Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 1)
gp: K = bnfinit(y^20 - 2*y^19 + 5*y^18 - 10*y^17 + 21*y^16 - 29*y^15 + 42*y^14 - 62*y^13 + 78*y^12 - 77*y^11 + 89*y^10 - 77*y^9 + 92*y^8 - 54*y^7 + 45*y^6 - 18*y^5 + 24*y^4 - 7*y^3 + 4*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 1)
\( x^{20} - 2 x^{19} + 5 x^{18} - 10 x^{17} + 21 x^{16} - 29 x^{15} + 42 x^{14} - 62 x^{13} + 78 x^{12} + \cdots + 1 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(157092982407310367598961\)
\(\medspace = 11^{16}\cdot 43^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.45\) | 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: | $11^{4/5}43^{1/2}\approx 44.65276699099921$ | ||
| Ramified primes: |
\(11\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11}a^{15}+\frac{2}{11}a^{14}+\frac{3}{11}a^{12}+\frac{3}{11}a^{11}+\frac{4}{11}a^{10}-\frac{5}{11}a^{9}+\frac{3}{11}a^{6}-\frac{4}{11}a^{5}-\frac{5}{11}a^{4}-\frac{1}{11}a^{3}+\frac{1}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{16}-\frac{4}{11}a^{14}+\frac{3}{11}a^{13}-\frac{3}{11}a^{12}-\frac{2}{11}a^{11}-\frac{2}{11}a^{10}-\frac{1}{11}a^{9}+\frac{3}{11}a^{7}+\frac{1}{11}a^{6}+\frac{3}{11}a^{5}-\frac{2}{11}a^{4}+\frac{2}{11}a^{3}+\frac{1}{11}a^{2}-\frac{3}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{17}-\frac{3}{11}a^{13}-\frac{1}{11}a^{12}-\frac{1}{11}a^{11}+\frac{4}{11}a^{10}+\frac{2}{11}a^{9}+\frac{3}{11}a^{8}+\frac{1}{11}a^{7}+\frac{4}{11}a^{6}+\frac{4}{11}a^{5}+\frac{4}{11}a^{4}-\frac{3}{11}a^{3}-\frac{3}{11}a^{2}-\frac{5}{11}a-\frac{4}{11}$, $\frac{1}{253}a^{18}-\frac{8}{253}a^{17}-\frac{10}{253}a^{16}+\frac{2}{253}a^{15}+\frac{41}{253}a^{14}+\frac{59}{253}a^{13}-\frac{89}{253}a^{12}-\frac{94}{253}a^{11}+\frac{108}{253}a^{10}-\frac{24}{253}a^{9}+\frac{76}{253}a^{8}-\frac{56}{253}a^{7}+\frac{56}{253}a^{6}+\frac{3}{23}a^{5}-\frac{47}{253}a^{4}-\frac{67}{253}a^{3}+\frac{75}{253}a^{2}-\frac{31}{253}a+\frac{65}{253}$, $\frac{1}{6985583}a^{19}+\frac{548}{635053}a^{18}+\frac{123213}{6985583}a^{17}+\frac{183258}{6985583}a^{16}-\frac{305402}{6985583}a^{15}+\frac{2564601}{6985583}a^{14}+\frac{1776630}{6985583}a^{13}-\frac{1259268}{6985583}a^{12}-\frac{2045233}{6985583}a^{11}+\frac{664468}{6985583}a^{10}-\frac{884077}{6985583}a^{9}-\frac{321894}{6985583}a^{8}-\frac{275659}{635053}a^{7}-\frac{1009544}{6985583}a^{6}-\frac{843380}{6985583}a^{5}-\frac{988157}{6985583}a^{4}-\frac{1126882}{6985583}a^{3}-\frac{2626801}{6985583}a^{2}-\frac{283766}{6985583}a-\frac{3230574}{6985583}$
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: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{8094}{6985583}a^{19}+\frac{3729480}{6985583}a^{18}-\frac{219500}{303721}a^{17}+\frac{14136553}{6985583}a^{16}-\frac{26914516}{6985583}a^{15}+\frac{56919798}{6985583}a^{14}-\frac{64513375}{6985583}a^{13}+\frac{4342244}{303721}a^{12}-\frac{153292346}{6985583}a^{11}+\frac{168670956}{6985583}a^{10}-\frac{142856570}{6985583}a^{9}+\frac{206162872}{6985583}a^{8}-\frac{130592090}{6985583}a^{7}+\frac{213631733}{6985583}a^{6}-\frac{52566312}{6985583}a^{5}+\frac{83514109}{6985583}a^{4}-\frac{26333673}{6985583}a^{3}+\frac{56297476}{6985583}a^{2}-\frac{36191}{6985583}a+\frac{2471087}{6985583}$, $\frac{2516310}{6985583}a^{19}-\frac{4894205}{6985583}a^{18}+\frac{10251037}{6985583}a^{17}-\frac{22333364}{6985583}a^{16}+\frac{45094020}{6985583}a^{15}-\frac{58944320}{6985583}a^{14}+\frac{77990929}{6985583}a^{13}-\frac{130437838}{6985583}a^{12}+\frac{156051482}{6985583}a^{11}-\frac{132937791}{6985583}a^{10}+\frac{169903639}{6985583}a^{9}-\frac{164097379}{6985583}a^{8}+\frac{171751843}{6985583}a^{7}-\frac{121448942}{6985583}a^{6}+\frac{44269425}{6985583}a^{5}-\frac{73104101}{6985583}a^{4}+\frac{49350644}{6985583}a^{3}-\frac{24091852}{6985583}a^{2}-\frac{10835508}{6985583}a-\frac{3166616}{6985583}$, $\frac{3166616}{6985583}a^{19}-\frac{3816922}{6985583}a^{18}+\frac{10938875}{6985583}a^{17}-\frac{21415123}{6985583}a^{16}+\frac{4015052}{635053}a^{15}-\frac{46737844}{6985583}a^{14}+\frac{74053552}{6985583}a^{13}-\frac{118339263}{6985583}a^{12}+\frac{116558210}{6985583}a^{11}-\frac{87777950}{6985583}a^{10}+\frac{148891033}{6985583}a^{9}-\frac{73925793}{6985583}a^{8}+\frac{127231293}{6985583}a^{7}+\frac{754579}{6985583}a^{6}+\frac{21048778}{6985583}a^{5}-\frac{12729663}{6985583}a^{4}+\frac{263153}{635053}a^{3}+\frac{27184332}{6985583}a^{2}-\frac{496756}{303721}a-\frac{697172}{635053}$, $\frac{5594659}{6985583}a^{19}-\frac{908249}{635053}a^{18}+\frac{2089484}{635053}a^{17}-\frac{47923394}{6985583}a^{16}+\frac{97482983}{6985583}a^{15}-\frac{124095727}{6985583}a^{14}+\frac{171212856}{6985583}a^{13}-\frac{274123428}{6985583}a^{12}+\frac{317349191}{6985583}a^{11}-\frac{271024014}{6985583}a^{10}+\frac{346994420}{6985583}a^{9}-\frac{286078777}{6985583}a^{8}+\frac{331195435}{6985583}a^{7}-\frac{176995020}{6985583}a^{6}+\frac{75449358}{6985583}a^{5}-\frac{7996718}{635053}a^{4}+\frac{53976861}{6985583}a^{3}-\frac{1133007}{303721}a^{2}-\frac{2112214}{635053}a-\frac{6911375}{6985583}$, $\frac{4241960}{6985583}a^{19}-\frac{4788923}{6985583}a^{18}+\frac{14673152}{6985583}a^{17}-\frac{27410824}{6985583}a^{16}+\frac{58092811}{6985583}a^{15}-\frac{59845102}{6985583}a^{14}+\frac{99032229}{6985583}a^{13}-\frac{154120336}{6985583}a^{12}+\frac{156831275}{6985583}a^{11}-\frac{11408659}{635053}a^{10}+\frac{9098052}{303721}a^{9}-\frac{9799412}{635053}a^{8}+\frac{19257859}{635053}a^{7}-\frac{17683585}{6985583}a^{6}+\frac{3393471}{303721}a^{5}-\frac{19108865}{6985583}a^{4}+\frac{53395095}{6985583}a^{3}+\frac{16830445}{6985583}a^{2}-\frac{6517202}{6985583}a+\frac{467493}{6985583}$, $\frac{1201894}{6985583}a^{19}-\frac{67180}{303721}a^{18}+\frac{470985}{635053}a^{17}-\frac{8543243}{6985583}a^{16}+\frac{19544477}{6985583}a^{15}-\frac{22148518}{6985583}a^{14}+\frac{36870447}{6985583}a^{13}-\frac{49531265}{6985583}a^{12}+\frac{2636092}{303721}a^{11}-\frac{57671600}{6985583}a^{10}+\frac{72395875}{6985583}a^{9}-\frac{1948671}{303721}a^{8}+\frac{95893687}{6985583}a^{7}-\frac{15191995}{6985583}a^{6}+\frac{64236806}{6985583}a^{5}+\frac{4057624}{6985583}a^{4}+\frac{43298690}{6985583}a^{3}-\frac{8309108}{6985583}a^{2}+\frac{7230700}{6985583}a+\frac{3587509}{6985583}$, $\frac{6915665}{6985583}a^{19}-\frac{9509584}{6985583}a^{18}+\frac{26000694}{6985583}a^{17}-\frac{49074986}{6985583}a^{16}+\frac{103202049}{6985583}a^{15}-\frac{114726058}{6985583}a^{14}+\frac{173252699}{6985583}a^{13}-\frac{264279881}{6985583}a^{12}+\frac{285431917}{6985583}a^{11}-\frac{224427232}{6985583}a^{10}+\frac{324082238}{6985583}a^{9}-\frac{16585152}{635053}a^{8}+\frac{331677127}{6985583}a^{7}-\frac{35164031}{6985583}a^{6}+\frac{88476334}{6985583}a^{5}+\frac{2103480}{6985583}a^{4}+\frac{64603977}{6985583}a^{3}+\frac{14251063}{6985583}a^{2}-\frac{8589270}{6985583}a+\frac{3570265}{6985583}$, $\frac{4701792}{6985583}a^{19}-\frac{5038416}{6985583}a^{18}+\frac{17853631}{6985583}a^{17}-\frac{120877}{27611}a^{16}+\frac{2962416}{303721}a^{15}-\frac{70917168}{6985583}a^{14}+\frac{124999106}{6985583}a^{13}-\frac{176915251}{6985583}a^{12}+\frac{17544831}{635053}a^{11}-\frac{171116577}{6985583}a^{10}+\frac{257372342}{6985583}a^{9}-\frac{127261851}{6985583}a^{8}+\frac{287026934}{6985583}a^{7}-\frac{1075414}{6985583}a^{6}+\frac{165001534}{6985583}a^{5}+\frac{24112536}{6985583}a^{4}+\frac{88909125}{6985583}a^{3}+\frac{41729847}{6985583}a^{2}+\frac{21326227}{6985583}a+\frac{3838990}{6985583}$, $\frac{99638}{635053}a^{19}-\frac{4629835}{6985583}a^{18}+\frac{8922680}{6985583}a^{17}-\frac{22557385}{6985583}a^{16}+\frac{44860814}{6985583}a^{15}-\frac{79166666}{6985583}a^{14}+\frac{108023826}{6985583}a^{13}-\frac{172426365}{6985583}a^{12}+\frac{234551087}{6985583}a^{11}-\frac{263416178}{6985583}a^{10}+\frac{290940897}{6985583}a^{9}-\frac{332458818}{6985583}a^{8}+\frac{283282761}{6985583}a^{7}-\frac{310223302}{6985583}a^{6}+\frac{159221636}{6985583}a^{5}-\frac{180521585}{6985583}a^{4}+\frac{71853051}{6985583}a^{3}-\frac{61084706}{6985583}a^{2}-\frac{4576279}{6985583}a-\frac{13183230}{6985583}$
| 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: | \( 1838.7824 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 1838.7824 \cdot 1}{2\cdot\sqrt{157092982407310367598961}}\cr\approx \mathstrut & 0.22244397 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 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 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 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 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 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 - 2*x^19 + 5*x^18 - 10*x^17 + 21*x^16 - 29*x^15 + 42*x^14 - 62*x^13 + 78*x^12 - 77*x^11 + 89*x^10 - 77*x^9 + 92*x^8 - 54*x^7 + 45*x^6 - 18*x^5 + 24*x^4 - 7*x^3 + 4*x^2 + x + 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
$C_2\wr C_5$ (as 20T41):
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 160 |
| The 16 conjugacy class representatives for $C_2\wr C_5$ |
| Character table for $C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.4.9217431883.1 x2, 10.2.396349570969.1 |
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 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.4.9217431883.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
|
\(43\)
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |