{94705}{1154176}a^{10}+\frac{636889}{9233408}a^{9}-\frac{154163}{2308352}a^{8}+\frac{325135}{2308352}a^{7}-\frac{395199}{577088}a^{6}-\frac{214115}{577088}a^{5}+\frac{233399}{144272}a^{4}+\frac{38911}{288544}a^{3}+\frac{81833}{144272}a^{2}-\frac{21615}{18034}a-\frac{54981}{36068}$, $\frac{93625}{2363752448}a^{35}+\frac{74341}{2363752448}a^{34}-\frac{173147}{2363752448}a^{33}-\frac{24977}{147734528}a^{32}-\frac{76933}{1181876224}a^{31}+\frac{456797}{2363752448}a^{30}+\frac{66229}{295469056}a^{29}+\frac{179291}{295469056}a^{28}+\frac{84369}{295469056}a^{27}-\frac{1380199}{1181876224}a^{26}-\frac{855979}{295469056}a^{25}-\frac{1458299}{1181876224}a^{24}+\frac{4804267}{1181876224}a^{23}+\frac{7753463}{1181876224}a^{22}+\frac{5268515}{2363752448}a^{21}-\frac{23035417}{2363752448}a^{20}-\frac{29530347}{2363752448}a^{19}-\frac{4504789}{590938112}a^{18}+\frac{4250039}{295469056}a^{17}+\frac{95179101}{2363752448}a^{16}+\frac{60343867}{1181876224}a^{15}-\frac{22201471}{590938112}a^{14}-\frac{6104811}{36933632}a^{13}-\frac{9413769}{73867264}a^{12}+\frac{900007}{9233408}a^{11}+\frac{1141873}{2308352}a^{10}+\frac{1685375}{4616704}a^{9}-\frac{51337}{288544}a^{8}-\frac{702501}{1154176}a^{7}-\frac{529727}{577088}a^{6}-\frac{455555}{577088}a^{5}+\frac{9221}{36068}a^{4}+\frac{220147}{72136}a^{3}+\frac{451867}{144272}a^{2}-\frac{189623}{72136}a-\frac{153813}{36068}$ 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1458299/1181876224a^24 + 4804267/1181876224a^23 + 7753463/1181876224a^22 + 5268515/2363752448a^21 - 23035417/2363752448a^20 - 29530347/2363752448a^19 - 4504789/590938112a^18 + 4250039/295469056a^17 + 95179101/2363752448a^16 + 60343867/1181876224a^15 - 22201471/590938112a^14 - 6104811/36933632a^13 - 9413769/73867264a^12 + 900007/9233408a^11 + 1141873/2308352a^10 + 1685375/4616704a^9 - 51337/288544a^8 - 702501/1154176a^7 - 529727/577088a^6 - 455555/577088a^5 + 9221/36068a^4 + 220147/72136a^3 + 451867/144272a^2 - 189623/72136a - 153813/36068 (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:	$ 549735585568699.8 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 549735585568699.8 \cdot 1029}{14\cdot\sqrt{111655439513580694005446762483626621692955058293302970913053089}}\cr\approx \mathstrut & 0.890715228551090 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144)

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^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144, 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^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144);

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^36 - 3*x^34 - x^33 + 3*x^32 + 6*x^31 + 2*x^30 + 6*x^29 - 6*x^28 - 48*x^27 - 24*x^26 + 66*x^25 + 140*x^24 + 66*x^23 - 207*x^22 - 350*x^21 - 93*x^20 + 333*x^19 + 751*x^18 + 666*x^17 - 372*x^16 - 2800*x^15 - 3312*x^14 + 2112*x^13 + 8960*x^12 + 8448*x^11 - 6144*x^10 - 24576*x^9 - 6144*x^8 + 12288*x^7 + 8192*x^6 + 49152*x^5 + 49152*x^4 - 32768*x^3 - 196608*x^2 + 262144);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^2:C_6^2$ (as 36T103):

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 144

The 48 conjugacy class representatives for $C_2^2:C_6^2$

Character table for $C_2^2:C_6^2$

Intermediate fields

$\Q(\sqrt{-7}) $, $\Q(\zeta_{9})^+$, 3.3.3969.1, 3.3.3969.2, $\Q(\zeta_{7})^+$, 6.0.1305639.1, 6.6.447834177.1, $\Q(\zeta_{7})$, 6.0.2250423.1, 6.0.110270727.2, 6.0.110270727.1, 9.9.62523502209.1, 12.0.200555450089267329.1, 18.0.1340851596668237962730583.1, 18.0.30806745632221922509786208919.1, 18.18.10566713751852119420856669659217.1

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 36 siblings:	deg 36, deg 36, some data not computed
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	${\href{/padicField/2.3.0.1}{3} }^{12}$	R	${\href{/padicField/5.6.0.1}{6} }^{6}$	R	${\href{/padicField/11.6.0.1}{6} }^{6}$	${\href{/padicField/13.6.0.1}{6} }^{6}$	${\href{/padicField/17.6.0.1}{6} }^{6}$	${\href{/padicField/19.6.0.1}{6} }^{6}$	${\href{/padicField/23.3.0.1}{3} }^{12}$	${\href{/padicField/29.3.0.1}{3} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{6}$	${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{8}$	${\href{/padicField/41.6.0.1}{6} }^{6}$	${\href{/padicField/43.3.0.1}{3} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{6}$	${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{6}$

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
$3$ 3		Deg $18$	$3$	$6$	$24$
$3$ 3		Deg $18$	$3$	$6$	$24$
$7$ 7	7.18.15.5	$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$	$6$	$3$	$15$	$C_6 \times C_3$	$[\ ]_{6}^{3}$
$7$ 7	7.18.15.5	$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98757 x^{12} + 161784 x^{11} + 246024 x^{10} + 264920 x^{9} + 530640 x^{8} + 156384 x^{7} - 1885725 x^{6} - 6133212 x^{5} - 3645540 x^{4} + 5968464 x^{3} + 5011344 x^{2} + 1820448 x + 2358791$	$6$	$3$	$15$	$C_6 \times C_3$	$[\ ]_{6}^{3}$
$199$ 199	199.6.0.1	$x^{6} + 90 x^{3} + 58 x^{2} + 79 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	199.6.0.1	$x^{6} + 90 x^{3} + 58 x^{2} + 79 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	199.6.0.1	$x^{6} + 90 x^{3} + 58 x^{2} + 79 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	199.6.0.1	$x^{6} + 90 x^{3} + 58 x^{2} + 79 x + 3$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	199.12.6.1	$x^{12} + 1194 x^{10} + 180 x^{9} + 594131 x^{8} + 158 x^{7} + 157527750 x^{6} - 43041618 x^{5} + 23482486229 x^{4} - 11375182526 x^{3} + 1869788221909 x^{2} - 840208559160 x + 62349113178683$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$

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