LMFDB - Number field 32.0.26255893369003065702238822824526479360000000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 65536)

gp: K = bnfinit(y^32 + 5*y^30 + 8*y^28 + 5*y^26 + y^24 - 40*y^22 - 152*y^20 - 240*y^18 - 336*y^16 - 960*y^14 - 2432*y^12 - 2560*y^10 + 256*y^8 + 5120*y^6 + 32768*y^4 + 81920*y^2 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 65536);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 65536)

$ x^{32} + 5 x^{30} + 8 x^{28} + 5 x^{26} + x^{24} - 40 x^{22} - 152 x^{20} - 240 x^{18} - 336 x^{16} + \cdots + 65536 $ x^32 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 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:	$26255893369003065702238822824526479360000000000000000$ 26255893369003065702238822824526479360000000000000000 $\medspace = 2^{40}\cdot 3^{16}\cdot 5^{16}\cdot 31^{4}\cdot 89^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$43.46$	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:	$2^{3/2}3^{1/2}5^{1/2}31^{1/2}89^{1/2}\approx 575.395516145199$
Ramified primes:	$2$, $3$, $5$, $31$, $89$ 2, 3, 5, 31, 89	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{3}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{3}{8}a^{6}$, $\frac{1}{16}a^{13}+\frac{5}{16}a^{7}-\frac{1}{2}a$, $\frac{1}{16}a^{14}-\frac{3}{16}a^{8}$, $\frac{1}{32}a^{15}+\frac{5}{32}a^{9}+\frac{1}{4}a^{3}$, $\frac{1}{64}a^{16}-\frac{1}{16}a^{12}-\frac{3}{64}a^{10}-\frac{1}{4}a^{8}+\frac{3}{16}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{17}-\frac{1}{32}a^{13}-\frac{1}{16}a^{12}-\frac{3}{128}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{3}{32}a^{7}+\frac{3}{16}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{18}-\frac{1}{32}a^{14}-\frac{3}{128}a^{12}-\frac{1}{8}a^{10}+\frac{3}{32}a^{8}-\frac{1}{2}a^{6}+\frac{3}{8}a^{4}$, $\frac{1}{128}a^{19}-\frac{3}{128}a^{13}-\frac{1}{2}a$, $\frac{1}{128}a^{20}-\frac{3}{128}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{256}a^{21}-\frac{1}{256}a^{19}+\frac{1}{256}a^{15}-\frac{1}{32}a^{14}+\frac{3}{256}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{8}a^{10}-\frac{3}{64}a^{9}+\frac{3}{32}a^{8}+\frac{1}{4}a^{7}+\frac{3}{16}a^{6}-\frac{5}{16}a^{5}+\frac{3}{8}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{256}a^{22}-\frac{1}{256}a^{20}+\frac{1}{256}a^{16}+\frac{3}{256}a^{14}-\frac{1}{16}a^{12}-\frac{3}{64}a^{10}-\frac{1}{4}a^{8}-\frac{5}{16}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{512}a^{23}-\frac{1}{512}a^{21}-\frac{1}{256}a^{19}+\frac{1}{512}a^{17}-\frac{5}{512}a^{15}-\frac{5}{256}a^{13}-\frac{1}{16}a^{12}+\frac{5}{128}a^{11}-\frac{1}{8}a^{10}+\frac{11}{64}a^{9}-\frac{1}{4}a^{8}+\frac{3}{32}a^{7}+\frac{3}{16}a^{6}-\frac{5}{16}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{5120}a^{24}-\frac{1}{1024}a^{22}-\frac{1}{512}a^{20}+\frac{1}{1024}a^{18}-\frac{5}{1024}a^{16}-\frac{13}{512}a^{14}-\frac{39}{640}a^{12}+\frac{3}{128}a^{10}-\frac{5}{32}a^{8}-\frac{1}{2}a^{7}+\frac{13}{32}a^{6}+\frac{7}{16}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{5}$, $\frac{1}{5120}a^{25}-\frac{1}{1024}a^{23}-\frac{1}{512}a^{21}+\frac{1}{1024}a^{19}+\frac{3}{1024}a^{17}+\frac{3}{512}a^{15}-\frac{19}{640}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{3}{16}a^{7}+\frac{3}{16}a^{6}-\frac{1}{16}a^{5}+\frac{3}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}+\frac{3}{10}a$, $\frac{1}{10240}a^{26}-\frac{1}{10240}a^{24}+\frac{1}{1024}a^{22}-\frac{7}{2048}a^{20}-\frac{1}{2048}a^{18}-\frac{3}{1024}a^{16}-\frac{3}{160}a^{14}+\frac{19}{1280}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{13}{64}a^{8}+\frac{5}{32}a^{6}+\frac{3}{8}a^{4}-\frac{1}{4}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}$, $\frac{1}{10240}a^{27}-\frac{1}{10240}a^{25}-\frac{1}{1024}a^{23}-\frac{3}{2048}a^{21}+\frac{7}{2048}a^{19}+\frac{3}{1024}a^{17}-\frac{23}{2560}a^{15}+\frac{1}{320}a^{13}-\frac{1}{16}a^{11}-\frac{3}{32}a^{9}-\frac{11}{32}a^{7}-\frac{3}{16}a^{5}+\frac{11}{40}a^{3}-\frac{2}{5}a$, $\frac{1}{20480}a^{28}-\frac{1}{20480}a^{26}-\frac{1}{10240}a^{24}+\frac{5}{4096}a^{22}-\frac{9}{4096}a^{20}+\frac{7}{2048}a^{18}+\frac{27}{5120}a^{16}+\frac{17}{1280}a^{14}-\frac{9}{320}a^{12}-\frac{3}{32}a^{10}-\frac{1}{4}a^{9}-\frac{11}{64}a^{8}-\frac{5}{32}a^{6}+\frac{21}{80}a^{4}-\frac{1}{4}a^{3}+\frac{3}{10}a^{2}-\frac{2}{5}$, $\frac{1}{40960}a^{29}+\frac{1}{40960}a^{27}-\frac{1}{10240}a^{25}-\frac{7}{8192}a^{23}-\frac{7}{8192}a^{21}-\frac{5}{2048}a^{19}-\frac{1}{1280}a^{17}+\frac{9}{1280}a^{15}-\frac{1}{32}a^{14}-\frac{67}{2560}a^{13}-\frac{1}{16}a^{12}-\frac{3}{128}a^{11}-\frac{1}{8}a^{10}-\frac{5}{32}a^{9}+\frac{3}{32}a^{8}+\frac{3}{32}a^{7}+\frac{3}{16}a^{6}+\frac{41}{160}a^{5}+\frac{3}{8}a^{4}-\frac{3}{20}a^{3}-\frac{2}{5}a$, $\frac{1}{10485760}a^{30}-\frac{47}{10485760}a^{28}+\frac{101}{2621440}a^{26}-\frac{523}{10485760}a^{24}-\frac{2343}{2097152}a^{22}-\frac{1799}{524288}a^{20}+\frac{891}{1310720}a^{18}-\frac{4621}{655360}a^{16}+\frac{10127}{655360}a^{14}+\frac{4567}{81920}a^{12}-\frac{1651}{16384}a^{10}-\frac{277}{2048}a^{8}-\frac{1}{2}a^{7}+\frac{10321}{40960}a^{6}-\frac{211}{1280}a^{4}-\frac{19}{40}a^{2}-\frac{1}{2}a-\frac{187}{640}$, $\frac{1}{20971520}a^{31}-\frac{47}{20971520}a^{29}+\frac{101}{5242880}a^{27}+\frac{305}{4194304}a^{25}+\frac{3801}{4194304}a^{23}-\frac{775}{1048576}a^{21}-\frac{8069}{2621440}a^{19}+\frac{4979}{1310720}a^{17}+\frac{6287}{1310720}a^{15}-\frac{1}{32}a^{14}+\frac{299}{32768}a^{13}-\frac{1}{16}a^{12}-\frac{755}{32768}a^{11}-\frac{1}{8}a^{10}-\frac{789}{4096}a^{9}+\frac{3}{32}a^{8}+\frac{26961}{81920}a^{7}+\frac{3}{16}a^{6}+\frac{829}{2560}a^{5}+\frac{3}{8}a^{4}+\frac{11}{80}a^{3}+\frac{65}{256}a$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a^2, 1/2*a^9 - 1/2*a^3, 1/4*a^10 + 1/4*a^4, 1/8*a^11 - 1/4*a^9 - 1/2*a^7 - 3/8*a^5 - 1/4*a^3 - 1/2*a, 1/8*a^12 - 3/8*a^6, 1/16*a^13 + 5/16*a^7 - 1/2*a, 1/16*a^14 - 3/16*a^8, 1/32*a^15 + 5/32*a^9 + 1/4*a^3, 1/64*a^16 - 1/16*a^12 - 3/64*a^10 - 1/4*a^8 + 3/16*a^6 - 1/4*a^2, 1/128*a^17 - 1/32*a^13 - 1/16*a^12 - 3/128*a^11 - 1/8*a^10 + 1/8*a^9 - 1/4*a^8 + 3/32*a^7 + 3/16*a^6 - 1/2*a^5 + 3/8*a^4 - 3/8*a^3 - 1/4*a^2, 1/128*a^18 - 1/32*a^14 - 3/128*a^12 - 1/8*a^10 + 3/32*a^8 - 1/2*a^6 + 3/8*a^4, 1/128*a^19 - 3/128*a^13 - 1/2*a, 1/128*a^20 - 3/128*a^14 - 1/2*a^2, 1/256*a^21 - 1/256*a^19 + 1/256*a^15 - 1/32*a^14 + 3/256*a^13 - 1/16*a^12 - 1/16*a^11 - 1/8*a^10 - 3/64*a^9 + 3/32*a^8 + 1/4*a^7 + 3/16*a^6 - 5/16*a^5 + 3/8*a^4 - 1/4*a^3, 1/256*a^22 - 1/256*a^20 + 1/256*a^16 + 3/256*a^14 - 1/16*a^12 - 3/64*a^10 - 1/4*a^8 - 5/16*a^6 - 1/4*a^4 - 1/2*a^2, 1/512*a^23 - 1/512*a^21 - 1/256*a^19 + 1/512*a^17 - 5/512*a^15 - 5/256*a^13 - 1/16*a^12 + 5/128*a^11 - 1/8*a^10 + 11/64*a^9 - 1/4*a^8 + 3/32*a^7 + 3/16*a^6 - 5/16*a^5 + 3/8*a^4 - 1/2*a^3 - 1/4*a^2 - 1/2*a, 1/5120*a^24 - 1/1024*a^22 - 1/512*a^20 + 1/1024*a^18 - 5/1024*a^16 - 13/512*a^14 - 39/640*a^12 + 3/128*a^10 - 5/32*a^8 - 1/2*a^7 + 13/32*a^6 + 7/16*a^4 - 1/2*a^2 - 1/2*a - 1/5, 1/5120*a^25 - 1/1024*a^23 - 1/512*a^21 + 1/1024*a^19 + 3/1024*a^17 + 3/512*a^15 - 19/640*a^13 - 1/16*a^12 - 1/8*a^10 + 1/8*a^9 - 1/4*a^8 - 3/16*a^7 + 3/16*a^6 - 1/16*a^5 + 3/8*a^4 + 3/8*a^3 - 1/4*a^2 + 3/10*a, 1/10240*a^26 - 1/10240*a^24 + 1/1024*a^22 - 7/2048*a^20 - 1/2048*a^18 - 3/1024*a^16 - 3/160*a^14 + 19/1280*a^12 - 1/8*a^10 - 1/4*a^9 + 13/64*a^8 + 5/32*a^6 + 3/8*a^4 - 1/4*a^3 + 2/5*a^2 - 2/5, 1/10240*a^27 - 1/10240*a^25 - 1/1024*a^23 - 3/2048*a^21 + 7/2048*a^19 + 3/1024*a^17 - 23/2560*a^15 + 1/320*a^13 - 1/16*a^11 - 3/32*a^9 - 11/32*a^7 - 3/16*a^5 + 11/40*a^3 - 2/5*a, 1/20480*a^28 - 1/20480*a^26 - 1/10240*a^24 + 5/4096*a^22 - 9/4096*a^20 + 7/2048*a^18 + 27/5120*a^16 + 17/1280*a^14 - 9/320*a^12 - 3/32*a^10 - 1/4*a^9 - 11/64*a^8 - 5/32*a^6 + 21/80*a^4 - 1/4*a^3 + 3/10*a^2 - 2/5, 1/40960*a^29 + 1/40960*a^27 - 1/10240*a^25 - 7/8192*a^23 - 7/8192*a^21 - 5/2048*a^19 - 1/1280*a^17 + 9/1280*a^15 - 1/32*a^14 - 67/2560*a^13 - 1/16*a^12 - 3/128*a^11 - 1/8*a^10 - 5/32*a^9 + 3/32*a^8 + 3/32*a^7 + 3/16*a^6 + 41/160*a^5 + 3/8*a^4 - 3/20*a^3 - 2/5*a, 1/10485760*a^30 - 47/10485760*a^28 + 101/2621440*a^26 - 523/10485760*a^24 - 2343/2097152*a^22 - 1799/524288*a^20 + 891/1310720*a^18 - 4621/655360*a^16 + 10127/655360*a^14 + 4567/81920*a^12 - 1651/16384*a^10 - 277/2048*a^8 - 1/2*a^7 + 10321/40960*a^6 - 211/1280*a^4 - 19/40*a^2 - 1/2*a - 187/640, 1/20971520*a^31 - 47/20971520*a^29 + 101/5242880*a^27 + 305/4194304*a^25 + 3801/4194304*a^23 - 775/1048576*a^21 - 8069/2621440*a^19 + 4979/1310720*a^17 + 6287/1310720*a^15 - 1/32*a^14 + 299/32768*a^13 - 1/16*a^12 - 755/32768*a^11 - 1/8*a^10 - 789/4096*a^9 + 3/32*a^8 + 26961/81920*a^7 + 3/16*a^6 + 829/2560*a^5 + 3/8*a^4 + 11/80*a^3 + 65/256*a

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

$C_{2}\times C_{6}\times C_{12}$, which has order $144$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Relative class number: $144$ (assuming GRH)

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{1409}{20971520} a^{31} + \frac{3409}{20971520} a^{29} + \frac{97}{1048576} a^{27} + \frac{373}{20971520} a^{25} - \frac{1959}{4194304} a^{23} - \frac{2951}{1048576} a^{21} - \frac{14341}{2621440} a^{19} - \frac{6413}{1310720} a^{17} - \frac{2813}{262144} a^{15} - \frac{6697}{163840} a^{13} - \frac{2419}{32768} a^{11} + \frac{43}{4096} a^{9} + \frac{5329}{81920} a^{7} + \frac{957}{2560} a^{5} + \frac{29}{16} a^{3} + \frac{2757}{1280} a $ (1409)/(20971520)a^(31) + (3409)/(20971520)a^(29) + (97)/(1048576)a^(27) + (373)/(20971520)a^(25) - (1959)/(4194304)a^(23) - (2951)/(1048576)a^(21) - (14341)/(2621440)a^(19) - (6413)/(1310720)a^(17) - (2813)/(262144)a^(15) - (6697)/(163840)a^(13) - (2419)/(32768)a^(11) + (43)/(4096)a^(9) + (5329)/(81920)a^(7) + (957)/(2560)a^(5) + (29)/(16)a^(3) + (2757)/(1280)a (order $12$)	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:	$\frac{2173}{10485760}a^{30}+\frac{13069}{10485760}a^{28}+\frac{605}{524288}a^{26}+\frac{161}{10485760}a^{24}+\frac{2549}{2097152}a^{22}-\frac{5099}{524288}a^{20}-\frac{41457}{1310720}a^{18}-\frac{18393}{655360}a^{16}-\frac{7689}{131072}a^{14}-\frac{14789}{81920}a^{12}-\frac{8615}{16384}a^{10}-\frac{577}{2048}a^{8}+\frac{13453}{40960}a^{6}+\frac{377}{1280}a^{4}+\frac{59}{8}a^{2}+\frac{10289}{640}$, $\frac{1409}{20971520}a^{31}+\frac{3409}{20971520}a^{29}+\frac{97}{1048576}a^{27}+\frac{373}{20971520}a^{25}-\frac{1959}{4194304}a^{23}-\frac{2951}{1048576}a^{21}-\frac{14341}{2621440}a^{19}-\frac{6413}{1310720}a^{17}-\frac{2813}{262144}a^{15}-\frac{6697}{163840}a^{13}-\frac{2419}{32768}a^{11}+\frac{43}{4096}a^{9}+\frac{5329}{81920}a^{7}+\frac{957}{2560}a^{5}+\frac{29}{16}a^{3}+\frac{2757}{1280}a+1$, $\frac{579}{1310720}a^{31}-\frac{1527}{10485760}a^{30}+\frac{1779}{1310720}a^{29}-\frac{4007}{10485760}a^{28}+\frac{51}{65536}a^{27}-\frac{371}{2621440}a^{26}+\frac{799}{1310720}a^{25}+\frac{185}{2097152}a^{24}+\frac{11}{262144}a^{23}+\frac{1953}{2097152}a^{22}-\frac{1093}{65536}a^{21}+\frac{3009}{524288}a^{20}-\frac{5391}{163840}a^{19}+\frac{14803}{1310720}a^{18}-\frac{2983}{81920}a^{17}+\frac{8459}{655360}a^{16}-\frac{1175}{16384}a^{15}+\frac{11783}{655360}a^{14}-\frac{2771}{10240}a^{13}+\frac{1347}{16384}a^{12}-\frac{1097}{2048}a^{11}+\frac{2293}{16384}a^{10}-\frac{15}{256}a^{9}-\frac{93}{2048}a^{8}+\frac{979}{5120}a^{7}-\frac{9767}{40960}a^{6}+\frac{237}{160}a^{5}-\frac{1131}{1280}a^{4}+11a^{3}-\frac{131}{40}a^{2}+\frac{1071}{80}a-\frac{567}{128}$, $\frac{579}{1310720}a^{31}+\frac{1527}{10485760}a^{30}+\frac{1779}{1310720}a^{29}+\frac{4007}{10485760}a^{28}+\frac{51}{65536}a^{27}+\frac{371}{2621440}a^{26}+\frac{799}{1310720}a^{25}-\frac{185}{2097152}a^{24}+\frac{11}{262144}a^{23}-\frac{1953}{2097152}a^{22}-\frac{1093}{65536}a^{21}-\frac{3009}{524288}a^{20}-\frac{5391}{163840}a^{19}-\frac{14803}{1310720}a^{18}-\frac{2983}{81920}a^{17}-\frac{8459}{655360}a^{16}-\frac{1175}{16384}a^{15}-\frac{11783}{655360}a^{14}-\frac{2771}{10240}a^{13}-\frac{1347}{16384}a^{12}-\frac{1097}{2048}a^{11}-\frac{2293}{16384}a^{10}-\frac{15}{256}a^{9}+\frac{93}{2048}a^{8}+\frac{979}{5120}a^{7}+\frac{9767}{40960}a^{6}+\frac{237}{160}a^{5}+\frac{1131}{1280}a^{4}+11a^{3}+\frac{131}{40}a^{2}+\frac{1071}{80}a+\frac{567}{128}$, $\frac{14667}{10485760}a^{30}+\frac{49979}{10485760}a^{28}+\frac{7831}{2621440}a^{26}+\frac{2651}{2097152}a^{24}-\frac{3437}{2097152}a^{22}-\frac{30221}{524288}a^{20}-\frac{161783}{1310720}a^{18}-\frac{85583}{655360}a^{16}-\frac{164123}{655360}a^{14}-\frac{14823}{16384}a^{12}-\frac{30257}{16384}a^{10}-\frac{487}{2048}a^{8}+\frac{47547}{40960}a^{6}+\frac{6847}{1280}a^{4}+\frac{1531}{40}a^{2}+\frac{6667}{128}$, $\frac{5653}{20971520}a^{31}-\frac{17}{5242880}a^{30}+\frac{18469}{20971520}a^{29}-\frac{1761}{5242880}a^{28}+\frac{501}{1048576}a^{27}-\frac{821}{1310720}a^{26}-\frac{251}{4194304}a^{25}-\frac{1349}{5242880}a^{24}-\frac{3123}{4194304}a^{23}-\frac{1641}{1048576}a^{22}-\frac{12819}{1048576}a^{21}-\frac{265}{262144}a^{20}-\frac{63977}{2621440}a^{19}+\frac{4053}{655360}a^{18}-\frac{35473}{1310720}a^{17}+\frac{1117}{327680}a^{16}-\frac{13633}{262144}a^{15}+\frac{4993}{327680}a^{14}-\frac{5689}{32768}a^{13}+\frac{1401}{40960}a^{12}-\frac{10991}{32768}a^{11}+\frac{1315}{8192}a^{10}+\frac{135}{4096}a^{9}+\frac{293}{1024}a^{8}+\frac{34213}{81920}a^{7}+\frac{2463}{20480}a^{6}+\frac{3057}{2560}a^{5}+\frac{387}{640}a^{4}+\frac{127}{16}a^{3}-\frac{3}{10}a^{2}+\frac{2645}{256}a-\frac{1621}{320}$, $\frac{817}{2097152}a^{31}-\frac{585}{2097152}a^{30}+\frac{11781}{10485760}a^{29}-\frac{4349}{10485760}a^{28}+\frac{341}{524288}a^{27}-\frac{129}{2621440}a^{26}+\frac{741}{2097152}a^{25}-\frac{2129}{10485760}a^{24}-\frac{1875}{2097152}a^{23}+\frac{4251}{2097152}a^{22}-\frac{8115}{524288}a^{21}+\frac{5243}{524288}a^{20}-\frac{8053}{262144}a^{19}+\frac{3053}{262144}a^{18}-\frac{19697}{655360}a^{17}+\frac{11273}{655360}a^{16}-\frac{8353}{131072}a^{15}+\frac{17437}{655360}a^{14}-\frac{3929}{16384}a^{13}+\frac{11061}{81920}a^{12}-\frac{6927}{16384}a^{11}+\frac{2167}{16384}a^{10}-\frac{25}{2048}a^{9}-\frac{335}{2048}a^{8}+\frac{2177}{8192}a^{7}-\frac{1561}{8192}a^{6}+\frac{2433}{1280}a^{5}-\frac{2217}{1280}a^{4}+\frac{81}{8}a^{3}-\frac{249}{40}a^{2}+\frac{1589}{128}a-\frac{1761}{640}$, $\frac{4259}{5242880}a^{31}+\frac{145}{2097152}a^{30}+\frac{12691}{5242880}a^{29}+\frac{11493}{10485760}a^{28}+\frac{419}{262144}a^{27}+\frac{3721}{2621440}a^{26}+\frac{9471}{5242880}a^{25}+\frac{5849}{10485760}a^{24}-\frac{85}{1048576}a^{23}+\frac{7821}{2097152}a^{22}-\frac{7317}{262144}a^{21}-\frac{1619}{524288}a^{20}-\frac{35951}{655360}a^{19}-\frac{5973}{262144}a^{18}-\frac{20807}{327680}a^{17}-\frac{10321}{655360}a^{16}-\frac{8599}{65536}a^{15}-\frac{26373}{655360}a^{14}-\frac{20299}{40960}a^{13}-\frac{10461}{81920}a^{12}-\frac{7929}{8192}a^{11}-\frac{8239}{16384}a^{10}-\frac{271}{1024}a^{9}-\frac{1145}{2048}a^{8}+\frac{19}{20480}a^{7}-\frac{1311}{8192}a^{6}+\frac{1563}{640}a^{5}-\frac{1551}{1280}a^{4}+\frac{75}{4}a^{3}+\frac{141}{40}a^{2}+\frac{7279}{320}a+\frac{8681}{640}$, $\frac{195}{4194304}a^{31}-\frac{4409}{10485760}a^{30}+\frac{7423}{20971520}a^{29}-\frac{14473}{10485760}a^{28}+\frac{2219}{5242880}a^{27}-\frac{2301}{2621440}a^{26}+\frac{4123}{20971520}a^{25}-\frac{5261}{10485760}a^{24}+\frac{3191}{4194304}a^{23}+\frac{1711}{2097152}a^{22}-\frac{1961}{1048576}a^{21}+\frac{9743}{524288}a^{20}-\frac{3663}{524288}a^{19}+\frac{49181}{1310720}a^{18}-\frac{4611}{1310720}a^{17}+\frac{26981}{655360}a^{16}-\frac{14687}{1310720}a^{15}+\frac{51753}{655360}a^{14}-\frac{9127}{163840}a^{13}+\frac{21409}{81920}a^{12}-\frac{6269}{32768}a^{11}+\frac{8091}{16384}a^{10}-\frac{699}{4096}a^{9}-\frac{51}{2048}a^{8}-\frac{333}{16384}a^{7}-\frac{12809}{40960}a^{6}-\frac{301}{2560}a^{5}-\frac{2069}{1280}a^{4}+\frac{169}{80}a^{3}-\frac{461}{40}a^{2}+\frac{6347}{1280}a-\frac{9309}{640}$, $\frac{41447}{20971520}a^{31}+\frac{9969}{10485760}a^{30}+\frac{23275}{4194304}a^{29}+\frac{43457}{10485760}a^{28}+\frac{14883}{5242880}a^{27}+\frac{8213}{2621440}a^{26}+\frac{48403}{20971520}a^{25}+\frac{3493}{10485760}a^{24}-\frac{16689}{4194304}a^{23}+\frac{2121}{2097152}a^{22}-\frac{77905}{1048576}a^{21}-\frac{22679}{524288}a^{20}-\frac{364803}{2621440}a^{19}-\frac{144821}{1310720}a^{18}-\frac{40255}{262144}a^{17}-\frac{71229}{655360}a^{16}-\frac{409079}{1310720}a^{15}-\frac{135009}{655360}a^{14}-\frac{194047}{163840}a^{13}-\frac{57817}{81920}a^{12}-\frac{69445}{32768}a^{11}-\frac{27587}{16384}a^{10}+\frac{141}{4096}a^{9}-\frac{1061}{2048}a^{8}+\frac{79127}{81920}a^{7}+\frac{57409}{40960}a^{6}+\frac{4175}{512}a^{5}+\frac{3981}{1280}a^{4}+\frac{3873}{80}a^{3}+\frac{1213}{40}a^{2}+\frac{69187}{1280}a+\frac{32757}{640}$, $\frac{22437}{10485760}a^{31}-\frac{3501}{10485760}a^{30}+\frac{72373}{10485760}a^{29}-\frac{21821}{10485760}a^{28}+\frac{11673}{2621440}a^{27}-\frac{5953}{2621440}a^{26}+\frac{36329}{10485760}a^{25}-\frac{10129}{10485760}a^{24}-\frac{1315}{2097152}a^{23}-\frac{9637}{2097152}a^{22}-\frac{41731}{524288}a^{21}+\frac{7355}{524288}a^{20}-\frac{217913}{1310720}a^{19}+\frac{61409}{1310720}a^{18}-\frac{120161}{655360}a^{17}+\frac{31817}{655360}a^{16}-\frac{241109}{655360}a^{15}+\frac{68189}{655360}a^{14}-\frac{112301}{81920}a^{13}+\frac{25781}{81920}a^{12}-\frac{45471}{16384}a^{11}+\frac{15415}{16384}a^{10}-\frac{1097}{2048}a^{9}+\frac{1457}{2048}a^{8}+\frac{33077}{40960}a^{7}-\frac{5821}{40960}a^{6}+\frac{9329}{1280}a^{5}+\frac{647}{1280}a^{4}+\frac{1079}{20}a^{3}-\frac{443}{40}a^{2}+\frac{44921}{640}a-\frac{16161}{640}$, $\frac{5367}{10485760}a^{31}-\frac{71}{10485760}a^{30}+\frac{17063}{10485760}a^{29}-\frac{5367}{10485760}a^{28}+\frac{1907}{2621440}a^{27}-\frac{2179}{2621440}a^{26}-\frac{157}{10485760}a^{25}-\frac{3827}{10485760}a^{24}-\frac{3233}{2097152}a^{23}-\frac{4143}{2097152}a^{22}-\frac{11713}{524288}a^{21}-\frac{399}{524288}a^{20}-\frac{58323}{1310720}a^{19}+\frac{8419}{1310720}a^{18}-\frac{32011}{655360}a^{17}+\frac{4379}{655360}a^{16}-\frac{62471}{655360}a^{15}+\frac{10327}{655360}a^{14}-\frac{26447}{81920}a^{13}+\frac{3103}{81920}a^{12}-\frac{10229}{16384}a^{11}+\frac{4197}{16384}a^{10}+\frac{157}{2048}a^{9}+\frac{659}{2048}a^{8}+\frac{28967}{40960}a^{7}+\frac{649}{40960}a^{6}+\frac{3019}{1280}a^{5}+\frac{1189}{1280}a^{4}+\frac{143}{10}a^{3}-\frac{29}{40}a^{2}+\frac{11987}{640}a-\frac{4003}{640}$, $\frac{17757}{20971520}a^{31}+\frac{5981}{10485760}a^{30}+\frac{45549}{20971520}a^{29}+\frac{14829}{10485760}a^{28}+\frac{4913}{5242880}a^{27}+\frac{2737}{2621440}a^{26}+\frac{13057}{20971520}a^{25}+\frac{3533}{2097152}a^{24}-\frac{12331}{4194304}a^{23}+\frac{2005}{2097152}a^{22}-\frac{33547}{1048576}a^{21}-\frac{8075}{524288}a^{20}-\frac{151633}{2621440}a^{19}-\frac{36689}{1310720}a^{18}-\frac{87353}{1310720}a^{17}-\frac{22073}{655360}a^{16}-\frac{164109}{1310720}a^{15}-\frac{51981}{655360}a^{14}-\frac{82213}{163840}a^{13}-\frac{5729}{16384}a^{12}-\frac{26951}{32768}a^{11}-\frac{9543}{16384}a^{10}+\frac{607}{4096}a^{9}-\frac{417}{2048}a^{8}+\frac{44397}{81920}a^{7}-\frac{18579}{40960}a^{6}+\frac{10297}{2560}a^{5}+\frac{1897}{1280}a^{4}+\frac{1653}{80}a^{3}+\frac{437}{40}a^{2}+\frac{27793}{1280}a+\frac{1309}{128}$, $\frac{9221}{20971520}a^{31}+\frac{39189}{20971520}a^{29}+\frac{7033}{5242880}a^{27}+\frac{501}{4194304}a^{25}+\frac{1085}{4194304}a^{23}-\frac{20899}{1048576}a^{21}-\frac{129689}{2621440}a^{19}-\frac{60353}{1310720}a^{17}-\frac{118069}{1310720}a^{15}-\frac{10217}{32768}a^{13}-\frac{24255}{32768}a^{11}-\frac{681}{4096}a^{9}+\frac{56981}{81920}a^{7}+\frac{4257}{2560}a^{5}+\frac{1133}{80}a^{3}+\frac{5957}{256}a$, $\frac{3417}{5242880}a^{30}+\frac{12201}{5242880}a^{28}+\frac{2269}{1310720}a^{26}+\frac{10029}{5242880}a^{24}+\frac{2417}{1048576}a^{22}-\frac{5551}{262144}a^{20}-\frac{28093}{655360}a^{18}-\frac{17157}{327680}a^{16}-\frac{36937}{327680}a^{14}-\frac{17761}{40960}a^{12}-\frac{8059}{8192}a^{10}-\frac{333}{1024}a^{8}-\frac{6103}{20480}a^{6}+\frac{653}{640}a^{4}+\frac{299}{20}a^{2}+\frac{6141}{320}$ 2173/10485760a^30 + 13069/10485760a^28 + 605/524288a^26 + 161/10485760a^24 + 2549/2097152a^22 - 5099/524288a^20 - 41457/1310720a^18 - 18393/655360a^16 - 7689/131072a^14 - 14789/81920a^12 - 8615/16384a^10 - 577/2048a^8 + 13453/40960a^6 + 377/1280a^4 + 59/8a^2 + 10289/640, 1409/20971520a^31 + 3409/20971520a^29 + 97/1048576a^27 + 373/20971520a^25 - 1959/4194304a^23 - 2951/1048576a^21 - 14341/2621440a^19 - 6413/1310720a^17 - 2813/262144a^15 - 6697/163840a^13 - 2419/32768a^11 + 43/4096a^9 + 5329/81920a^7 + 957/2560a^5 + 29/16a^3 + 2757/1280a + 1, 579/1310720a^31 - 1527/10485760a^30 + 1779/1310720a^29 - 4007/10485760a^28 + 51/65536a^27 - 371/2621440a^26 + 799/1310720a^25 + 185/2097152a^24 + 11/262144a^23 + 1953/2097152a^22 - 1093/65536a^21 + 3009/524288a^20 - 5391/163840a^19 + 14803/1310720a^18 - 2983/81920a^17 + 8459/655360a^16 - 1175/16384a^15 + 11783/655360a^14 - 2771/10240a^13 + 1347/16384a^12 - 1097/2048a^11 + 2293/16384a^10 - 15/256a^9 - 93/2048a^8 + 979/5120a^7 - 9767/40960a^6 + 237/160a^5 - 1131/1280a^4 + 11a^3 - 131/40a^2 + 1071/80a - 567/128, 579/1310720a^31 + 1527/10485760a^30 + 1779/1310720a^29 + 4007/10485760a^28 + 51/65536a^27 + 371/2621440a^26 + 799/1310720a^25 - 185/2097152a^24 + 11/262144a^23 - 1953/2097152a^22 - 1093/65536a^21 - 3009/524288a^20 - 5391/163840a^19 - 14803/1310720a^18 - 2983/81920a^17 - 8459/655360a^16 - 1175/16384a^15 - 11783/655360a^14 - 2771/10240a^13 - 1347/16384a^12 - 1097/2048a^11 - 2293/16384a^10 - 15/256a^9 + 93/2048a^8 + 979/5120a^7 + 9767/40960a^6 + 237/160a^5 + 1131/1280a^4 + 11a^3 + 131/40a^2 + 1071/80a + 567/128, 14667/10485760a^30 + 49979/10485760a^28 + 7831/2621440a^26 + 2651/2097152a^24 - 3437/2097152a^22 - 30221/524288a^20 - 161783/1310720a^18 - 85583/655360a^16 - 164123/655360a^14 - 14823/16384a^12 - 30257/16384a^10 - 487/2048a^8 + 47547/40960a^6 + 6847/1280a^4 + 1531/40a^2 + 6667/128, 5653/20971520a^31 - 17/5242880a^30 + 18469/20971520a^29 - 1761/5242880a^28 + 501/1048576a^27 - 821/1310720a^26 - 251/4194304a^25 - 1349/5242880a^24 - 3123/4194304a^23 - 1641/1048576a^22 - 12819/1048576a^21 - 265/262144a^20 - 63977/2621440a^19 + 4053/655360a^18 - 35473/1310720a^17 + 1117/327680a^16 - 13633/262144a^15 + 4993/327680a^14 - 5689/32768a^13 + 1401/40960a^12 - 10991/32768a^11 + 1315/8192a^10 + 135/4096a^9 + 293/1024a^8 + 34213/81920a^7 + 2463/20480a^6 + 3057/2560a^5 + 387/640a^4 + 127/16a^3 - 3/10a^2 + 2645/256a - 1621/320, 817/2097152a^31 - 585/2097152a^30 + 11781/10485760a^29 - 4349/10485760a^28 + 341/524288a^27 - 129/2621440a^26 + 741/2097152a^25 - 2129/10485760a^24 - 1875/2097152a^23 + 4251/2097152a^22 - 8115/524288a^21 + 5243/524288a^20 - 8053/262144a^19 + 3053/262144a^18 - 19697/655360a^17 + 11273/655360a^16 - 8353/131072a^15 + 17437/655360a^14 - 3929/16384a^13 + 11061/81920a^12 - 6927/16384a^11 + 2167/16384a^10 - 25/2048a^9 - 335/2048a^8 + 2177/8192a^7 - 1561/8192a^6 + 2433/1280a^5 - 2217/1280a^4 + 81/8a^3 - 249/40a^2 + 1589/128a - 1761/640, 4259/5242880a^31 + 145/2097152a^30 + 12691/5242880a^29 + 11493/10485760a^28 + 419/262144a^27 + 3721/2621440a^26 + 9471/5242880a^25 + 5849/10485760a^24 - 85/1048576a^23 + 7821/2097152a^22 - 7317/262144a^21 - 1619/524288a^20 - 