LMFDB - Number field 32.0.1039812137443144915085734490774568960000000000000000.1

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 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*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 - 29*y^24 + 15*y^22 + 88*y^20 - 255*y^18 - 1131*y^16 - 1020*y^14 + 1408*y^12 + 960*y^10 - 7424*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 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*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 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*x^8 - 5120*x^6 + 32768*x^4 + 81920*x^2 + 65536)

$ x^{32} + 5 x^{30} + 8 x^{28} - 5 x^{26} - 29 x^{24} + 15 x^{22} + 88 x^{20} - 255 x^{18} - 1131 x^{16} + \cdots + 65536 $ x^32 + 5*x^30 + 8*x^28 - 5*x^26 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*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:	$1039812137443144915085734490774568960000000000000000$ 1039812137443144915085734490774568960000000000000000 $\medspace = 2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 29^{8}\cdot 521^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$39.29$	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:	$2\cdot 3^{1/2}5^{1/2}29^{1/2}521^{1/2}\approx 952.1239415118181$
Ramified primes:	$2$, $3$, $5$, $29$, $521$ 2, 3, 5, 29, 521	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}+\frac{1}{8}a^{17}-\frac{1}{2}a^{15}+\frac{3}{8}a^{13}-\frac{1}{8}a^{11}+\frac{3}{8}a^{9}-\frac{1}{2}a^{7}+\frac{1}{8}a^{5}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{20}+\frac{1}{16}a^{18}+\frac{1}{4}a^{16}-\frac{5}{16}a^{14}+\frac{7}{16}a^{12}+\frac{3}{16}a^{10}-\frac{1}{4}a^{8}+\frac{1}{16}a^{6}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{21}+\frac{1}{32}a^{19}+\frac{1}{8}a^{17}+\frac{11}{32}a^{15}-\frac{9}{32}a^{13}-\frac{13}{32}a^{11}+\frac{3}{8}a^{9}-\frac{15}{32}a^{7}-\frac{15}{32}a^{5}$, $\frac{1}{64}a^{22}+\frac{1}{64}a^{20}+\frac{1}{16}a^{18}-\frac{21}{64}a^{16}-\frac{9}{64}a^{14}-\frac{13}{64}a^{12}+\frac{3}{16}a^{10}+\frac{17}{64}a^{8}+\frac{17}{64}a^{6}$, $\frac{1}{128}a^{23}+\frac{1}{128}a^{21}+\frac{1}{32}a^{19}-\frac{21}{128}a^{17}+\frac{55}{128}a^{15}+\frac{51}{128}a^{13}+\frac{3}{32}a^{11}-\frac{47}{128}a^{9}-\frac{47}{128}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{16640}a^{24}+\frac{25}{3328}a^{22}-\frac{3}{104}a^{20}+\frac{319}{3328}a^{18}+\frac{1039}{3328}a^{16}-\frac{885}{3328}a^{14}-\frac{81}{520}a^{12}-\frac{1395}{3328}a^{10}+\frac{1193}{3328}a^{8}-\frac{5}{832}a^{6}+\frac{51}{104}a^{4}+\frac{5}{26}a^{2}+\frac{16}{65}$, $\frac{1}{33280}a^{25}+\frac{25}{6656}a^{23}-\frac{3}{208}a^{21}+\frac{319}{6656}a^{19}+\frac{1039}{6656}a^{17}+\frac{2443}{6656}a^{15}-\frac{81}{1040}a^{13}+\frac{1933}{6656}a^{11}-\frac{2135}{6656}a^{9}-\frac{5}{1664}a^{7}+\frac{51}{208}a^{5}-\frac{21}{52}a^{3}+\frac{8}{65}a$, $\frac{1}{66560}a^{26}+\frac{1}{66560}a^{24}-\frac{19}{3328}a^{22}+\frac{367}{13312}a^{20}-\frac{1077}{13312}a^{18}+\frac{1111}{13312}a^{16}+\frac{3667}{16640}a^{14}+\frac{3473}{66560}a^{12}-\frac{3043}{13312}a^{10}-\frac{27}{208}a^{8}-\frac{69}{208}a^{6}+\frac{49}{104}a^{4}-\frac{2}{5}a^{2}+\frac{24}{65}$, $\frac{1}{133120}a^{27}+\frac{1}{133120}a^{25}-\frac{19}{6656}a^{23}+\frac{367}{26624}a^{21}-\frac{1077}{26624}a^{19}+\frac{1111}{26624}a^{17}-\frac{12973}{33280}a^{15}+\frac{3473}{133120}a^{13}+\frac{10269}{26624}a^{11}-\frac{27}{416}a^{9}-\frac{69}{416}a^{7}-\frac{55}{208}a^{5}-\frac{1}{5}a^{3}+\frac{12}{65}a$, $\frac{1}{54046720}a^{28}+\frac{193}{54046720}a^{26}-\frac{9}{1930240}a^{24}+\frac{4633}{1544192}a^{22}-\frac{212341}{10809344}a^{20}+\frac{482775}{10809344}a^{18}-\frac{193121}{1039360}a^{16}+\frac{11198289}{54046720}a^{14}-\frac{26929711}{54046720}a^{12}+\frac{6407}{12992}a^{10}+\frac{19741}{42224}a^{8}-\frac{121}{464}a^{6}+\frac{11253}{30160}a^{4}+\frac{7491}{26390}a^{2}-\frac{5477}{13195}$, $\frac{1}{108093440}a^{29}+\frac{193}{108093440}a^{27}-\frac{9}{3860480}a^{25}+\frac{4633}{3088384}a^{23}-\frac{212341}{21618688}a^{21}+\frac{482775}{21618688}a^{19}-\frac{193121}{2078720}a^{17}-\frac{42848431}{108093440}a^{15}+\frac{27117009}{108093440}a^{13}+\frac{6407}{25984}a^{11}+\frac{19741}{84448}a^{9}-\frac{121}{928}a^{7}+\frac{11253}{60320}a^{5}+\frac{7491}{52780}a^{3}-\frac{5477}{26390}a$, $\frac{1}{4107550720}a^{30}-\frac{31}{4107550720}a^{28}-\frac{1089}{146698240}a^{26}-\frac{9603}{586792960}a^{24}+\frac{3657259}{821510144}a^{22}-\frac{20511625}{821510144}a^{20}-\frac{71655733}{1026887680}a^{18}-\frac{574052783}{4107550720}a^{16}+\frac{76213}{315965440}a^{14}+\frac{30330581}{128360960}a^{12}+\frac{498941}{6418048}a^{10}+\frac{887149}{1833728}a^{8}-\frac{555107}{2292160}a^{6}-\frac{1642499}{4011280}a^{4}+\frac{177529}{501410}a^{2}+\frac{17078}{35815}$, $\frac{1}{8215101440}a^{31}-\frac{31}{8215101440}a^{29}-\frac{1089}{293396480}a^{27}-\frac{9603}{1173585920}a^{25}+\frac{3657259}{1643020288}a^{23}-\frac{20511625}{1643020288}a^{21}-\frac{71655733}{2053775360}a^{19}-\frac{574052783}{8215101440}a^{17}+\frac{76213}{631930880}a^{15}+\frac{30330581}{256721920}a^{13}+\frac{498941}{12836096}a^{11}+\frac{887149}{3667456}a^{9}-\frac{555107}{4584320}a^{7}-\frac{1642499}{8022560}a^{5}-\frac{323881}{1002820}a^{3}+\frac{8539}{35815}a$ 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, 1/2*a^17 - 1/2*a^15 - 1/2*a^11 - 1/2*a^9 - 1/2*a^7 - 1/2*a^3 - 1/2*a, 1/4*a^18 + 1/4*a^16 - 1/4*a^12 - 1/4*a^10 - 1/4*a^8 + 1/4*a^4 + 1/4*a^2, 1/8*a^19 + 1/8*a^17 - 1/2*a^15 + 3/8*a^13 - 1/8*a^11 + 3/8*a^9 - 1/2*a^7 + 1/8*a^5 + 1/8*a^3, 1/16*a^20 + 1/16*a^18 + 1/4*a^16 - 5/16*a^14 + 7/16*a^12 + 3/16*a^10 - 1/4*a^8 + 1/16*a^6 + 1/16*a^4, 1/32*a^21 + 1/32*a^19 + 1/8*a^17 + 11/32*a^15 - 9/32*a^13 - 13/32*a^11 + 3/8*a^9 - 15/32*a^7 - 15/32*a^5, 1/64*a^22 + 1/64*a^20 + 1/16*a^18 - 21/64*a^16 - 9/64*a^14 - 13/64*a^12 + 3/16*a^10 + 17/64*a^8 + 17/64*a^6, 1/128*a^23 + 1/128*a^21 + 1/32*a^19 - 21/128*a^17 + 55/128*a^15 + 51/128*a^13 + 3/32*a^11 - 47/128*a^9 - 47/128*a^7 - 1/2*a^5 - 1/2*a, 1/16640*a^24 + 25/3328*a^22 - 3/104*a^20 + 319/3328*a^18 + 1039/3328*a^16 - 885/3328*a^14 - 81/520*a^12 - 1395/3328*a^10 + 1193/3328*a^8 - 5/832*a^6 + 51/104*a^4 + 5/26*a^2 + 16/65, 1/33280*a^25 + 25/6656*a^23 - 3/208*a^21 + 319/6656*a^19 + 1039/6656*a^17 + 2443/6656*a^15 - 81/1040*a^13 + 1933/6656*a^11 - 2135/6656*a^9 - 5/1664*a^7 + 51/208*a^5 - 21/52*a^3 + 8/65*a, 1/66560*a^26 + 1/66560*a^24 - 19/3328*a^22 + 367/13312*a^20 - 1077/13312*a^18 + 1111/13312*a^16 + 3667/16640*a^14 + 3473/66560*a^12 - 3043/13312*a^10 - 27/208*a^8 - 69/208*a^6 + 49/104*a^4 - 2/5*a^2 + 24/65, 1/133120*a^27 + 1/133120*a^25 - 19/6656*a^23 + 367/26624*a^21 - 1077/26624*a^19 + 1111/26624*a^17 - 12973/33280*a^15 + 3473/133120*a^13 + 10269/26624*a^11 - 27/416*a^9 - 69/416*a^7 - 55/208*a^5 - 1/5*a^3 + 12/65*a, 1/54046720*a^28 + 193/54046720*a^26 - 9/1930240*a^24 + 4633/1544192*a^22 - 212341/10809344*a^20 + 482775/10809344*a^18 - 193121/1039360*a^16 + 11198289/54046720*a^14 - 26929711/54046720*a^12 + 6407/12992*a^10 + 19741/42224*a^8 - 121/464*a^6 + 11253/30160*a^4 + 7491/26390*a^2 - 5477/13195, 1/108093440*a^29 + 193/108093440*a^27 - 9/3860480*a^25 + 4633/3088384*a^23 - 212341/21618688*a^21 + 482775/21618688*a^19 - 193121/2078720*a^17 - 42848431/108093440*a^15 + 27117009/108093440*a^13 + 6407/25984*a^11 + 19741/84448*a^9 - 121/928*a^7 + 11253/60320*a^5 + 7491/52780*a^3 - 5477/26390*a, 1/4107550720*a^30 - 31/4107550720*a^28 - 1089/146698240*a^26 - 9603/586792960*a^24 + 3657259/821510144*a^22 - 20511625/821510144*a^20 - 71655733/1026887680*a^18 - 574052783/4107550720*a^16 + 76213/315965440*a^14 + 30330581/128360960*a^12 + 498941/6418048*a^10 + 887149/1833728*a^8 - 555107/2292160*a^6 - 1642499/4011280*a^4 + 177529/501410*a^2 + 17078/35815, 1/8215101440*a^31 - 31/8215101440*a^29 - 1089/293396480*a^27 - 9603/1173585920*a^25 + 3657259/1643020288*a^23 - 20511625/1643020288*a^21 - 71655733/2053775360*a^19 - 574052783/8215101440*a^17 + 76213/631930880*a^15 + 30330581/256721920*a^13 + 498941/12836096*a^11 + 887149/3667456*a^9 - 555107/4584320*a^7 - 1642499/8022560*a^5 - 323881/1002820*a^3 + 8539/35815*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

$C_{24}$, which has order $24$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Relative class number: $24$ (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{520561}{8215101440} a^{31} - \frac{602079}{8215101440} a^{29} - \frac{112407}{410755072} a^{27} + \frac{18017}{90275840} a^{25} + \frac{1624171}{1643020288} a^{23} - \frac{165961}{1643020288} a^{21} - \frac{18913353}{2053775360} a^{19} + \frac{6385793}{8215101440} a^{17} + \frac{51380029}{1643020288} a^{15} - \frac{6387149}{513443840} a^{13} - \frac{827831}{12836096} a^{11} - \frac{138347}{3209024} a^{9} + \frac{887729}{2292160} a^{7} + \frac{2520299}{8022560} a^{5} - \frac{84403}{100282} a^{3} + \frac{124489}{250705} a $ (520561)/(8215101440)a^(31) - (602079)/(8215101440)a^(29) - (112407)/(410755072)a^(27) + (18017)/(90275840)a^(25) + (1624171)/(1643020288)a^(23) - (165961)/(1643020288)a^(21) - (18913353)/(2053775360)a^(19) + (6385793)/(8215101440)a^(17) + (51380029)/(1643020288)a^(15) - (6387149)/(513443840)a^(13) - (827831)/(12836096)a^(11) - (138347)/(3209024)a^(9) + (887729)/(2292160)a^(7) + (2520299)/(8022560)a^(5) - (84403)/(100282)a^(3) + (124489)/(250705)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:	$\frac{1949}{7454720}a^{30}+\frac{2185}{1490944}a^{28}+\frac{487}{266240}a^{26}-\frac{875}{212992}a^{24}-\frac{12049}{1490944}a^{22}+\frac{21963}{1490944}a^{20}+\frac{51843}{1863680}a^{18}-\frac{170295}{1490944}a^{16}-\frac{2523267}{7454720}a^{14}+\frac{1957}{93184}a^{12}+\frac{8133}{11648}a^{10}-\frac{13}{32}a^{8}-\frac{11483}{4160}a^{6}+\frac{739}{1456}a^{4}+\frac{5869}{455}a^{2}+\frac{191}{13}$, $\frac{687}{7454720}a^{30}-\frac{16649}{216186880}a^{28}-\frac{979}{1930240}a^{26}+\frac{9591}{30883840}a^{24}+\frac{75373}{43237376}a^{22}+\frac{28545}{43237376}a^{20}-\frac{196711}{13511680}a^{18}-\frac{1109517}{216186880}a^{16}+\frac{10842471}{216186880}a^{14}+\frac{1162169}{54046720}a^{12}-\frac{34935}{337792}a^{10}-\frac{113}{928}a^{8}+\frac{56539}{120640}a^{6}+\frac{156109}{211120}a^{4}-\frac{14797}{13195}a^{2}-\frac{1641}{1885}$, $\frac{520561}{8215101440}a^{31}-\frac{602079}{8215101440}a^{29}-\frac{112407}{410755072}a^{27}+\frac{18017}{90275840}a^{25}+\frac{1624171}{1643020288}a^{23}-\frac{165961}{1643020288}a^{21}-\frac{18913353}{2053775360}a^{19}+\frac{6385793}{8215101440}a^{17}+\frac{51380029}{1643020288}a^{15}-\frac{6387149}{513443840}a^{13}-\frac{827831}{12836096}a^{11}-\frac{138347}{3209024}a^{9}+\frac{887729}{2292160}a^{7}+\frac{2520299}{8022560}a^{5}-\frac{84403}{100282}a^{3}+\frac{124489}{250705}a+1$, $\frac{142513}{586792960}a^{30}+\frac{3487707}{4107550720}a^{28}+\frac{378047}{256721920}a^{26}-\frac{1353509}{586792960}a^{24}-\frac{754241}{117358592}a^{22}+\frac{6454829}{821510144}a^{20}+\frac{4725573}{256721920}a^{18}-\frac{255141529}{4107550720}a^{16}-\frac{1033911029}{4107550720}a^{14}-\frac{101982971}{1026887680}a^{12}+\frac{27702557}{51344384}a^{10}-\frac{936443}{12836096}a^{8}-\frac{4237197}{2292160}a^{6}-\frac{277421}{573040}a^{4}+\frac{4380801}{501410}a^{2}+\frac{3813403}{250705}$, $\frac{47343}{586792960}a^{30}+\frac{97359}{586792960}a^{28}-\frac{156369}{146698240}a^{26}-\frac{1069819}{586792960}a^{24}+\frac{31545}{9027584}a^{22}+\frac{797241}{117358592}a^{20}-\frac{1975699}{146698240}a^{18}-\frac{30077953}{586792960}a^{16}+\frac{1056203}{20234240}a^{14}+\frac{2157917}{9168640}a^{12}-\frac{463183}{3667456}a^{10}-\frac{890489}{1833728}a^{8}+\frac{353733}{2292160}a^{6}+\frac{1387961}{573040}a^{4}+\frac{37617}{71630}a^{2}-\frac{247446}{35815}$, $\frac{390227}{1643020288}a^{31}+\frac{390941}{4107550720}a^{30}+\frac{1478615}{1643020288}a^{29}-\frac{484483}{4107550720}a^{28}+\frac{36873}{513443840}a^{27}-\frac{168271}{1026887680}a^{26}-\frac{3523693}{1173585920}a^{25}+\frac{218809}{586792960}a^{24}-\frac{2635967}{1643020288}a^{23}+\frac{606367}{821510144}a^{22}+\frac{20244837}{1643020288}a^{21}+\frac{84619}{821510144}a^{20}+\frac{35353}{14669824}a^{19}-\frac{8129053}{1026887680}a^{18}-\frac{22404235}{234717184}a^{17}+\frac{17099261}{4107550720}a^{16}-\frac{1025213981}{8215101440}a^{15}+\frac{2654577}{141639680}a^{14}+\frac{295035813}{2053775360}a^{13}-\frac{10560831}{256721920}a^{12}+\frac{6950193}{25672192}a^{11}-\frac{1016581}{51344384}a^{10}-\frac{7172175}{12836096}a^{9}-\frac{310785}{12836096}a^{8}-\frac{1099925}{916864}a^{7}+\frac{567273}{2292160}a^{6}+\frac{1723741}{802256}a^{5}+\frac{382203}{4011280}a^{4}+\frac{6619919}{1002820}a^{3}-\frac{134077}{143260}a^{2}+\frac{1038458}{250705}a+\frac{705742}{250705}$, $\frac{2874441}{8215101440}a^{31}+\frac{289493}{410755072}a^{30}+\frac{7879121}{8215101440}a^{29}+\frac{4682219}{2053775360}a^{28}-\frac{54059}{293396480}a^{27}+\frac{284887}{1026887680}a^{26}-\frac{4117147}{1173585920}a^{25}-\frac{2278859}{293396480}a^{24}-\frac{1318501}{1643020288}a^{23}-\frac{1436415}{410755072}a^{22}+\frac{24904583}{1643020288}a^{21}+\frac{13259901}{410755072}a^{20}-\frac{17650023}{2053775360}a^{19}-\frac{83327}{205377536}a^{18}-\frac{941042727}{8215101440}a^{17}-\frac{496810263}{2053775360}a^{16}-\frac{833472351}{8215101440}a^{15}-\frac{642814533}{2053775360}a^{14}+\frac{6465009}{35409920}a^{13}+\frac{335317173}{1026887680}a^{12}+\frac{10999565}{51344384}a^{11}+\frac{32389621}{51344384}a^{10}-\frac{2832627}{3667456}a^{9}-\frac{9499761}{6418048}a^{8}-\frac{4499107}{4584320}a^{7}-\frac{663355}{229216}a^{6}+\frac{11942087}{4011280}a^{5}+\frac{20405517}{4011280}a^{4}+\frac{12552993}{2005640}a^{3}+\frac{16722673}{1002820}a^{2}+\frac{152721}{35815}a+\frac{3307606}{250705}$, $\frac{8189}{16629760}a^{31}+\frac{359}{7454720}a^{30}+\frac{287993}{216186880}a^{29}-\frac{12977}{216186880