LMFDB - Number field 32.0.97424180758632937220739951448553101324025212174336.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 65536)

gp: K = bnfinit(y^32 - 10*y^28 + 27*y^24 + 150*y^20 - 1215*y^16 + 2400*y^12 + 6912*y^8 - 40960*y^4 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 65536);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 65536)

$ x^{32} - 10x^{28} + 27x^{24} + 150x^{20} - 1215x^{16} + 2400x^{12} + 6912x^{8} - 40960x^{4} + 65536 $ x^32 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 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:	$97424180758632937220739951448553101324025212174336$ 97424180758632937220739951448553101324025212174336 $\medspace = 2^{72}\cdot 223^{8}\cdot 241^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$36.49$	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:	$2^{9/4}223^{1/2}241^{1/2}\approx 1102.753495447589$
Ramified primes:	$2$, $223$, $241$ 2, 223, 241	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{2}a^{8}+\frac{3}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{12}+\frac{3}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{4}-\frac{1}{2}a$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{9}-\frac{1}{2}a^{8}-\frac{3}{8}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}-\frac{3}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{15}-\frac{1}{2}a^{8}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{16}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{17}-\frac{1}{2}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{18}-\frac{1}{16}a^{17}-\frac{1}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}+\frac{1}{16}a^{10}-\frac{3}{16}a^{8}-\frac{3}{16}a^{7}-\frac{1}{2}a^{6}-\frac{5}{16}a^{5}-\frac{1}{16}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{19}-\frac{1}{16}a^{17}-\frac{1}{16}a^{16}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}+\frac{1}{16}a^{10}-\frac{3}{16}a^{9}-\frac{1}{2}a^{8}-\frac{3}{16}a^{7}+\frac{3}{16}a^{6}-\frac{3}{8}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{20}-\frac{1}{16}a^{12}+\frac{1}{8}a^{8}-\frac{1}{2}a^{5}+\frac{7}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{21}-\frac{1}{16}a^{17}-\frac{1}{32}a^{13}+\frac{1}{16}a^{9}-\frac{1}{2}a^{8}+\frac{13}{32}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{22}-\frac{1}{32}a^{18}+\frac{3}{64}a^{14}-\frac{1}{32}a^{10}-\frac{1}{2}a^{7}-\frac{31}{64}a^{6}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{23}-\frac{1}{64}a^{19}-\frac{1}{16}a^{17}-\frac{1}{16}a^{16}-\frac{5}{128}a^{15}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}+\frac{3}{64}a^{11}+\frac{1}{16}a^{10}-\frac{3}{16}a^{9}-\frac{7}{128}a^{7}-\frac{5}{16}a^{6}-\frac{3}{8}a^{5}-\frac{5}{16}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{2048}a^{24}-\frac{1}{256}a^{23}-\frac{1}{64}a^{21}-\frac{29}{1024}a^{20}-\frac{3}{128}a^{19}+\frac{1}{32}a^{17}-\frac{5}{2048}a^{16}+\frac{5}{256}a^{15}-\frac{3}{64}a^{13}-\frac{61}{1024}a^{12}-\frac{3}{128}a^{11}-\frac{7}{32}a^{9}-\frac{223}{2048}a^{8}+\frac{31}{256}a^{7}-\frac{1}{2}a^{6}-\frac{17}{64}a^{5}+\frac{51}{128}a^{4}+\frac{5}{16}a^{3}-\frac{1}{2}a+\frac{1}{8}$, $\frac{1}{4096}a^{25}-\frac{1}{128}a^{22}-\frac{29}{2048}a^{21}+\frac{1}{64}a^{18}+\frac{251}{4096}a^{17}-\frac{1}{16}a^{15}-\frac{3}{128}a^{14}-\frac{61}{2048}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{7}{64}a^{10}+\frac{801}{4096}a^{9}+\frac{5}{16}a^{8}+\frac{5}{16}a^{7}+\frac{47}{128}a^{6}-\frac{61}{256}a^{5}+\frac{7}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{7}{16}a-\frac{1}{2}$, $\frac{1}{8192}a^{26}-\frac{1}{256}a^{23}-\frac{29}{4096}a^{22}-\frac{3}{128}a^{19}-\frac{5}{8192}a^{18}-\frac{1}{16}a^{16}-\frac{11}{256}a^{15}-\frac{61}{4096}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{3}{128}a^{11}+\frac{801}{8192}a^{10}+\frac{1}{16}a^{9}-\frac{3}{16}a^{8}+\frac{31}{256}a^{7}+\frac{243}{512}a^{6}+\frac{3}{16}a^{5}-\frac{3}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{32}a^{2}-\frac{1}{4}a$, $\frac{1}{16384}a^{27}-\frac{29}{8192}a^{23}-\frac{1}{32}a^{20}+\frac{507}{16384}a^{19}-\frac{1}{16}a^{16}+\frac{451}{8192}a^{15}+\frac{1}{32}a^{12}-\frac{223}{16384}a^{11}-\frac{1}{8}a^{10}-\frac{1}{16}a^{8}-\frac{45}{1024}a^{7}-\frac{3}{8}a^{6}+\frac{15}{32}a^{4}-\frac{15}{64}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}$, $\frac{1}{294912}a^{28}-\frac{7}{49152}a^{24}-\frac{1}{256}a^{23}+\frac{1225}{98304}a^{20}-\frac{3}{128}a^{19}-\frac{1}{16}a^{17}-\frac{637}{16384}a^{16}+\frac{5}{256}a^{15}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1607}{32768}a^{12}+\frac{5}{128}a^{11}+\frac{1}{16}a^{10}+\frac{1}{16}a^{9}-\frac{193}{1536}a^{8}-\frac{49}{256}a^{7}+\frac{3}{16}a^{6}-\frac{1}{8}a^{5}-\frac{37}{96}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{31}{72}$, $\frac{1}{589824}a^{29}-\frac{7}{98304}a^{25}+\frac{1225}{196608}a^{21}-\frac{1}{32}a^{20}-\frac{1661}{32768}a^{17}-\frac{1}{16}a^{16}-\frac{1607}{65536}a^{13}+\frac{1}{32}a^{12}-\frac{1}{8}a^{10}-\frac{193}{3072}a^{9}-\frac{1}{16}a^{8}+\frac{1}{8}a^{6}-\frac{19}{192}a^{5}-\frac{1}{32}a^{4}-\frac{1}{8}a^{2}-\frac{31}{144}a-\frac{1}{2}$, $\frac{1}{2359296}a^{30}-\frac{1}{1179648}a^{29}-\frac{1}{32768}a^{27}+\frac{17}{393216}a^{26}+\frac{7}{196608}a^{25}-\frac{1}{4096}a^{24}+\frac{29}{16384}a^{23}-\frac{1559}{786432}a^{22}+\frac{4919}{393216}a^{21}-\frac{35}{2048}a^{20}-\frac{507}{32768}a^{19}-\frac{1701}{131072}a^{18}-\frac{387}{65536}a^{17}-\frac{251}{4096}a^{16}-\frac{451}{16384}a^{15}+\frac{4633}{262144}a^{14}-\frac{441}{131072}a^{13}-\frac{3}{2048}a^{12}-\frac{1825}{32768}a^{11}-\frac{6049}{49152}a^{10}-\frac{1151}{6144}a^{9}-\frac{1825}{4096}a^{8}+\frac{685}{2048}a^{7}-\frac{19}{3072}a^{6}+\frac{1}{384}a^{5}-\frac{11}{256}a^{4}+\frac{7}{128}a^{3}+\frac{133}{288}a^{2}-\frac{41}{288}a-\frac{1}{16}$, $\frac{1}{4718592}a^{31}-\frac{1}{1179648}a^{29}-\frac{1}{589824}a^{28}-\frac{7}{786432}a^{27}-\frac{1}{16384}a^{26}-\frac{17}{196608}a^{25}+\frac{7}{98304}a^{24}-\frac{4919}{1572864}a^{23}-\frac{35}{8192}a^{22}+\frac{1559}{393216}a^{21}+\frac{4919}{196608}a^{20}-\frac{3709}{262144}a^{19}-\frac{251}{16384}a^{18}-\frac{2395}{65536}a^{17}-\frac{387}{32768}a^{16}+\frac{16825}{524288}a^{15}+\frac{381}{8192}a^{14}-\frac{4633}{131072}a^{13}+\frac{3655}{65536}a^{12}+\frac{383}{24576}a^{11}-\frac{1569}{16384}a^{10}+\frac{6049}{24576}a^{9}+\frac{961}{3072}a^{8}+\frac{119}{1536}a^{7}+\frac{421}{1024}a^{6}-\frac{461}{1536}a^{5}-\frac{83}{192}a^{4}-\frac{391}{1152}a^{3}-\frac{25}{64}a^{2}-\frac{61}{144}a+\frac{31}{144}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/2*a^9 - 1/2*a^5 - 1/2*a, 1/4*a^10 - 1/4*a^6 + 1/4*a^2, 1/8*a^11 - 1/2*a^8 + 3/8*a^7 - 1/2*a^6 - 1/2*a^4 + 1/8*a^3, 1/8*a^12 + 3/8*a^8 - 1/2*a^7 + 1/8*a^4 - 1/2*a, 1/8*a^13 - 1/8*a^9 - 1/2*a^8 - 3/8*a^5 - 1/2*a^2 - 1/2*a, 1/8*a^14 - 1/8*a^10 - 3/8*a^6 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/8*a^15 - 1/2*a^8 - 3/8*a^3 - 1/2*a^2, 1/8*a^16 - 1/2*a^5 - 3/8*a^4 - 1/2*a^3 - 1/2*a, 1/8*a^17 - 1/2*a^6 - 3/8*a^5 - 1/2*a^4 - 1/2*a^2, 1/16*a^18 - 1/16*a^17 - 1/16*a^15 - 1/16*a^14 - 1/16*a^12 - 1/16*a^11 + 1/16*a^10 - 3/16*a^8 - 3/16*a^7 - 1/2*a^6 - 5/16*a^5 - 1/16*a^4 + 1/8*a^3 + 1/4*a^2 - 1/2*a, 1/16*a^19 - 1/16*a^17 - 1/16*a^16 - 1/16*a^14 - 1/16*a^13 + 1/16*a^10 - 3/16*a^9 - 1/2*a^8 - 3/16*a^7 + 3/16*a^6 - 3/8*a^5 + 3/16*a^4 + 1/4*a^2, 1/16*a^20 - 1/16*a^12 + 1/8*a^8 - 1/2*a^5 + 7/16*a^4 - 1/2*a^3 - 1/2*a, 1/32*a^21 - 1/16*a^17 - 1/32*a^13 + 1/16*a^9 - 1/2*a^8 + 13/32*a^5 - 1/2*a^2, 1/64*a^22 - 1/32*a^18 + 3/64*a^14 - 1/32*a^10 - 1/2*a^7 - 31/64*a^6 + 1/4*a^2 - 1/2*a, 1/128*a^23 - 1/64*a^19 - 1/16*a^17 - 1/16*a^16 - 5/128*a^15 - 1/16*a^14 - 1/16*a^13 + 3/64*a^11 + 1/16*a^10 - 3/16*a^9 - 7/128*a^7 - 5/16*a^6 - 3/8*a^5 - 5/16*a^4 + 3/8*a^3 + 1/4*a^2, 1/2048*a^24 - 1/256*a^23 - 1/64*a^21 - 29/1024*a^20 - 3/128*a^19 + 1/32*a^17 - 5/2048*a^16 + 5/256*a^15 - 3/64*a^13 - 61/1024*a^12 - 3/128*a^11 - 7/32*a^9 - 223/2048*a^8 + 31/256*a^7 - 1/2*a^6 - 17/64*a^5 + 51/128*a^4 + 5/16*a^3 - 1/2*a + 1/8, 1/4096*a^25 - 1/128*a^22 - 29/2048*a^21 + 1/64*a^18 + 251/4096*a^17 - 1/16*a^15 - 3/128*a^14 - 61/2048*a^13 - 1/16*a^12 - 1/16*a^11 - 7/64*a^10 + 801/4096*a^9 + 5/16*a^8 + 5/16*a^7 + 47/128*a^6 - 61/256*a^5 + 7/16*a^4 - 3/8*a^3 - 1/4*a^2 - 7/16*a - 1/2, 1/8192*a^26 - 1/256*a^23 - 29/4096*a^22 - 3/128*a^19 - 5/8192*a^18 - 1/16*a^16 - 11/256*a^15 - 61/4096*a^14 - 1/16*a^13 - 1/16*a^12 - 3/128*a^11 + 801/8192*a^10 + 1/16*a^9 - 3/16*a^8 + 31/256*a^7 + 243/512*a^6 + 3/16*a^5 - 3/8*a^4 - 1/2*a^3 + 1/32*a^2 - 1/4*a, 1/16384*a^27 - 29/8192*a^23 - 1/32*a^20 + 507/16384*a^19 - 1/16*a^16 + 451/8192*a^15 + 1/32*a^12 - 223/16384*a^11 - 1/8*a^10 - 1/16*a^8 - 45/1024*a^7 - 3/8*a^6 + 15/32*a^4 - 15/64*a^3 + 3/8*a^2 - 1/2, 1/294912*a^28 - 7/49152*a^24 - 1/256*a^23 + 1225/98304*a^20 - 3/128*a^19 - 1/16*a^17 - 637/16384*a^16 + 5/256*a^15 - 1/16*a^14 - 1/16*a^13 - 1607/32768*a^12 + 5/128*a^11 + 1/16*a^10 + 1/16*a^9 - 193/1536*a^8 - 49/256*a^7 + 3/16*a^6 - 1/8*a^5 - 37/96*a^4 + 3/8*a^3 - 1/4*a^2 - 1/4*a - 31/72, 1/589824*a^29 - 7/98304*a^25 + 1225/196608*a^21 - 1/32*a^20 - 1661/32768*a^17 - 1/16*a^16 - 1607/65536*a^13 + 1/32*a^12 - 1/8*a^10 - 193/3072*a^9 - 1/16*a^8 + 1/8*a^6 - 19/192*a^5 - 1/32*a^4 - 1/8*a^2 - 31/144*a - 1/2, 1/2359296*a^30 - 1/1179648*a^29 - 1/32768*a^27 + 17/393216*a^26 + 7/196608*a^25 - 1/4096*a^24 + 29/16384*a^23 - 1559/786432*a^22 + 4919/393216*a^21 - 35/2048*a^20 - 507/32768*a^19 - 1701/131072*a^18 - 387/65536*a^17 - 251/4096*a^16 - 451/16384*a^15 + 4633/262144*a^14 - 441/131072*a^13 - 3/2048*a^12 - 1825/32768*a^11 - 6049/49152*a^10 - 1151/6144*a^9 - 1825/4096*a^8 + 685/2048*a^7 - 19/3072*a^6 + 1/384*a^5 - 11/256*a^4 + 7/128*a^3 + 133/288*a^2 - 41/288*a - 1/16, 1/4718592*a^31 - 1/1179648*a^29 - 1/589824*a^28 - 7/786432*a^27 - 1/16384*a^26 - 17/196608*a^25 + 7/98304*a^24 - 4919/1572864*a^23 - 35/8192*a^22 + 1559/393216*a^21 + 4919/196608*a^20 - 3709/262144*a^19 - 251/16384*a^18 - 2395/65536*a^17 - 387/32768*a^16 + 16825/524288*a^15 + 381/8192*a^14 - 4633/131072*a^13 + 3655/65536*a^12 + 383/24576*a^11 - 1569/16384*a^10 + 6049/24576*a^9 + 961/3072*a^8 + 119/1536*a^7 + 421/1024*a^6 - 461/1536*a^5 - 83/192*a^4 - 391/1152*a^3 - 25/64*a^2 - 61/144*a + 31/144

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$, $3$

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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{559}{294912} a^{29} + \frac{577}{49152} a^{25} + \frac{185}{98304} a^{21} - \frac{4949}{16384} a^{17} + \frac{34025}{32768} a^{13} + \frac{3451}{6144} a^{9} - \frac{4691}{384} a^{5} + \frac{821}{36} a $ -(559)/(294912)a^(29) + (577)/(49152)a^(25) + (185)/(98304)a^(21) - (4949)/(16384)a^(17) + (34025)/(32768)a^(13) + (3451)/(6144)a^(9) - (4691)/(384)a^(5) + (821)/(36)a (order $8$)	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:	$\frac{2779}{4718592}a^{31}-\frac{31}{1179648}a^{30}-\frac{59}{1179648}a^{29}+\frac{21}{65536}a^{28}-\frac{2413}{786432}a^{27}+\frac{133}{196608}a^{26}+\frac{725}{196608}a^{25}-\frac{57}{32768}a^{24}-\frac{3917}{1572864}a^{23}-\frac{583}{393216}a^{22}-\frac{4019}{393216}a^{21}-\frac{137}{65536}a^{20}+\frac{23073}{262144}a^{19}-\frac{593}{65536}a^{18}-\frac{2873}{65536}a^{17}+\frac{1591}{32768}a^{16}-\frac{131709}{524288}a^{15}+\frac{10569}{131072}a^{14}+\frac{59197}{131072}a^{13}-\frac{8299}{65536}a^{12}-\frac{14189}{49152}a^{11}-\frac{3517}{49152}a^{10}-\frac{13069}{24576}a^{9}-\frac{179}{1024}a^{8}+\frac{10237}{3072}a^{7}-\frac{1615}{3072}a^{6}-\frac{4639}{1536}a^{5}+\frac{111}{64}a^{4}-\frac{5455}{1152}a^{3}+\frac{1427}{576}a^{2}+\frac{2017}{144}a-\frac{35}{16}$, $\frac{2779}{4718592}a^{31}+\frac{31}{1179648}a^{30}-\frac{59}{1179648}a^{29}-\frac{21}{65536}a^{28}-\frac{2413}{786432}a^{27}-\frac{133}{196608}a^{26}+\frac{725}{196608}a^{25}+\frac{57}{32768}a^{24}-\frac{3917}{1572864}a^{23}+\frac{583}{393216}a^{22}-\frac{4019}{393216}a^{21}+\frac{137}{65536}a^{20}+\frac{23073}{262144}a^{19}+\frac{593}{65536}a^{18}-\frac{2873}{65536}a^{17}-\frac{1591}{32768}a^{16}-\frac{131709}{524288}a^{15}-\frac{10569}{131072}a^{14}+\frac{59197}{131072}a^{13}+\frac{8299}{65536}a^{12}-\frac{14189}{49152}a^{11}+\frac{3517}{49152}a^{10}-\frac{13069}{24576}a^{9}+\frac{179}{1024}a^{8}+\frac{10237}{3072}a^{7}+\frac{1615}{3072}a^{6}-\frac{4639}{1536}a^{5}-\frac{111}{64}a^{4}-\frac{5455}{1152}a^{3}-\frac{1427}{576}a^{2}+\frac{2017}{144}a+\frac{35}{16}$, $\frac{2449}{4718592}a^{31}-\frac{31}{1179648}a^{30}-\frac{163}{393216}a^{29}-\frac{21}{65536}a^{28}-\frac{3847}{786432}a^{27}+\frac{133}{196608}a^{26}+\frac{141}{65536}a^{25}+\frac{57}{32768}a^{24}+\frac{7705}{1572864}a^{23}-\frac{583}{393216}a^{22}+\frac{293}{131072}a^{21}+\frac{137}{65536}a^{20}+\frac{26531}{262144}a^{19}-\frac{593}{65536}a^{18}-\frac{4019}{65536}a^{17}-\frac{1591}{32768}a^{16}-\frac{265591}{524288}a^{15}+\frac{10569}{131072}a^{14}+\frac{20159}{131072}a^{13}+\frac{8299}{65536}a^{12}+\frac{6781}{49152}a^{11}-\frac{3517}{49152}a^{10}+\frac{1903}{8192}a^{9}+\frac{179}{1024}a^{8}+\frac{14647}{3072}a^{7}-\frac{1615}{3072}a^{6}-\frac{1083}{512}a^{5}-\frac{111}{64}a^{4}-\frac{15493}{1152}a^{3}+\frac{1427}{576}a^{2}+\frac{131}{48}a+\frac{51}{16}$, $\frac{259}{1572864}a^{31}+\frac{31}{1179648}a^{30}+\frac{59}{1179648}a^{29}-\frac{21}{65536}a^{28}-\frac{261}{262144}a^{27}-\frac{133}{196608}a^{26}-\frac{725}{196608}a^{25}+\frac{57}{32768}a^{24}+\frac{27}{524288}a^{23}+\frac{583}{393216}a^{22}+\frac{4019}{393216}a^{21}+\frac{137}{65536}a^{20}+\frac{6219}{262144}a^{19}+\frac{593}{65536}a^{18}+\frac{2873}{65536}a^{17}-\frac{1591}{32768}a^{16}-\frac{48319}{524288}a^{15}-\frac{10569}{131072}a^{14}-\frac{59197}{131072}a^{13}+\frac{8299}{65536}a^{12}-\frac{133}{16384}a^{11}+\frac{3517}{49152}a^{10}+\frac{13069}{24576}a^{9}+\frac{179}{1024}a^{8}+\frac{1045}{1024}a^{7}+\frac{1615}{3072}a^{6}+\frac{4639}{1536}a^{5}-\frac{111}{64}a^{4}-\frac{871}{384}a^{3}-\frac{1427}{576}a^{2}-\frac{2017}{144}a+\frac{51}{16}$, $\frac{277}{262144}a^{31}-\frac{143}{262144}a^{30}+\frac{979}{1179648}a^{29}+\frac{17}{32768}a^{28}-\frac{965}{131072}a^{27}+\frac{915}{131072}a^{26}-\frac{901}{196608}a^{25}-\frac{57}{16384}a^{24}+\frac{487}{262144}a^{23}-\frac{2741}{262144}a^{22}-\frac{1109}{393216}a^{21}+\frac{27}{32768}a^{20}+\frac{23027}{131072}a^{19}-\frac{16589}{131072}a^{18}+\frac{8505}{65536}a^{17}+\frac{1391}{16384}a^{16}-\frac{180731}{262144}a^{15}+\frac{199953}{262144}a^{14}-\frac{52357}{131072}a^{13}-\frac{10847}{32768}a^{12}-\frac{5275}{32768}a^{11}-\frac{6641}{16384}a^{10}-\frac{2137}{6144}a^{9}-\frac{237}{4096}a^{8}+\frac{15231}{2048}a^{7}-\frac{6715}{1024}a^{6}+\frac{1979}{384}a^{5}+\frac{897}{256}a^{4}-\frac{2081}{128}a^{3}+\frac{347}{16}a^{2}-\frac{2413}{288}a-\frac{123}{16}$, $\frac{143}{786432}a^{31}-\frac{43}{147456}a^{30}-\frac{593}{131072}a^{27}+\frac{91}{24576}a^{26}+\frac{2919}{262144}a^{23}-\frac{283}{49152}a^{22}+\frac{8335}{131072}a^{19}-\frac{547}{8192}a^{18}-\frac{139739}{262144}a^{15}+\frac{6693}{16384}a^{14}+\frac{9047}{16384}a^{11}-\frac{2969}{12288}a^{10}+\frac{3949}{1024}a^{7}-\frac{2693}{768}a^{6}-\frac{385}{24}a^{3}+\frac{1669}{144}a^{2}+\frac{1}{2}$, $\frac{889}{1179648}a^{31}+\frac{43}{73728}a^{30}-\frac{559}{294912}a^{29}-\frac{799}{196608}a^{27}-\frac{91}{12288}a^{26}+\frac{577}{49152}a^{25}-\frac{959}{393216}a^{23}+\frac{283}{24576}a^{22}+\frac{185}{98304}a^{21}+\frac{7323}{65536}a^{19}+\frac{547}{4096}a^{18}-\frac{4949}{16384}a^{17}-\frac{45007}{131072}a^{15}-\frac{6693}{8192}a^{14}+\frac{34025}{32768}a^{13}-\frac{3647}{12288}a^{11}+\frac{2969}{6144}a^{10}+\frac{3451}{6144}a^{9}+\frac{3343}{768}a^{7}+\frac{2693}{384}a^{6}-\frac{4691}{384}a^{5}-\frac{2017}{288}a^{3}-\frac{1669}{72}a^{2}+\frac{821}{36}a$, $\frac{773}{2359296}a^{31}-\frac{17}{2359296}a^{30}+\frac{139}{1179648}a^{29}-\frac{25}{36864}a^{28}-\frac{1631}{393216}a^{27}+\frac{47}{393216}a^{26}-\frac{253}{196608}a^{25}+\frac{47}{12288}a^{24}+\frac{4637}{786432}a^{23}-\frac{185}{786432}a^{22}+\frac{739}{393216}a^{21}+\frac{5}{12288}a^{20}+\frac{10227}{131072}a^{19}-\frac{315}{131072}a^{18}