Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 102*x^10 + 199*x^9 + 3833*x^8 - 6937*x^7 - 66192*x^6 + 105589*x^5 + 514762*x^4 - 647862*x^3 - 1393261*x^2 + 752713*x + 494507)
gp: K = bnfinit(y^12 - 2*y^11 - 102*y^10 + 199*y^9 + 3833*y^8 - 6937*y^7 - 66192*y^6 + 105589*y^5 + 514762*y^4 - 647862*y^3 - 1393261*y^2 + 752713*y + 494507, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 102*x^10 + 199*x^9 + 3833*x^8 - 6937*x^7 - 66192*x^6 + 105589*x^5 + 514762*x^4 - 647862*x^3 - 1393261*x^2 + 752713*x + 494507);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 102*x^10 + 199*x^9 + 3833*x^8 - 6937*x^7 - 66192*x^6 + 105589*x^5 + 514762*x^4 - 647862*x^3 - 1393261*x^2 + 752713*x + 494507)
\( x^{12} - 2 x^{11} - 102 x^{10} + 199 x^{9} + 3833 x^{8} - 6937 x^{7} - 66192 x^{6} + 105589 x^{5} + \cdots + 494507 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(16551311845247868759889\)
\(\medspace = 7^{8}\cdot 13^{6}\cdot 29^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(71.05\) | 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: | $7^{2/3}13^{1/2}29^{1/2}\approx 71.05086481765254$ | ||
| Ramified primes: |
\(7\), \(13\), \(29\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{80\!\cdots\!88}a^{11}-\frac{92\!\cdots\!37}{80\!\cdots\!88}a^{10}+\frac{18\!\cdots\!21}{80\!\cdots\!88}a^{9}+\frac{10\!\cdots\!35}{10\!\cdots\!86}a^{8}-\frac{89\!\cdots\!19}{80\!\cdots\!88}a^{7}+\frac{21\!\cdots\!60}{50\!\cdots\!43}a^{6}+\frac{24\!\cdots\!14}{50\!\cdots\!43}a^{5}-\frac{30\!\cdots\!39}{80\!\cdots\!88}a^{4}-\frac{28\!\cdots\!69}{80\!\cdots\!88}a^{3}-\frac{24\!\cdots\!75}{80\!\cdots\!88}a^{2}-\frac{69\!\cdots\!71}{20\!\cdots\!72}a-\frac{83\!\cdots\!95}{80\!\cdots\!88}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (assuming GRH)
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 $2$)
| 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: |
$\frac{750185696004763}{15\!\cdots\!56}a^{11}-\frac{45\!\cdots\!23}{15\!\cdots\!56}a^{10}-\frac{57\!\cdots\!05}{15\!\cdots\!56}a^{9}+\frac{93\!\cdots\!45}{37\!\cdots\!64}a^{8}+\frac{12\!\cdots\!03}{15\!\cdots\!56}a^{7}-\frac{12\!\cdots\!31}{18\!\cdots\!82}a^{6}-\frac{90\!\cdots\!23}{18\!\cdots\!82}a^{5}+\frac{96\!\cdots\!39}{15\!\cdots\!56}a^{4}-\frac{36\!\cdots\!59}{15\!\cdots\!56}a^{3}-\frac{23\!\cdots\!61}{15\!\cdots\!56}a^{2}+\frac{58\!\cdots\!95}{94\!\cdots\!91}a+\frac{55\!\cdots\!51}{15\!\cdots\!56}$, $\frac{142313445355377}{75\!\cdots\!28}a^{11}-\frac{13\!\cdots\!03}{75\!\cdots\!28}a^{10}-\frac{94\!\cdots\!99}{75\!\cdots\!28}a^{9}+\frac{56\!\cdots\!51}{37\!\cdots\!64}a^{8}+\frac{12\!\cdots\!31}{75\!\cdots\!28}a^{7}-\frac{77\!\cdots\!63}{18\!\cdots\!82}a^{6}+\frac{42\!\cdots\!77}{18\!\cdots\!82}a^{5}+\frac{31\!\cdots\!65}{75\!\cdots\!28}a^{4}-\frac{33\!\cdots\!39}{75\!\cdots\!28}a^{3}-\frac{93\!\cdots\!71}{75\!\cdots\!28}a^{2}+\frac{23\!\cdots\!39}{37\!\cdots\!64}a+\frac{35\!\cdots\!43}{75\!\cdots\!28}$, $\frac{16\!\cdots\!27}{80\!\cdots\!88}a^{11}+\frac{69\!\cdots\!37}{80\!\cdots\!88}a^{10}-\frac{16\!\cdots\!81}{80\!\cdots\!88}a^{9}-\frac{70\!\cdots\!09}{10\!\cdots\!86}a^{8}+\frac{62\!\cdots\!83}{80\!\cdots\!88}a^{7}+\frac{89\!\cdots\!78}{50\!\cdots\!43}a^{6}-\frac{70\!\cdots\!06}{50\!\cdots\!43}a^{5}-\frac{99\!\cdots\!37}{80\!\cdots\!88}a^{4}+\frac{98\!\cdots\!97}{80\!\cdots\!88}a^{3}-\frac{34\!\cdots\!17}{80\!\cdots\!88}a^{2}-\frac{68\!\cdots\!81}{20\!\cdots\!72}a+\frac{18\!\cdots\!59}{80\!\cdots\!88}$, $\frac{52\!\cdots\!85}{80\!\cdots\!88}a^{11}-\frac{35\!\cdots\!77}{80\!\cdots\!88}a^{10}-\frac{39\!\cdots\!39}{80\!\cdots\!88}a^{9}+\frac{73\!\cdots\!63}{20\!\cdots\!72}a^{8}+\frac{83\!\cdots\!73}{80\!\cdots\!88}a^{7}-\frac{98\!\cdots\!77}{10\!\cdots\!86}a^{6}-\frac{41\!\cdots\!93}{10\!\cdots\!86}a^{5}+\frac{78\!\cdots\!25}{80\!\cdots\!88}a^{4}-\frac{38\!\cdots\!05}{80\!\cdots\!88}a^{3}-\frac{21\!\cdots\!59}{80\!\cdots\!88}a^{2}+\frac{47\!\cdots\!98}{50\!\cdots\!43}a+\frac{70\!\cdots\!41}{80\!\cdots\!88}$, $\frac{13\!\cdots\!15}{80\!\cdots\!88}a^{11}-\frac{14\!\cdots\!75}{80\!\cdots\!88}a^{10}-\frac{84\!\cdots\!73}{80\!\cdots\!88}a^{9}+\frac{31\!\cdots\!41}{20\!\cdots\!72}a^{8}+\frac{70\!\cdots\!43}{80\!\cdots\!88}a^{7}-\frac{42\!\cdots\!55}{10\!\cdots\!86}a^{6}+\frac{36\!\cdots\!33}{10\!\cdots\!86}a^{5}+\frac{35\!\cdots\!27}{80\!\cdots\!88}a^{4}-\frac{45\!\cdots\!91}{80\!\cdots\!88}a^{3}-\frac{95\!\cdots\!49}{80\!\cdots\!88}a^{2}+\frac{48\!\cdots\!32}{50\!\cdots\!43}a+\frac{19\!\cdots\!19}{80\!\cdots\!88}$, $\frac{27\!\cdots\!73}{80\!\cdots\!88}a^{11}-\frac{35\!\cdots\!97}{80\!\cdots\!88}a^{10}-\frac{25\!\cdots\!83}{80\!\cdots\!88}a^{9}+\frac{72\!\cdots\!25}{20\!\cdots\!72}a^{8}+\frac{84\!\cdots\!85}{80\!\cdots\!88}a^{7}-\frac{87\!\cdots\!65}{10\!\cdots\!86}a^{6}-\frac{15\!\cdots\!29}{10\!\cdots\!86}a^{5}+\frac{57\!\cdots\!33}{80\!\cdots\!88}a^{4}+\frac{71\!\cdots\!07}{80\!\cdots\!88}a^{3}-\frac{85\!\cdots\!47}{80\!\cdots\!88}a^{2}-\frac{17\!\cdots\!07}{10\!\cdots\!86}a-\frac{63\!\cdots\!87}{80\!\cdots\!88}$, $\frac{11\!\cdots\!59}{80\!\cdots\!88}a^{11}-\frac{11\!\cdots\!63}{80\!\cdots\!88}a^{10}-\frac{76\!\cdots\!33}{80\!\cdots\!88}a^{9}+\frac{24\!\cdots\!53}{20\!\cdots\!72}a^{8}+\frac{12\!\cdots\!11}{80\!\cdots\!88}a^{7}-\frac{33\!\cdots\!61}{10\!\cdots\!86}a^{6}+\frac{22\!\cdots\!53}{10\!\cdots\!86}a^{5}+\frac{28\!\cdots\!19}{80\!\cdots\!88}a^{4}-\frac{33\!\cdots\!63}{80\!\cdots\!88}a^{3}-\frac{10\!\cdots\!57}{80\!\cdots\!88}a^{2}-\frac{40\!\cdots\!26}{50\!\cdots\!43}a+\frac{13\!\cdots\!79}{80\!\cdots\!88}$, $\frac{23\!\cdots\!67}{80\!\cdots\!88}a^{11}-\frac{15\!\cdots\!95}{80\!\cdots\!88}a^{10}-\frac{18\!\cdots\!09}{80\!\cdots\!88}a^{9}+\frac{78\!\cdots\!81}{50\!\cdots\!43}a^{8}+\frac{45\!\cdots\!91}{80\!\cdots\!88}a^{7}-\frac{21\!\cdots\!54}{50\!\cdots\!43}a^{6}-\frac{21\!\cdots\!82}{50\!\cdots\!43}a^{5}+\frac{34\!\cdots\!23}{80\!\cdots\!88}a^{4}+\frac{89\!\cdots\!61}{80\!\cdots\!88}a^{3}-\frac{10\!\cdots\!77}{80\!\cdots\!88}a^{2}-\frac{15\!\cdots\!19}{20\!\cdots\!72}a+\frac{54\!\cdots\!11}{80\!\cdots\!88}$, $\frac{25\!\cdots\!69}{20\!\cdots\!72}a^{11}-\frac{62\!\cdots\!77}{10\!\cdots\!86}a^{10}-\frac{20\!\cdots\!41}{20\!\cdots\!72}a^{9}+\frac{10\!\cdots\!23}{20\!\cdots\!72}a^{8}+\frac{24\!\cdots\!21}{10\!\cdots\!86}a^{7}-\frac{13\!\cdots\!81}{10\!\cdots\!86}a^{6}-\frac{19\!\cdots\!43}{10\!\cdots\!86}a^{5}+\frac{25\!\cdots\!63}{20\!\cdots\!72}a^{4}-\frac{16\!\cdots\!34}{50\!\cdots\!43}a^{3}-\frac{62\!\cdots\!07}{20\!\cdots\!72}a^{2}+\frac{19\!\cdots\!97}{20\!\cdots\!72}a+\frac{46\!\cdots\!08}{50\!\cdots\!43}$, $\frac{74\!\cdots\!43}{40\!\cdots\!44}a^{11}+\frac{56\!\cdots\!69}{40\!\cdots\!44}a^{10}-\frac{70\!\cdots\!93}{40\!\cdots\!44}a^{9}-\frac{71\!\cdots\!42}{50\!\cdots\!43}a^{8}+\frac{23\!\cdots\!87}{40\!\cdots\!44}a^{7}+\frac{28\!\cdots\!96}{50\!\cdots\!43}a^{6}-\frac{42\!\cdots\!88}{50\!\cdots\!43}a^{5}-\frac{36\!\cdots\!69}{40\!\cdots\!44}a^{4}+\frac{18\!\cdots\!85}{40\!\cdots\!44}a^{3}+\frac{20\!\cdots\!95}{40\!\cdots\!44}a^{2}-\frac{41\!\cdots\!63}{10\!\cdots\!86}a-\frac{89\!\cdots\!45}{40\!\cdots\!44}$, $\frac{94\!\cdots\!85}{80\!\cdots\!88}a^{11}-\frac{10\!\cdots\!69}{80\!\cdots\!88}a^{10}-\frac{58\!\cdots\!43}{80\!\cdots\!88}a^{9}+\frac{21\!\cdots\!19}{20\!\cdots\!72}a^{8}+\frac{51\!\cdots\!37}{80\!\cdots\!88}a^{7}-\frac{29\!\cdots\!81}{10\!\cdots\!86}a^{6}+\frac{22\!\cdots\!97}{10\!\cdots\!86}a^{5}+\frac{23\!\cdots\!45}{80\!\cdots\!88}a^{4}-\frac{27\!\cdots\!45}{80\!\cdots\!88}a^{3}-\frac{69\!\cdots\!11}{80\!\cdots\!88}a^{2}+\frac{25\!\cdots\!06}{50\!\cdots\!43}a+\frac{20\!\cdots\!65}{80\!\cdots\!88}$
| 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: | \( 2120183.06022 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 2120183.06022 \cdot 8}{2\cdot\sqrt{16551311845247868759889}}\cr\approx \mathstrut & 0.270008284088 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 102*x^10 + 199*x^9 + 3833*x^8 - 6937*x^7 - 66192*x^6 + 105589*x^5 + 514762*x^4 - 647862*x^3 - 1393261*x^2 + 752713*x + 494507)
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^12 - 2*x^11 - 102*x^10 + 199*x^9 + 3833*x^8 - 6937*x^7 - 66192*x^6 + 105589*x^5 + 514762*x^4 - 647862*x^3 - 1393261*x^2 + 752713*x + 494507, 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^12 - 2*x^11 - 102*x^10 + 199*x^9 + 3833*x^8 - 6937*x^7 - 66192*x^6 + 105589*x^5 + 514762*x^4 - 647862*x^3 - 1393261*x^2 + 752713*x + 494507);
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^12 - 2*x^11 - 102*x^10 + 199*x^9 + 3833*x^8 - 6937*x^7 - 66192*x^6 + 105589*x^5 + 514762*x^4 - 647862*x^3 - 1393261*x^2 + 752713*x + 494507);
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
$C_2\times A_4$ (as 12T7):
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 24 |
| The 8 conjugacy class representatives for $A_4 \times C_2$ |
| Character table for $A_4 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{377}) \), \(\Q(\zeta_{7})^+\), 6.6.905177.1, 6.6.128651901833.1, 6.6.341251729.1 |
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 24 |
| Degree 6 sibling: | 6.6.905177.1 |
| Degree 8 sibling: | 8.8.48501766991041.4 |
| Degree 12 sibling: | 12.12.116452742545489441.3 |
| Minimal sibling: | 6.6.905177.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(13\)
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(29\)
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |