Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400)
gp: K = bnfinit(y^16 - 6*y^15 + 31*y^14 - 14*y^13 - 160*y^12 - 1310*y^11 + 11738*y^10 - 22602*y^9 + 2030*y^8 + 30444*y^7 + 300929*y^6 - 1078216*y^5 + 4034535*y^4 - 9955312*y^3 + 4560832*y^2 + 10864280*y + 34932400, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400)
\( x^{16} - 6 x^{15} + 31 x^{14} - 14 x^{13} - 160 x^{12} - 1310 x^{11} + 11738 x^{10} - 22602 x^{9} + \cdots + 34932400 \)
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: |
\(1766969156799962667099298073761\)
\(\medspace = 23^{8}\cdot 41^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(77.71\) | 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: | $23^{1/2}41^{3/4}\approx 77.70550917341174$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}$, $\frac{1}{10}a^{11}-\frac{1}{10}a^{10}+\frac{1}{10}a^{7}+\frac{1}{10}a^{6}-\frac{3}{10}a^{5}-\frac{1}{10}a^{2}+\frac{1}{5}a$, $\frac{1}{60}a^{12}+\frac{1}{15}a^{10}+\frac{1}{10}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{3}a^{4}-\frac{1}{10}a^{3}+\frac{11}{60}a^{2}+\frac{1}{5}a+\frac{1}{3}$, $\frac{1}{60}a^{13}-\frac{1}{30}a^{11}+\frac{1}{10}a^{10}+\frac{1}{10}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{10}a^{6}-\frac{11}{30}a^{5}-\frac{1}{10}a^{4}+\frac{11}{60}a^{3}-\frac{1}{5}a^{2}+\frac{2}{15}a$, $\frac{1}{1106760}a^{14}-\frac{351}{73784}a^{13}+\frac{1021}{184460}a^{12}-\frac{2839}{92230}a^{11}-\frac{17677}{276690}a^{10}-\frac{12173}{184460}a^{9}+\frac{81}{802}a^{8}-\frac{23267}{184460}a^{7}-\frac{43303}{276690}a^{6}-\frac{1922}{46115}a^{5}+\frac{6767}{368920}a^{4}+\frac{16263}{73784}a^{3}-\frac{6835}{18446}a^{2}-\frac{776}{2005}a-\frac{421}{1203}$, $\frac{1}{24\!\cdots\!80}a^{15}-\frac{12\!\cdots\!37}{31\!\cdots\!35}a^{14}+\frac{58\!\cdots\!11}{10\!\cdots\!60}a^{13}+\frac{30\!\cdots\!27}{62\!\cdots\!87}a^{12}-\frac{83\!\cdots\!69}{20\!\cdots\!90}a^{11}-\frac{59\!\cdots\!53}{41\!\cdots\!80}a^{10}+\frac{83\!\cdots\!79}{83\!\cdots\!16}a^{9}+\frac{10\!\cdots\!31}{41\!\cdots\!80}a^{8}+\frac{67\!\cdots\!19}{12\!\cdots\!40}a^{7}-\frac{65\!\cdots\!32}{31\!\cdots\!35}a^{6}-\frac{17\!\cdots\!27}{49\!\cdots\!96}a^{5}+\frac{37\!\cdots\!41}{12\!\cdots\!40}a^{4}+\frac{29\!\cdots\!53}{24\!\cdots\!80}a^{3}+\frac{45\!\cdots\!01}{12\!\cdots\!40}a^{2}+\frac{21\!\cdots\!24}{45\!\cdots\!15}a+\frac{52\!\cdots\!82}{90\!\cdots\!23}$
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
$C_{6}\times C_{6}\times C_{12}$, which has order $432$ (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: |
$\frac{15\!\cdots\!32}{28\!\cdots\!05}a^{15}+\frac{85\!\cdots\!77}{11\!\cdots\!20}a^{14}-\frac{70\!\cdots\!07}{12\!\cdots\!35}a^{13}+\frac{10\!\cdots\!61}{28\!\cdots\!05}a^{12}-\frac{97\!\cdots\!50}{18\!\cdots\!67}a^{11}-\frac{16\!\cdots\!67}{56\!\cdots\!10}a^{10}-\frac{10\!\cdots\!23}{18\!\cdots\!70}a^{9}+\frac{22\!\cdots\!21}{18\!\cdots\!70}a^{8}-\frac{20\!\cdots\!23}{56\!\cdots\!10}a^{7}+\frac{61\!\cdots\!11}{28\!\cdots\!05}a^{6}+\frac{10\!\cdots\!31}{11\!\cdots\!02}a^{5}-\frac{34\!\cdots\!71}{11\!\cdots\!20}a^{4}-\frac{37\!\cdots\!49}{56\!\cdots\!10}a^{3}+\frac{26\!\cdots\!49}{56\!\cdots\!10}a^{2}-\frac{63\!\cdots\!51}{41\!\cdots\!45}a+\frac{41\!\cdots\!63}{24\!\cdots\!87}$, $\frac{19\!\cdots\!81}{36\!\cdots\!80}a^{15}-\frac{22\!\cdots\!97}{10\!\cdots\!40}a^{14}+\frac{18\!\cdots\!63}{15\!\cdots\!60}a^{13}+\frac{50\!\cdots\!67}{21\!\cdots\!48}a^{12}-\frac{75\!\cdots\!69}{90\!\cdots\!95}a^{11}-\frac{65\!\cdots\!64}{90\!\cdots\!95}a^{10}+\frac{81\!\cdots\!91}{18\!\cdots\!79}a^{9}-\frac{28\!\cdots\!91}{90\!\cdots\!95}a^{8}-\frac{16\!\cdots\!09}{18\!\cdots\!90}a^{7}+\frac{65\!\cdots\!69}{10\!\cdots\!74}a^{6}+\frac{31\!\cdots\!83}{36\!\cdots\!80}a^{5}-\frac{46\!\cdots\!31}{10\!\cdots\!40}a^{4}+\frac{84\!\cdots\!11}{36\!\cdots\!80}a^{3}-\frac{14\!\cdots\!81}{10\!\cdots\!40}a^{2}-\frac{48\!\cdots\!47}{78\!\cdots\!30}a-\frac{61\!\cdots\!14}{78\!\cdots\!73}$, $\frac{69\!\cdots\!93}{12\!\cdots\!40}a^{15}-\frac{94\!\cdots\!19}{24\!\cdots\!80}a^{14}+\frac{49\!\cdots\!63}{24\!\cdots\!80}a^{13}-\frac{11\!\cdots\!28}{62\!\cdots\!87}a^{12}-\frac{19\!\cdots\!87}{20\!\cdots\!90}a^{11}-\frac{19\!\cdots\!64}{31\!\cdots\!35}a^{10}+\frac{29\!\cdots\!69}{41\!\cdots\!80}a^{9}-\frac{36\!\cdots\!63}{20\!\cdots\!90}a^{8}+\frac{17\!\cdots\!35}{24\!\cdots\!48}a^{7}+\frac{40\!\cdots\!26}{13\!\cdots\!45}a^{6}+\frac{33\!\cdots\!29}{24\!\cdots\!48}a^{5}-\frac{17\!\cdots\!91}{24\!\cdots\!80}a^{4}+\frac{13\!\cdots\!01}{49\!\cdots\!96}a^{3}-\frac{17\!\cdots\!39}{24\!\cdots\!48}a^{2}+\frac{86\!\cdots\!73}{18\!\cdots\!46}a+\frac{30\!\cdots\!17}{27\!\cdots\!69}$, $\frac{65\!\cdots\!93}{62\!\cdots\!70}a^{15}-\frac{52\!\cdots\!81}{83\!\cdots\!60}a^{14}+\frac{70\!\cdots\!03}{24\!\cdots\!80}a^{13}+\frac{99\!\cdots\!09}{12\!\cdots\!40}a^{12}-\frac{35\!\cdots\!69}{12\!\cdots\!74}a^{11}-\frac{17\!\cdots\!23}{13\!\cdots\!45}a^{10}+\frac{10\!\cdots\!07}{83\!\cdots\!16}a^{9}-\frac{34\!\cdots\!33}{20\!\cdots\!29}a^{8}-\frac{47\!\cdots\!29}{12\!\cdots\!40}a^{7}+\frac{99\!\cdots\!29}{10\!\cdots\!45}a^{6}+\frac{56\!\cdots\!67}{31\!\cdots\!35}a^{5}-\frac{28\!\cdots\!19}{24\!\cdots\!80}a^{4}+\frac{95\!\cdots\!09}{24\!\cdots\!80}a^{3}-\frac{18\!\cdots\!41}{31\!\cdots\!35}a^{2}-\frac{85\!\cdots\!37}{67\!\cdots\!90}a+\frac{12\!\cdots\!88}{27\!\cdots\!69}$, $\frac{39\!\cdots\!77}{31\!\cdots\!35}a^{15}-\frac{20\!\cdots\!89}{24\!\cdots\!80}a^{14}+\frac{19\!\cdots\!35}{49\!\cdots\!96}a^{13}-\frac{51\!\cdots\!41}{62\!\cdots\!70}a^{12}-\frac{41\!\cdots\!01}{20\!\cdots\!90}a^{11}-\frac{14\!\cdots\!23}{62\!\cdots\!70}a^{10}+\frac{39\!\cdots\!87}{18\!\cdots\!60}a^{9}-\frac{82\!\cdots\!93}{20\!\cdots\!90}a^{8}-\frac{56\!\cdots\!39}{12\!\cdots\!40}a^{7}-\frac{72\!\cdots\!09}{12\!\cdots\!74}a^{6}+\frac{91\!\cdots\!87}{62\!\cdots\!87}a^{5}-\frac{40\!\cdots\!73}{24\!\cdots\!80}a^{4}-\frac{97\!\cdots\!79}{24\!\cdots\!80}a^{3}+\frac{12\!\cdots\!91}{54\!\cdots\!80}a^{2}+\frac{20\!\cdots\!48}{90\!\cdots\!23}a-\frac{14\!\cdots\!02}{27\!\cdots\!69}$, $\frac{21\!\cdots\!41}{24\!\cdots\!48}a^{15}-\frac{15\!\cdots\!99}{24\!\cdots\!80}a^{14}+\frac{52\!\cdots\!99}{24\!\cdots\!80}a^{13}-\frac{78\!\cdots\!16}{31\!\cdots\!35}a^{12}-\frac{79\!\cdots\!47}{20\!\cdots\!90}a^{11}-\frac{18\!\cdots\!63}{10\!\cdots\!45}a^{10}+\frac{45\!\cdots\!01}{41\!\cdots\!80}a^{9}-\frac{33\!\cdots\!23}{20\!\cdots\!90}a^{8}-\frac{10\!\cdots\!33}{24\!\cdots\!48}a^{7}-\frac{86\!\cdots\!66}{31\!\cdots\!35}a^{6}+\frac{23\!\cdots\!37}{24\!\cdots\!48}a^{5}-\frac{36\!\cdots\!63}{24\!\cdots\!80}a^{4}+\frac{46\!\cdots\!93}{24\!\cdots\!80}a^{3}-\frac{20\!\cdots\!07}{12\!\cdots\!40}a^{2}-\frac{25\!\cdots\!59}{90\!\cdots\!30}a-\frac{28\!\cdots\!85}{90\!\cdots\!23}$, $\frac{51\!\cdots\!81}{10\!\cdots\!80}a^{15}-\frac{59\!\cdots\!45}{31\!\cdots\!74}a^{14}+\frac{58\!\cdots\!91}{45\!\cdots\!60}a^{13}+\frac{43\!\cdots\!43}{31\!\cdots\!40}a^{12}+\frac{64\!\cdots\!54}{25\!\cdots\!45}a^{11}-\frac{23\!\cdots\!42}{25\!\cdots\!45}a^{10}+\frac{96\!\cdots\!18}{25\!\cdots\!45}a^{9}-\frac{11\!\cdots\!01}{25\!\cdots\!45}a^{8}+\frac{10\!\cdots\!99}{51\!\cdots\!90}a^{7}-\frac{91\!\cdots\!69}{15\!\cdots\!70}a^{6}+\frac{13\!\cdots\!91}{10\!\cdots\!80}a^{5}-\frac{68\!\cdots\!53}{15\!\cdots\!70}a^{4}+\frac{95\!\cdots\!23}{10\!\cdots\!80}a^{3}-\frac{62\!\cdots\!25}{62\!\cdots\!48}a^{2}-\frac{11\!\cdots\!15}{45\!\cdots\!46}a-\frac{50\!\cdots\!11}{22\!\cdots\!23}$
| 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: | \( 1119246.56118 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1119246.56118 \cdot 432}{2\cdot\sqrt{1766969156799962667099298073761}}\cr\approx \mathstrut & 0.441777895832 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400)
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 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400, 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 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400);
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 - 6*x^15 + 31*x^14 - 14*x^13 - 160*x^12 - 1310*x^11 + 11738*x^10 - 22602*x^9 + 2030*x^8 + 30444*x^7 + 300929*x^6 - 1078216*x^5 + 4034535*x^4 - 9955312*x^3 + 4560832*x^2 + 10864280*x + 34932400);
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
$\SD_{16}$ (as 16T12):
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 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{-943}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-23}, \sqrt{41})\), 4.2.38663.1 x2, 4.0.21689.1 x2, 8.0.790763784001.2, 8.2.57794518300247.1 x4 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(41\)
| 41.4.3.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 41.4.3.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 41.4.3.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 41.4.3.1 | $x^{4} + 41$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |