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Properties

Label 32.0.480...776.1
Degree $32$
Signature $[0, 16]$
Discriminant $4.810\times 10^{50}$
Root discriminant $38.35$
Ramified primes $2, 3, 13, 17$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2^2\times C_2^2\wr C_2$ (as 32T1369)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096)
 
gp: K = bnfinit(x^32 + 20*x^30 + 236*x^28 + 1948*x^26 + 12276*x^24 + 58148*x^22 + 212422*x^20 + 567784*x^18 + 1013620*x^16 + 772588*x^14 + 788932*x^12 + 1345276*x^10 + 364625*x^8 - 228564*x^6 + 145744*x^4 - 25856*x^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, -25856, 0, 145744, 0, -228564, 0, 364625, 0, 1345276, 0, 788932, 0, 772588, 0, 1013620, 0, 567784, 0, 212422, 0, 58148, 0, 12276, 0, 1948, 0, 236, 0, 20, 0, 1]);
 

\( x^{32} + 20 x^{30} + 236 x^{28} + 1948 x^{26} + 12276 x^{24} + 58148 x^{22} + 212422 x^{20} + 567784 x^{18} + 1013620 x^{16} + 772588 x^{14} + 788932 x^{12} + 1345276 x^{10} + 364625 x^{8} - 228564 x^{6} + 145744 x^{4} - 25856 x^{2} + 4096 \)

