Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867)
gp: K = bnfinit(y^16 - 2*y^15 + 6*y^14 - 16*y^13 + 102*y^12 - 104*y^11 + 967*y^10 - 3083*y^9 + 10279*y^8 - 13606*y^7 + 67069*y^6 - 182893*y^5 + 423774*y^4 - 115408*y^3 + 333897*y^2 + 960577*y + 385867, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867)
\( x^{16} - 2 x^{15} + 6 x^{14} - 16 x^{13} + 102 x^{12} - 104 x^{11} + 967 x^{10} - 3083 x^{9} + \cdots + 385867 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10614152867452364890755536401\)
\(\medspace = 23^{4}\cdot 41^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(56.44\) | 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: | $23^{1/2}41^{7/8}\approx 123.60891451503595$ | ||
| Ramified primes: |
\(23\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{472}a^{12}-\frac{15}{118}a^{11}+\frac{89}{472}a^{10}+\frac{105}{236}a^{9}-\frac{75}{236}a^{8}-\frac{93}{236}a^{7}-\frac{77}{236}a^{6}+\frac{69}{472}a^{5}+\frac{113}{472}a^{4}-\frac{67}{472}a^{3}+\frac{39}{118}a^{2}+\frac{99}{472}a+\frac{201}{472}$, $\frac{1}{472}a^{13}+\frac{29}{472}a^{11}-\frac{57}{236}a^{10}+\frac{89}{236}a^{9}+\frac{9}{236}a^{8}-\frac{111}{236}a^{7}-\frac{203}{472}a^{6}-\frac{231}{472}a^{5}+\frac{105}{472}a^{4}+\frac{37}{118}a^{3}-\frac{217}{472}a^{2}-\frac{231}{472}a+\frac{3}{59}$, $\frac{1}{78352}a^{14}-\frac{1}{78352}a^{13}+\frac{37}{39176}a^{12}+\frac{19577}{78352}a^{11}+\frac{4533}{78352}a^{10}+\frac{2167}{39176}a^{9}-\frac{17419}{39176}a^{8}-\frac{35019}{78352}a^{7}+\frac{10799}{39176}a^{6}-\frac{12607}{78352}a^{5}-\frac{1125}{4897}a^{4}-\frac{1229}{4897}a^{3}+\frac{13651}{39176}a^{2}-\frac{4843}{39176}a-\frac{369}{944}$, $\frac{1}{98\!\cdots\!12}a^{15}+\frac{60\!\cdots\!79}{98\!\cdots\!12}a^{14}-\frac{40\!\cdots\!38}{61\!\cdots\!57}a^{13}+\frac{33\!\cdots\!17}{98\!\cdots\!12}a^{12}+\frac{16\!\cdots\!95}{98\!\cdots\!12}a^{11}+\frac{63\!\cdots\!65}{49\!\cdots\!56}a^{10}+\frac{15\!\cdots\!55}{49\!\cdots\!56}a^{9}+\frac{11\!\cdots\!21}{98\!\cdots\!12}a^{8}-\frac{26\!\cdots\!87}{49\!\cdots\!56}a^{7}-\frac{21\!\cdots\!65}{98\!\cdots\!12}a^{6}+\frac{66\!\cdots\!31}{49\!\cdots\!56}a^{5}-\frac{21\!\cdots\!81}{49\!\cdots\!56}a^{4}-\frac{22\!\cdots\!69}{49\!\cdots\!56}a^{3}-\frac{44\!\cdots\!93}{24\!\cdots\!28}a^{2}-\frac{19\!\cdots\!37}{98\!\cdots\!12}a-\frac{30\!\cdots\!44}{74\!\cdots\!79}$
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
$C_{6}$, which has order $6$ (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: | $7$ | 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: |
$\frac{32\!\cdots\!19}{98\!\cdots\!12}a^{15}-\frac{35\!\cdots\!43}{49\!\cdots\!56}a^{14}+\frac{60\!\cdots\!39}{98\!\cdots\!12}a^{13}-\frac{37\!\cdots\!09}{98\!\cdots\!12}a^{12}+\frac{14\!\cdots\!25}{49\!\cdots\!56}a^{11}-\frac{24\!\cdots\!95}{98\!\cdots\!12}a^{10}+\frac{48\!\cdots\!23}{24\!\cdots\!28}a^{9}-\frac{10\!\cdots\!23}{98\!\cdots\!12}a^{8}+\frac{23\!\cdots\!63}{98\!\cdots\!12}a^{7}-\frac{15\!\cdots\!73}{98\!\cdots\!12}a^{6}+\frac{12\!\cdots\!81}{98\!\cdots\!12}a^{5}-\frac{14\!\cdots\!91}{24\!\cdots\!28}a^{4}+\frac{94\!\cdots\!47}{12\!\cdots\!14}a^{3}+\frac{73\!\cdots\!31}{49\!\cdots\!56}a^{2}-\frac{21\!\cdots\!15}{98\!\cdots\!12}a-\frac{23\!\cdots\!07}{11\!\cdots\!64}$, $\frac{67\!\cdots\!60}{19\!\cdots\!47}a^{15}-\frac{44\!\cdots\!55}{19\!\cdots\!47}a^{14}-\frac{16\!\cdots\!23}{19\!\cdots\!47}a^{13}-\frac{25\!\cdots\!41}{19\!\cdots\!47}a^{12}+\frac{75\!\cdots\!77}{19\!\cdots\!47}a^{11}-\frac{26\!\cdots\!42}{19\!\cdots\!47}a^{10}+\frac{93\!\cdots\!38}{19\!\cdots\!47}a^{9}-\frac{52\!\cdots\!85}{19\!\cdots\!47}a^{8}+\frac{58\!\cdots\!85}{19\!\cdots\!47}a^{7}-\frac{20\!\cdots\!90}{19\!\cdots\!47}a^{6}+\frac{13\!\cdots\!98}{19\!\cdots\!47}a^{5}-\frac{30\!\cdots\!67}{19\!\cdots\!47}a^{4}+\frac{28\!\cdots\!73}{19\!\cdots\!47}a^{3}+\frac{23\!\cdots\!13}{19\!\cdots\!47}a^{2}-\frac{51\!\cdots\!87}{19\!\cdots\!47}a-\frac{68\!\cdots\!51}{23\!\cdots\!09}$, $\frac{80\!\cdots\!67}{49\!\cdots\!56}a^{15}-\frac{55\!\cdots\!57}{98\!\cdots\!12}a^{14}+\frac{97\!\cdots\!59}{98\!\cdots\!12}a^{13}-\frac{14\!\cdots\!09}{49\!\cdots\!56}a^{12}+\frac{19\!\cdots\!71}{98\!\cdots\!12}a^{11}-\frac{34\!\cdots\!37}{98\!\cdots\!12}a^{10}+\frac{64\!\cdots\!59}{49\!\cdots\!56}a^{9}-\frac{17\!\cdots\!81}{24\!\cdots\!28}a^{8}+\frac{20\!\cdots\!13}{98\!\cdots\!12}a^{7}-\frac{18\!\cdots\!55}{61\!\cdots\!57}a^{6}+\frac{10\!\cdots\!31}{98\!\cdots\!12}a^{5}-\frac{22\!\cdots\!77}{49\!\cdots\!56}a^{4}+\frac{39\!\cdots\!13}{49\!\cdots\!56}a^{3}-\frac{20\!\cdots\!63}{49\!\cdots\!56}a^{2}+\frac{71\!\cdots\!33}{49\!\cdots\!56}a+\frac{80\!\cdots\!69}{11\!\cdots\!64}$, $\frac{57\!\cdots\!21}{19\!\cdots\!47}a^{15}-\frac{53\!\cdots\!62}{19\!\cdots\!47}a^{14}+\frac{24\!\cdots\!02}{19\!\cdots\!47}a^{13}-\frac{93\!\cdots\!84}{19\!\cdots\!47}a^{12}+\frac{35\!\cdots\!10}{19\!\cdots\!47}a^{11}+\frac{12\!\cdots\!70}{19\!\cdots\!47}a^{10}+\frac{53\!\cdots\!81}{19\!\cdots\!47}a^{9}-\frac{11\!\cdots\!95}{19\!\cdots\!47}a^{8}+\frac{33\!\cdots\!96}{19\!\cdots\!47}a^{7}-\frac{66\!\cdots\!02}{19\!\cdots\!47}a^{6}+\frac{25\!\cdots\!09}{19\!\cdots\!47}a^{5}-\frac{35\!\cdots\!80}{19\!\cdots\!47}a^{4}+\frac{10\!\cdots\!97}{19\!\cdots\!47}a^{3}+\frac{11\!\cdots\!80}{19\!\cdots\!47}a^{2}-\frac{20\!\cdots\!35}{19\!\cdots\!47}a-\frac{92\!\cdots\!64}{23\!\cdots\!09}$, $\frac{49\!\cdots\!79}{49\!\cdots\!56}a^{15}-\frac{83\!\cdots\!23}{12\!\cdots\!14}a^{14}-\frac{19\!\cdots\!41}{49\!\cdots\!56}a^{13}-\frac{51\!\cdots\!01}{49\!\cdots\!56}a^{12}-\frac{39\!\cdots\!33}{24\!\cdots\!28}a^{11}-\frac{82\!\cdots\!69}{49\!\cdots\!56}a^{10}-\frac{72\!\cdots\!46}{61\!\cdots\!57}a^{9}-\frac{36\!\cdots\!63}{49\!\cdots\!56}a^{8}-\frac{99\!\cdots\!15}{49\!\cdots\!56}a^{7}-\frac{75\!\cdots\!75}{49\!\cdots\!56}a^{6}-\frac{22\!\cdots\!23}{49\!\cdots\!56}a^{5}-\frac{65\!\cdots\!81}{24\!\cdots\!28}a^{4}-\frac{65\!\cdots\!64}{61\!\cdots\!57}a^{3}-\frac{97\!\cdots\!07}{12\!\cdots\!14}a^{2}-\frac{12\!\cdots\!05}{49\!\cdots\!56}a-\frac{18\!\cdots\!37}{59\!\cdots\!32}$, $\frac{72\!\cdots\!95}{49\!\cdots\!56}a^{15}+\frac{43\!\cdots\!83}{24\!\cdots\!28}a^{14}-\frac{11\!\cdots\!35}{49\!\cdots\!56}a^{13}+\frac{26\!\cdots\!01}{49\!\cdots\!56}a^{12}-\frac{13\!\cdots\!17}{12\!\cdots\!14}a^{11}+\frac{77\!\cdots\!47}{49\!\cdots\!56}a^{10}+\frac{49\!\cdots\!52}{61\!\cdots\!57}a^{9}+\frac{66\!\cdots\!65}{49\!\cdots\!56}a^{8}-\frac{21\!\cdots\!73}{49\!\cdots\!56}a^{7}+\frac{58\!\cdots\!97}{49\!\cdots\!56}a^{6}+\frac{11\!\cdots\!33}{49\!\cdots\!56}a^{5}+\frac{11\!\cdots\!53}{12\!\cdots\!14}a^{4}-\frac{70\!\cdots\!59}{24\!\cdots\!28}a^{3}+\frac{53\!\cdots\!75}{12\!\cdots\!14}a^{2}+\frac{43\!\cdots\!09}{49\!\cdots\!56}a+\frac{18\!\cdots\!47}{59\!\cdots\!32}$, $\frac{29\!\cdots\!35}{49\!\cdots\!56}a^{15}-\frac{54\!\cdots\!03}{24\!\cdots\!28}a^{14}-\frac{77\!\cdots\!19}{49\!\cdots\!56}a^{13}+\frac{85\!\cdots\!01}{49\!\cdots\!56}a^{12}+\frac{74\!\cdots\!43}{12\!\cdots\!14}a^{11}-\frac{10\!\cdots\!33}{49\!\cdots\!56}a^{10}-\frac{28\!\cdots\!41}{10\!\cdots\!23}a^{9}+\frac{48\!\cdots\!85}{49\!\cdots\!56}a^{8}+\frac{17\!\cdots\!27}{49\!\cdots\!56}a^{7}-\frac{36\!\cdots\!15}{49\!\cdots\!56}a^{6}+\frac{11\!\cdots\!61}{49\!\cdots\!56}a^{5}-\frac{80\!\cdots\!51}{12\!\cdots\!14}a^{4}+\frac{64\!\cdots\!79}{24\!\cdots\!28}a^{3}-\frac{18\!\cdots\!20}{61\!\cdots\!57}a^{2}+\frac{23\!\cdots\!41}{49\!\cdots\!56}a+\frac{24\!\cdots\!95}{59\!\cdots\!32}$
| 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: | \( 3403166.59806 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3403166.59806 \cdot 6}{2\cdot\sqrt{10614152867452364890755536401}}\cr\approx \mathstrut & 0.240713663450 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867)
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^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867, 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^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867);
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^16 - 2*x^15 + 6*x^14 - 16*x^13 + 102*x^12 - 104*x^11 + 967*x^10 - 3083*x^9 + 10279*x^8 - 13606*x^7 + 67069*x^6 - 182893*x^5 + 423774*x^4 - 115408*x^3 + 333897*x^2 + 960577*x + 385867);
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
$C_2^2:C_8$ (as 16T24):
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 32 |
| The 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.2.38663.1, 4.4.68921.1, 4.2.1585183.1, 8.0.194754273881.1, 8.4.103025010883049.3, 8.4.2512805143489.1 |
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 32 |
| Degree 16 sibling: | 16.8.2970275152580737243393920061992241.2 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(41\)
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |