Properties

Label 32.0.146...416.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.468\times 10^{46}$
Root discriminant $27.71$
Ramified primes $2, 3, 13, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 32T12882

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536)
 
gp: K = bnfinit(x^32 - 4*x^30 + 10*x^28 - 32*x^26 + 83*x^24 - 208*x^22 + 458*x^20 - 956*x^18 + 2113*x^16 - 3824*x^14 + 7328*x^12 - 13312*x^10 + 21248*x^8 - 32768*x^6 + 40960*x^4 - 65536*x^2 + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, -65536, 0, 40960, 0, -32768, 0, 21248, 0, -13312, 0, 7328, 0, -3824, 0, 2113, 0, -956, 0, 458, 0, -208, 0, 83, 0, -32, 0, 10, 0, -4, 0, 1]);
 

\( x^{32} - 4 x^{30} + 10 x^{28} - 32 x^{26} + 83 x^{24} - 208 x^{22} + 458 x^{20} - 956 x^{18} + 2113 x^{16} - 3824 x^{14} + 7328 x^{12} - 13312 x^{10} + 21248 x^{8} - 32768 x^{6} + 40960 x^{4} - 65536 x^{2} + 65536 \)

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:  \(14680252274932068764306928624388060814210236416\)\(\medspace = 2^{72}\cdot 3^{16}\cdot 13^{8}\cdot 97^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $27.71$
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:  $2, 3, 13, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{12} a^{16} + \frac{1}{12} a^{14} - \frac{1}{12} a^{10} + \frac{1}{6} a^{8} + \frac{5}{12} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{12} a^{17} + \frac{1}{12} a^{15} - \frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{1}{12} a^{7} - \frac{1}{4} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{14} - \frac{1}{12} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{24} a^{19} + \frac{1}{12} a^{15} - \frac{1}{6} a^{13} + \frac{1}{8} a^{11} - \frac{1}{12} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{48} a^{20} - \frac{1}{24} a^{16} + \frac{1}{12} a^{14} + \frac{1}{16} a^{12} - \frac{1}{6} a^{10} + \frac{1}{24} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{7}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{96} a^{21} + \frac{1}{48} a^{17} + \frac{1}{12} a^{15} - \frac{7}{32} a^{13} - \frac{1}{8} a^{11} - \frac{7}{48} a^{9} + \frac{1}{24} a^{7} - \frac{5}{32} a^{5} + \frac{11}{24} a^{3} - \frac{1}{2} a$, $\frac{1}{384} a^{22} - \frac{1}{192} a^{21} - \frac{1}{96} a^{20} + \frac{5}{192} a^{18} + \frac{1}{32} a^{17} - \frac{1}{8} a^{15} - \frac{13}{384} a^{14} + \frac{15}{64} a^{13} - \frac{5}{24} a^{12} - \frac{11}{48} a^{11} - \frac{43}{192} a^{10} + \frac{1}{32} a^{9} - \frac{5}{32} a^{8} + \frac{3}{16} a^{7} + \frac{97}{384} a^{6} - \frac{11}{64} a^{5} + \frac{7}{24} a^{4} - \frac{7}{16} a^{3} - \frac{1}{12} a^{2} + \frac{5}{12} a + \frac{1}{6}$, $\frac{1}{768} a^{23} - \frac{1}{96} a^{20} + \frac{5}{384} a^{19} + \frac{1}{96} a^{17} - \frac{1}{48} a^{16} - \frac{77}{768} a^{15} - \frac{1}{12} a^{14} + \frac{31}{192} a^{13} + \frac{7}{32} a^{12} - \frac{67}{384} a^{11} + \frac{1}{8} a^{10} - \frac{5}{192} a^{9} + \frac{7}{48} a^{8} - \frac{79}{768} a^{7} - \frac{1}{24} a^{6} + \frac{37}{192} a^{5} + \frac{5}{32} a^{4} - \frac{3}{16} a^{3} - \frac{11}{24} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{1536} a^{24} - \frac{1}{192} a^{21} + \frac{5}{768} a^{20} - \frac{7}{192} a^{18} + \frac{1}{32} a^{17} - \frac{13}{1536} a^{16} - \frac{1}{8} a^{15} - \frac{11}{128} a^{14} + \frac{15}{64} a^{13} + \frac{157}{768} a^{12} - \frac{11}{48} a^{11} - \frac{23}{128} a^{10} + \frac{1}{32} a^{9} - \frac{143}{1536} a^{8} + \frac{3}{16} a^{7} - \frac{139}{384} a^{6} - \frac{11}{64} a^{5} + \frac{47}{96} a^{4} - \frac{7}{16} a^{3} - \frac{1}{6} a^{2} + \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{3072} a^{25} - \frac{1}{512} a^{21} - \frac{1}{96} a^{20} - \frac{7}{384} a^{19} - \frac{15}{1024} a^{17} - \frac{1}{48} a^{16} - \frac{65}{768} a^{15} - \frac{1}{12} a^{14} + \frac{325}{1536} a^{13} + \frac{7}{32} a^{12} - \frac{7}{256} a^{11} + \frac{1}{8} a^{10} - \frac{229}{1024} a^{9} + \frac{7}{48} a^{8} + \frac{37}{768} a^{7} - \frac{1}{24} a^{6} + \frac{7}{96} a^{5} + \frac{5}{32} a^{4} + \frac{3}{16} a^{3} - \frac{11}{24} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{6144} a^{26} - \frac{1}{1024} a^{22} - \frac{1}{192} a^{21} - \frac{7}{768} a^{20} + \frac{211}{6144} a^{18} + \frac{1}{32} a^{17} - \frac{1}{1536} a^{16} - \frac{1}{8} a^{15} + \frac{325}{3072} a^{14} + \frac{15}{64} a^{13} + \frac{299}{1536} a^{12} - \frac{11}{48} a^{11} - \frac{175}{6144} a^{10} + \frac{1}{32} a^{9} + \frac{119}{512} a^{8} + \frac{3}{16} a^{7} - \frac{17}{192} a^{6} - \frac{11}{64} a^{5} - \frac{23}{96} a^{4} - \frac{7}{16} a^{3} - \frac{1}{8} a^{2} + \frac{5}{12} a$, $\frac{1}{12288} a^{27} - \frac{1}{2048} a^{23} + \frac{1}{1536} a^{21} - \frac{1}{96} a^{20} + \frac{211}{12288} a^{19} + \frac{31}{3072} a^{17} - \frac{1}{48} a^{16} + \frac{581}{6144} a^{15} - \frac{1}{12} a^{14} + \frac{731}{3072} a^{13} + \frac{7}{32} a^{12} + \frac{2129}{12288} a^{11} + \frac{1}{8} a^{10} + \frac{133}{3072} a^{9} + \frac{7}{48} a^{8} - \frac{3}{128} a^{7} + \frac{11}{24} a^{6} + \frac{29}{96} a^{5} - \frac{11}{32} a^{4} + \frac{5}{12} a^{3} + \frac{1}{24} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{19218432} a^{28} - \frac{1}{1201152} a^{26} + \frac{815}{3203072} a^{24} + \frac{2053}{2402304} a^{22} - \frac{1}{192} a^{21} - \frac{94573}{19218432} a^{20} - \frac{111245}{4804608} a^{18} + \frac{1}{32} a^{17} - \frac{199147}{9609216} a^{16} - \frac{1}{8} a^{15} - \frac{310685}{4804608} a^{14} + \frac{15}{64} a^{13} - \frac{1111855}{19218432} a^{12} - \frac{11}{48} a^{11} - \frac{788735}{4804608} a^{10} + \frac{1}{32} a^{9} - \frac{58601}{300288} a^{8} + \frac{3}{16} a^{7} - \frac{110645}{300288} a^{6} + \frac{21}{64} a^{5} + \frac{6845}{18768} a^{4} + \frac{1}{16} a^{3} + \frac{5801}{18768} a^{2} - \frac{1}{12} a - \frac{129}{1564}$, $\frac{1}{38436864} a^{29} - \frac{1}{2402304} a^{27} + \frac{815}{6406144} a^{25} + \frac{2053}{4804608} a^{23} + \frac{105619}{38436864} a^{21} - \frac{1}{96} a^{20} - \frac{111245}{9609216} a^{19} + \frac{1045}{19218432} a^{17} - \frac{1}{48} a^{16} + \frac{89699}{9609216} a^{15} - \frac{1}{12} a^{14} + \frac{4293329}{38436864} a^{13} + \frac{7}{32} a^{12} - \frac{1389311}{9609216} a^{11} + \frac{1}{8} a^{10} + \frac{15917}{200192} a^{9} + \frac{7}{48} a^{8} - \frac{32711}{200192} a^{7} - \frac{1}{24} a^{6} - \frac{10943}{75072} a^{5} + \frac{5}{32} a^{4} - \frac{1455}{12512} a^{3} - \frac{11}{24} a^{2} + \frac{653}{3128} a - \frac{1}{2}$, $\frac{1}{10992943104} a^{30} + \frac{35}{1374117888} a^{28} - \frac{12505}{499679232} a^{26} - \frac{111049}{1374117888} a^{24} - \frac{11198381}{10992943104} a^{22} - \frac{1}{192} a^{21} + \frac{17488945}{2748235776} a^{20} + \frac{74336183}{1832157184} a^{18} + \frac{1}{32} a^{17} + \frac{4961383}{2748235776} a^{16} - \frac{1}{8} a^{15} - \frac{15625}{477954048} a^{14} + \frac{15}{64} a^{13} - \frac{167246741}{2748235776} a^{12} - \frac{11}{48} a^{11} + \frac{29578729}{343529472} a^{10} + \frac{1}{32} a^{9} + \frac{10835437}{171764736} a^{8} + \frac{3}{16} a^{7} + \frac{172087}{447304} a^{6} - \frac{11}{64} a^{5} - \frac{447989}{975936} a^{4} - \frac{7}{16} a^{3} - \frac{868451}{2683824} a^{2} + \frac{5}{12} a + \frac{50122}{167739}$, $\frac{1}{21985886208} a^{31} + \frac{35}{2748235776} a^{29} - \frac{12505}{999358464} a^{27} - \frac{111049}{2748235776} a^{25} - \frac{11198381}{21985886208} a^{23} - \frac{3712837}{1832157184} a^{21} - \frac{1}{96} a^{20} + \frac{74336183}{3664314368} a^{19} + \frac{176726119}{5496471552} a^{17} - \frac{1}{48} a^{16} - \frac{15625}{955908096} a^{15} - \frac{1}{12} a^{14} + \frac{433929835}{5496471552} a^{13} + \frac{7}{32} a^{12} - \frac{42624093}{229019648} a^{11} + \frac{1}{8} a^{10} + \frac{64511917}{343529472} a^{9} + \frac{7}{48} a^{8} + \frac{58087}{447304} a^{7} + \frac{11}{24} a^{6} + \frac{436453}{1951872} a^{5} - \frac{11}{32} a^{4} - \frac{1203929}{5367648} a^{3} + \frac{1}{24} a^{2} + \frac{14777}{223652} a - \frac{1}{2}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{1215085}{10992943104} a^{31} - \frac{1394149}{5496471552} a^{29} + \frac{309323}{499679232} a^{27} - \frac{97963}{39829504} a^{25} + \frac{18384557}{3664314368} a^{23} - \frac{26777331}{1832157184} a^{21} + \frac{8784421}{323321856} a^{19} - \frac{82237741}{1374117888} a^{17} + \frac{1497652933}{10992943104} a^{15} - \frac{1103988403}{5496471552} a^{13} + \frac{220163729}{458039296} a^{11} - \frac{118625219}{171764736} a^{9} + \frac{104765123}{85882368} a^{7} - \frac{1638191}{975936} a^{5} + \frac{2897783}{1789216} a^{3} - \frac{5762987}{1341912} a \) (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:  \( 32664023933.600838 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 32664023933.600838 \cdot 4}{24\sqrt{14680252274932068764306928624388060814210236416}}\approx 0.265112111724684$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), 4.0.29952.1, 4.4.7488.1, 4.0.7488.1, 4.4.29952.1, \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.8.348083453952.1, 8.0.5438803968.1, 8.8.5438803968.1, Deg 8, \(\Q(\zeta_{24})\), 8.0.3588489216.16, 8.0.3588489216.5, 8.0.3588489216.11, 8.8.3588489216.1, 8.0.897122304.10, 8.0.56070144.2, 16.0.12877254853348294656.1, 16.0.121162090915154104418304.2, Deg 16, Deg 16, 16.0.121162090915154104418304.1, 16.0.29580588602332545024.3, 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
$97$97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.1$x^{2} - 97$$2$$1$$1$$C_2$$[\ ]_{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$