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Properties

Label 20.4.436...184.1
Degree $20$
Signature $[4, 8]$
Discriminant $4.360\times 10^{23}$
Root discriminant \(15.20\)
Ramified primes $2,67$
Class number $1$
Class group trivial
Galois group $C_2\times A_5$ (as 20T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^16 - 22*x^14 - 6*x^12 + 22*x^10 + 33*x^8 - 22*x^6 + 19*x^4 - 2*x^2 + 1)
 
gp: K = bnfinit(y^20 - 8*y^16 - 22*y^14 - 6*y^12 + 22*y^10 + 33*y^8 - 22*y^6 + 19*y^4 - 2*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^16 - 22*x^14 - 6*x^12 + 22*x^10 + 33*x^8 - 22*x^6 + 19*x^4 - 2*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 8*x^16 - 22*x^14 - 6*x^12 + 22*x^10 + 33*x^8 - 22*x^6 + 19*x^4 - 2*x^2 + 1)
 

\( x^{20} - 8x^{16} - 22x^{14} - 6x^{12} + 22x^{10} + 33x^{8} - 22x^{6} + 19x^{4} - 2x^{2} + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(436011848767111570653184\) \(\medspace = 2^{30}\cdot 67^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(15.20\)
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:  $2^{3/2}67^{1/2}\approx 23.15167380558045$
Ramified primes:   \(2\), \(67\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{1}{16}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{16}a^{8}+\frac{1}{4}a^{7}-\frac{3}{8}a^{6}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{5}{16}$, $\frac{1}{16}a^{17}-\frac{1}{8}a^{15}-\frac{1}{16}a^{13}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}+\frac{3}{16}a^{9}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{5}{16}a-\frac{1}{4}$, $\frac{1}{18208}a^{18}-\frac{315}{18208}a^{16}+\frac{1349}{18208}a^{14}-\frac{1621}{18208}a^{12}-\frac{1491}{18208}a^{10}+\frac{831}{18208}a^{8}-\frac{1}{2}a^{7}-\frac{1137}{2276}a^{6}-\frac{3547}{9104}a^{4}-\frac{1}{2}a^{3}+\frac{8701}{18208}a^{2}-\frac{1}{2}a+\frac{8591}{18208}$, $\frac{1}{36416}a^{19}-\frac{1}{36416}a^{18}+\frac{823}{36416}a^{17}-\frac{823}{36416}a^{16}-\frac{927}{36416}a^{15}+\frac{927}{36416}a^{14}+\frac{1793}{36416}a^{13}-\frac{1793}{36416}a^{12}+\frac{5337}{36416}a^{11}-\frac{5337}{36416}a^{10}-\frac{4859}{36416}a^{9}-\frac{4245}{36416}a^{8}-\frac{3981}{9104}a^{7}+\frac{1705}{9104}a^{6}+\frac{2143}{18208}a^{5}-\frac{6695}{18208}a^{4}-\frac{4955}{36416}a^{3}-\frac{13253}{36416}a^{2}+\frac{625}{36416}a-\frac{9729}{36416}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $a$, $\frac{477}{4552}a^{19}+\frac{1231}{18208}a^{18}-\frac{76}{569}a^{17}-\frac{845}{18208}a^{16}-\frac{4049}{4552}a^{15}-\frac{9965}{18208}a^{14}-\frac{1409}{1138}a^{13}-\frac{19883}{18208}a^{12}+\frac{12561}{4552}a^{11}+\frac{12691}{18208}a^{10}+\frac{4785}{1138}a^{9}+\frac{30625}{18208}a^{8}+\frac{1339}{2276}a^{7}+\frac{1231}{2276}a^{6}-\frac{19059}{2276}a^{5}-\frac{32853}{9104}a^{4}+\frac{13747}{4552}a^{3}+\frac{45595}{18208}a^{2}-\frac{644}{569}a+\frac{1225}{18208}$, $\frac{2881}{18208}a^{19}-\frac{2675}{9104}a^{18}-\frac{1667}{18208}a^{17}+\frac{505}{9104}a^{16}-\frac{23699}{18208}a^{15}+\frac{21645}{9104}a^{14}-\frac{49821}{18208}a^{13}+\frac{55019}{9104}a^{12}+\frac{24277}{18208}a^{11}+\frac{3149}{9104}a^{10}+\frac{86247}{18208}a^{9}-\frac{69829}{9104}a^{8}+\frac{7433}{2276}a^{7}-\frac{20713}{2276}a^{6}-\frac{65671}{9104}a^{5}+\frac{40549}{4552}a^{4}+\frac{72549}{18208}a^{3}-\frac{46319}{9104}a^{2}-\frac{7657}{18208}a+\frac{6675}{9104}$, $\frac{477}{4552}a^{19}-\frac{1231}{18208}a^{18}-\frac{76}{569}a^{17}+\frac{845}{18208}a^{16}-\frac{4049}{4552}a^{15}+\frac{9965}{18208}a^{14}-\frac{1409}{1138}a^{13}+\frac{19883}{18208}a^{12}+\frac{12561}{4552}a^{11}-\frac{12691}{18208}a^{10}+\frac{4785}{1138}a^{9}-\frac{30625}{18208}a^{8}+\frac{1339}{2276}a^{7}-\frac{1231}{2276}a^{6}-\frac{19059}{2276}a^{5}+\frac{32853}{9104}a^{4}+\frac{13747}{4552}a^{3}-\frac{45595}{18208}a^{2}-\frac{644}{569}a-\frac{1225}{18208}$, $\frac{77}{4552}a^{19}+\frac{3317}{18208}a^{18}+\frac{53}{1138}a^{17}-\frac{171}{18208}a^{16}-\frac{823}{4552}a^{15}-\frac{27295}{18208}a^{14}-\frac{905}{1138}a^{13}-\frac{71501}{18208}a^{12}-\frac{3283}{4552}a^{11}-\frac{11279}{18208}a^{10}+\frac{3259}{2276}a^{9}+\frac{91231}{18208}a^{8}+\frac{2999}{1138}a^{7}+\frac{14697}{2276}a^{6}+\frac{285}{1138}a^{5}-\frac{41723}{9104}a^{4}-\frac{11685}{4552}a^{3}+\frac{28849}{18208}a^{2}+\frac{509}{1138}a-\frac{6001}{18208}$, $\frac{13455}{36416}a^{19}-\frac{2135}{36416}a^{18}-\frac{6103}{36416}a^{17}+\frac{2243}{36416}a^{16}-\frac{109553}{36416}a^{15}+\frac{17233}{36416}a^{14}-\frac{246593}{36416}a^{13}+\frac{29765}{36416}a^{12}+\frac{69815}{36416}a^{11}-\frac{37255}{36416}a^{10}+\frac{371243}{36416}a^{9}-\frac{70599}{36416}a^{8}+\frac{74137}{9104}a^{7}-\frac{9391}{9104}a^{6}-\frac{285079}{18208}a^{5}+\frac{56355}{18208}a^{4}+\frac{335915}{36416}a^{3}-\frac{81859}{36416}a^{2}-\frac{39137}{36416}a+\frac{42605}{36416}$, $\frac{9125}{36416}a^{19}+\frac{13089}{36416}a^{18}+\frac{3627}{36416}a^{17}-\frac{2337}{36416}a^{16}-\frac{74091}{36416}a^{15}-\frac{107119}{36416}a^{14}-\frac{230915}{36416}a^{13}-\frac{270103}{36416}a^{12}-\frac{124627}{36416}a^{11}-\frac{8103}{36416}a^{10}+\frac{212049}{36416}a^{9}+\frac{360701}{36416}a^{8}+\frac{100755}{9104}a^{7}+\frac{104131}{9104}a^{6}-\frac{59693}{18208}a^{5}-\frac{204537}{18208}a^{4}+\frac{23401}{36416}a^{3}+\frac{219397}{36416}a^{2}+\frac{44989}{36416}a-\frac{17527}{36416}$, $\frac{3627}{36416}a^{19}+\frac{4597}{36416}a^{18}-\frac{1091}{36416}a^{17}-\frac{3933}{36416}a^{16}-\frac{30165}{36416}a^{15}-\frac{37163}{36416}a^{14}-\frac{69877}{36416}a^{13}-\frac{69531}{36416}a^{12}+\frac{11299}{36416}a^{11}+\frac{62637}{36416}a^{10}+\frac{101895}{36416}a^{9}+\frac{131849}{36416}a^{8}+\frac{20341}{9104}a^{7}+\frac{14315}{9104}a^{6}-\frac{74987}{18208}a^{5}-\frac{122061}{18208}a^{4}+\frac{63239}{36416}a^{3}+\frac{173049}{36416}a^{2}+\frac{27291}{36416}a-\frac{40155}{36416}$, $\frac{4597}{36416}a^{19}-\frac{10967}{36416}a^{18}-\frac{3933}{36416}a^{17}-\frac{3777}{36416}a^{16}-\frac{37163}{36416}a^{15}+\frac{88281}{36416}a^{14}-\frac{69531}{36416}a^{13}+\frac{273929}{36416}a^{12}+\frac{62637}{36416}a^{11}+\frac{144401}{36416}a^{10}+\frac{131849}{36416}a^{9}-\frac{251971}{36416}a^{8}+\frac{14315}{9104}a^{7}-\frac{123785}{9104}a^{6}-\frac{122061}{18208}a^{5}+\frac{58871}{18208}a^{4}+\frac{173049}{36416}a^{3}-\frac{45811}{36416}a^{2}-\frac{40155}{36416}a-\frac{17271}{36416}$, $\frac{7693}{36416}a^{19}-\frac{6711}{36416}a^{18}-\frac{2769}{36416}a^{17}+\frac{699}{36416}a^{16}-\frac{62155}{36416}a^{15}+\frac{53137}{36416}a^{14}-\frac{146951}{36416}a^{13}+\frac{141501}{36416}a^{12}+\frac{21261}{36416}a^{11}+\frac{30393}{36416}a^{10}+\frac{198749}{36416}a^{9}-\frac{136063}{36416}a^{8}+\frac{44405}{9104}a^{7}-\frac{50407}{9104}a^{6}-\frac{153737}{18208}a^{5}+\frac{77755}{18208}a^{4}+\frac{190817}{36416}a^{3}-\frac{167779}{36416}a^{2}-\frac{23823}{36416}a+\frac{18501}{36416}$, $\frac{13455}{36416}a^{19}+\frac{2135}{36416}a^{18}-\frac{6103}{36416}a^{17}-\frac{2243}{36416}a^{16}-\frac{109553}{36416}a^{15}-\frac{17233}{36416}a^{14}-\frac{246593}{36416}a^{13}-\frac{29765}{36416}a^{12}+\frac{69815}{36416}a^{11}+\frac{37255}{36416}a^{10}+\frac{371243}{36416}a^{9}+\frac{70599}{36416}a^{8}+\frac{74137}{9104}a^{7}+\frac{9391}{9104}a^{6}-\frac{285079}{18208}a^{5}-\frac{56355}{18208}a^{4}+\frac{335915}{36416}a^{3}+\frac{81859}{36416}a^{2}-\frac{39137}{36416}a-\frac{42605}{36416}$ Copy content Toggle raw display
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:  \( 7917.88619089 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 7917.88619089 \cdot 1}{2\cdot\sqrt{436011848767111570653184}}\cr\approx \mathstrut & 0.233017770773 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^16 - 22*x^14 - 6*x^12 + 22*x^10 + 33*x^8 - 22*x^6 + 19*x^4 - 2*x^2 + 1)
 
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^20 - 8*x^16 - 22*x^14 - 6*x^12 + 22*x^10 + 33*x^8 - 22*x^6 + 19*x^4 - 2*x^2 + 1, 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^20 - 8*x^16 - 22*x^14 - 6*x^12 + 22*x^10 + 33*x^8 - 22*x^6 + 19*x^4 - 2*x^2 + 1);
 
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^20 - 8*x^16 - 22*x^14 - 6*x^12 + 22*x^10 + 33*x^8 - 22*x^6 + 19*x^4 - 2*x^2 + 1);
 
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

$C_2\times A_5$ (as 20T36):

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 non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

10.2.82538991616.1

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 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 10.0.5530112438272.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{10}$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.3.0.1}{3} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.12.16$x^{8} + 8 x^{7} + 32 x^{6} + 78 x^{5} + 137 x^{4} + 186 x^{3} + 128 x^{2} - 10 x + 7$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
2.12.18.48$x^{12} + 6 x^{11} + 18 x^{10} + 32 x^{9} + 34 x^{8} + 16 x^{7} - 8 x^{6} - 16 x^{5} - 12 x^{4} - 8 x^{3} - 8 x^{2} + 72$$4$$3$$18$$A_4\times C_2$$[2, 2]^{6}$
\(67\) Copy content Toggle raw display 67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} + 134$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$