+\frac{1393}{65536}a^{17}-\frac{413}{4096}a^{16}-\frac{120147}{262144}a^{15}+\frac{4503}{262144}a^{14}-\frac{15693}{131072}a^{13}+\frac{1297}{4096}a^{12}+\frac{21161}{98304}a^{11}-\frac{229}{49152}a^{10}+\frac{823}{12288}a^{9}+\frac{2353}{12288}a^{8}+\frac{24971}{6144}a^{7}-\frac{751}{3072}a^{6}+\frac{733}{768}a^{5}-\frac{2909}{768}a^{4}-\frac{14851}{1152}a^{3}+\frac{107}{144}a^{2}-\frac{871}{288}a+\frac{763}{144}$, $\frac{211}{262144}a^{31}-\frac{295}{262144}a^{30}+\frac{1409}{1179648}a^{29}+\frac{17}{32768}a^{28}-\frac{611}{131072}a^{27}+\frac{347}{131072}a^{26}-\frac{599}{196608}a^{25}-\frac{57}{16384}a^{24}-\frac{495}{262144}a^{23}+\frac{3843}{262144}a^{22}-\frac{6007}{393216}a^{21}+\frac{27}{32768}a^{20}+\frac{16421}{131072}a^{19}-\frac{17253}{131072}a^{18}+\frac{9651}{65536}a^{17}+\frac{1391}{16384}a^{16}-\frac{107933}{262144}a^{15}+\frac{16441}{262144}a^{14}-\frac{13319}{131072}a^{13}-\frac{10847}{32768}a^{12}-\frac{9389}{32768}a^{11}+\frac{17305}{16384}a^{10}-\frac{13663}{12288}a^{9}-\frac{237}{4096}a^{8}+\frac{10273}{2048}a^{7}-\frac{3781}{1024}a^{6}+\frac{3263}{768}a^{5}+\frac{897}{256}a^{4}-\frac{1143}{128}a^{3}-\frac{129}{32}a^{2}+\frac{1123}{288}a-\frac{131}{16}$, $\frac{5657}{4718592}a^{31}+\frac{2773}{2359296}a^{30}+\frac{31}{9216}a^{29}-\frac{767}{196608}a^{28}-\frac{8039}{786432}a^{27}-\frac{3331}{393216}a^{26}-\frac{559}{24576}a^{25}+\frac{1041}{32768}a^{24}+\frac{12545}{1572864}a^{23}+\frac{1789}{786432}a^{22}+\frac{31}{12288}a^{21}-\frac{1303}{65536}a^{20}+\frac{57939}{262144}a^{19}+\frac{26447}{131072}a^{18}+\frac{4569}{8192}a^{17}-\frac{23023}{32768}a^{16}-\frac{537039}{524288}a^{15}-\frac{207827}{262144}a^{14}-\frac{8707}{4096}a^{13}+\frac{205131}{65536}a^{12}+\frac{12325}{98304}a^{11}-\frac{4637}{24576}a^{10}-\frac{16511}{24576}a^{9}-\frac{705}{4096}a^{8}+\frac{61615}{6144}a^{7}+\frac{13261}{1536}a^{6}+\frac{36043}{1536}a^{5}-\frac{8059}{256}a^{4}-\frac{7547}{288}a^{3}-\frac{10669}{576}a^{2}-\frac{14333}{288}a+\frac{1897}{24}$, $\frac{23}{131072}a^{31}-\frac{149}{393216}a^{30}+\frac{79}{36864}a^{29}+\frac{1109}{294912}a^{28}-\frac{147}{65536}a^{27}-\frac{85}{65536}a^{26}-\frac{65}{3072}a^{25}-\frac{1523}{49152}a^{24}+\frac{493}{131072}a^{23}+\frac{1571}{131072}a^{22}+\frac{283}{12288}a^{21}+\frac{1853}{98304}a^{20}+\frac{2653}{65536}a^{19}-\frac{1461}{65536}a^{18}+\frac{219}{512}a^{17}+\frac{11231}{16384}a^{16}-\frac{33705}{131072}a^{15}-\frac{34535}{131072}a^{14}-\frac{9101}{4096}a^{13}-\frac{100371}{32768}a^{12}+\frac{683}{4096}a^{11}+\frac{6183}{8192}a^{10}+\frac{4327}{6144}a^{9}+\frac{71}{768}a^{8}+\frac{537}{256}a^{7}+\frac{213}{512}a^{6}+\frac{7915}{384}a^{5}+\frac{92}{3}a^{4}-\frac{237}{32}a^{3}-\frac{259}{24}a^{2}-\frac{4265}{72}a-\frac{5615}{72}$, $\frac{103}{49152}a^{31}-\frac{349}{196608}a^{30}+\frac{1135}{589824}a^{29}-\frac{403}{98304}a^{28}-\frac{243}{16384}a^{27}+\frac{359}{32768}a^{26}-\frac{1201}{98304}a^{25}+\frac{549}{16384}a^{24}+\frac{57}{16384}a^{23}+\frac{75}{65536}a^{22}-\frac{185}{196608}a^{21}-\frac{747}{32768}a^{20}+\frac{5843}{16384}a^{19}-\frac{9233}{32768}a^{18}+\frac{10213}{32768}a^{17}-\frac{12107}{16384}a^{16}-\frac{23065}{16384}a^{15}+\frac{65713}{65536}a^{14}-\frac{72553}{65536}a^{13}+\frac{109263}{32768}a^{12}-\frac{5169}{16384}a^{11}+\frac{3903}{8192}a^{10}-\frac{6673}{12288}a^{9}-\frac{37}{128}a^{8}+\frac{15661}{1024}a^{7}-\frac{5959}{512}a^{6}+\frac{9749}{768}a^{5}-\frac{133}{4}a^{4}-\frac{6499}{192}a^{3}+\frac{2171}{96}a^{2}-\frac{455}{18}a+\frac{2029}{24}$, $\frac{9295}{4718592}a^{31}-\frac{151}{294912}a^{30}+\frac{549}{131072}a^{29}-\frac{877}{589824}a^{28}-\frac{11689}{786432}a^{27}+\frac{19}{3072}a^{26}-\frac{2097}{65536}a^{25}+\frac{1627}{98304}a^{24}+\frac{9223}{1572864}a^{23}-\frac{859}{98304}a^{22}+\frac{1863}{131072}a^{21}-\frac{4117}{196608}a^{20}+\frac{90093}{262144}a^{19}-\frac{943}{8192}a^{18}+\frac{48431}{65536}a^{17}-\frac{10343}{32768}a^{16}-\frac{748009}{524288}a^{15}+\frac{21789}{32768}a^{14}-\frac{407099}{131072}a^{13}+\frac{115387}{65536}a^{12}-\frac{3595}{24576}a^{11}-\frac{16177}{49152}a^{10}-\frac{1985}{8192}a^{9}-\frac{19}{24}a^{8}+\frac{22967}{1536}a^{7}-\frac{17779}{3072}a^{6}+\frac{16669}{512}a^{5}-\frac{1513}{96}a^{4}-\frac{39961}{1152}a^{3}+\frac{10349}{576}a^{2}-\frac{305}{4}a+\frac{6991}{144}$, $\frac{215}{524288}a^{31}-\frac{191}{98304}a^{30}-\frac{3017}{1179648}a^{29}+\frac{49}{196608}a^{28}-\frac{371}{262144}a^{27}+\frac{99}{8192}a^{26}+\frac{4583}{196608}a^{25}+\frac{169}{32768}a^{24}-\frac{1875}{524288}a^{23}+\frac{37}{32768}a^{22}-\frac{8177}{393216}a^{21}-\frac{1415}{65536}a^{20}+\frac{13373}{262144}a^{19}-\frac{639}{2048}a^{18}-\frac{32051}{65536}a^{17}-\frac{967}{32768}a^{16}-\frac{49257}{524288}a^{15}+\frac{35975}{32768}a^{14}+\frac{311391}{131072}a^{13}+\frac{46395}{65536}a^{12}-\frac{4391}{16384}a^{11}+\frac{8717}{16384}a^{10}-\frac{11599}{24576}a^{9}-\frac{1249}{1024}a^{8}+\frac{1723}{1024}a^{7}-\frac{13177}{1024}a^{6}-\frac{34909}{1536}a^{5}-\frac{245}{64}a^{4}-\frac{123}{128}a^{3}+\frac{4711}{192}a^{2}+\frac{8923}{144}a+\frac{1097}{48}$, $\frac{65}{589824}a^{31}+\frac{29}{262144}a^{30}-\frac{479}{1179648}a^{29}+\frac{691}{294912}a^{28}-\frac{13}{3072}a^{27}+\frac{631}{131072}a^{26}-\frac{487}{196608}a^{25}-\frac{409}{49152}a^{24}+\frac{2141}{196608}a^{23}-\frac{4497}{262144}a^{22}+\frac{6121}{393216}a^{21}-\frac{2021}{98304}a^{20}+\frac{897}{16384}a^{19}-\frac{5545}{131072}a^{18}-\frac{797}{65536}a^{17}+\frac{5061}{16384}a^{16}-\frac{33131}{65536}a^{15}+\frac{170941}{262144}a^{14}-\frac{56615}{131072}a^{13}-\frac{17525}{32768}a^{12}+\frac{52967}{98304}a^{11}-\frac{15201}{16384}a^{10}+\frac{5927}{6144}a^{9}-\frac{20453}{12288}a^{8}+\frac{21341}{6144}a^{7}-\frac{4051}{1024}a^{6}+\frac{605}{384}a^{5}+\frac{8185}{768}a^{4}-\frac{17875}{1152}a^{3}+\frac{171}{8}a^{2}-\frac{4807}{288}a-\frac{641}{144}$ 2779/4718592a^31 - 31/1179648a^30 - 59/1179648a^29 + 21/65536a^28 - 2413/786432a^27 + 133/196608a^26 + 725/196608a^25 - 57/32768a^24 - 3917/1572864a^23 - 583/393216a^22 - 4019/393216a^21 - 137/65536a^20 + 23073/262144a^19 - 593/65536a^18 - 2873/65536a^17 + 1591/32768a^16 - 131709/524288a^15 + 10569/131072a^14 + 59197/131072a^13 - 8299/65536a^12 - 14189/49152a^11 - 3517/49152a^10 - 13069/24576a^9 - 179/1024a^8 + 10237/3072a^7 - 1615/3072a^6 - 4639/1536a^5 + 111/64a^4 - 5455/1152a^3 + 1427/576a^2 + 2017/144a - 35/16, 2779/4718592a^31 + 31/1179648a^30 - 59/1179648a^29 - 21/65536a^28 - 2413/786432a^27 - 133/196608a^26 + 725/196608a^25 + 57/32768a^24 - 3917/1572864a^23 + 583/393216a^22 - 4019/393216a^21 + 137/65536a^20 + 23073/262144a^19 + 593/65536a^18 - 2873/65536a^17 - 1591/32768a^16 - 131709/524288a^15 - 10569/131072a^14 + 59197/131072a^13 + 8299/65536a^12 - 14189/49152a^11 + 3517/49152a^10 - 13069/24576a^9 + 179/1024a^8 + 10237/3072a^7 + 1615/3072a^6 - 4639/1536a^5 - 111/64a^4 - 5455/1152a^3 - 1427/576a^2 + 2017/144a + 35/16, 2449/4718592a^31 - 31/1179648a^30 - 163/393216a^29 - 21/65536a^28 - 3847/786432a^27 + 133/196608a^26 + 141/65536a^25 + 57/32768a^24 + 7705/1572864a^23 - 583/393216a^22 + 293/131072a^21 + 137/65536a^20 + 26531/262144a^19 - 593/65536a^18 - 4019/65536a^17 - 1591/32768a^16 - 265591/524288a^15 + 10569/131072a^14 + 20159/131072a^13 + 8299/65536a^12 + 6781/49152a^11 - 3517/49152a^10 + 1903/8192a^9 + 179/1024a^8 + 14647/3072a^7 - 1615/3072a^6 - 1083/512a^5 - 111/64a^4 - 15493/1152a^3 + 1427/576a^2 + 131/48a + 51/16, 259/1572864a^31 + 31/1179648a^30 + 59/1179648a^29 - 21/65536a^28 - 261/262144a^27 - 133/196608a^26 - 725/196608a^25 + 57/32768a^24 + 27/524288a^23 + 583/393216a^22 + 4019/393216a^21 + 137/65536a^20 + 6219/262144a^19 + 593/65536a^18 + 2873/65536a^17 - 1591/32768a^16 - 48319/524288a^15 - 10569/131072a^14 - 59197/131072a^13 + 8299/65536a^12 - 133/16384a^11 + 3517/49152a^10 + 13069/24576a^9 + 179/1024a^8 + 1045/1024a^7 + 1615/3072a^6 + 4639/1536a^5 - 111/64a^4 - 871/384a^3 - 1427/576a^2 - 2017/144a + 51/16, 277/262144a^31 - 143/262144a^30 + 979/1179648a^29 + 17/32768a^28 - 965/131072a^27 + 915/131072a^26 - 901/196608a^25 - 57/16384a^24 + 487/262144a^23 - 2741/262144a^22 - 1109/393216a^21 + 27/32768a^20 + 23027/131072a^19 - 16589/131072a^18 + 8505/65536a^17 + 1391/16384a^16 - 180731/262144a^15 + 199953/262144a^14 - 52357/131072a^13 - 10847/32768a^12 - 5275/32768a^11 - 6641/16384a^10 - 2137/6144a^9 - 237/4096a^8 + 15231/2048a^7 - 6715/1024a^6 + 1979/384a^5 + 897/256a^4 - 2081/128a^3 + 347/16a^2 - 2413/288a - 123/16, 143/786432a^31 - 43/147456a^30 - 593/131072a^27 + 91/24576a^26 + 2919/262144a^23 - 283/49152a^22 + 8335/131072a^19 - 547/8192a^18 - 139739/262144a^15 + 6693/16384a^14 + 9047/16384a^11 - 2969/12288a^10 + 3949/1024a^7 - 2693/768a^6 - 385/24a^3 + 1669/144a^2 + 1/2, 889/1179648a^31 + 43/73728a^30 - 559/294912a^29 - 799/196608a^27 - 91/12288a^26 + 577/49152a^25 - 959/393216a^23 + 283/24576a^22 + 185/98304a^21 + 7323/65536a^19 + 547/4096a^18 - 4949/16384a^17 - 45007/131072a^15 - 6693/8192a^14 + 34025/32768a^13 - 3647/12288a^11 + 2969/6144a^10 + 3451/6144a^9 + 3343/768a^7 + 2693/384a^6 - 4691/384a^5 - 2017/288a^3 - 1669/72a^2 + 821/36a, 773/2359296a^31 - 17/2359296a^30 + 139/1179648a^29 - 25/36864a^28 - 1631/393216a^27 + 47/393216a^26 - 253/196608a^25 + 47/12288a^24 + 4637/786432a^23 - 185/786432a^22 + 739/393216a^21 + 5/12288a^20 + 10227/131072a^19 - 315/131072a^18 + 1393/65536a^17 - 413/4096a^16 - 120147/262144a^15 + 4503/262144a^14 - 15693/131072a^13 + 1297/4096a^12 + 21161/98304a^11 - 229/49152a^10 + 823/12288a^9 + 2353/12288a^8 + 24971/6144a^7 - 751/3072a^6 + 733/768a^5 - 2909/768a^4 - 14851/1152a^3 + 107/144a^2 - 871/288a + 763/144, 211/262144a^31 - 295/262144a^30 + 1409/1179648a^29 + 17/32768a^28 - 611/131072a^27 + 347/131072a^26 - 599/196608a^25 - 57/16384a^24 - 495/262144a^23 + 3843/262144a^22 - 6007/393216a^21 + 27/32768a^20 + 16421/131072a^19 - 17253/131072a^18 + 9651/65536a^17 + 1391/16384a^16 - 107933/262144a^15 + 16441/262144a^14 - 13319/131072a^13 - 10847/32768a^12 - 9389/32768a^11 + 17305/16384a^10 - 13663/12288a^9 - 237/4096a^8 + 10273/2048a^7 - 3781/1024a^6 + 3263/768a^5 + 897/256a^4 - 1143/128a^3 - 129/32a^2 + 1123/288a - 131/16, 5657/4718592a^31 + 2773/2359296a^30 + 31/9216a^29 - 767/196608a^28 - 8039/786432a^27 - 3331/393216a^26 - 559/24576a^25 + 1041/32768a^24 + 12545/1572864a^23 + 1789/786432a^22 + 31/12288a^21 - 1303/65536a^20 + 57939/262144a^19 + 26447/131072a^18 + 4569/8192a^17 - 23023/32768a^16 - 537039/524288a^15 - 207827/262144a^14 - 8707/4096a^13 + 205131/65536a^12 + 12325/98304a^11 - 4637/24576a^10 - 16511/24576a^9 - 705/4096a^8 + 61615/6144a^7 + 13261/1536a^6 + 36043/1536a^5 - 8059/256a^4 - 7547/288a^3 - 10669/576a^2 - 14333/288a + 1897/24, 23/131072a^31 - 149/393216a^30 + 79/36864a^29 + 1109/294912a^28 - 147/65536a^27 - 85/65536a^26 - 65/3072a^25 - 1523/49152a^24 + 493/131072a^23 + 1571/131072a^22 + 283/12288a^21 + 1853/98304a^20 + 2653/65536a^19 - 1461/65536a^18 + 219/512a^17 + 11231/16384a^16 - 33705/131072a^15 - 34535/131072a^14 - 9101/4096a^13 - 100371/32768a^12 + 683/4096a^11 + 6183/8192a^10 + 4327/6144a^9 + 71/768a^8 + 537/256a^7 + 213/512a^6 + 7915/384a^5 + 92/3a^4 - 237/32a^3 - 259/24a^2 - 4265/72a - 5615/72, 103/49152a^31 - 349/196608a^30 + 1135/589824a^29 - 403/98304a^28 - 243/16384a^27 + 359/32768a^26 - 1201/98304a^25 + 549/16384a^24 + 57/16384a^23 + 75/65536a^22 - 185/196608a^21 - 747/32768a^20 + 5843/16384a^19 - 9233/32768a^18 + 10213/32768a^17 - 12107/16384a^16 - 23065/16384a^15 + 65713/65536a^14 - 72553/65536a^13 + 109263/32768a^12 - 5169/16384a^11 + 3903/8192a^10 - 6673/12288a^9 - 37/128a^8 + 15661/1024a^7 - 5959/512a^6 + 9749/768a^5 - 133/4a^4 - 6499/192a^3 + 2171/96a^2 - 455/18a + 2029/24, 9295/4718592a^31 - 151/294912a^30 + 549/131072a^29 - 877/589824a^28 - 11689/786432a^27 + 19/3072a^26 - 2097/65536a^25 + 1627/98304a^24 + 9223/1572864a^23 - 859/98304a^22 + 1863/131072a^21 - 4117/196608a^20 + 90093/262144a^19 - 943/8192a^18 + 48431/65536a^17 - 10343/32768a^16 - 748009/524288a^15 + 21789/32768a^14 - 407099/131072a^13 + 115387/65536a^12 - 3595/24576a^11 - 16177/49152a^10 - 1985/8192a^9 - 19/24a^8 + 22967/1536a^7 - 17779/3072a^6 + 16669/512a^5 - 1513/96a^4 - 39961/1152a^3 + 10349/576a^2 - 305/4a + 6991/144, 215/524288a^31 - 191/98304a^30 - 3017/1179648a^29 + 49/196608a^28 - 371/262144a^27 + 99/8192a^26 + 4583/196608a^25 + 169/32768a^24 - 1875/524288a^23 + 37/32768a^22 - 8177/393216a^21 - 1415/65536a^20 + 13373/262144a^19 - 639/2048a^18 - 32051/65536a^17 - 967/32768a^16 - 49257/524288a^15 + 35975/32768a^14 + 311391/131072a^13 + 46395/65536a^12 - 4391/16384a^11 + 8717/16384a^10 - 11599/24576a^9 - 1249/1024a^8 + 1723/1024a^7 - 13177/1024a^6 - 34909/1536a^5 - 245/64a^4 - 123/128a^3 + 4711/192a^2 + 8923/144a + 1097/48, 65/589824a^31 + 29/262144a^30 - 479/1179648a^29 + 691/294912a^28 - 13/3072a^27 + 631/131072a^26 - 487/196608a^25 - 409/49152a^24 + 2141/196608a^23 - 4497/262144a^22 + 6121/393216a^21 - 2021/98304a^20 + 897/16384a^19 - 5545/131072a^18 - 797/65536a^17 + 5061/16384a^16 - 33131/65536a^15 + 170941/262144a^14 - 56615/131072a^13 - 17525/32768a^12 + 52967/98304a^11 - 15201/16384a^10 + 5927/6144a^9 - 20453/12288a^8 + 21341/6144a^7 - 4051/1024a^6 + 605/384a^5 + 8185/768a^4 - 17875/1152a^3 + 171/8a^2 - 4807/288*a - 641/144 (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:	$ 423122700646.94574 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 423122700646.94574 \cdot 16}{8\cdot\sqrt{97424180758632937220739951448553101324025212174336}}\cr\approx \mathstrut & 0.505872223915853 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^32 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 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 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 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 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 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 - 10*x^28 + 27*x^24 + 150*x^20 - 1215*x^16 + 2400*x^12 + 6912*x^8 - 40960*x^4 + 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

$C_2^6:S_4$ (as 32T96908):

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 1536

The 80 conjugacy class representatives for $C_2^6:S_4$

Character table for $C_2^6:S_4$

Intermediate fields

$\Q(\sqrt{2}) $, $\Q(\sqrt{-2}) $, $\Q(\sqrt{-1}) $, 4.4.14272.1, $\Q(\zeta_{8})$, 8.0.49089286144.1, 8.0.785428578304.1, 8.8.785428578304.1, 8.8.785428578304.2, 8.8.3259039744.1, 8.0.3259039744.4, 8.0.3259039744.9, 16.0.169941440847545368576.2, 16.0.616898051616642659516416.2, 16.16.9870368825866282552262656.1, 16.0.616898051616642659516416.3, 16.0.9870368825866282552262656.2, 16.0.616898051616642659516416.1, 16.0.9870368825866282552262656.1

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:	not computed

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$	${\href{/padicField/5.4.0.1}{4} }^{8}$	${\href{/padicField/7.8.0.1}{8} }^{4}$	${\href{/padicField/11.6.0.1}{6} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	${\href{/padicField/13.2.0.1}{2} }^{16}$	${\href{/padicField/17.3.0.1}{3} }^{8}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$	${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$	${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$	${\href{/padicField/29.6.0.1}{6} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.8.0.1}{8} }^{4}$	${\href{/padicField/37.6.0.1}{6} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$	${\href{/padicField/41.3.0.1}{3} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$	${\href{/padicField/43.8.0.1}{8} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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

$p$	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	Deg $16$	$8$	$2$	$36$
$2$ 2	Deg $16$	$8$	$2$	$36$
$223$ 223	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	Deg $4$	$2$	$2$	$2$
	Deg $4$	$2$	$2$	$2$
	Deg $4$	$2$	$2$	$2$
	Deg $4$	$2$	$2$	$2$
$241$ 241	Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
	Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$

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