}a^{28}-\frac{11989}{54046720}a^{27}-\frac{673}{1930240}a^{26}-\frac{28503}{6176768}a^{25}+\frac{8943}{30883840}a^{24}-\frac{36333}{43237376}a^{23}+\frac{52277}{43237376}a^{22}+\frac{127761}{6176768}a^{21}+\frac{15209}{43237376}a^{20}-\frac{543271}{54046720}a^{19}-\frac{101277}{13511680}a^{18}-\frac{34350111}{216186880}a^{17}-\frac{591541}{216186880}a^{16}-\frac{32944663}{216186880}a^{15}+\frac{7656767}{216186880}a^{14}+\frac{1293805}{5404672}a^{13}+\frac{367257}{54046720}a^{12}+\frac{231375}{772096}a^{11}-\frac{24671}{337792}a^{10}-\frac{366789}{337792}a^{9}-\frac{73}{928}a^{8}-\frac{349113}{241280}a^{7}+\frac{6509}{60320}a^{6}+\frac{873007}{211120}a^{5}+\frac{103317}{211120}a^{4}+\frac{482319}{52780}a^{3}-\frac{10869}{13195}a^{2}+\frac{15910}{2639}a-\frac{2003}{1885}$, $\frac{65879}{821510144}a^{31}-\frac{1941061}{4107550720}a^{30}+\frac{1172453}{4107550720}a^{29}-\frac{6135589}{4107550720}a^{28}+\frac{636203}{2053775360}a^{27}-\frac{56779}{146698240}a^{26}-\frac{44477}{45137920}a^{25}+\frac{564611}{117358592}a^{24}-\frac{1491681}{821510144}a^{23}+\frac{2168553}{821510144}a^{22}+\frac{2795939}{821510144}a^{21}-\frac{16093011}{821510144}a^{20}+\frac{88497}{31596544}a^{19}+\frac{3943673}{1026887680}a^{18}-\frac{102840541}{4107550720}a^{17}+\frac{630762603}{4107550720}a^{16}-\frac{238810827}{4107550720}a^{15}+\frac{840075339}{4107550720}a^{14}+\frac{16051957}{2053775360}a^{13}-\frac{4397283}{25672192}a^{12}+\frac{12981147}{102688768}a^{11}-\frac{5132531}{12836096}a^{10}-\frac{3009275}{25672192}a^{9}+\frac{1622199}{1833728}a^{8}-\frac{209455}{458432}a^{7}+\frac{3977447}{2292160}a^{6}+\frac{217733}{1002820}a^{5}-\frac{3252569}{1002820}a^{4}+\frac{718631}{286520}a^{3}-\frac{10987387}{1002820}a^{2}+\frac{2102059}{501410}a-\frac{73430}{7163}$, $\frac{428677}{513443840}a^{30}+\frac{376009}{73349120}a^{28}+\frac{799923}{128360960}a^{26}-\frac{1108813}{73349120}a^{24}-\frac{2859715}{102688768}a^{22}+\frac{5266291}{102688768}a^{20}+\frac{1035673}{9873920}a^{18}-\frac{7047029}{17704960}a^{16}-\frac{46509173}{39495680}a^{14}+\frac{7885979}{64180480}a^{12}+\frac{127170375}{51344384}a^{10}-\frac{17509031}{12836096}a^{8}-\frac{429781}{44080}a^{6}+\frac{8785211}{4011280}a^{4}+\frac{11422049}{250705}a^{2}+\frac{12468538}{250705}$, $\frac{378085}{821510144}a^{31}-\frac{88343}{586792960}a^{30}+\frac{914471}{586792960}a^{29}-\frac{4541417}{4107550720}a^{28}+\frac{353263}{1026887680}a^{27}-\frac{457335}{205377536}a^{26}-\frac{229943}{45137920}a^{25}+\frac{1766019}{586792960}a^{24}-\frac{2840741}{821510144}a^{23}+\frac{1138427}{117358592}a^{22}+\frac{17458279}{821510144}a^{21}-\frac{6533327}{821510144}a^{20}+\frac{770745}{205377536}a^{19}-\frac{2832029}{78991360}a^{18}-\frac{677045399}{4107550720}a^{17}+\frac{245517999}{4107550720}a^{16}-\frac{960630239}{4107550720}a^{15}+\frac{21842823}{63193088}a^{14}+\frac{106940787}{513443840}a^{13}+\frac{1974507}{17704960}a^{12}+\frac{106801}{221312}a^{11}-\frac{1221873}{1604512}a^{10}-\frac{1587575}{1604512}a^{9}-\frac{746091}{12836096}a^{8}-\frac{1945643}{916864}a^{7}+\frac{463919}{176320}a^{6}+\frac{14134683}{4011280}a^{5}+\frac{809871}{573040}a^{4}+\frac{3005673}{250705}a^{3}-\frac{2381031}{200564}a^{2}+\frac{4562371}{501410}a-\frac{4642938}{250705}$, $\frac{78349}{102688768}a^{31}-\frac{12969}{102688768}a^{30}+\frac{4429687}{2053775360}a^{29}-\frac{176261}{73349120}a^{28}-\frac{386677}{410755072}a^{27}-\frac{249227}{64180480}a^{26}-\frac{568067}{73349120}a^{25}+\frac{517677}{73349120}a^{24}+\frac{374809}{410755072}a^{23}+\frac{1743775}{102688768}a^{22}+\frac{14274329}{410755072}a^{21}-\frac{150827}{7899136}a^{20}-\frac{7834303}{410755072}a^{19}-\frac{38365}{493696}a^{18}-\frac{133605551}{513443840}a^{17}+\frac{80109249}{513443840}a^{16}-\frac{11428803}{58679296}a^{15}+\frac{325978677}{513443840}a^{14}+\frac{136406001}{293396480}a^{13}-\frac{2197257}{128360960}a^{12}+\frac{34547913}{102688768}a^{11}-\frac{70723971}{51344384}a^{10}-\frac{12099401}{6418048}a^{9}+\frac{1137301}{6418048}a^{8}-\frac{1029271}{458432}a^{7}+\frac{1197151}{229216}a^{6}+\frac{3567327}{501410}a^{5}+\frac{3276321}{4011280}a^{4}+\frac{107899}{7714}a^{3}-\frac{23048383}{1002820}a^{2}+\frac{105724}{19285}a-\frac{5967267}{250705}$, $\frac{129193}{1643020288}a^{31}+\frac{130583}{216186880}a^{30}-\frac{5541671}{8215101440}a^{29}+\frac{354287}{216186880}a^{28}-\frac{563201}{513443840}a^{27}-\frac{71263}{54046720}a^{26}+\frac{541157}{234717184}a^{25}-\frac{202741}{30883840}a^{24}+\frac{8212483}{1643020288}a^{23}+\frac{6329}{3325952}a^{22}-\frac{8499073}{1643020288}a^{21}+\frac{1265737}{43237376}a^{20}-\frac{453685}{14669824}a^{19}-\frac{1123949}{54046720}a^{18}+\frac{48879131}{1173585920}a^{17}-\frac{46640809}{216186880}a^{16}+\frac{1501538217}{8215101440}a^{15}-\frac{24200481}{216186880}a^{14}-\frac{21766013}{410755072}a^{13}+\frac{12098013}{27023360}a^{12}-\frac{40241197}{102688768}a^{11}+\frac{529297}{2702336}a^{10}+\frac{242507}{6418048}a^{9}-\frac{1098613}{675584}a^{8}+\frac{1548217}{916864}a^{7}-\frac{186691}{120640}a^{6}+\frac{3233611}{8022560}a^{5}+\frac{1409523}{211120}a^{4}-\frac{6784693}{1002820}a^{3}+\frac{284799}{26390}a^{2}-\frac{188606}{50141}a+\frac{17847}{13195}$, $\frac{309385}{1643020288}a^{31}+\frac{830003}{1026887680}a^{30}+\frac{1622409}{1643020288}a^{29}+\frac{283027}{102688768}a^{28}+\frac{390665}{410755072}a^{27}+\frac{1016179}{1026887680}a^{26}-\frac{680283}{234717184}a^{25}-\frac{259213}{29339648}a^{24}-\frac{7487185}{1643020288}a^{23}-\frac{723613}{102688768}a^{22}+\frac{17275723}{1643020288}a^{21}+\frac{3666777}{102688768}a^{20}+\frac{7038475}{410755072}a^{19}+\frac{12035449}{1026887680}a^{18}-\frac{133767143}{1643020288}a^{17}-\frac{57598753}{205377536}a^{16}-\frac{337871207}{1643020288}a^{15}-\frac{1121803}{2529280}a^{14}+\frac{1191423}{25672192}a^{13}+\frac{9062205}{29339648}a^{12}+\frac{11229745}{25672192}a^{11}+\frac{45057603}{51344384}a^{10}-\frac{4042319}{12836096}a^{9}-\frac{10028159}{6418048}a^{8}-\frac{414035}{229216}a^{7}-\frac{4575527}{1146080}a^{6}+\frac{74591}{100282}a^{5}+\frac{542385}{100282}a^{4}+\frac{1645135}{200564}a^{3}+\frac{21994521}{1002820}a^{2}+\frac{431558}{50141}a+\frac{947249}{50141}$, $\frac{256691}{4107550720}a^{31}+\frac{204489}{293396480}a^{30}-\frac{376763}{4107550720}a^{29}+\frac{4600553}{2053775360}a^{28}+\frac{65043}{410755072}a^{27}+\frac{1062339}{1026887680}a^{26}+\frac{448031}{586792960}a^{25}-\frac{2104581}{293396480}a^{24}-\frac{747169}{821510144}a^{23}-\frac{387971}{58679296}a^{22}-\frac{1584965}{821510144}a^{21}+\frac{11971991}{410755072}a^{20}-\frac{1468771}{2053775360}a^{19}+\frac{9014137}{1026887680}a^{18}+\frac{5165927}{315965440}a^{17}-\frac{452291321}{2053775360}a^{16}-\frac{1469433}{117358592}a^{15}-\frac{781895111}{2053775360}a^{14}-\frac{36017201}{293396480}a^{13}+\frac{186956597}{1026887680}a^{12}+\frac{533997}{12836096}a^{11}+\frac{10391161}{12836096}a^{10}+\frac{5630493}{25672192}a^{9}-\frac{7529351}{6418048}a^{8}+\frac{236411}{4584320}a^{7}-\frac{3697441}{1146080}a^{6}-\frac{6423399}{8022560}a^{5}+\frac{2045047}{573040}a^{4}-\frac{264839}{401128}a^{3}+\frac{17625341}{1002820}a^{2}+\frac{2494221}{501410}a+\frac{4829074}{250705}$ 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353263/1026887680a^27 - 457335/205377536a^26 - 229943/45137920a^25 + 1766019/586792960a^24 - 2840741/821510144a^23 + 1138427/117358592a^22 + 17458279/821510144a^21 - 6533327/821510144a^20 + 770745/205377536a^19 - 2832029/78991360a^18 - 677045399/4107550720a^17 + 245517999/4107550720a^16 - 960630239/4107550720a^15 + 21842823/63193088a^14 + 106940787/513443840a^13 + 1974507/17704960a^12 + 106801/221312a^11 - 1221873/1604512a^10 - 1587575/1604512a^9 - 746091/12836096a^8 - 1945643/916864a^7 + 463919/176320a^6 + 14134683/4011280a^5 + 809871/573040a^4 + 3005673/250705a^3 - 2381031/200564a^2 + 4562371/501410a - 4642938/250705, 78349/102688768a^31 - 12969/102688768a^30 + 4429687/2053775360a^29 - 176261/73349120a^28 - 386677/410755072a^27 - 249227/64180480a^26 - 568067/73349120a^25 + 517677/73349120a^24 + 374809/410755072a^23 + 1743775/102688768a^22 + 14274329/410755072a^21 - 150827/7899136a^20 - 7834303/410755072a^19 - 38365/493696a^18 - 133605551/513443840a^17 + 80109249/513443840a^16 - 11428803/58679296a^15 + 325978677/513443840a^14 + 136406001/293396480a^13 - 2197257/128360960a^12 + 34547913/102688768a^11 - 70723971/51344384a^10 - 12099401/6418048a^9 + 1137301/6418048a^8 - 1029271/458432a^7 + 1197151/229216a^6 + 3567327/501410a^5 + 3276321/4011280a^4 + 107899/7714a^3 - 23048383/1002820a^2 + 105724/19285a - 5967267/250705, 129193/1643020288a^31 + 130583/216186880a^30 - 5541671/8215101440a^29 + 354287/216186880a^28 - 563201/513443840a^27 - 71263/54046720a^26 + 541157/234717184a^25 - 202741/30883840a^24 + 8212483/1643020288a^23 + 6329/3325952a^22 - 8499073/1643020288a^21 + 1265737/43237376a^20 - 453685/14669824a^19 - 1123949/54046720a^18 + 48879131/1173585920a^17 - 46640809/216186880a^16 + 1501538217/8215101440a^15 - 24200481/216186880a^14 - 21766013/410755072a^13 + 12098013/27023360a^12 - 40241197/102688768a^11 + 529297/2702336a^10 + 242507/6418048a^9 - 1098613/675584a^8 + 1548217/916864a^7 - 186691/120640a^6 + 3233611/8022560a^5 + 1409523/211120a^4 - 6784693/1002820a^3 + 284799/26390a^2 - 188606/50141a + 17847/13195, 309385/1643020288a^31 + 830003/1026887680a^30 + 1622409/1643020288a^29 + 283027/102688768a^28 + 390665/410755072a^27 + 1016179/1026887680a^26 - 680283/234717184a^25 - 259213/29339648a^24 - 7487185/1643020288a^23 - 723613/102688768a^22 + 17275723/1643020288a^21 + 3666777/102688768a^20 + 7038475/410755072a^19 + 12035449/1026887680a^18 - 133767143/1643020288a^17 - 57598753/205377536a^16 - 337871207/1643020288a^15 - 1121803/2529280a^14 + 1191423/25672192a^13 + 9062205/29339648a^12 + 11229745/25672192a^11 + 45057603/51344384a^10 - 4042319/12836096a^9 - 10028159/6418048a^8 - 414035/229216a^7 - 4575527/1146080a^6 + 74591/100282a^5 + 542385/100282a^4 + 1645135/200564a^3 + 21994521/1002820a^2 + 431558/50141a + 947249/50141, 256691/4107550720a^31 + 204489/293396480a^30 - 376763/4107550720a^29 + 4600553/2053775360a^28 + 65043/410755072a^27 + 1062339/1026887680a^26 + 448031/586792960a^25 - 2104581/293396480a^24 - 747169/821510144a^23 - 387971/58679296a^22 - 1584965/821510144a^21 + 11971991/410755072a^20 - 1468771/2053775360a^19 + 9014137/1026887680a^18 + 5165927/315965440a^17 - 452291321/2053775360a^16 - 1469433/117358592a^15 - 781895111/2053775360a^14 - 36017201/293396480a^13 + 186956597/1026887680a^12 + 533997/12836096a^11 + 10391161/12836096a^10 + 5630493/25672192a^9 - 7529351/6418048a^8 + 236411/4584320a^7 - 3697441/1146080a^6 - 6423399/8022560a^5 + 2045047/573040a^4 - 264839/401128a^3 + 17625341/1002820a^2 + 2494221/501410a + 4829074/250705 (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:	$ 716680193852.9397 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 716680193852.9397 \cdot 24}{12\cdot\sqrt{1039812137443144915085734490774568960000000000000000}}\cr\approx \mathstrut & 0.262274368864858 \end{aligned}\] (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 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*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 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*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 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*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 - 29*x^24 + 15*x^22 + 88*x^20 - 255*x^18 - 1131*x^16 - 1020*x^14 + 1408*x^12 + 960*x^10 - 7424*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{5}) $, $\Q(\sqrt{-15}) $, $\Q(\sqrt{-1}) $, $\Q(\sqrt{15}) $, $\Q(\sqrt{-5}) $, $\Q(\sqrt{-3}) $, 4.0.6525.1, 4.0.11600.1, 4.4.104400.1, 4.4.725.1, $\Q(\sqrt{-3}, \sqrt{-5})$, $\Q(\sqrt{3}, \sqrt{5})$, $\Q(i, \sqrt{15})$, $\Q(\sqrt{3}, \sqrt{-5})$, $\Q(i, \sqrt{5})$, $\Q(\zeta_{12})$, $\Q(\sqrt{-3}, \sqrt{5})$, 8.0.273850625.1, 8.0.5678566560000.1, 8.8.70105760000.2, 8.8.22181900625.1, 8.0.12960000.1, 8.0.10899360000.14, 8.8.10899360000.1, 8.0.10899360000.2, 8.0.134560000.4, 8.0.42575625.1, 8.0.10899360000.6, 16.0.118796048409600000000.1, 16.0.32246118176350233600000000.2, 16.16.32246118176350233600000000.1, 16.0.4914817585177600000000.1, 16.0.32246118176350233600000000.1, 16.0.32246118176350233600000000.3, 16.0.492036715337375390625.1

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	R	R	${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$	${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$	${\href{/padicField/13.2.0.1}{2} }^{16}$	${\href{/padicField/17.8.0.1}{8} }^{4}$	${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$	R	${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$	${\href{/padicField/43.8.0.1}{8} }^{4}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{8}$	${\href{/padicField/59.4.0.1}{4} }^{8}$

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	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
	2.8.8.1	$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
$3$ 3	3.16.8.1	$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$	$2$	$8$	$8$	$C_8\times C_2$	$[\ ]_{2}^{8}$
$3$ 3	3.16.8.1	$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$	$2$	$8$	$8$	$C_8\times C_2$	$[\ ]_{2}^{8}$
$5$ 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	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$29$ 29	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.2.0.1	$x^{2} + 24 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	29.4.2.1	$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$521$ 521		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 $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		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 $4$	$2$	$2$	$2$

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