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:  \(480960519379403029833827263813614000556122650443776\)\(\medspace = 2^{48}\cdot 3^{16}\cdot 13^{8}\cdot 17^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $38.35$
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:  $2, 3, 13, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $16$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{136} a^{20} + \frac{3}{68} a^{18} - \frac{1}{8} a^{17} + \frac{5}{68} a^{16} + \frac{3}{34} a^{14} - \frac{1}{4} a^{13} + \frac{2}{17} a^{12} - \frac{1}{4} a^{11} + \frac{11}{68} a^{10} + \frac{9}{136} a^{8} - \frac{1}{4} a^{7} - \frac{21}{68} a^{6} + \frac{3}{8} a^{5} + \frac{3}{17} a^{4} - \frac{1}{2} a^{3} - \frac{27}{68} a^{2} - \frac{1}{2} a - \frac{8}{17}$, $\frac{1}{136} a^{21} + \frac{3}{68} a^{19} + \frac{5}{68} a^{17} - \frac{5}{136} a^{15} + \frac{2}{17} a^{13} - \frac{3}{34} a^{11} - \frac{25}{136} a^{9} + \frac{13}{68} a^{7} + \frac{29}{68} a^{5} - \frac{3}{136} a^{3} - \frac{8}{17} a$, $\frac{1}{136} a^{22} + \frac{1}{17} a^{18} + \frac{3}{136} a^{16} + \frac{3}{34} a^{14} + \frac{7}{34} a^{12} - \frac{21}{136} a^{10} - \frac{7}{34} a^{8} + \frac{1}{34} a^{6} + \frac{57}{136} a^{4} + \frac{7}{17} a^{2} - \frac{3}{17}$, $\frac{1}{272} a^{23} - \frac{1}{272} a^{22} + \frac{1}{34} a^{19} - \frac{1}{34} a^{18} + \frac{3}{272} a^{17} - \frac{3}{272} a^{16} + \frac{3}{68} a^{15} - \frac{3}{68} a^{14} + \frac{7}{68} a^{13} - \frac{7}{68} a^{12} + \frac{47}{272} a^{11} - \frac{47}{272} a^{10} + \frac{5}{34} a^{9} - \frac{5}{34} a^{8} + \frac{1}{68} a^{7} + \frac{33}{68} a^{6} - \frac{11}{272} a^{5} + \frac{11}{272} a^{4} - \frac{3}{68} a^{3} + \frac{3}{68} a^{2} - \frac{3}{34} a - \frac{7}{17}$, $\frac{1}{272} a^{24} - \frac{1}{272} a^{22} + \frac{15}{272} a^{18} - \frac{3}{272} a^{16} - \frac{3}{68} a^{14} + \frac{27}{272} a^{12} - \frac{47}{272} a^{10} + \frac{7}{68} a^{8} - \frac{19}{272} a^{6} - \frac{125}{272} a^{4} - \frac{7}{34} a^{2} + \frac{8}{17}$, $\frac{1}{1088} a^{25} + \frac{1}{1088} a^{23} - \frac{1}{272} a^{22} + \frac{31}{1088} a^{19} - \frac{1}{34} a^{18} + \frac{71}{1088} a^{17} - \frac{3}{272} a^{16} - \frac{7}{136} a^{15} - \frac{3}{68} a^{14} - \frac{189}{1088} a^{13} - \frac{7}{68} a^{12} + \frac{183}{1088} a^{11} - \frac{47}{272} a^{10} - \frac{7}{272} a^{9} - \frac{5}{34} a^{8} + \frac{261}{1088} a^{7} + \frac{33}{68} a^{6} - \frac{351}{1088} a^{5} + \frac{11}{272} a^{4} - \frac{37}{272} a^{3} + \frac{3}{68} a^{2} - \frac{29}{68} a - \frac{7}{17}$, $\frac{1}{1088} a^{26} + \frac{1}{1088} a^{24} - \frac{1}{272} a^{22} - \frac{1}{1088} a^{20} - \frac{1}{64} a^{18} - \frac{29}{272} a^{16} - \frac{1}{8} a^{15} - \frac{77}{1088} a^{14} - \frac{169}{1088} a^{12} - \frac{1}{4} a^{11} + \frac{21}{136} a^{10} - \frac{1}{4} a^{9} + \frac{5}{64} a^{8} + \frac{25}{1088} a^{6} + \frac{1}{4} a^{5} + \frac{61}{136} a^{4} + \frac{3}{8} a^{3} + \frac{31}{68} a^{2} - \frac{1}{2} a + \frac{8}{17}$, $\frac{1}{1088} a^{27} - \frac{1}{1088} a^{23} - \frac{1}{1088} a^{21} - \frac{1}{68} a^{19} + \frac{97}{1088} a^{17} - \frac{1}{8} a^{16} + \frac{27}{1088} a^{15} + \frac{33}{272} a^{13} - \frac{1}{4} a^{12} + \frac{173}{1088} a^{11} - \frac{1}{4} a^{10} - \frac{271}{1088} a^{9} - \frac{55}{272} a^{7} + \frac{1}{4} a^{6} + \frac{523}{1088} a^{5} + \frac{3}{8} a^{4} + \frac{13}{272} a^{3} - \frac{1}{2} a^{2} - \frac{13}{68} a$, $\frac{1}{239206592} a^{28} + \frac{12879}{239206592} a^{26} - \frac{193411}{119603296} a^{24} + \frac{181391}{239206592} a^{22} - \frac{725967}{239206592} a^{20} + \frac{173463}{119603296} a^{18} - \frac{1}{8} a^{17} - \frac{28584061}{239206592} a^{16} + \frac{21811377}{239206592} a^{14} - \frac{1}{4} a^{13} + \frac{23518525}{119603296} a^{12} - \frac{1}{4} a^{11} + \frac{2659581}{239206592} a^{10} + \frac{46079727}{239206592} a^{8} - \frac{1}{4} a^{7} - \frac{28592741}{119603296} a^{6} + \frac{3}{8} a^{5} - \frac{1159657}{29900824} a^{4} - \frac{1}{2} a^{3} - \frac{326555}{14950412} a^{2} - \frac{1}{2} a - \frac{958939}{3737603}$, $\frac{1}{239206592} a^{29} + \frac{12879}{239206592} a^{27} + \frac{1653}{7475206} a^{25} - \frac{258327}{239206592} a^{23} - \frac{1}{272} a^{22} - \frac{725967}{239206592} a^{21} + \frac{867837}{29900824} a^{19} - \frac{1}{34} a^{18} - \frac{2391}{239206592} a^{17} - \frac{3}{272} a^{16} + \frac{972633}{14070976} a^{15} - \frac{3}{68} a^{14} + \frac{14727359}{59801648} a^{13} - \frac{7}{68} a^{12} - \frac{1059245}{14070976} a^{11} - \frac{47}{272} a^{10} + \frac{58391831}{239206592} a^{9} - \frac{5}{34} a^{8} + \frac{13515793}{59801648} a^{7} + \frac{33}{68} a^{6} - \frac{47071415}{119603296} a^{5} + \frac{11}{272} a^{4} + \frac{2804819}{7475206} a^{3} + \frac{3}{68} a^{2} - \frac{79503}{3737603} a - \frac{7}{17}$, $\frac{1}{84567116008217470262422553261312} a^{30} + \frac{41150504136136911876065}{21141779002054367565605638315328} a^{28} - \frac{54048236396413467409215199}{154319554759520931135807578944} a^{26} - \frac{1106558163114461918813664727}{1243634058944374562682684606784} a^{24} - \frac{31177622024325403788928677871}{21141779002054367565605638315328} a^{22} + \frac{77479106297715728212230243523}{21141779002054367565605638315328} a^{20} - \frac{2555301804277694842055361798913}{42283558004108735131211276630656} a^{18} - \frac{1}{8} a^{17} + \frac{977383509659628230511066289871}{10570889501027183782802819157664} a^{16} + \frac{1813891406256474595605181680519}{21141779002054367565605638315328} a^{14} - \frac{1}{4} a^{13} - \frac{3288666259101704025906587070103}{21141779002054367565605638315328} a^{12} - \frac{1}{4} a^{11} - \frac{42701074337579909975234192779}{1243634058944374562682684606784} a^{10} - \frac{3695869201339368929836237801475}{21141779002054367565605638315328} a^{8} - \frac{1}{4} a^{7} - \frac{41569318085026221409568349196263}{84567116008217470262422553261312} a^{6} + \frac{3}{8} a^{5} + \frac{5883223821574092979410963097331}{21141779002054367565605638315328} a^{4} - \frac{1}{2} a^{3} + \frac{183770892128376852209166928765}{5285444750513591891401409578832} a^{2} - \frac{1}{2} a + \frac{7319939122720848462054573968}{330340296907099493212588098677}$, $\frac{1}{338268464032869881049690213045248} a^{31} + \frac{41150504136136911876065}{84567116008217470262422553261312} a^{29} + \frac{229627415735058832472784027}{617278219038083724543230315776} a^{27} + \frac{20052075569065852464001593603}{84567116008217470262422553261312} a^{25} - \frac{31177622024325403788928677871}{84567116008217470262422553261312} a^{23} - \frac{1}{272} a^{22} - \frac{116838715412342797206939226287}{84567116008217470262422553261312} a^{21} + \frac{7316043538593278249238447267435}{169134232016434940524845106522624} a^{19} - \frac{1}{34} a^{18} - \frac{1820793122965214535524974075393}{42283558004108735131211276630656} a^{17} + \frac{31}{272} a^{16} + \frac{4106841702435165195551381424277}{84567116008217470262422553261312} a^{15} - \frac{3}{68} a^{14} - \frac{7991157544485120341050488239505}{84567116008217470262422553261312} a^{13} + \frac{5}{34} a^{12} + \frac{9689516979920278492888502304573}{84567116008217470262422553261312} a^{11} + \frac{21}{272} a^{10} - \frac{6144273754886106350117773121081}{84567116008217470262422553261312} a^{9} - \frac{5}{34} a^{8} + \frac{54345958711058666737333701101953}{338268464032869881049690213045248} a^{7} + \frac{4}{17} a^{6} + \frac{40549523214648533914190796511435}{84567116008217470262422553261312} a^{5} - \frac{91}{272} a^{4} + \frac{2671039010017125977574536142333}{21141779002054367565605638315328} a^{3} - \frac{31}{68} a^{2} - \frac{245293229100355234582865736785}{1321361187628397972850352394708} a - \frac{7}{17}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( \frac{31733846347289050006707609339}{338268464032869881049690213045248} a^{31} + \frac{159066371668502704786194320775}{84567116008217470262422553261312} a^{29} + \frac{13722982620419582052588929353}{617278219038083724543230315776} a^{27} + \frac{15544719819385212136664947501213}{84567116008217470262422553261312} a^{25} + \frac{98127949168048442982113604227447}{84567116008217470262422553261312} a^{23} + \frac{465912278340159774396442411472379}{84567116008217470262422553261312} a^{21} + \frac{3413485261243466901479279119063505}{169134232016434940524845106522624} a^{19} + \frac{2291173245929342645683680194076411}{42283558004108735131211276630656} a^{17} + \frac{8247282890832293527114323360526327}{84567116008217470262422553261312} a^{15} + \frac{151071612981110777014047569238667}{1966677116470173727033082633984} a^{13} + \frac{6552031141224874223212789373396867}{84567116008217470262422553261312} a^{11} + \frac{11119375391339031974950856247776221}{84567116008217470262422553261312} a^{9} + \frac{13520292253964077881824453407256171}{338268464032869881049690213045248} a^{7} - \frac{1617629871718382246042141050801871}{84567116008217470262422553261312} a^{5} + \frac{18716742129237302188519841608487}{1243634058944374562682684606784} a^{3} - \frac{3516851695558134935161606145415}{1321361187628397972850352394708} a \) (order $12$)
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:  \( 2205277237958.931 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 2205277237958.931 \cdot 16}{12\sqrt{480960519379403029833827263813614000556122650443776}}\approx 0.791088572881245$ (assuming GRH)

Galois group

$C_2^2\times C_2^2\wr C_2$ (as 32T1369):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 128
The 56 conjugacy class representatives for $C_2^2\times C_2^2\wr C_2$ are not computed
Character table for $C_2^2\times C_2^2\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{51}) \), \(\Q(\sqrt{-51}) \), 4.0.120224.1, 4.4.20808.1, 4.4.30056.1, 4.0.83232.1, 4.0.2312.1, 4.4.1082016.1, 4.4.9248.1, 4.0.270504.1, \(\Q(i, \sqrt{51})\), 4.0.541008.2, \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{17})\), 4.4.60112.1, \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{17})\), 4.0.3757.1, \(\Q(\sqrt{3}, \sqrt{-17})\), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.33813.1, 8.0.292689656064.35, 8.0.292689656064.9, 8.0.3613452544.3, 8.0.292689656064.39, 8.0.1731891456.1, 8.8.292689656064.1, 8.0.1143318969.1, 8.0.4683034497024.8, 8.0.4683034497024.5, 8.0.4683034497024.90, 8.0.4683034497024.31, 8.0.57815240704.2, 8.0.27710263296.14, 8.0.342102016.5, 8.0.4683034497024.9, 8.0.57815240704.21, 8.8.4683034497024.2, 8.8.57815240704.3, 8.0.4683034497024.3, 8.0.4683034497024.66, 8.8.27710263296.2, 8.8.4683034497024.6, 8.0.27710263296.13, 8.0.14453810176.1, 8.0.903363136.1, 8.0.73172414016.5, 8.0.1170758624256.10, 8.0.1170758624256.4, 8.0.432972864.2, 8.0.73172414016.6, 8.0.6927565824.3, 8.0.1170758624256.17, 8.8.73172414016.1, 8.0.73172414016.14, 8.8.1170758624256.2, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, 16.0.3342602057661458415616.4, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, 16.0.1370675756269801791553536.1, 16.0.5354202172928913248256.2

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{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}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$