Properties

Label 32.0.103...616.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.034\times 10^{48}$
Root discriminant $31.66$
Ramified primes $2, 3, 41, 113$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group 32T12882

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 65536)
 
gp: K = bnfinit(x^32 - 4*x^31 + 8*x^30 - 16*x^29 + 25*x^28 - 24*x^27 + 24*x^26 - 20*x^25 - 36*x^23 + 152*x^22 - 256*x^21 + 425*x^20 - 584*x^19 + 296*x^18 + 4*x^17 + 113*x^16 + 8*x^15 + 1184*x^14 - 4672*x^13 + 6800*x^12 - 8192*x^11 + 9728*x^10 - 4608*x^9 - 10240*x^7 + 24576*x^6 - 49152*x^5 + 102400*x^4 - 131072*x^3 + 131072*x^2 - 131072*x + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, -131072, 131072, -131072, 102400, -49152, 24576, -10240, 0, -4608, 9728, -8192, 6800, -4672, 1184, 8, 113, 4, 296, -584, 425, -256, 152, -36, 0, -20, 24, -24, 25, -16, 8, -4, 1]);
 

\( x^{32} - 4 x^{31} + 8 x^{30} - 16 x^{29} + 25 x^{28} - 24 x^{27} + 24 x^{26} - 20 x^{25} - 36 x^{23} + 152 x^{22} - 256 x^{21} + 425 x^{20} - 584 x^{19} + 296 x^{18} + 4 x^{17} + 113 x^{16} + 8 x^{15} + 1184 x^{14} - 4672 x^{13} + 6800 x^{12} - 8192 x^{11} + 9728 x^{10} - 4608 x^{9} - 10240 x^{7} + 24576 x^{6} - 49152 x^{5} + 102400 x^{4} - 131072 x^{3} + 131072 x^{2} - 131072 x + 65536 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1033818800357475102568684143000990752577071087616\)\(\medspace = 2^{64}\cdot 3^{16}\cdot 41^{8}\cdot 113^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.66$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 41, 113$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{16} - \frac{1}{4} a^{13} - \frac{1}{2} a^{10} - \frac{1}{4} a^{7} - \frac{7}{16} a^{4}$, $\frac{1}{16} a^{17} - \frac{1}{2} a^{11} - \frac{1}{4} a^{8} - \frac{7}{16} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{18} - \frac{1}{2} a^{12} - \frac{1}{4} a^{9} - \frac{7}{16} a^{6} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{19} - \frac{1}{4} a^{10} - \frac{7}{16} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{16} a^{20} - \frac{1}{4} a^{11} - \frac{7}{16} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{21} - \frac{1}{32} a^{17} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{9}{32} a^{9} + \frac{1}{8} a^{8} - \frac{3}{8} a^{6} - \frac{9}{32} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{22} + \frac{1}{64} a^{18} - \frac{1}{16} a^{15} - \frac{1}{16} a^{13} + \frac{1}{8} a^{12} - \frac{1}{2} a^{11} - \frac{23}{64} a^{10} - \frac{1}{16} a^{9} - \frac{3}{16} a^{7} - \frac{23}{64} a^{6} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{23} + \frac{1}{128} a^{19} - \frac{1}{32} a^{18} - \frac{1}{32} a^{16} - \frac{1}{32} a^{14} + \frac{1}{16} a^{13} + \frac{41}{128} a^{11} + \frac{15}{32} a^{10} + \frac{1}{8} a^{9} - \frac{3}{32} a^{8} + \frac{41}{128} a^{7} - \frac{9}{32} a^{6} + \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{256} a^{24} - \frac{7}{256} a^{20} - \frac{1}{64} a^{19} - \frac{1}{64} a^{17} - \frac{1}{64} a^{15} + \frac{1}{32} a^{14} + \frac{41}{256} a^{12} + \frac{23}{64} a^{11} + \frac{1}{16} a^{10} - \frac{3}{64} a^{9} - \frac{31}{256} a^{8} + \frac{23}{64} a^{7} + \frac{1}{32} a^{6} - \frac{1}{16} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{25} - \frac{7}{512} a^{21} + \frac{3}{128} a^{20} - \frac{1}{128} a^{18} - \frac{1}{32} a^{17} - \frac{1}{128} a^{16} + \frac{1}{64} a^{15} - \frac{1}{8} a^{14} + \frac{41}{512} a^{13} - \frac{41}{128} a^{12} + \frac{5}{32} a^{11} + \frac{61}{128} a^{10} + \frac{225}{512} a^{9} + \frac{11}{128} a^{8} + \frac{1}{64} a^{7} + \frac{15}{32} a^{6} - \frac{15}{32} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{5120} a^{26} - \frac{1}{1280} a^{24} + \frac{1}{640} a^{23} + \frac{9}{5120} a^{22} - \frac{1}{256} a^{21} + \frac{7}{1280} a^{20} - \frac{3}{1280} a^{19} - \frac{1}{40} a^{18} - \frac{21}{1280} a^{17} - \frac{3}{640} a^{16} - \frac{3}{64} a^{15} + \frac{617}{5120} a^{14} - \frac{21}{256} a^{13} + \frac{619}{1280} a^{12} - \frac{105}{256} a^{11} - \frac{623}{5120} a^{10} + \frac{191}{1280} a^{9} - \frac{151}{1280} a^{8} - \frac{99}{640} a^{7} + \frac{17}{40} a^{6} - \frac{15}{32} a^{5} - \frac{27}{80} a^{4} + \frac{11}{40} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{10240} a^{27} - \frac{1}{2560} a^{25} + \frac{1}{1280} a^{24} + \frac{9}{10240} a^{23} - \frac{1}{512} a^{22} + \frac{7}{2560} a^{21} + \frac{77}{2560} a^{20} - \frac{1}{80} a^{19} - \frac{21}{2560} a^{18} - \frac{3}{1280} a^{17} - \frac{3}{128} a^{16} + \frac{617}{10240} a^{15} - \frac{21}{512} a^{14} + \frac{619}{2560} a^{13} - \frac{105}{512} a^{12} - \frac{1903}{10240} a^{11} - \frac{1089}{2560} a^{10} - \frac{151}{2560} a^{9} - \frac{379}{1280} a^{8} - \frac{23}{80} a^{7} - \frac{15}{64} a^{6} - \frac{7}{160} a^{5} - \frac{29}{80} a^{4} + \frac{3}{10} a^{3} - \frac{1}{4} a^{2} - \frac{3}{10} a$, $\frac{1}{2293760} a^{28} - \frac{11}{573440} a^{27} + \frac{1}{286720} a^{26} + \frac{89}{143360} a^{25} - \frac{4471}{2293760} a^{24} + \frac{5}{7168} a^{23} + \frac{1847}{286720} a^{22} - \frac{8861}{573440} a^{21} + \frac{4241}{143360} a^{20} + \frac{7031}{573440} a^{19} - \frac{8793}{286720} a^{18} + \frac{113}{8960} a^{17} - \frac{5591}{2293760} a^{16} + \frac{431}{20480} a^{15} - \frac{5419}{57344} a^{14} + \frac{8687}{81920} a^{13} - \frac{124991}{2293760} a^{12} + \frac{12083}{71680} a^{11} - \frac{10929}{35840} a^{10} + \frac{26591}{71680} a^{9} + \frac{419}{2240} a^{8} - \frac{341}{1120} a^{7} - \frac{1033}{4480} a^{6} - \frac{11}{224} a^{5} - \frac{241}{560} a^{4} - \frac{249}{1120} a^{3} - \frac{3}{35} a^{2} - \frac{43}{140} a - \frac{199}{560}$, $\frac{1}{4587520} a^{29} - \frac{17}{573440} a^{27} - \frac{1}{286720} a^{26} - \frac{2743}{4587520} a^{25} + \frac{279}{229376} a^{24} + \frac{273}{81920} a^{23} - \frac{2677}{1146880} a^{22} - \frac{807}{143360} a^{21} - \frac{26073}{1146880} a^{20} + \frac{685}{114688} a^{19} - \frac{303}{28672} a^{18} - \frac{58839}{4587520} a^{17} - \frac{20761}{1146880} a^{16} + \frac{289}{573440} a^{15} + \frac{79073}{1146880} a^{14} + \frac{74749}{917504} a^{13} + \frac{113863}{229376} a^{12} - \frac{7267}{71680} a^{11} + \frac{49711}{143360} a^{10} - \frac{10439}{35840} a^{9} - \frac{11}{448} a^{8} - \frac{681}{8960} a^{7} - \frac{93}{560} a^{6} - \frac{9}{140} a^{5} - \frac{1073}{2240} a^{4} + \frac{191}{560} a^{3} - \frac{67}{280} a^{2} - \frac{151}{1120} a + \frac{107}{280}$, $\frac{1}{211658997760} a^{30} + \frac{8699}{105829498880} a^{29} - \frac{10923}{52914749440} a^{28} - \frac{55279}{1889812480} a^{27} - \frac{1823313}{30236999680} a^{26} - \frac{55653543}{105829498880} a^{25} + \frac{16121887}{52914749440} a^{24} - \frac{53452033}{52914749440} a^{23} - \frac{13262617}{3779624960} a^{22} - \frac{21535173}{3112632320} a^{21} + \frac{86307243}{13228687360} a^{20} - \frac{74738773}{6614343680} a^{19} + \frac{5183540233}{211658997760} a^{18} - \frac{1892741887}{105829498880} a^{17} + \frac{477832011}{52914749440} a^{16} - \frac{14377399}{622526464} a^{15} + \frac{3348841897}{211658997760} a^{14} - \frac{6050051447}{105829498880} a^{13} - \frac{13135916631}{52914749440} a^{12} - \frac{2401429377}{6614343680} a^{11} - \frac{194399689}{3307171840} a^{10} - \frac{31941589}{97269760} a^{9} - \frac{12753041}{59056640} a^{8} - \frac{57050797}{206698240} a^{7} - \frac{1779657}{103349120} a^{6} - \frac{51178157}{103349120} a^{5} + \frac{1845089}{7382080} a^{4} + \frac{1195667}{3691040} a^{3} - \frac{9384191}{51674560} a^{2} + \frac{8731809}{25837280} a - \frac{2329967}{12918640}$, $\frac{1}{423317995520} a^{31} + \frac{3}{54132736} a^{29} + \frac{4203}{26457374720} a^{28} - \frac{12312599}{423317995520} a^{27} - \frac{3208253}{105829498880} a^{26} - \frac{41543407}{105829498880} a^{25} - \frac{156940673}{105829498880} a^{24} + \frac{21846471}{6614343680} a^{23} + \frac{584132363}{105829498880} a^{22} + \frac{93143943}{7559249920} a^{21} - \frac{12326463}{1150320640} a^{20} - \frac{2236834379}{84663599104} a^{19} + \frac{227616731}{21165899776} a^{18} - \frac{901330683}{105829498880} a^{17} - \frac{1979489851}{105829498880} a^{16} + \frac{173101137}{423317995520} a^{15} + \frac{7756402319}{105829498880} a^{14} + \frac{22206553323}{105829498880} a^{13} - \frac{1982054257}{5291474944} a^{12} + \frac{172213997}{3307171840} a^{11} - \frac{1715089}{472453120} a^{10} + \frac{67881517}{413396480} a^{9} + \frac{83802111}{206698240} a^{8} + \frac{47205517}{206698240} a^{7} + \frac{80913831}{206698240} a^{6} + \frac{124569}{875840} a^{5} - \frac{18928597}{51674560} a^{4} + \frac{3985041}{103349120} a^{3} + \frac{10828623}{25837280} a^{2} + \frac{1766133}{3691040} a + \frac{1098613}{6459320}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1593236667}{423317995520} a^{31} - \frac{1971102443}{211658997760} a^{30} + \frac{209909977}{13228687360} a^{29} - \frac{1902864259}{52914749440} a^{28} + \frac{16602969107}{423317995520} a^{27} - \frac{381405713}{12450529280} a^{26} + \frac{333853813}{7559249920} a^{25} - \frac{54278569}{6225264640} a^{24} - \frac{679321413}{52914749440} a^{23} - \frac{16448134747}{105829498880} a^{22} + \frac{8844704229}{26457374720} a^{21} - \frac{11905849059}{26457374720} a^{20} + \frac{385813198163}{423317995520} a^{19} - \frac{24537871407}{30236999680} a^{18} - \frac{6243208907}{52914749440} a^{17} - \frac{19043198463}{105829498880} a^{16} + \frac{73902225651}{423317995520} a^{15} + \frac{12898997451}{42331799552} a^{14} + \frac{129265098757}{26457374720} a^{13} - \frac{534848452331}{52914749440} a^{12} + \frac{66945785597}{6614343680} a^{11} - \frac{12736096263}{826792960} a^{10} + \frac{958595899}{71895040} a^{9} + \frac{232102257}{82679296} a^{8} + \frac{289011}{66080} a^{7} - \frac{946273007}{29528320} a^{6} + \frac{263104557}{6079360} a^{5} - \frac{3052709039}{25837280} a^{4} + \frac{4238847323}{20669824} a^{3} - \frac{1872027245}{10334912} a^{2} + \frac{281827617}{1291864} a - \frac{2113804967}{12918640} \) (order $24$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 137041291644.51082 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 137041291644.51082 \cdot 18}{24\sqrt{1033818800357475102568684143000990752577071087616}}\approx 0.596442233123376$ (assuming GRH)

Galois group

32T12882:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 512
The 80 conjugacy class representatives for t32n12882 are not computed
Character table for t32n12882 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), 4.4.94464.1, 4.0.10496.2, 4.4.2624.1, 4.0.23616.1, \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), 8.8.63021846528.1, 8.0.778047488.1, 8.8.12448759808.1, Deg 8, \(\Q(\zeta_{24})\), 8.0.8923447296.2, 8.0.557715456.2, 8.8.8923447296.1, 8.0.8923447296.11, 8.0.8923447296.3, 8.0.110166016.2, Deg 16, 16.0.3971753139798785654784.1, Deg 16, Deg 16, Deg 16, 16.0.154971620757276196864.1, Deg 16

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 32 siblings: data 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
3Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$