Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 3*y^18 - 4*y^17 + 9*y^16 - 10*y^15 + y^14 + 10*y^13 - 14*y^12 + 8*y^11 + 16*y^10 - 34*y^9 + 36*y^8 - 29*y^7 + 24*y^6 - 12*y^5 + 8*y^4 - 6*y^3 + 3*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - x + 1)
\( x^{20} - 3 x^{19} + 3 x^{18} - 4 x^{17} + 9 x^{16} - 10 x^{15} + x^{14} + 10 x^{13} - 14 x^{12} + \cdots + 1 \)
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: |
\(116354803848679984543641\)
\(\medspace = 3^{10}\cdot 7^{10}\cdot 17^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.23\) | 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: | $3^{1/2}7^{1/2}17^{1/2}\approx 18.894443627691185$ | ||
| Ramified primes: |
\(3\), \(7\), \(17\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13}a^{16}-\frac{3}{13}a^{15}-\frac{2}{13}a^{14}-\frac{1}{13}a^{13}-\frac{3}{13}a^{12}-\frac{2}{13}a^{11}+\frac{1}{13}a^{10}-\frac{3}{13}a^{9}-\frac{3}{13}a^{8}-\frac{3}{13}a^{7}-\frac{4}{13}a^{6}-\frac{4}{13}a^{5}+\frac{6}{13}a^{4}+\frac{5}{13}a^{3}+\frac{1}{13}a^{2}+\frac{6}{13}a-\frac{2}{13}$, $\frac{1}{13}a^{17}+\frac{2}{13}a^{15}+\frac{6}{13}a^{14}-\frac{6}{13}a^{13}+\frac{2}{13}a^{12}-\frac{5}{13}a^{11}+\frac{1}{13}a^{9}+\frac{1}{13}a^{8}-\frac{3}{13}a^{6}-\frac{6}{13}a^{5}-\frac{3}{13}a^{4}+\frac{3}{13}a^{3}-\frac{4}{13}a^{2}+\frac{3}{13}a-\frac{6}{13}$, $\frac{1}{13}a^{18}-\frac{1}{13}a^{15}-\frac{2}{13}a^{14}+\frac{4}{13}a^{13}+\frac{1}{13}a^{12}+\frac{4}{13}a^{11}-\frac{1}{13}a^{10}-\frac{6}{13}a^{9}+\frac{6}{13}a^{8}+\frac{3}{13}a^{7}+\frac{2}{13}a^{6}+\frac{5}{13}a^{5}+\frac{4}{13}a^{4}-\frac{1}{13}a^{3}+\frac{1}{13}a^{2}-\frac{5}{13}a+\frac{4}{13}$, $\frac{1}{924443}a^{19}+\frac{25134}{924443}a^{18}-\frac{27874}{924443}a^{17}-\frac{12059}{924443}a^{16}+\frac{90230}{924443}a^{15}-\frac{187178}{924443}a^{14}-\frac{176292}{924443}a^{13}-\frac{2139}{71111}a^{12}+\frac{249890}{924443}a^{11}+\frac{23090}{54379}a^{10}-\frac{51290}{924443}a^{9}+\frac{2829}{71111}a^{8}+\frac{9571}{54379}a^{7}-\frac{125557}{924443}a^{6}+\frac{206561}{924443}a^{5}-\frac{130264}{924443}a^{4}+\frac{16853}{54379}a^{3}-\frac{346449}{924443}a^{2}+\frac{346771}{924443}a+\frac{280690}{924443}$
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: |
\( -\frac{64590}{924443} a^{19} - \frac{367596}{924443} a^{18} + \frac{1415582}{924443} a^{17} - \frac{81126}{71111} a^{16} + \frac{1012970}{924443} a^{15} - \frac{3736306}{924443} a^{14} + \frac{343099}{71111} a^{13} + \frac{572491}{924443} a^{12} - \frac{5949199}{924443} a^{11} + \frac{375092}{54379} a^{10} - \frac{2302609}{924443} a^{9} - \frac{9563460}{924443} a^{8} + \frac{922391}{54379} a^{7} - \frac{12287346}{924443} a^{6} + \frac{6470820}{924443} a^{5} - \frac{6430639}{924443} a^{4} + \frac{178323}{54379} a^{3} - \frac{2699677}{924443} a^{2} + \frac{1670555}{924443} a - \frac{373205}{924443} \)
(order $6$)
| 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: |
$\frac{720490}{924443}a^{19}-\frac{1896042}{924443}a^{18}+\frac{1368136}{924443}a^{17}-\frac{145832}{71111}a^{16}+\frac{5043159}{924443}a^{15}-\frac{4905709}{924443}a^{14}-\frac{2065985}{924443}a^{13}+\frac{8538808}{924443}a^{12}-\frac{7601459}{924443}a^{11}+\frac{21924}{54379}a^{10}+\frac{15467913}{924443}a^{9}-\frac{20257836}{924443}a^{8}+\frac{870498}{54379}a^{7}-\frac{6953156}{924443}a^{6}+\frac{7802973}{924443}a^{5}-\frac{3958223}{924443}a^{4}+\frac{227269}{54379}a^{3}-\frac{2194472}{924443}a^{2}+\frac{379284}{924443}a+\frac{698535}{924443}$, $\frac{663084}{924443}a^{19}-\frac{1185146}{924443}a^{18}-\frac{414517}{924443}a^{17}-\frac{195783}{924443}a^{16}+\frac{2891689}{924443}a^{15}-\frac{6649}{71111}a^{14}-\frac{7058279}{924443}a^{13}+\frac{7247724}{924443}a^{12}+\frac{470774}{924443}a^{11}-\frac{458804}{54379}a^{10}+\frac{16324030}{924443}a^{9}-\frac{6752729}{924443}a^{8}-\frac{25705}{4183}a^{7}+\frac{776829}{71111}a^{6}-\frac{2589638}{924443}a^{5}+\frac{5117520}{924443}a^{4}-\frac{55338}{54379}a^{3}+\frac{292338}{924443}a^{2}-\frac{538956}{924443}a+\frac{663218}{924443}$, $\frac{506469}{924443}a^{19}-\frac{1197151}{924443}a^{18}+\frac{421035}{924443}a^{17}-\frac{496991}{924443}a^{16}+\frac{2627048}{924443}a^{15}-\frac{1922604}{924443}a^{14}-\frac{3799119}{924443}a^{13}+\frac{7403389}{924443}a^{12}-\frac{276705}{71111}a^{11}-\frac{244674}{54379}a^{10}+\frac{13919935}{924443}a^{9}-\frac{11924909}{924443}a^{8}+\frac{158582}{54379}a^{7}+\frac{5332398}{924443}a^{6}-\frac{2328979}{924443}a^{5}+\frac{1752429}{924443}a^{4}+\frac{2069}{4183}a^{3}+\frac{429475}{924443}a^{2}+\frac{591574}{924443}a+\frac{650180}{924443}$, $\frac{373205}{924443}a^{19}-\frac{1184205}{924443}a^{18}+\frac{752019}{924443}a^{17}-\frac{77238}{924443}a^{16}+\frac{2304207}{924443}a^{15}-\frac{209160}{71111}a^{14}-\frac{3363101}{924443}a^{13}+\frac{8192337}{924443}a^{12}-\frac{4652379}{924443}a^{11}-\frac{174327}{54379}a^{10}+\frac{12347844}{924443}a^{9}-\frac{14991579}{924443}a^{8}+\frac{17520}{4183}a^{7}+\frac{4857702}{924443}a^{6}-\frac{3330426}{924443}a^{5}+\frac{1992360}{924443}a^{4}-\frac{202647}{54379}a^{3}+\frac{792261}{924443}a^{2}-\frac{1580062}{924443}a+\frac{1297350}{924443}$, $\frac{30303}{71111}a^{19}-\frac{872813}{924443}a^{18}+\frac{376045}{924443}a^{17}-\frac{1220044}{924443}a^{16}+\frac{3198171}{924443}a^{15}-\frac{2002686}{924443}a^{14}-\frac{936033}{924443}a^{13}+\frac{1525553}{924443}a^{12}-\frac{1096140}{924443}a^{11}+\frac{17618}{54379}a^{10}+\frac{4970445}{924443}a^{9}-\frac{5167196}{924443}a^{8}+\frac{306648}{54379}a^{7}-\frac{8151850}{924443}a^{6}+\frac{10000908}{924443}a^{5}-\frac{5501180}{924443}a^{4}+\frac{377804}{54379}a^{3}-\frac{3470300}{924443}a^{2}+\frac{1394415}{924443}a-\frac{50973}{71111}$, $\frac{587000}{924443}a^{19}-\frac{1163390}{924443}a^{18}+\frac{34212}{924443}a^{17}-\frac{670726}{924443}a^{16}+\frac{2888309}{924443}a^{15}-\frac{733232}{924443}a^{14}-\frac{370010}{71111}a^{13}+\frac{6296597}{924443}a^{12}-\frac{2340800}{924443}a^{11}-\frac{17843}{4183}a^{10}+\frac{14465157}{924443}a^{9}-\frac{10231697}{924443}a^{8}+\frac{81726}{54379}a^{7}+\frac{4308056}{924443}a^{6}-\frac{3259095}{924443}a^{5}+\frac{5454737}{924443}a^{4}-\frac{12889}{54379}a^{3}+\frac{1651}{71111}a^{2}-\frac{589389}{924443}a-\frac{579220}{924443}$, $\frac{20086}{54379}a^{19}-\frac{71674}{54379}a^{18}+\frac{75948}{54379}a^{17}-\frac{71570}{54379}a^{16}+\frac{221435}{54379}a^{15}-\frac{290633}{54379}a^{14}+\frac{17146}{54379}a^{13}+\frac{246287}{54379}a^{12}-\frac{234166}{54379}a^{11}+\frac{109507}{54379}a^{10}+\frac{266927}{54379}a^{9}-\frac{52957}{4183}a^{8}+\frac{661829}{54379}a^{7}-\frac{684848}{54379}a^{6}+\frac{874649}{54379}a^{5}-\frac{606007}{54379}a^{4}+\frac{239842}{54379}a^{3}-\frac{170062}{54379}a^{2}+\frac{87176}{54379}a+\frac{4097}{54379}$, $\frac{26007}{71111}a^{19}-\frac{1038638}{924443}a^{18}+\frac{1458961}{924443}a^{17}-\frac{2379069}{924443}a^{16}+\frac{3669701}{924443}a^{15}-\frac{4433849}{924443}a^{14}+\frac{3283849}{924443}a^{13}+\frac{1426649}{924443}a^{12}-\frac{6823546}{924443}a^{11}+\frac{411557}{54379}a^{10}+\frac{1934401}{924443}a^{9}-\frac{11518958}{924443}a^{8}+\frac{1201074}{54379}a^{7}-\frac{21029093}{924443}a^{6}+\frac{15510813}{924443}a^{5}-\frac{8088159}{924443}a^{4}+\frac{26329}{4183}a^{3}-\frac{2156318}{924443}a^{2}+\frac{1012903}{924443}a-\frac{786707}{924443}$, $\frac{502855}{924443}a^{19}-\frac{1436013}{924443}a^{18}+\frac{1175605}{924443}a^{17}-\frac{1502362}{924443}a^{16}+\frac{318785}{71111}a^{15}-\frac{4429140}{924443}a^{14}-\frac{705507}{924443}a^{13}+\frac{5640269}{924443}a^{12}-\frac{5154367}{924443}a^{11}+\frac{39177}{54379}a^{10}+\frac{9344514}{924443}a^{9}-\frac{13992747}{924443}a^{8}+\frac{681090}{54379}a^{7}-\frac{9241650}{924443}a^{6}+\frac{10909491}{924443}a^{5}-\frac{7188281}{924443}a^{4}+\frac{208687}{54379}a^{3}-\frac{1617435}{924443}a^{2}+\frac{1048110}{924443}a-\frac{218397}{924443}$
| 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: | \( 4706.41314047 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 4706.41314047 \cdot 1}{6\cdot\sqrt{116354803848679984543641}}\cr\approx \mathstrut & 0.220518648409 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*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 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*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 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*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 - 3*x^19 + 3*x^18 - 4*x^17 + 9*x^16 - 10*x^15 + x^14 + 10*x^13 - 14*x^12 + 8*x^11 + 16*x^10 - 34*x^9 + 36*x^8 - 29*x^7 + 24*x^6 - 12*x^5 + 8*x^4 - 6*x^3 + 3*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
$C_2\times D_{10}$ (as 20T8):
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 40 |
| The 16 conjugacy class representatives for $C_2\times D_{10}$ |
| Character table for $C_2\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 5.1.14161.1, 10.0.48729742803.1, 10.2.341108199621.1, 10.0.1403737447.1 |
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
| Galois closure: | deg 40 |
| Degree 20 siblings: | 20.0.686255883923847255777801.1, 20.4.33626538312268515533112249.1, deg 20 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | ${\href{/padicField/13.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |