LMFDB - Number field 36.0.48908816365067043970916287981601635325839249495639564072729.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 262144)

gp: K = bnfinit(y^36 - y^35 - y^34 + 3*y^33 - y^32 - 5*y^31 + 7*y^30 + 3*y^29 - 17*y^28 + 11*y^27 + 23*y^26 - 45*y^25 - y^24 + 91*y^23 - 89*y^22 - 93*y^21 + 271*y^20 - 85*y^19 - 457*y^18 - 170*y^17 + 1084*y^16 - 744*y^15 - 1424*y^14 + 2912*y^13 - 64*y^12 - 5760*y^11 + 5888*y^10 + 5632*y^9 - 17408*y^8 + 6144*y^7 + 28672*y^6 - 40960*y^5 - 16384*y^4 + 98304*y^3 - 65536*y^2 - 131072*y + 262144, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 262144);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 262144)

$ x^{36} - x^{35} - x^{34} + 3 x^{33} - x^{32} - 5 x^{31} + 7 x^{30} + 3 x^{29} - 17 x^{28} + \cdots + 262144 $ x^36 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 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:	$48908816365067043970916287981601635325839249495639564072729$ 48908816365067043970916287981601635325839249495639564072729 $\medspace = 7^{18}\cdot 19^{34}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$42.68$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$7^{1/2}19^{17/18}\approx 42.68357178143818$
Ramified primes:	$7$, $19$ 7, 19	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$36$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$133=7\cdot 19$
Dirichlet character group:	$\lbrace$$\chi_{133}(1,·)$, $\chi_{133}(132,·)$, $\chi_{133}(6,·)$, $\chi_{133}(8,·)$, $\chi_{133}(13,·)$, $\chi_{133}(15,·)$, $\chi_{133}(20,·)$, $\chi_{133}(22,·)$, $\chi_{133}(27,·)$, $\chi_{133}(29,·)$, $\chi_{133}(34,·)$, $\chi_{133}(36,·)$, $\chi_{133}(41,·)$, $\chi_{133}(43,·)$, $\chi_{133}(48,·)$, $\chi_{133}(50,·)$, $\chi_{133}(55,·)$, $\chi_{133}(62,·)$, $\chi_{133}(64,·)$, $\chi_{133}(69,·)$, $\chi_{133}(71,·)$, $\chi_{133}(78,·)$, $\chi_{133}(83,·)$, $\chi_{133}(85,·)$, $\chi_{133}(90,·)$, $\chi_{133}(92,·)$, $\chi_{133}(97,·)$, $\chi_{133}(99,·)$, $\chi_{133}(104,·)$, $\chi_{133}(106,·)$, $\chi_{133}(111,·)$, $\chi_{133}(113,·)$, $\chi_{133}(118,·)$, $\chi_{133}(120,·)$, $\chi_{133}(125,·)$, $\chi_{133}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{914}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{85}{457}$, $\frac{1}{1828}a^{20}-\frac{1}{1828}a^{19}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{85}{914}a-\frac{186}{457}$, $\frac{1}{3656}a^{21}-\frac{1}{3656}a^{20}-\frac{1}{3656}a^{19}+\frac{3}{8}a^{18}-\frac{1}{8}a^{17}+\frac{3}{8}a^{16}-\frac{1}{8}a^{15}+\frac{3}{8}a^{14}-\frac{1}{8}a^{13}+\frac{3}{8}a^{12}-\frac{1}{8}a^{11}+\frac{3}{8}a^{10}-\frac{1}{8}a^{9}+\frac{3}{8}a^{8}-\frac{1}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{85}{1828}a^{2}+\frac{271}{914}a-\frac{93}{457}$, $\frac{1}{7312}a^{22}-\frac{1}{7312}a^{21}-\frac{1}{7312}a^{20}+\frac{3}{7312}a^{19}+\frac{7}{16}a^{18}+\frac{3}{16}a^{17}-\frac{1}{16}a^{16}-\frac{5}{16}a^{15}+\frac{7}{16}a^{14}+\frac{3}{16}a^{13}-\frac{1}{16}a^{12}-\frac{5}{16}a^{11}+\frac{7}{16}a^{10}+\frac{3}{16}a^{9}-\frac{1}{16}a^{8}-\frac{5}{16}a^{7}+\frac{7}{16}a^{6}+\frac{3}{16}a^{5}-\frac{1}{16}a^{4}-\frac{85}{3656}a^{3}+\frac{271}{1828}a^{2}-\frac{93}{914}a-\frac{89}{457}$, $\frac{1}{14624}a^{23}-\frac{1}{14624}a^{22}-\frac{1}{14624}a^{21}+\frac{3}{14624}a^{20}-\frac{1}{14624}a^{19}+\frac{3}{32}a^{18}+\frac{15}{32}a^{17}+\frac{11}{32}a^{16}-\frac{9}{32}a^{15}-\frac{13}{32}a^{14}-\frac{1}{32}a^{13}-\frac{5}{32}a^{12}+\frac{7}{32}a^{11}+\frac{3}{32}a^{10}+\frac{15}{32}a^{9}+\frac{11}{32}a^{8}-\frac{9}{32}a^{7}-\frac{13}{32}a^{6}-\frac{1}{32}a^{5}-\frac{85}{7312}a^{4}+\frac{271}{3656}a^{3}-\frac{93}{1828}a^{2}-\frac{89}{914}a+\frac{91}{457}$, $\frac{1}{29248}a^{24}-\frac{1}{29248}a^{23}-\frac{1}{29248}a^{22}+\frac{3}{29248}a^{21}-\frac{1}{29248}a^{20}-\frac{5}{29248}a^{19}+\frac{15}{64}a^{18}-\frac{21}{64}a^{17}-\frac{9}{64}a^{16}-\frac{13}{64}a^{15}+\frac{31}{64}a^{14}-\frac{5}{64}a^{13}+\frac{7}{64}a^{12}+\frac{3}{64}a^{11}-\frac{17}{64}a^{10}+\frac{11}{64}a^{9}+\frac{23}{64}a^{8}+\frac{19}{64}a^{7}-\frac{1}{64}a^{6}-\frac{85}{14624}a^{5}+\frac{271}{7312}a^{4}-\frac{93}{3656}a^{3}-\frac{89}{1828}a^{2}+\frac{91}{914}a-\frac{1}{457}$, $\frac{1}{58496}a^{25}-\frac{1}{58496}a^{24}-\frac{1}{58496}a^{23}+\frac{3}{58496}a^{22}-\frac{1}{58496}a^{21}-\frac{5}{58496}a^{20}+\frac{7}{58496}a^{19}-\frac{21}{128}a^{18}-\frac{9}{128}a^{17}+\frac{51}{128}a^{16}-\frac{33}{128}a^{15}+\frac{59}{128}a^{14}+\frac{7}{128}a^{13}+\frac{3}{128}a^{12}-\frac{17}{128}a^{11}+\frac{11}{128}a^{10}+\frac{23}{128}a^{9}-\frac{45}{128}a^{8}-\frac{1}{128}a^{7}-\frac{85}{29248}a^{6}+\frac{271}{14624}a^{5}-\frac{93}{7312}a^{4}-\frac{89}{3656}a^{3}+\frac{91}{1828}a^{2}-\frac{1}{914}a-\frac{45}{457}$, $\frac{1}{116992}a^{26}-\frac{1}{116992}a^{25}-\frac{1}{116992}a^{24}+\frac{3}{116992}a^{23}-\frac{1}{116992}a^{22}-\frac{5}{116992}a^{21}+\frac{7}{116992}a^{20}+\frac{3}{116992}a^{19}-\frac{9}{256}a^{18}+\frac{51}{256}a^{17}-\frac{33}{256}a^{16}-\frac{69}{256}a^{15}-\frac{121}{256}a^{14}+\frac{3}{256}a^{13}-\frac{17}{256}a^{12}+\frac{11}{256}a^{11}+\frac{23}{256}a^{10}-\frac{45}{256}a^{9}-\frac{1}{256}a^{8}-\frac{85}{58496}a^{7}+\frac{271}{29248}a^{6}-\frac{93}{14624}a^{5}-\frac{89}{7312}a^{4}+\frac{91}{3656}a^{3}-\frac{1}{1828}a^{2}-\frac{45}{914}a+\frac{23}{457}$, $\frac{1}{233984}a^{27}-\frac{1}{233984}a^{26}-\frac{1}{233984}a^{25}+\frac{3}{233984}a^{24}-\frac{1}{233984}a^{23}-\frac{5}{233984}a^{22}+\frac{7}{233984}a^{21}+\frac{3}{233984}a^{20}-\frac{17}{233984}a^{19}-\frac{205}{512}a^{18}+\frac{223}{512}a^{17}+\frac{187}{512}a^{16}-\frac{121}{512}a^{15}-\frac{253}{512}a^{14}-\frac{17}{512}a^{13}+\frac{11}{512}a^{12}+\frac{23}{512}a^{11}-\frac{45}{512}a^{10}-\frac{1}{512}a^{9}-\frac{85}{116992}a^{8}+\frac{271}{58496}a^{7}-\frac{93}{29248}a^{6}-\frac{89}{14624}a^{5}+\frac{91}{7312}a^{4}-\frac{1}{3656}a^{3}-\frac{45}{1828}a^{2}+\frac{23}{914}a+\frac{11}{457}$, $\frac{1}{467968}a^{28}-\frac{1}{467968}a^{27}-\frac{1}{467968}a^{26}+\frac{3}{467968}a^{25}-\frac{1}{467968}a^{24}-\frac{5}{467968}a^{23}+\frac{7}{467968}a^{22}+\frac{3}{467968}a^{21}-\frac{17}{467968}a^{20}+\frac{11}{467968}a^{19}-\frac{289}{1024}a^{18}-\frac{325}{1024}a^{17}-\frac{121}{1024}a^{16}-\frac{253}{1024}a^{15}+\frac{495}{1024}a^{14}+\frac{11}{1024}a^{13}+\frac{23}{1024}a^{12}-\frac{45}{1024}a^{11}-\frac{1}{1024}a^{10}-\frac{85}{233984}a^{9}+\frac{271}{116992}a^{8}-\frac{93}{58496}a^{7}-\frac{89}{29248}a^{6}+\frac{91}{14624}a^{5}-\frac{1}{7312}a^{4}-\frac{45}{3656}a^{3}+\frac{23}{1828}a^{2}+\frac{11}{914}a-\frac{17}{457}$, $\frac{1}{935936}a^{29}-\frac{1}{935936}a^{28}-\frac{1}{935936}a^{27}+\frac{3}{935936}a^{26}-\frac{1}{935936}a^{25}-\frac{5}{935936}a^{24}+\frac{7}{935936}a^{23}+\frac{3}{935936}a^{22}-\frac{17}{935936}a^{21}+\frac{11}{935936}a^{20}+\frac{23}{935936}a^{19}+\frac{699}{2048}a^{18}-\frac{121}{2048}a^{17}+\frac{771}{2048}a^{16}-\frac{529}{2048}a^{15}-\frac{1013}{2048}a^{14}+\frac{23}{2048}a^{13}-\frac{45}{2048}a^{12}-\frac{1}{2048}a^{11}-\frac{85}{467968}a^{10}+\frac{271}{233984}a^{9}-\frac{93}{116992}a^{8}-\frac{89}{58496}a^{7}+\frac{91}{29248}a^{6}-\frac{1}{14624}a^{5}-\frac{45}{7312}a^{4}+\frac{23}{3656}a^{3}+\frac{11}{1828}a^{2}-\frac{17}{914}a+\frac{3}{457}$, $\frac{1}{1871872}a^{30}-\frac{1}{1871872}a^{29}-\frac{1}{1871872}a^{28}+\frac{3}{1871872}a^{27}-\frac{1}{1871872}a^{26}-\frac{5}{1871872}a^{25}+\frac{7}{1871872}a^{24}+\frac{3}{1871872}a^{23}-\frac{17}{1871872}a^{22}+\frac{11}{1871872}a^{21}+\frac{23}{1871872}a^{20}-\frac{45}{1871872}a^{19}+\frac{1927}{4096}a^{18}+\frac{771}{4096}a^{17}-\frac{529}{4096}a^{16}-\frac{1013}{4096}a^{15}-\frac{2025}{4096}a^{14}-\frac{45}{4096}a^{13}-\frac{1}{4096}a^{12}-\frac{85}{935936}a^{11}+\frac{271}{467968}a^{10}-\frac{93}{233984}a^{9}-\frac{89}{116992}a^{8}+\frac{91}{58496}a^{7}-\frac{1}{29248}a^{6}-\frac{45}{14624}a^{5}+\frac{23}{7312}a^{4}+\frac{11}{3656}a^{3}-\frac{17}{1828}a^{2}+\frac{3}{914}a+\frac{7}{457}$, $\frac{1}{3743744}a^{31}-\frac{1}{3743744}a^{30}-\frac{1}{3743744}a^{29}+\frac{3}{3743744}a^{28}-\frac{1}{3743744}a^{27}-\frac{5}{3743744}a^{26}+\frac{7}{3743744}a^{25}+\frac{3}{3743744}a^{24}-\frac{17}{3743744}a^{23}+\frac{11}{3743744}a^{22}+\frac{23}{3743744}a^{21}-\frac{45}{3743744}a^{20}-\frac{1}{3743744}a^{19}+\frac{771}{8192}a^{18}+\frac{3567}{8192}a^{17}+\frac{3083}{8192}a^{16}-\frac{2025}{8192}a^{15}+\frac{4051}{8192}a^{14}-\frac{1}{8192}a^{13}-\frac{85}{1871872}a^{12}+\frac{271}{935936}a^{11}-\frac{93}{467968}a^{10}-\frac{89}{233984}a^{9}+\frac{91}{116992}a^{8}-\frac{1}{58496}a^{7}-\frac{45}{29248}a^{6}+\frac{23}{14624}a^{5}+\frac{11}{7312}a^{4}-\frac{17}{3656}a^{3}+\frac{3}{1828}a^{2}+\frac{7}{914}a-\frac{5}{457}$, $\frac{1}{7487488}a^{32}-\frac{1}{7487488}a^{31}-\frac{1}{7487488}a^{30}+\frac{3}{7487488}a^{29}-\frac{1}{7487488}a^{28}-\frac{5}{7487488}a^{27}+\frac{7}{7487488}a^{26}+\frac{3}{7487488}a^{25}-\frac{17}{7487488}a^{24}+\frac{11}{7487488}a^{23}+\frac{23}{7487488}a^{22}-\frac{45}{7487488}a^{21}-\frac{1}{7487488}a^{20}+\frac{91}{7487488}a^{19}-\frac{4625}{16384}a^{18}+\frac{3083}{16384}a^{17}+\frac{6167}{16384}a^{16}+\frac{4051}{16384}a^{15}-\frac{1}{16384}a^{14}-\frac{85}{3743744}a^{13}+\frac{271}{1871872}a^{12}-\frac{93}{935936}a^{11}-\frac{89}{467968}a^{10}+\frac{91}{233984}a^{9}-\frac{1}{116992}a^{8}-\frac{45}{58496}a^{7}+\frac{23}{29248}a^{6}+\frac{11}{14624}a^{5}-\frac{17}{7312}a^{4}+\frac{3}{3656}a^{3}+\frac{7}{1828}a^{2}-\frac{5}{914}a-\frac{1}{457}$, $\frac{1}{14974976}a^{33}-\frac{1}{14974976}a^{32}-\frac{1}{14974976}a^{31}+\frac{3}{14974976}a^{30}-\frac{1}{14974976}a^{29}-\frac{5}{14974976}a^{28}+\frac{7}{14974976}a^{27}+\frac{3}{14974976}a^{26}-\frac{17}{14974976}a^{25}+\frac{11}{14974976}a^{24}+\frac{23}{14974976}a^{23}-\frac{45}{14974976}a^{22}-\frac{1}{14974976}a^{21}+\frac{91}{14974976}a^{20}-\frac{89}{14974976}a^{19}+\frac{3083}{32768}a^{18}+\frac{6167}{32768}a^{17}-\frac{12333}{32768}a^{16}-\frac{1}{32768}a^{15}-\frac{85}{7487488}a^{14}+\frac{271}{3743744}a^{13}-\frac{93}{1871872}a^{12}-\frac{89}{935936}a^{11}+\frac{91}{467968}a^{10}-\frac{1}{233984}a^{9}-\frac{45}{116992}a^{8}+\frac{23}{58496}a^{7}+\frac{11}{29248}a^{6}-\frac{17}{14624}a^{5}+\frac{3}{7312}a^{4}+\frac{7}{3656}a^{3}-\frac{5}{1828}a^{2}-\frac{1}{914}a+\frac{3}{457}$, $\frac{1}{29949952}a^{34}-\frac{1}{29949952}a^{33}-\frac{1}{29949952}a^{32}+\frac{3}{29949952}a^{31}-\frac{1}{29949952}a^{30}-\frac{5}{29949952}a^{29}+\frac{7}{29949952}a^{28}+\frac{3}{29949952}a^{27}-\frac{17}{29949952}a^{26}+\frac{11}{29949952}a^{25}+\frac{23}{29949952}a^{24}-\frac{45}{29949952}a^{23}-\frac{1}{29949952}a^{22}+\frac{91}{29949952}a^{21}-\frac{89}{29949952}a^{20}-\frac{93}{29949952}a^{19}+\frac{6167}{65536}a^{18}-\frac{12333}{65536}a^{17}-\frac{1}{65536}a^{16}-\frac{85}{14974976}a^{15}+\frac{271}{7487488}a^{14}-\frac{93}{3743744}a^{13}-\frac{89}{1871872}a^{12}+\frac{91}{935936}a^{11}-\frac{1}{467968}a^{10}-\frac{45}{233984}a^{9}+\frac{23}{116992}a^{8}+\frac{11}{58496}a^{7}-\frac{17}{29248}a^{6}+\frac{3}{14624}a^{5}+\frac{7}{7312}a^{4}-\frac{5}{3656}a^{3}-\frac{1}{1828}a^{2}+\frac{3}{914}a-\frac{1}{457}$, $\frac{1}{59899904}a^{35}-\frac{1}{59899904}a^{34}-\frac{1}{59899904}a^{33}+\frac{3}{59899904}a^{32}-\frac{1}{59899904}a^{31}-\frac{5}{59899904}a^{30}+\frac{7}{59899904}a^{29}+\frac{3}{59899904}a^{28}-\frac{17}{59899904}a^{27}+\frac{11}{59899904}a^{26}+\frac{23}{59899904}a^{25}-\frac{45}{59899904}a^{24}-\frac{1}{59899904}a^{23}+\frac{91}{59899904}a^{22}-\frac{89}{59899904}a^{21}-\frac{93}{59899904}a^{20}+\frac{271}{59899904}a^{19}-\frac{12333}{131072}a^{18}-\frac{1}{131072}a^{17}-\frac{85}{29949952}a^{16}+\frac{271}{14974976}a^{15}-\frac{93}{7487488}a^{14}-\frac{89}{3743744}a^{13}+\frac{91}{1871872}a^{12}-\frac{1}{935936}a^{11}-\frac{45}{467968}a^{10}+\frac{23}{233984}a^{9}+\frac{11}{116992}a^{8}-\frac{17}{58496}a^{7}+\frac{3}{29248}a^{6}+\frac{7}{14624}a^{5}-\frac{5}{7312}a^{4}-\frac{1}{3656}a^{3}+\frac{3}{1828}a^{2}-\frac{1}{914}a-\frac{1}{457}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, 1/914*a^19 - 1/2*a^18 - 1/2*a^17 - 1/2*a^16 - 1/2*a^15 - 1/2*a^14 - 1/2*a^13 - 1/2*a^12 - 1/2*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2 - 1/2*a - 85/457, 1/1828*a^20 - 1/1828*a^19 - 1/4*a^18 - 1/4*a^17 - 1/4*a^16 - 1/4*a^15 - 1/4*a^14 - 1/4*a^13 - 1/4*a^12 - 1/4*a^11 - 1/4*a^10 - 1/4*a^9 - 1/4*a^8 - 1/4*a^7 - 1/4*a^6 - 1/4*a^5 - 1/4*a^4 - 1/4*a^3 - 1/4*a^2 - 85/914*a - 186/457, 1/3656*a^21 - 1/3656*a^20 - 1/3656*a^19 + 3/8*a^18 - 1/8*a^17 + 3/8*a^16 - 1/8*a^15 + 3/8*a^14 - 1/8*a^13 + 3/8*a^12 - 1/8*a^11 + 3/8*a^10 - 1/8*a^9 + 3/8*a^8 - 1/8*a^7 + 3/8*a^6 - 1/8*a^5 + 3/8*a^4 - 1/8*a^3 - 85/1828*a^2 + 271/914*a - 93/457, 1/7312*a^22 - 1/7312*a^21 - 1/7312*a^20 + 3/7312*a^19 + 7/16*a^18 + 3/16*a^17 - 1/16*a^16 - 5/16*a^15 + 7/16*a^14 + 3/16*a^13 - 1/16*a^12 - 5/16*a^11 + 7/16*a^10 + 3/16*a^9 - 1/16*a^8 - 5/16*a^7 + 7/16*a^6 + 3/16*a^5 - 1/16*a^4 - 85/3656*a^3 + 271/1828*a^2 - 93/914*a - 89/457, 1/14624*a^23 - 1/14624*a^22 - 1/14624*a^21 + 3/14624*a^20 - 1/14624*a^19 + 3/32*a^18 + 15/32*a^17 + 11/32*a^16 - 9/32*a^15 - 13/32*a^14 - 1/32*a^13 - 5/32*a^12 + 7/32*a^11 + 3/32*a^10 + 15/32*a^9 + 11/32*a^8 - 9/32*a^7 - 13/32*a^6 - 1/32*a^5 - 85/7312*a^4 + 271/3656*a^3 - 93/1828*a^2 - 89/914*a + 91/457, 1/29248*a^24 - 1/29248*a^23 - 1/29248*a^22 + 3/29248*a^21 - 1/29248*a^20 - 5/29248*a^19 + 15/64*a^18 - 21/64*a^17 - 9/64*a^16 - 13/64*a^15 + 31/64*a^14 - 5/64*a^13 + 7/64*a^12 + 3/64*a^11 - 17/64*a^10 + 11/64*a^9 + 23/64*a^8 + 19/64*a^7 - 1/64*a^6 - 85/14624*a^5 + 271/7312*a^4 - 93/3656*a^3 - 89/1828*a^2 + 91/914*a - 1/457, 1/58496*a^25 - 1/58496*a^24 - 1/58496*a^23 + 3/58496*a^22 - 1/58496*a^21 - 5/58496*a^20 + 7/58496*a^19 - 21/128*a^18 - 9/128*a^17 + 51/128*a^16 - 33/128*a^15 + 59/128*a^14 + 7/128*a^13 + 3/128*a^12 - 17/128*a^11 + 11/128*a^10 + 23/128*a^9 - 45/128*a^8 - 1/128*a^7 - 85/29248*a^6 + 271/14624*a^5 - 93/7312*a^4 - 89/3656*a^3 + 91/1828*a^2 - 1/914*a - 45/457, 1/116992*a^26 - 1/116992*a^25 - 1/116992*a^24 + 3/116992*a^23 - 1/116992*a^22 - 5/116992*a^21 + 7/116992*a^20 + 3/116992*a^19 - 9/256*a^18 + 51/256*a^17 - 33/256*a^16 - 69/256*a^15 - 121/256*a^14 + 3/256*a^13 - 17/256*a^12 + 11/256*a^11 + 23/256*a^10 - 45/256*a^9 - 1/256*a^8 - 85/58496*a^7 + 271/29248*a^6 - 93/14624*a^5 - 89/7312*a^4 + 91/3656*a^3 - 1/1828*a^2 - 45/914*a + 23/457, 1/233984*a^27 - 1/233984*a^26 - 1/233984*a^25 + 3/233984*a^24 - 1/233984*a^23 - 5/233984*a^22 + 7/233984*a^21 + 3/233984*a^20 - 17/233984*a^19 - 205/512*a^18 + 223/512*a^17 + 187/512*a^16 - 121/512*a^15 - 253/512*a^14 - 17/512*a^13 + 11/512*a^12 + 23/512*a^11 - 45/512*a^10 - 1/512*a^9 - 85/116992*a^8 + 271/58496*a^7 - 93/29248*a^6 - 89/14624*a^5 + 91/7312*a^4 - 1/3656*a^3 - 45/1828*a^2 + 23/914*a + 11/457, 1/467968*a^28 - 1/467968*a^27 - 1/467968*a^26 + 3/467968*a^25 - 1/467968*a^24 - 5/467968*a^23 + 7/467968*a^22 + 3/467968*a^21 - 17/467968*a^20 + 11/467968*a^19 - 289/1024*a^18 - 325/1024*a^17 - 121/1024*a^16 - 253/1024*a^15 + 495/1024*a^14 + 11/1024*a^13 + 23/1024*a^12 - 45/1024*a^11 - 1/1024*a^10 - 85/233984*a^9 + 271/116992*a^8 - 93/58496*a^7 - 89/29248*a^6 + 91/14624*a^5 - 1/7312*a^4 - 45/3656*a^3 + 23/1828*a^2 + 11/914*a - 17/457, 1/935936*a^29 - 1/935936*a^28 - 1/935936*a^27 + 3/935936*a^26 - 1/935936*a^25 - 5/935936*a^24 + 7/935936*a^23 + 3/935936*a^22 - 17/935936*a^21 + 11/935936*a^20 + 23/935936*a^19 + 699/2048*a^18 - 121/2048*a^17 + 771/2048*a^16 - 529/2048*a^15 - 1013/2048*a^14 + 23/2048*a^13 - 45/2048*a^12 - 1/2048*a^11 - 85/467968*a^10 + 271/233984*a^9 - 93/116992*a^8 - 89/58496*a^7 + 91/29248*a^6 - 1/14624*a^5 - 45/7312*a^4 + 23/3656*a^3 + 11/1828*a^2 - 17/914*a + 3/457, 1/1871872*a^30 - 1/1871872*a^29 - 1/1871872*a^28 + 3/1871872*a^27 - 1/1871872*a^26 - 5/1871872*a^25 + 7/1871872*a^24 + 3/1871872*a^23 - 17/1871872*a^22 + 11/1871872*a^21 + 23/1871872*a^20 - 45/1871872*a^19 + 1927/4096*a^18 + 771/4096*a^17 - 529/4096*a^16 - 1013/4096*a^15 - 2025/4096*a^14 - 45/4096*a^13 - 1/4096*a^12 - 85/935936*a^11 + 271/467968*a^10 - 93/233984*a^9 - 89/116992*a^8 + 91/58496*a^7 - 1/29248*a^6 - 45/14624*a^5 + 23/7312*a^4 + 11/3656*a^3 - 17/1828*a^2 + 3/914*a + 7/457, 1/3743744*a^31 - 1/3743744*a^30 - 1/3743744*a^29 + 3/3743744*a^28 - 1/3743744*a^27 - 5/3743744*a^26 + 7/3743744*a^25 + 3/3743744*a^24 - 17/3743744*a^23 + 11/3743744*a^22 + 23/3743744*a^21 - 45/3743744*a^20 - 1/3743744*a^19 + 771/8192*a^18 + 3567/8192*a^17 + 3083/8192*a^16 - 2025/8192*a^15 + 4051/8192*a^14 - 1/8192*a^13 - 85/1871872*a^12 + 271/935936*a^11 - 93/467968*a^10 - 89/233984*a^9 + 91/116992*a^8 - 1/58496*a^7 - 45/29248*a^6 + 23/14624*a^5 + 11/7312*a^4 - 17/3656*a^3 + 3/1828*a^2 + 7/914*a - 5/457, 1/7487488*a^32 - 1/7487488*a^31 - 1/7487488*a^30 + 3/7487488*a^29 - 1/7487488*a^28 - 5/7487488*a^27 + 7/7487488*a^26 + 3/7487488*a^25 - 17/7487488*a^24 + 11/7487488*a^23 + 23/7487488*a^22 - 45/7487488*a^21 - 1/7487488*a^20 + 91/7487488*a^19 - 4625/16384*a^18 + 3083/16384*a^17 + 6167/16384*a^16 + 4051/16384*a^15 - 1/16384*a^14 - 85/3743744*a^13 + 271/1871872*a^12 - 93/935936*a^11 - 89/467968*a^10 + 91/233984*a^9 - 1/116992*a^8 - 45/58496*a^7 + 23/29248*a^6 + 11/14624*a^5 - 17/7312*a^4 + 3/3656*a^3 + 7/1828*a^2 - 5/914*a - 1/457, 1/14974976*a^33 - 1/14974976*a^32 - 1/14974976*a^31 + 3/14974976*a^30 - 1/14974976*a^29 - 5/14974976*a^28 + 7/14974976*a^27 + 3/14974976*a^26 - 17/14974976*a^25 + 11/14974976*a^24 + 23/14974976*a^23 - 45/14974976*a^22 - 1/14974976*a^21 + 91/14974976*a^20 - 89/14974976*a^19 + 3083/32768*a^18 + 6167/32768*a^17 - 12333/32768*a^16 - 1/32768*a^15 - 85/7487488*a^14 + 271/3743744*a^13 - 93/1871872*a^12 - 89/935936*a^11 + 91/467968*a^10 - 1/233984*a^9 - 45/116992*a^8 + 23/58496*a^7 + 11/29248*a^6 - 17/14624*a^5 + 3/7312*a^4 + 7/3656*a^3 - 5/1828*a^2 - 1/914*a + 3/457, 1/29949952*a^34 - 1/29949952*a^33 - 1/29949952*a^32 + 3/29949952*a^31 - 1/29949952*a^30 - 5/29949952*a^29 + 7/29949952*a^28 + 3/29949952*a^27 - 17/29949952*a^26 + 11/29949952*a^25 + 23/29949952*a^24 - 45/29949952*a^23 - 1/29949952*a^22 + 91/29949952*a^21 - 89/29949952*a^20 - 93/29949952*a^19 + 6167/65536*a^18 - 12333/65536*a^17 - 1/65536*a^16 - 85/14974976*a^15 + 271/7487488*a^14 - 93/3743744*a^13 - 89/1871872*a^12 + 91/935936*a^11 - 1/467968*a^10 - 45/233984*a^9 + 23/116992*a^8 + 11/58496*a^7 - 17/29248*a^6 + 3/14624*a^5 + 7/7312*a^4 - 5/3656*a^3 - 1/1828*a^2 + 3/914*a - 1/457, 1/59899904*a^35 - 1/59899904*a^34 - 1/59899904*a^33 + 3/59899904*a^32 - 1/59899904*a^31 - 5/59899904*a^30 + 7/59899904*a^29 + 3/59899904*a^28 - 17/59899904*a^27 + 11/59899904*a^26 + 23/59899904*a^25 - 45/59899904*a^24 - 1/59899904*a^23 + 91/59899904*a^22 - 89/59899904*a^21 - 93/59899904*a^20 + 271/59899904*a^19 - 12333/131072*a^18 - 1/131072*a^17 - 85/29949952*a^16 + 271/14974976*a^15 - 93/7487488*a^14 - 89/3743744*a^13 + 91/1871872*a^12 - 1/935936*a^11 - 45/467968*a^10 + 23/233984*a^9 + 11/116992*a^8 - 17/58496*a^7 + 3/29248*a^6 + 7/14624*a^5 - 5/7312*a^4 - 1/3656*a^3 + 3/1828*a^2 - 1/914*a - 1/457

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (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{3}{7312} a^{23} + \frac{967}{7312} a^{4} $ -(3)/(7312)a^(23) + (967)/(7312)a^(4) (order $38$)	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:	$\frac{23}{935936}a^{30}-\frac{24475}{935936}a^{11}+1$, $\frac{89}{14974976}a^{34}+\frac{27371}{14974976}a^{15}-1$, $\frac{23}{935936}a^{30}-\frac{1}{3656}a^{22}-\frac{24475}{935936}a^{11}+\frac{1541}{3656}a^{3}+1$, $\frac{93}{29949952}a^{35}+\frac{1}{3743744}a^{32}-\frac{11}{467968}a^{29}-\frac{7}{58496}a^{26}-\frac{139657}{29949952}a^{16}-\frac{41757}{3743744}a^{13}-\frac{8641}{467968}a^{10}-\frac{181}{58496}a^{7}$, $\frac{7}{58496}a^{26}+\frac{3}{7312}a^{23}+\frac{181}{58496}a^{7}-\frac{967}{7312}a^{4}$, $\frac{271}{59899904}a^{35}+\frac{271}{59899904}a^{34}-\frac{85}{59899904}a^{33}+\frac{271}{59899904}a^{32}+\frac{1355}{59899904}a^{31}-\frac{1897}{59899904}a^{30}-\frac{813}{59899904}a^{29}+\frac{255}{59899904}a^{28}-\frac{2981}{59899904}a^{27}-\frac{6233}{59899904}a^{26}+\frac{1955}{59899904}a^{25}+\frac{271}{59899904}a^{24}-\frac{24661}{59899904}a^{23}+\frac{7735}{59899904}a^{22}+\frac{25203}{59899904}a^{21}-\frac{7905}{59899904}a^{20}+\frac{23035}{59899904}a^{19}+\frac{271}{131072}a^{18}-\frac{85}{131072}a^{17}-\frac{73441}{14974976}a^{16}+\frac{25203}{7487488}a^{15}-\frac{7905}{7487488}a^{14}-\frac{24661}{1871872}a^{13}+\frac{271}{935936}a^{12}+\frac{12195}{467968}a^{11}-\frac{6233}{233984}a^{10}+\frac{1955}{233984}a^{9}+\frac{4607}{58496}a^{8}-\frac{813}{29248}a^{7}+\frac{255}{29248}a^{6}+\frac{1355}{7312}a^{5}+\frac{271}{3656}a^{4}-\frac{85}{3656}a^{3}+\frac{271}{914}a^{2}-\frac{85}{914}a-\frac{542}{457}$, $\frac{1}{3656}a^{22}+\frac{1}{1828}a^{21}-\frac{1}{914}a^{20}-\frac{1541}{3656}a^{3}+\frac{287}{1828}a^{2}+\frac{627}{914}a$, $\frac{1}{1828}a^{21}+\frac{287}{1828}a^{2}-1$, $\frac{91}{7487488}a^{33}+\frac{17}{233984}a^{29}+\frac{7}{29248}a^{25}+\frac{1}{1828}a^{21}-\frac{56143}{7487488}a^{14}-\frac{7917}{233984}a^{10}+\frac{181}{29248}a^{6}+\frac{287}{1828}a^{2}$, $\frac{23}{935936}a^{30}-\frac{3}{116992}a^{28}+\frac{3}{58496}a^{26}+\frac{1}{14624}a^{24}-\frac{24475}{935936}a^{11}+\frac{8279}{116992}a^{9}-\frac{8279}{58496}a^{7}+\frac{2115}{14624}a^{5}$, $\frac{1}{3743744}a^{33}+\frac{3}{58496}a^{26}-\frac{1}{1828}a^{21}-\frac{41757}{3743744}a^{14}-\frac{8279}{58496}a^{7}-\frac{287}{1828}a^{2}-1$, $\frac{89}{14974976}a^{35}-\frac{93}{29949952}a^{34}-\frac{271}{29949952}a^{33}+\frac{93}{29949952}a^{32}-\frac{991}{29949952}a^{31}-\frac{619}{29949952}a^{30}+\frac{3369}{29949952}a^{29}+\frac{813}{29949952}a^{28}-\frac{5375}{29949952}a^{27}+\frac{6565}{29949952}a^{26}+\frac{6233}{29949952}a^{25}-\frac{22435}{29949952}a^{24}-\frac{271}{29949952}a^{23}+\frac{32853}{29949952}a^{22}-\frac{24119}{29949952}a^{21}-\frac{57971}{29949952}a^{20}+\frac{73441}{29949952}a^{19}+\frac{93}{65536}a^{18}-\frac{271}{65536}a^{17}+\frac{93587}{29949952}a^{16}+\frac{174253}{14974976}a^{15}-\frac{25203}{3743744}a^{14}-\frac{8463}{935936}a^{13}+\frac{56515}{1871872}a^{12}-\frac{25017}{935936}a^{11}-\frac{48865}{467968}a^{10}+\frac{6233}{116992}a^{9}+\frac{14241}{116992}a^{8}-\frac{9033}{58496}a^{7}+\frac{813}{14624}a^{6}+\frac{7843}{14624}a^{5}-\frac{1355}{3656}a^{4}-\frac{2083}{3656}a^{3}+\frac{813}{914}a^{2}+\frac{85}{914}a-\frac{542}{457}$, $\frac{91}{7487488}a^{34}+\frac{1}{3743744}a^{32}+\frac{3}{116992}a^{27}+\frac{1}{3656}a^{22}-\frac{1}{914}a^{20}-\frac{56143}{7487488}a^{15}-\frac{41757}{3743744}a^{13}-\frac{8279}{116992}a^{8}-\frac{1541}{3656}a^{3}-\frac{287}{914}a$, $\frac{441}{59899904}a^{35}-\frac{1}{131072}a^{34}-\frac{797}{59899904}a^{33}+\frac{1711}{59899904}a^{32}-\frac{117}{59899904}a^{31}-\frac{4777}{59899904}a^{30}+\frac{595}{59899904}a^{29}+\frac{3071}{59899904}a^{28}-\frac{10149}{59899904}a^{27}+\frac{4007}{59899904}a^{26}+\frac{12195}{59899904}a^{25}-\frac{28401}{59899904}a^{24}-\frac{8277}{59899904}a^{23}+\frac{56887}{59899904}a^{22}-\frac{40333}{59899904}a^{21}-\frac{73441}{59899904}a^{20}+\frac{154107}{59899904}a^{19}+\frac{271}{131072}a^{18}-\frac{85}{131072}a^{17}+\frac{47517}{29949952}a^{16}+\frac{89}{8192}a^{15}-\frac{8819}{1871872}a^{14}-\frac{15927}{935936}a^{13}+\frac{12373}{467968}a^{12}+\frac{31583}{935936}a^{11}-\frac{3825}{467968}a^{10}+\frac{2649}{58496}a^{9}+\frac{2213}{29248}a^{8}-\frac{2431}{14624}a^{7}-\frac{1897}{14624}a^{6}+\frac{2529}{14624}a^{5}-\frac{635}{1828}a^{4}-\frac{263}{914}a^{3}+\frac{449}{457}a^{2}-\frac{186}{457}a-\frac{1169}{457}$, $\frac{271}{29949952}a^{35}-\frac{1}{3743744}a^{32}-\frac{3}{116992}a^{27}-\frac{3}{7312}a^{24}-\frac{84915}{29949952}a^{16}+\frac{41757}{3743744}a^{13}+\frac{8279}{116992}a^{8}+\frac{967}{7312}a^{5}$, $\frac{23}{935936}a^{30}-\frac{23}{467968}a^{29}+\frac{3}{116992}a^{28}-\frac{3}{116992}a^{27}-\frac{24475}{935936}a^{11}+\frac{24475}{467968}a^{10}-\frac{8279}{116992}a^{9}+\frac{8279}{116992}a^{8}$, $\frac{93}{29949952}a^{35}+\frac{45}{1871872}a^{31}-\frac{3}{116992}a^{27}+\frac{5}{29248}a^{25}+\frac{5}{14624}a^{24}+\frac{1}{3656}a^{23}+\frac{1}{3656}a^{22}-\frac{1}{914}a^{20}-\frac{139657}{29949952}a^{16}-\frac{7193}{1871872}a^{12}+\frac{8279}{116992}a^{8}-\frac{4049}{29248}a^{6}-\frac{4049}{14624}a^{5}-\frac{1541}{3656}a^{4}-\frac{1541}{3656}a^{3}+\frac{627}{914}a$ 23/935936a^30 - 24475/935936a^11 + 1, 89/14974976a^34 + 27371/14974976a^15 - 1, 23/935936a^30 - 1/3656a^22 - 24475/935936a^11 + 1541/3656a^3 + 1, 93/29949952a^35 + 1/3743744a^32 - 11/467968a^29 - 7/58496a^26 - 139657/29949952a^16 - 41757/3743744a^13 - 8641/467968a^10 - 181/58496a^7, 7/58496a^26 + 3/7312a^23 + 181/58496a^7 - 967/7312a^4, 271/59899904a^35 + 271/59899904a^34 - 85/59899904a^33 + 271/59899904a^32 + 1355/59899904a^31 - 1897/59899904a^30 - 813/59899904a^29 + 255/59899904a^28 - 2981/59899904a^27 - 6233/59899904a^26 + 1955/59899904a^25 + 271/59899904a^24 - 24661/59899904a^23 + 7735/59899904a^22 + 25203/59899904a^21 - 7905/59899904a^20 + 23035/59899904a^19 + 271/131072a^18 - 85/131072a^17 - 73441/14974976a^16 + 25203/7487488a^15 - 7905/7487488a^14 - 24661/1871872a^13 + 271/935936a^12 + 12195/467968a^11 - 6233/233984a^10 + 1955/233984a^9 + 4607/58496a^8 - 813/29248a^7 + 255/29248a^6 + 1355/7312a^5 + 271/3656a^4 - 85/3656a^3 + 271/914a^2 - 85/914a - 542/457, 1/3656a^22 + 1/1828a^21 - 1/914a^20 - 1541/3656a^3 + 287/1828a^2 + 627/914a, 1/1828a^21 + 287/1828a^2 - 1, 91/7487488a^33 + 17/233984a^29 + 7/29248a^25 + 1/1828a^21 - 56143/7487488a^14 - 7917/233984a^10 + 181/29248a^6 + 287/1828a^2, 23/935936a^30 - 3/116992a^28 + 3/58496a^26 + 1/14624a^24 - 24475/935936a^11 + 8279/116992a^9 - 8279/58496a^7 + 2115/14624a^5, 1/3743744a^33 + 3/58496a^26 - 1/1828a^21 - 41757/3743744a^14 - 8279/58496a^7 - 287/1828a^2 - 1, 89/14974976a^35 - 93/29949952a^34 - 271/29949952a^33 + 93/29949952a^32 - 991/29949952a^31 - 619/29949952a^30 + 3369/29949952a^29 + 813/29949952a^28 - 5375/29949952a^27 + 6565/29949952a^26 + 6233/29949952a^25 - 22435/29949952a^24 - 271/29949952a^23 + 32853/29949952a^22 - 24119/29949952a^21 - 57971/29949952a^20 + 73441/29949952a^19 + 93/65536a^18 - 271/65536a^17 + 93587/29949952a^16 + 174253/14974976a^15 - 25203/3743744a^14 - 8463/935936a^13 + 56515/1871872a^12 - 25017/935936a^11 - 48865/467968a^10 + 6233/116992a^9 + 14241/116992a^8 - 9033/58496a^7 + 813/14624a^6 + 7843/14624a^5 - 1355/3656a^4 - 2083/3656a^3 + 813/914a^2 + 85/914a - 542/457, 91/7487488a^34 + 1/3743744a^32 + 3/116992a^27 + 1/3656a^22 - 1/914a^20 - 56143/7487488a^15 - 41757/3743744a^13 - 8279/116992a^8 - 1541/3656a^3 - 287/914a, 441/59899904a^35 - 1/131072a^34 - 797/59899904a^33 + 1711/59899904a^32 - 117/59899904a^31 - 4777/59899904a^30 + 595/59899904a^29 + 3071/59899904a^28 - 10149/59899904a^27 + 4007/59899904a^26 + 12195/59899904a^25 - 28401/59899904a^24 - 8277/59899904a^23 + 56887/59899904a^22 - 40333/59899904a^21 - 73441/59899904a^20 + 154107/59899904a^19 + 271/131072a^18 - 85/131072a^17 + 47517/29949952a^16 + 89/8192a^15 - 8819/1871872a^14 - 15927/935936a^13 + 12373/467968a^12 + 31583/935936a^11 - 3825/467968a^10 + 2649/58496a^9 + 2213/29248a^8 - 2431/14624a^7 - 1897/14624a^6 + 2529/14624a^5 - 635/1828a^4 - 263/914a^3 + 449/457a^2 - 186/457a - 1169/457, 271/29949952a^35 - 1/3743744a^32 - 3/116992a^27 - 3/7312a^24 - 84915/29949952a^16 + 41757/3743744a^13 + 8279/116992a^8 + 967/7312a^5, 23/935936a^30 - 23/467968a^29 + 3/116992a^28 - 3/116992a^27 - 24475/935936a^11 + 24475/467968a^10 - 8279/116992a^9 + 8279/116992a^8, 93/29949952a^35 + 45/1871872a^31 - 3/116992a^27 + 5/29248a^25 + 5/14624a^24 + 1/3656a^23 + 1/3656a^22 - 1/914a^20 - 139657/29949952a^16 - 7193/1871872a^12 + 8279/116992a^8 - 4049/29248a^6 - 4049/14624a^5 - 1541/3656a^4 - 1541/3656a^3 + 627/914*a (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:	$ 28076820524481.113 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 28076820524481.113 \cdot 148}{38\cdot\sqrt{48908816365067043970916287981601635325839249495639564072729}}\cr\approx \mathstrut & 0.115178150048102 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 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 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 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 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 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 - x^35 - x^34 + 3*x^33 - x^32 - 5*x^31 + 7*x^30 + 3*x^29 - 17*x^28 + 11*x^27 + 23*x^26 - 45*x^25 - x^24 + 91*x^23 - 89*x^22 - 93*x^21 + 271*x^20 - 85*x^19 - 457*x^18 - 170*x^17 + 1084*x^16 - 744*x^15 - 1424*x^14 + 2912*x^13 - 64*x^12 - 5760*x^11 + 5888*x^10 + 5632*x^9 - 17408*x^8 + 6144*x^7 + 28672*x^6 - 40960*x^5 - 16384*x^4 + 98304*x^3 - 65536*x^2 - 131072*x + 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

$C_2\times C_{18}$ (as 36T2):

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)

An abelian group of order 36

The 36 conjugacy class representatives for $C_2\times C_{18}$

Character table for $C_2\times C_{18}$

Intermediate fields

$\Q(\sqrt{-19}) $, $\Q(\sqrt{133}) $, $\Q(\sqrt{-7}) $, 3.3.361.1, $\Q(\sqrt{-7}, \sqrt{-19})$, 6.0.2476099.1, 6.6.849301957.1, 6.0.44700103.1, $\Q(\zeta_{19})^+$, 12.0.721313814164029849.1, $\Q(\zeta_{19})$, 18.18.221153377467012797984123331973.1, 18.0.11639651445632252525480175367.1

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]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$18^{2}$	$18^{2}$	$18^{2}$	R	${\href{/padicField/11.3.0.1}{3} }^{12}$	$18^{2}$	$18^{2}$	R	${\href{/padicField/23.9.0.1}{9} }^{4}$	$18^{2}$	${\href{/padicField/31.6.0.1}{6} }^{6}$	${\href{/padicField/37.2.0.1}{2} }^{18}$	$18^{2}$	${\href{/padicField/43.9.0.1}{9} }^{4}$	$18^{2}$	$18^{2}$	$18^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$7$ 7	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
	7.6.3.2	$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$19$ 19		Deg $36$	$18$	$2$	$34$

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