35951/655360a^19 - 5973/262144a^18 - 20807/327680a^17 - 10321/655360a^16 - 8599/65536a^15 - 26373/655360a^14 - 20299/40960a^13 - 10461/81920a^12 - 7929/8192a^11 - 8239/16384a^10 - 271/1024a^9 - 1145/2048a^8 + 19/20480a^7 - 1311/8192a^6 + 1563/640a^5 - 1551/1280a^4 + 75/4a^3 + 141/40a^2 + 7279/320a + 8681/640, 195/4194304a^31 - 4409/10485760a^30 + 7423/20971520a^29 - 14473/10485760a^28 + 2219/5242880a^27 - 2301/2621440a^26 + 4123/20971520a^25 - 5261/10485760a^24 + 3191/4194304a^23 + 1711/2097152a^22 - 1961/1048576a^21 + 9743/524288a^20 - 3663/524288a^19 + 49181/1310720a^18 - 4611/1310720a^17 + 26981/655360a^16 - 14687/1310720a^15 + 51753/655360a^14 - 9127/163840a^13 + 21409/81920a^12 - 6269/32768a^11 + 8091/16384a^10 - 699/4096a^9 - 51/2048a^8 - 333/16384a^7 - 12809/40960a^6 - 301/2560a^5 - 2069/1280a^4 + 169/80a^3 - 461/40a^2 + 6347/1280a - 9309/640, 41447/20971520a^31 + 9969/10485760a^30 + 23275/4194304a^29 + 43457/10485760a^28 + 14883/5242880a^27 + 8213/2621440a^26 + 48403/20971520a^25 + 3493/10485760a^24 - 16689/4194304a^23 + 2121/2097152a^22 - 77905/1048576a^21 - 22679/524288a^20 - 364803/2621440a^19 - 144821/1310720a^18 - 40255/262144a^17 - 71229/655360a^16 - 409079/1310720a^15 - 135009/655360a^14 - 194047/163840a^13 - 57817/81920a^12 - 69445/32768a^11 - 27587/16384a^10 + 141/4096a^9 - 1061/2048a^8 + 79127/81920a^7 + 57409/40960a^6 + 4175/512a^5 + 3981/1280a^4 + 3873/80a^3 + 1213/40a^2 + 69187/1280a + 32757/640, 22437/10485760a^31 - 3501/10485760a^30 + 72373/10485760a^29 - 21821/10485760a^28 + 11673/2621440a^27 - 5953/2621440a^26 + 36329/10485760a^25 - 10129/10485760a^24 - 1315/2097152a^23 - 9637/2097152a^22 - 41731/524288a^21 + 7355/524288a^20 - 217913/1310720a^19 + 61409/1310720a^18 - 120161/655360a^17 + 31817/655360a^16 - 241109/655360a^15 + 68189/655360a^14 - 112301/81920a^13 + 25781/81920a^12 - 45471/16384a^11 + 15415/16384a^10 - 1097/2048a^9 + 1457/2048a^8 + 33077/40960a^7 - 5821/40960a^6 + 9329/1280a^5 + 647/1280a^4 + 1079/20a^3 - 443/40a^2 + 44921/640a - 16161/640, 5367/10485760a^31 - 71/10485760a^30 + 17063/10485760a^29 - 5367/10485760a^28 + 1907/2621440a^27 - 2179/2621440a^26 - 157/10485760a^25 - 3827/10485760a^24 - 3233/2097152a^23 - 4143/2097152a^22 - 11713/524288a^21 - 399/524288a^20 - 58323/1310720a^19 + 8419/1310720a^18 - 32011/655360a^17 + 4379/655360a^16 - 62471/655360a^15 + 10327/655360a^14 - 26447/81920a^13 + 3103/81920a^12 - 10229/16384a^11 + 4197/16384a^10 + 157/2048a^9 + 659/2048a^8 + 28967/40960a^7 + 649/40960a^6 + 3019/1280a^5 + 1189/1280a^4 + 143/10a^3 - 29/40a^2 + 11987/640a - 4003/640, 17757/20971520a^31 + 5981/10485760a^30 + 45549/20971520a^29 + 14829/10485760a^28 + 4913/5242880a^27 + 2737/2621440a^26 + 13057/20971520a^25 + 3533/2097152a^24 - 12331/4194304a^23 + 2005/2097152a^22 - 33547/1048576a^21 - 8075/524288a^20 - 151633/2621440a^19 - 36689/1310720a^18 - 87353/1310720a^17 - 22073/655360a^16 - 164109/1310720a^15 - 51981/655360a^14 - 82213/163840a^13 - 5729/16384a^12 - 26951/32768a^11 - 9543/16384a^10 + 607/4096a^9 - 417/2048a^8 + 44397/81920a^7 - 18579/40960a^6 + 10297/2560a^5 + 1897/1280a^4 + 1653/80a^3 + 437/40a^2 + 27793/1280a + 1309/128, 9221/20971520a^31 + 39189/20971520a^29 + 7033/5242880a^27 + 501/4194304a^25 + 1085/4194304a^23 - 20899/1048576a^21 - 129689/2621440a^19 - 60353/1310720a^17 - 118069/1310720a^15 - 10217/32768a^13 - 24255/32768a^11 - 681/4096a^9 + 56981/81920a^7 + 4257/2560a^5 + 1133/80a^3 + 5957/256a, 3417/5242880a^30 + 12201/5242880a^28 + 2269/1310720a^26 + 10029/5242880a^24 + 2417/1048576a^22 - 5551/262144a^20 - 28093/655360a^18 - 17157/327680a^16 - 36937/327680a^14 - 17761/40960a^12 - 8059/8192a^10 - 333/1024a^8 - 6103/20480a^6 + 653/640a^4 + 299/20*a^2 + 6141/320 (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:	$ 2519992645123.5127 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 2519992645123.5127 \cdot 144}{12\cdot\sqrt{26255893369003065702238822824526479360000000000000000}}\cr\approx \mathstrut & 1.10114627264127 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^32 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 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 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 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 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 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 + 5*x^30 + 8*x^28 + 5*x^26 + x^24 - 40*x^22 - 152*x^20 - 240*x^18 - 336*x^16 - 960*x^14 - 2432*x^12 - 2560*x^10 + 256*x^8 + 5120*x^6 + 32768*x^4 + 81920*x^2 + 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

$D_4^2:C_2^3$ (as 32T12882):

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 512

The 80 conjugacy class representatives for $D_4^2:C_2^3$

Character table for $D_4^2:C_2^3$

Intermediate fields

$\Q(\sqrt{-3}) $, $\Q(\sqrt{-1}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{15}) $, $\Q(\sqrt{3}) $, $\Q(\sqrt{-5}) $, $\Q(\sqrt{-15}) $, $\Q(\sqrt{3}, \sqrt{-5})$, $\Q(\zeta_{12})$, $\Q(\sqrt{3}, \sqrt{5})$, 4.4.320400.1, 4.0.35600.3, $\Q(i, \sqrt{5})$, $\Q(\sqrt{-3}, \sqrt{-5})$, $\Q(\sqrt{-3}, \sqrt{5})$, $\Q(i, \sqrt{15})$, 4.0.20025.1, 4.4.2225.1, 8.8.102656160000.1, 8.0.102656160000.21, 8.0.12960000.1, 8.0.102656160000.10, 8.0.102656160000.5, 8.0.1267360000.3, 8.0.401000625.1, 8.0.3182340960000.1, 8.0.2455510000.1, 8.8.198896310000.1, 8.8.39288160000.1, 16.0.10538287185945600000000.2, 16.0.162036703771099545600000000.2, 16.16.162036703771099545600000000.1, 16.0.162036703771099545600000000.1, 16.0.24696952258969600000000.1, 16.0.39559742131616100000000.1, 16.0.10127293985693721600000000.1

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 32 siblings:	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	R	R	R	${\href{/padicField/7.8.0.1}{8} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{8}$	${\href{/padicField/13.8.0.1}{8} }^{4}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$	${\href{/padicField/19.2.0.1}{2} }^{16}$	${\href{/padicField/23.4.0.1}{4} }^{8}$	${\href{/padicField/29.2.0.1}{2} }^{16}$	R	${\href{/padicField/37.8.0.1}{8} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$	${\href{/padicField/59.2.0.1}{2} }^{16}$

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
$2$ 2	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.8.12.13	$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$	$4$	$2$	$12$	$D_4$	$[2, 2]^{2}$
	2.8.12.13	$x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 87 x^{4} + 98 x^{3} + 58 x^{2} - 2 x + 1$	$4$	$2$	$12$	$D_4$	$[2, 2]^{2}$
$3$ 3	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$5$ 5	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$31$ 31	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.2.0.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.4.2.1	$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	31.4.2.1	$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	31.4.0.1	$x^{4} + 3 x^{2} + 16 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	31.4.0.1	$x^{4} + 3 x^{2} + 16 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	31.4.0.1	$x^{4} + 3 x^{2} + 16 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	31.4.0.1	$x^{4} + 3 x^{2} + 16 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$89$ 89	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	89.4.2.1	$x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	89.4.2.1	$x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	89.4.2.1	$x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	89.4.2.1	$x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	89.4.0.1	$x^{4} + 4 x^{2} + 72 x + 3$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$

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