LMFDB - Number field 32.0.272292877590407567597186433188495333554574218609249.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536)

gp: K = bnfinit(y^32 - y^31 - y^30 + 3*y^29 - y^28 - 5*y^27 + 7*y^26 + 3*y^25 - 17*y^24 + 11*y^23 + 23*y^22 - 45*y^21 - y^20 + 91*y^19 - 89*y^18 - 93*y^17 + 271*y^16 - 186*y^15 - 356*y^14 + 728*y^13 - 16*y^12 - 1440*y^11 + 1472*y^10 + 1408*y^9 - 4352*y^8 + 1536*y^7 + 7168*y^6 - 10240*y^5 - 4096*y^4 + 24576*y^3 - 16384*y^2 - 32768*y + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536)

$ x^{32} - x^{31} - x^{30} + 3 x^{29} - x^{28} - 5 x^{27} + 7 x^{26} + 3 x^{25} - 17 x^{24} + 11 x^{23} + \cdots + 65536 $ x^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$272292877590407567597186433188495333554574218609249$ 272292877590407567597186433188495333554574218609249 $\medspace = 7^{16}\cdot 17^{30}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$37.68$	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:	$7^{1/2}17^{15/16}\approx 37.67861094758833$
Ramified primes:	$7$, $17$ 7, 17	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$32$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$119=7\cdot 17$
Dirichlet character group:	$\lbrace$$\chi_{119}(1,·)$, $\chi_{119}(6,·)$, $\chi_{119}(8,·)$, $\chi_{119}(13,·)$, $\chi_{119}(15,·)$, $\chi_{119}(20,·)$, $\chi_{119}(22,·)$, $\chi_{119}(27,·)$, $\chi_{119}(29,·)$, $\chi_{119}(36,·)$, $\chi_{119}(41,·)$, $\chi_{119}(43,·)$, $\chi_{119}(48,·)$, $\chi_{119}(50,·)$, $\chi_{119}(55,·)$, $\chi_{119}(57,·)$, $\chi_{119}(62,·)$, $\chi_{119}(64,·)$, $\chi_{119}(69,·)$, $\chi_{119}(71,·)$, $\chi_{119}(76,·)$, $\chi_{119}(78,·)$, $\chi_{119}(83,·)$, $\chi_{119}(90,·)$, $\chi_{119}(92,·)$, $\chi_{119}(97,·)$, $\chi_{119}(99,·)$, $\chi_{119}(104,·)$, $\chi_{119}(106,·)$, $\chi_{119}(111,·)$, $\chi_{119}(113,·)$, $\chi_{119}(118,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{542}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{93}{271}$, $\frac{1}{1084}a^{18}-\frac{1}{1084}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}+\frac{1}{4}a^{11}+\frac{1}{4}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{93}{542}a-\frac{89}{271}$, $\frac{1}{2168}a^{19}-\frac{1}{2168}a^{18}-\frac{1}{2168}a^{17}-\frac{3}{8}a^{16}+\frac{1}{8}a^{15}-\frac{3}{8}a^{14}+\frac{1}{8}a^{13}-\frac{3}{8}a^{12}+\frac{1}{8}a^{11}-\frac{3}{8}a^{10}+\frac{1}{8}a^{9}-\frac{3}{8}a^{8}+\frac{1}{8}a^{7}-\frac{3}{8}a^{6}+\frac{1}{8}a^{5}-\frac{3}{8}a^{4}+\frac{1}{8}a^{3}-\frac{93}{1084}a^{2}-\frac{89}{542}a+\frac{91}{271}$, $\frac{1}{4336}a^{20}-\frac{1}{4336}a^{19}-\frac{1}{4336}a^{18}+\frac{3}{4336}a^{17}+\frac{1}{16}a^{16}+\frac{5}{16}a^{15}-\frac{7}{16}a^{14}-\frac{3}{16}a^{13}+\frac{1}{16}a^{12}+\frac{5}{16}a^{11}-\frac{7}{16}a^{10}-\frac{3}{16}a^{9}+\frac{1}{16}a^{8}+\frac{5}{16}a^{7}-\frac{7}{16}a^{6}-\frac{3}{16}a^{5}+\frac{1}{16}a^{4}-\frac{93}{2168}a^{3}-\frac{89}{1084}a^{2}+\frac{91}{542}a-\frac{1}{271}$, $\frac{1}{8672}a^{21}-\frac{1}{8672}a^{20}-\frac{1}{8672}a^{19}+\frac{3}{8672}a^{18}-\frac{1}{8672}a^{17}-\frac{11}{32}a^{16}+\frac{9}{32}a^{15}+\frac{13}{32}a^{14}+\frac{1}{32}a^{13}+\frac{5}{32}a^{12}-\frac{7}{32}a^{11}-\frac{3}{32}a^{10}-\frac{15}{32}a^{9}-\frac{11}{32}a^{8}+\frac{9}{32}a^{7}+\frac{13}{32}a^{6}+\frac{1}{32}a^{5}-\frac{93}{4336}a^{4}-\frac{89}{2168}a^{3}+\frac{91}{1084}a^{2}-\frac{1}{542}a-\frac{45}{271}$, $\frac{1}{17344}a^{22}-\frac{1}{17344}a^{21}-\frac{1}{17344}a^{20}+\frac{3}{17344}a^{19}-\frac{1}{17344}a^{18}-\frac{5}{17344}a^{17}+\frac{9}{64}a^{16}+\frac{13}{64}a^{15}-\frac{31}{64}a^{14}+\frac{5}{64}a^{13}-\frac{7}{64}a^{12}-\frac{3}{64}a^{11}+\frac{17}{64}a^{10}-\frac{11}{64}a^{9}-\frac{23}{64}a^{8}-\frac{19}{64}a^{7}+\frac{1}{64}a^{6}-\frac{93}{8672}a^{5}-\frac{89}{4336}a^{4}+\frac{91}{2168}a^{3}-\frac{1}{1084}a^{2}-\frac{45}{542}a+\frac{23}{271}$, $\frac{1}{34688}a^{23}-\frac{1}{34688}a^{22}-\frac{1}{34688}a^{21}+\frac{3}{34688}a^{20}-\frac{1}{34688}a^{19}-\frac{5}{34688}a^{18}+\frac{7}{34688}a^{17}-\frac{51}{128}a^{16}+\frac{33}{128}a^{15}-\frac{59}{128}a^{14}-\frac{7}{128}a^{13}-\frac{3}{128}a^{12}+\frac{17}{128}a^{11}-\frac{11}{128}a^{10}-\frac{23}{128}a^{9}+\frac{45}{128}a^{8}+\frac{1}{128}a^{7}-\frac{93}{17344}a^{6}-\frac{89}{8672}a^{5}+\frac{91}{4336}a^{4}-\frac{1}{2168}a^{3}-\frac{45}{1084}a^{2}+\frac{23}{542}a+\frac{11}{271}$, $\frac{1}{69376}a^{24}-\frac{1}{69376}a^{23}-\frac{1}{69376}a^{22}+\frac{3}{69376}a^{21}-\frac{1}{69376}a^{20}-\frac{5}{69376}a^{19}+\frac{7}{69376}a^{18}+\frac{3}{69376}a^{17}+\frac{33}{256}a^{16}+\frac{69}{256}a^{15}+\frac{121}{256}a^{14}-\frac{3}{256}a^{13}+\frac{17}{256}a^{12}-\frac{11}{256}a^{11}-\frac{23}{256}a^{10}+\frac{45}{256}a^{9}+\frac{1}{256}a^{8}-\frac{93}{34688}a^{7}-\frac{89}{17344}a^{6}+\frac{91}{8672}a^{5}-\frac{1}{4336}a^{4}-\frac{45}{2168}a^{3}+\frac{23}{1084}a^{2}+\frac{11}{542}a-\frac{17}{271}$, $\frac{1}{138752}a^{25}-\frac{1}{138752}a^{24}-\frac{1}{138752}a^{23}+\frac{3}{138752}a^{22}-\frac{1}{138752}a^{21}-\frac{5}{138752}a^{20}+\frac{7}{138752}a^{19}+\frac{3}{138752}a^{18}-\frac{17}{138752}a^{17}-\frac{187}{512}a^{16}+\frac{121}{512}a^{15}+\frac{253}{512}a^{14}+\frac{17}{512}a^{13}-\frac{11}{512}a^{12}-\frac{23}{512}a^{11}+\frac{45}{512}a^{10}+\frac{1}{512}a^{9}-\frac{93}{69376}a^{8}-\frac{89}{34688}a^{7}+\frac{91}{17344}a^{6}-\frac{1}{8672}a^{5}-\frac{45}{4336}a^{4}+\frac{23}{2168}a^{3}+\frac{11}{1084}a^{2}-\frac{17}{542}a+\frac{3}{271}$, $\frac{1}{277504}a^{26}-\frac{1}{277504}a^{25}-\frac{1}{277504}a^{24}+\frac{3}{277504}a^{23}-\frac{1}{277504}a^{22}-\frac{5}{277504}a^{21}+\frac{7}{277504}a^{20}+\frac{3}{277504}a^{19}-\frac{17}{277504}a^{18}+\frac{11}{277504}a^{17}+\frac{121}{1024}a^{16}+\frac{253}{1024}a^{15}-\frac{495}{1024}a^{14}-\frac{11}{1024}a^{13}-\frac{23}{1024}a^{12}+\frac{45}{1024}a^{11}+\frac{1}{1024}a^{10}-\frac{93}{138752}a^{9}-\frac{89}{69376}a^{8}+\frac{91}{34688}a^{7}-\frac{1}{17344}a^{6}-\frac{45}{8672}a^{5}+\frac{23}{4336}a^{4}+\frac{11}{2168}a^{3}-\frac{17}{1084}a^{2}+\frac{3}{542}a+\frac{7}{271}$, $\frac{1}{555008}a^{27}-\frac{1}{555008}a^{26}-\frac{1}{555008}a^{25}+\frac{3}{555008}a^{24}-\frac{1}{555008}a^{23}-\frac{5}{555008}a^{22}+\frac{7}{555008}a^{21}+\frac{3}{555008}a^{20}-\frac{17}{555008}a^{19}+\frac{11}{555008}a^{18}+\frac{23}{555008}a^{17}-\frac{771}{2048}a^{16}+\frac{529}{2048}a^{15}+\frac{1013}{2048}a^{14}-\frac{23}{2048}a^{13}+\frac{45}{2048}a^{12}+\frac{1}{2048}a^{11}-\frac{93}{277504}a^{10}-\frac{89}{138752}a^{9}+\frac{91}{69376}a^{8}-\frac{1}{34688}a^{7}-\frac{45}{17344}a^{6}+\frac{23}{8672}a^{5}+\frac{11}{4336}a^{4}-\frac{17}{2168}a^{3}+\frac{3}{1084}a^{2}+\frac{7}{542}a-\frac{5}{271}$, $\frac{1}{1110016}a^{28}-\frac{1}{1110016}a^{27}-\frac{1}{1110016}a^{26}+\frac{3}{1110016}a^{25}-\frac{1}{1110016}a^{24}-\frac{5}{1110016}a^{23}+\frac{7}{1110016}a^{22}+\frac{3}{1110016}a^{21}-\frac{17}{1110016}a^{20}+\frac{11}{1110016}a^{19}+\frac{23}{1110016}a^{18}-\frac{45}{1110016}a^{17}+\frac{529}{4096}a^{16}+\frac{1013}{4096}a^{15}+\frac{2025}{4096}a^{14}+\frac{45}{4096}a^{13}+\frac{1}{4096}a^{12}-\frac{93}{555008}a^{11}-\frac{89}{277504}a^{10}+\frac{91}{138752}a^{9}-\frac{1}{69376}a^{8}-\frac{45}{34688}a^{7}+\frac{23}{17344}a^{6}+\frac{11}{8672}a^{5}-\frac{17}{4336}a^{4}+\frac{3}{2168}a^{3}+\frac{7}{1084}a^{2}-\frac{5}{542}a-\frac{1}{271}$, $\frac{1}{2220032}a^{29}-\frac{1}{2220032}a^{28}-\frac{1}{2220032}a^{27}+\frac{3}{2220032}a^{26}-\frac{1}{2220032}a^{25}-\frac{5}{2220032}a^{24}+\frac{7}{2220032}a^{23}+\frac{3}{2220032}a^{22}-\frac{17}{2220032}a^{21}+\frac{11}{2220032}a^{20}+\frac{23}{2220032}a^{19}-\frac{45}{2220032}a^{18}-\frac{1}{2220032}a^{17}-\frac{3083}{8192}a^{16}+\frac{2025}{8192}a^{15}-\frac{4051}{8192}a^{14}+\frac{1}{8192}a^{13}-\frac{93}{1110016}a^{12}-\frac{89}{555008}a^{11}+\frac{91}{277504}a^{10}-\frac{1}{138752}a^{9}-\frac{45}{69376}a^{8}+\frac{23}{34688}a^{7}+\frac{11}{17344}a^{6}-\frac{17}{8672}a^{5}+\frac{3}{4336}a^{4}+\frac{7}{2168}a^{3}-\frac{5}{1084}a^{2}-\frac{1}{542}a+\frac{3}{271}$, $\frac{1}{4440064}a^{30}-\frac{1}{4440064}a^{29}-\frac{1}{4440064}a^{28}+\frac{3}{4440064}a^{27}-\frac{1}{4440064}a^{26}-\frac{5}{4440064}a^{25}+\frac{7}{4440064}a^{24}+\frac{3}{4440064}a^{23}-\frac{17}{4440064}a^{22}+\frac{11}{4440064}a^{21}+\frac{23}{4440064}a^{20}-\frac{45}{4440064}a^{19}-\frac{1}{4440064}a^{18}+\frac{91}{4440064}a^{17}-\frac{6167}{16384}a^{16}-\frac{4051}{16384}a^{15}+\frac{1}{16384}a^{14}-\frac{93}{2220032}a^{13}-\frac{89}{1110016}a^{12}+\frac{91}{555008}a^{11}-\frac{1}{277504}a^{10}-\frac{45}{138752}a^{9}+\frac{23}{69376}a^{8}+\frac{11}{34688}a^{7}-\frac{17}{17344}a^{6}+\frac{3}{8672}a^{5}+\frac{7}{4336}a^{4}-\frac{5}{2168}a^{3}-\frac{1}{1084}a^{2}+\frac{3}{542}a-\frac{1}{271}$, $\frac{1}{8880128}a^{31}-\frac{1}{8880128}a^{30}-\frac{1}{8880128}a^{29}+\frac{3}{8880128}a^{28}-\frac{1}{8880128}a^{27}-\frac{5}{8880128}a^{26}+\frac{7}{8880128}a^{25}+\frac{3}{8880128}a^{24}-\frac{17}{8880128}a^{23}+\frac{11}{8880128}a^{22}+\frac{23}{8880128}a^{21}-\frac{45}{8880128}a^{20}-\frac{1}{8880128}a^{19}+\frac{91}{8880128}a^{18}-\frac{89}{8880128}a^{17}+\frac{12333}{32768}a^{16}+\frac{1}{32768}a^{15}-\frac{93}{4440064}a^{14}-\frac{89}{2220032}a^{13}+\frac{91}{1110016}a^{12}-\frac{1}{555008}a^{11}-\frac{45}{277504}a^{10}+\frac{23}{138752}a^{9}+\frac{11}{69376}a^{8}-\frac{17}{34688}a^{7}+\frac{3}{17344}a^{6}+\frac{7}{8672}a^{5}-\frac{5}{4336}a^{4}-\frac{1}{2168}a^{3}+\frac{3}{1084}a^{2}-\frac{1}{542}a-\frac{1}{271}$ 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, 1/542*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 - 93/271, 1/1084*a^18 - 1/1084*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 - 93/542*a - 89/271, 1/2168*a^19 - 1/2168*a^18 - 1/2168*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 - 93/1084*a^2 - 89/542*a + 91/271, 1/4336*a^20 - 1/4336*a^19 - 1/4336*a^18 + 3/4336*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 - 93/2168*a^3 - 89/1084*a^2 + 91/542*a - 1/271, 1/8672*a^21 - 1/8672*a^20 - 1/8672*a^19 + 3/8672*a^18 - 1/8672*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 - 93/4336*a^4 - 89/2168*a^3 + 91/1084*a^2 - 1/542*a - 45/271, 1/17344*a^22 - 1/17344*a^21 - 1/17344*a^20 + 3/17344*a^19 - 1/17344*a^18 - 5/17344*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 - 93/8672*a^5 - 89/4336*a^4 + 91/2168*a^3 - 1/1084*a^2 - 45/542*a + 23/271, 1/34688*a^23 - 1/34688*a^22 - 1/34688*a^21 + 3/34688*a^20 - 1/34688*a^19 - 5/34688*a^18 + 7/34688*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 - 93/17344*a^6 - 89/8672*a^5 + 91/4336*a^4 - 1/2168*a^3 - 45/1084*a^2 + 23/542*a + 11/271, 1/69376*a^24 - 1/69376*a^23 - 1/69376*a^22 + 3/69376*a^21 - 1/69376*a^20 - 5/69376*a^19 + 7/69376*a^18 + 3/69376*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 - 93/34688*a^7 - 89/17344*a^6 + 91/8672*a^5 - 1/4336*a^4 - 45/2168*a^3 + 23/1084*a^2 + 11/542*a - 17/271, 1/138752*a^25 - 1/138752*a^24 - 1/138752*a^23 + 3/138752*a^22 - 1/138752*a^21 - 5/138752*a^20 + 7/138752*a^19 + 3/138752*a^18 - 17/138752*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 - 93/69376*a^8 - 89/34688*a^7 + 91/17344*a^6 - 1/8672*a^5 - 45/4336*a^4 + 23/2168*a^3 + 11/1084*a^2 - 17/542*a + 3/271, 1/277504*a^26 - 1/277504*a^25 - 1/277504*a^24 + 3/277504*a^23 - 1/277504*a^22 - 5/277504*a^21 + 7/277504*a^20 + 3/277504*a^19 - 17/277504*a^18 + 11/277504*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 - 93/138752*a^9 - 89/69376*a^8 + 91/34688*a^7 - 1/17344*a^6 - 45/8672*a^5 + 23/4336*a^4 + 11/2168*a^3 - 17/1084*a^2 + 3/542*a + 7/271, 1/555008*a^27 - 1/555008*a^26 - 1/555008*a^25 + 3/555008*a^24 - 1/555008*a^23 - 5/555008*a^22 + 7/555008*a^21 + 3/555008*a^20 - 17/555008*a^19 + 11/555008*a^18 + 23/555008*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 - 93/277504*a^10 - 89/138752*a^9 + 91/69376*a^8 - 1/34688*a^7 - 45/17344*a^6 + 23/8672*a^5 + 11/4336*a^4 - 17/2168*a^3 + 3/1084*a^2 + 7/542*a - 5/271, 1/1110016*a^28 - 1/1110016*a^27 - 1/1110016*a^26 + 3/1110016*a^25 - 1/1110016*a^24 - 5/1110016*a^23 + 7/1110016*a^22 + 3/1110016*a^21 - 17/1110016*a^20 + 11/1110016*a^19 + 23/1110016*a^18 - 45/1110016*a^17 + 529/4096*a^16 + 1013/4096*a^15 + 2025/4096*a^14 + 45/4096*a^13 + 1/4096*a^12 - 93/555008*a^11 - 89/277504*a^10 + 91/138752*a^9 - 1/69376*a^8 - 45/34688*a^7 + 23/17344*a^6 + 11/8672*a^5 - 17/4336*a^4 + 3/2168*a^3 + 7/1084*a^2 - 5/542*a - 1/271, 1/2220032*a^29 - 1/2220032*a^28 - 1/2220032*a^27 + 3/2220032*a^26 - 1/2220032*a^25 - 5/2220032*a^24 + 7/2220032*a^23 + 3/2220032*a^22 - 17/2220032*a^21 + 11/2220032*a^20 + 23/2220032*a^19 - 45/2220032*a^18 - 1/2220032*a^17 - 3083/8192*a^16 + 2025/8192*a^15 - 4051/8192*a^14 + 1/8192*a^13 - 93/1110016*a^12 - 89/555008*a^11 + 91/277504*a^10 - 1/138752*a^9 - 45/69376*a^8 + 23/34688*a^7 + 11/17344*a^6 - 17/8672*a^5 + 3/4336*a^4 + 7/2168*a^3 - 5/1084*a^2 - 1/542*a + 3/271, 1/4440064*a^30 - 1/4440064*a^29 - 1/4440064*a^28 + 3/4440064*a^27 - 1/4440064*a^26 - 5/4440064*a^25 + 7/4440064*a^24 + 3/4440064*a^23 - 17/4440064*a^22 + 11/4440064*a^21 + 23/4440064*a^20 - 45/4440064*a^19 - 1/4440064*a^18 + 91/4440064*a^17 - 6167/16384*a^16 - 4051/16384*a^15 + 1/16384*a^14 - 93/2220032*a^13 - 89/1110016*a^12 + 91/555008*a^11 - 1/277504*a^10 - 45/138752*a^9 + 23/69376*a^8 + 11/34688*a^7 - 17/17344*a^6 + 3/8672*a^5 + 7/4336*a^4 - 5/2168*a^3 - 1/1084*a^2 + 3/542*a - 1/271, 1/8880128*a^31 - 1/8880128*a^30 - 1/8880128*a^29 + 3/8880128*a^28 - 1/8880128*a^27 - 5/8880128*a^26 + 7/8880128*a^25 + 3/8880128*a^24 - 17/8880128*a^23 + 11/8880128*a^22 + 23/8880128*a^21 - 45/8880128*a^20 - 1/8880128*a^19 + 91/8880128*a^18 - 89/8880128*a^17 + 12333/32768*a^16 + 1/32768*a^15 - 93/4440064*a^14 - 89/2220032*a^13 + 91/1110016*a^12 - 1/555008*a^11 - 45/277504*a^10 + 23/138752*a^9 + 11/69376*a^8 - 17/34688*a^7 + 3/17344*a^6 + 7/8672*a^5 - 5/4336*a^4 - 1/2168*a^3 + 3/1084*a^2 - 1/542*a - 1/271

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

$C_{3}\times C_{15}$, which has order $45$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{45}{1110016} a^{29} + \frac{8641}{1110016} a^{12} $ -(45)/(1110016)a^(29) + (8641)/(1110016)a^(12) (order $34$)	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:	$\frac{3}{4336}a^{21}-\frac{287}{4336}a^{4}-1$, $\frac{1}{1084}a^{19}-\frac{457}{1084}a^{2}+1$, $\frac{23}{555008}a^{28}+\frac{11}{277504}a^{27}-\frac{17}{138752}a^{26}+\frac{3}{69376}a^{25}+\frac{7}{34688}a^{24}-\frac{5}{17344}a^{23}-\frac{1}{8672}a^{22}+\frac{3}{4336}a^{21}+\frac{7917}{555008}a^{11}-\frac{8279}{277504}a^{10}+\frac{181}{138752}a^{9}+\frac{4049}{69376}a^{8}-\frac{2115}{34688}a^{7}-\frac{967}{17344}a^{6}+\frac{1541}{8672}a^{5}-\frac{287}{4336}a^{4}$, $\frac{11}{277504}a^{27}-\frac{17}{138752}a^{26}+\frac{1}{542}a^{18}-\frac{8279}{277504}a^{10}+\frac{181}{138752}a^{9}+\frac{85}{542}a$, $\frac{17}{138752}a^{26}+\frac{1}{8672}a^{22}-\frac{1}{542}a^{18}-\frac{181}{138752}a^{9}-\frac{1541}{8672}a^{5}-\frac{85}{542}a$, $\frac{93}{8880128}a^{31}-\frac{89}{8880128}a^{30}+\frac{267}{8880128}a^{29}+\frac{279}{8880128}a^{28}-\frac{93}{8880128}a^{27}-\frac{465}{8880128}a^{26}+\frac{267}{8880128}a^{25}+\frac{279}{8880128}a^{24}-\frac{1581}{8880128}a^{23}+\frac{1023}{8880128}a^{22}-\frac{4005}{8880128}a^{21}-\frac{89}{8880128}a^{20}-\frac{93}{8880128}a^{19}+\frac{8463}{8880128}a^{18}-\frac{8277}{8880128}a^{17}+\frac{89}{32768}a^{16}+\frac{93}{32768}a^{15}-\frac{8649}{4440064}a^{14}+\frac{8099}{1110016}a^{13}-\frac{89}{555008}a^{12}-\frac{93}{555008}a^{11}-\frac{4185}{277504}a^{10}+\frac{2139}{138752}a^{9}-\frac{1513}{34688}a^{8}-\frac{1581}{34688}a^{7}+\frac{279}{17344}a^{6}+\frac{651}{8672}a^{5}-\frac{89}{2168}a^{4}+\frac{267}{1084}a^{3}+\frac{279}{1084}a^{2}-\frac{93}{542}a+\frac{178}{271}$, $\frac{45}{1110016}a^{29}+\frac{17}{138752}a^{26}+\frac{5}{17344}a^{23}-\frac{8641}{1110016}a^{12}-\frac{181}{138752}a^{9}+\frac{967}{17344}a^{6}$, $\frac{1}{2220032}a^{31}+\frac{1}{2220032}a^{30}-\frac{5}{17344}a^{23}-\frac{5}{8672}a^{22}+\frac{24475}{2220032}a^{14}+\frac{24475}{2220032}a^{13}-\frac{967}{17344}a^{6}-\frac{967}{8672}a^{5}$, $\frac{93}{8880128}a^{31}-\frac{93}{8880128}a^{30}-\frac{93}{8880128}a^{29}+\frac{647}{8880128}a^{28}-\frac{93}{8880128}a^{27}-\frac{465}{8880128}a^{26}+\frac{651}{8880128}a^{25}+\frac{2839}{8880128}a^{24}-\frac{1581}{8880128}a^{23}+\frac{1023}{8880128}a^{22}+\frac{2139}{8880128}a^{21}+\frac{8103}{8880128}a^{20}-\frac{93}{8880128}a^{19}+\frac{8463}{8880128}a^{18}-\frac{8277}{8880128}a^{17}+\frac{89}{32768}a^{16}+\frac{93}{32768}a^{15}-\frac{8649}{4440064}a^{14}-\frac{8277}{2220032}a^{13}+\frac{8463}{1110016}a^{12}+\frac{489}{34688}a^{11}-\frac{4185}{277504}a^{10}+\frac{2139}{138752}a^{9}+\frac{1023}{69376}a^{8}+\frac{353}{34688}a^{7}+\frac{279}{17344}a^{6}+\frac{651}{8672}a^{5}-\frac{465}{4336}a^{4}-\frac{95}{542}a^{3}+\frac{279}{1084}a^{2}-\frac{93}{542}a-\frac{93}{271}$, $\frac{23}{555008}a^{28}-\frac{3}{69376}a^{26}+\frac{3}{34688}a^{24}+\frac{1}{8672}a^{22}+\frac{7917}{555008}a^{11}-\frac{4049}{69376}a^{9}+\frac{4049}{34688}a^{7}-\frac{1541}{8672}a^{5}$, $\frac{45}{1110016}a^{29}-\frac{23}{555008}a^{28}-\frac{11}{277504}a^{27}+\frac{17}{138752}a^{26}-\frac{3}{69376}a^{25}-\frac{1}{8672}a^{24}+\frac{3}{4336}a^{23}-\frac{5}{8672}a^{22}-\frac{1}{4336}a^{21}+\frac{3}{2168}a^{20}-\frac{1}{542}a^{19}-\frac{8641}{1110016}a^{12}-\frac{7917}{555008}a^{11}+\frac{8279}{277504}a^{10}-\frac{181}{138752}a^{9}-\frac{4049}{69376}a^{8}+\frac{1541}{8672}a^{7}-\frac{287}{4336}a^{6}-\frac{967}{8672}a^{5}+\frac{1541}{4336}a^{4}-\frac{287}{2168}a^{3}-\frac{85}{542}a^{2}+a$, $\frac{17}{138752}a^{27}-\frac{7}{34688}a^{24}-\frac{1}{2168}a^{20}+\frac{1}{271}a^{17}-\frac{181}{138752}a^{10}+\frac{2115}{34688}a^{7}-\frac{627}{2168}a^{3}+\frac{85}{271}$, $\frac{89}{8880128}a^{31}+\frac{89}{8880128}a^{30}-\frac{267}{8880128}a^{29}+\frac{89}{8880128}a^{28}+\frac{93}{8880128}a^{27}-\frac{623}{8880128}a^{26}-\frac{267}{8880128}a^{25}+\frac{2281}{8880128}a^{24}-\frac{979}{8880128}a^{23}-\frac{2047}{8880128}a^{22}+\frac{10149}{8880128}a^{21}+\frac{89}{8880128}a^{20}-\frac{8099}{8880128}a^{19}+\frac{7921}{8880128}a^{18}+\frac{8277}{8880128}a^{17}-\frac{89}{32768}a^{16}-\frac{93}{32768}a^{15}+\frac{7921}{2220032}a^{14}-\frac{8099}{1110016}a^{13}+\frac{89}{555008}a^{12}+\frac{4005}{277504}a^{11}+\frac{4185}{277504}a^{10}-\frac{979}{69376}a^{9}+\frac{1513}{34688}a^{8}+\frac{3515}{34688}a^{7}-\frac{623}{8672}a^{6}+\frac{445}{4336}a^{5}-\frac{109}{4336}a^{4}-\frac{267}{1084}a^{3}+\frac{89}{542}a^{2}-\frac{182}{271}a-\frac{178}{271}$, $\frac{91}{4440064}a^{31}+\frac{5}{17344}a^{23}+\frac{3}{2168}a^{20}+\frac{1}{271}a^{17}+\frac{7193}{4440064}a^{14}+\frac{967}{17344}a^{6}-\frac{287}{2168}a^{3}-\frac{186}{271}$, $\frac{7}{34688}a^{24}-\frac{7}{17344}a^{23}+\frac{3}{4336}a^{22}-\frac{3}{4336}a^{21}-\frac{2115}{34688}a^{7}+\frac{2115}{17344}a^{6}-\frac{287}{4336}a^{5}+\frac{287}{4336}a^{4}$ 3/4336a^21 - 287/4336a^4 - 1, 1/1084a^19 - 457/1084a^2 + 1, 23/555008a^28 + 11/277504a^27 - 17/138752a^26 + 3/69376a^25 + 7/34688a^24 - 5/17344a^23 - 1/8672a^22 + 3/4336a^21 + 7917/555008a^11 - 8279/277504a^10 + 181/138752a^9 + 4049/69376a^8 - 2115/34688a^7 - 967/17344a^6 + 1541/8672a^5 - 287/4336a^4, 11/277504a^27 - 17/138752a^26 + 1/542a^18 - 8279/277504a^10 + 181/138752a^9 + 85/542a, 17/138752a^26 + 1/8672a^22 - 1/542a^18 - 181/138752a^9 - 1541/8672a^5 - 85/542a, 93/8880128a^31 - 89/8880128a^30 + 267/8880128a^29 + 279/8880128a^28 - 93/8880128a^27 - 465/8880128a^26 + 267/8880128a^25 + 279/8880128a^24 - 1581/8880128a^23 + 1023/8880128a^22 - 4005/8880128a^21 - 89/8880128a^20 - 93/8880128a^19 + 8463/8880128a^18 - 8277/8880128a^17 + 89/32768a^16 + 93/32768a^15 - 8649/4440064a^14 + 8099/1110016a^13 - 89/555008a^12 - 93/555008a^11 - 4185/277504a^10 + 2139/138752a^9 - 1513/34688a^8 - 1581/34688a^7 + 279/17344a^6 + 651/8672a^5 - 89/2168a^4 + 267/1084a^3 + 279/1084a^2 - 93/542a + 178/271, 45/1110016a^29 + 17/138752a^26 + 5/17344a^23 - 8641/1110016a^12 - 181/138752a^9 + 967/17344a^6, 1/2220032a^31 + 1/2220032a^30 - 5/17344a^23 - 5/8672a^22 + 24475/2220032a^14 + 24475/2220032a^13 - 967/17344a^6 - 967/8672a^5, 93/8880128a^31 - 93/8880128a^30 - 93/8880128a^29 + 647/8880128a^28 - 93/8880128a^27 - 465/8880128a^26 + 651/8880128a^25 + 2839/8880128a^24 - 1581/8880128a^23 + 1023/8880128a^22 + 2139/8880128a^21 + 8103/8880128a^20 - 93/8880128a^19 + 8463/8880128a^18 - 8277/8880128a^17 + 89/32768a^16 + 93/32768a^15 - 8649/4440064a^14 - 8277/2220032a^13 + 8463/1110016a^12 + 489/34688a^11 - 4185/277504a^10 + 2139/138752a^9 + 1023/69376a^8 + 353/34688a^7 + 279/17344a^6 + 651/8672a^5 - 465/4336a^4 - 95/542a^3 + 279/1084a^2 - 93/542a - 93/271, 23/555008a^28 - 3/69376a^26 + 3/34688a^24 + 1/8672a^22 + 7917/555008a^11 - 4049/69376a^9 + 4049/34688a^7 - 1541/8672a^5, 45/1110016a^29 - 23/555008a^28 - 11/277504a^27 + 17/138752a^26 - 3/69376a^25 - 1/8672a^24 + 3/4336a^23 - 5/8672a^22 - 1/4336a^21 + 3/2168a^20 - 1/542a^19 - 8641/1110016a^12 - 7917/555008a^11 + 8279/277504a^10 - 181/138752a^9 - 4049/69376a^8 + 1541/8672a^7 - 287/4336a^6 - 967/8672a^5 + 1541/4336a^4 - 287/2168a^3 - 85/542a^2 + a, 17/138752a^27 - 7/34688a^24 - 1/2168a^20 + 1/271a^17 - 181/138752a^10 + 2115/34688a^7 - 627/2168a^3 + 85/271, 89/8880128a^31 + 89/8880128a^30 - 267/8880128a^29 + 89/8880128a^28 + 93/8880128a^27 - 623/8880128a^26 - 267/8880128a^25 + 2281/8880128a^24 - 979/8880128a^23 - 2047/8880128a^22 + 10149/8880128a^21 + 89/8880128a^20 - 8099/8880128a^19 + 7921/8880128a^18 + 8277/8880128a^17 - 89/32768a^16 - 93/32768a^15 + 7921/2220032a^14 - 8099/1110016a^13 + 89/555008a^12 + 4005/277504a^11 + 4185/277504a^10 - 979/69376a^9 + 1513/34688a^8 + 3515/34688a^7 - 623/8672a^6 + 445/4336a^5 - 109/4336a^4 - 267/1084a^3 + 89/542a^2 - 182/271a - 178/271, 91/4440064a^31 + 5/17344a^23 + 3/2168a^20 + 1/271a^17 + 7193/4440064a^14 + 967/17344a^6 - 287/2168a^3 - 186/271, 7/34688a^24 - 7/17344a^23 + 3/4336a^22 - 3/4336a^21 - 2115/34688a^7 + 2115/17344a^6 - 287/4336a^5 + 287/4336*a^4 (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:	$ 308154877506.37555 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 308154877506.37555 \cdot 45}{34\cdot\sqrt{272292877590407567597186433188495333554574218609249}}\cr\approx \mathstrut & 0.145835211511193 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536)

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^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536, 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^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536);

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^32 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536);

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

$C_2\times C_{16}$ (as 32T32):

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 32

The 32 conjugacy class representatives for $C_2\times C_{16}$

Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

$\Q(\sqrt{-119}) $, $\Q(\sqrt{17}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{-7}, \sqrt{17})$, 4.4.4913.1, 4.0.240737.1, 8.0.57954303169.1, $\Q(\zeta_{17})^+$, 8.0.985223153873.1, 16.0.970664662927461034900129.1, 16.16.16501299269766837593302193.1, $\Q(\zeta_{17})$

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	${\href{/padicField/2.8.0.1}{8} }^{4}$	$16^{2}$	$16^{2}$	R	$16^{2}$	${\href{/padicField/13.4.0.1}{4} }^{8}$	R	${\href{/padicField/19.8.0.1}{8} }^{4}$	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	${\href{/padicField/43.8.0.1}{8} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.8.0.1}{8} }^{4}$	${\href{/padicField/59.8.0.1}{8} }^{4}$

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		Deg $32$	$2$	$16$	$16$
$17$ 17		Deg $32$	$16$	$2$	$30$

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