Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278)
gp: K = bnfinit(y^16 - 8*y^15 + 172*y^14 - 1064*y^13 + 12790*y^12 - 63272*y^11 + 545756*y^10 - 2173192*y^9 + 14717225*y^8 - 46483984*y^7 + 257684608*y^6 - 618835120*y^5 + 2865222448*y^4 - 4749263824*y^3 + 18507232360*y^2 - 16228594896*y + 53156361278, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278)
\( x^{16} - 8 x^{15} + 172 x^{14} - 1064 x^{13} + 12790 x^{12} - 63272 x^{11} + 545756 x^{10} + \cdots + 53156361278 \)
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: |
\(6996429487079101204569997541638144\)
\(\medspace = 2^{62}\cdot 79^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(130.41\) | 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: | $2^{31/8}79^{1/2}\approx 130.40816347948132$ | ||
| Ramified primes: |
\(2\), \(79\)
| 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 and abelian over $\Q$. | |||
| Conductor: | \(2528=2^{5}\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2528}(1,·)$, $\chi_{2528}(2053,·)$, $\chi_{2528}(2369,·)$, $\chi_{2528}(1737,·)$, $\chi_{2528}(1421,·)$, $\chi_{2528}(1105,·)$, $\chi_{2528}(789,·)$, $\chi_{2528}(473,·)$, $\chi_{2528}(157,·)$, $\chi_{2528}(2213,·)$, $\chi_{2528}(1897,·)$, $\chi_{2528}(1581,·)$, $\chi_{2528}(1265,·)$, $\chi_{2528}(949,·)$, $\chi_{2528}(633,·)$, $\chi_{2528}(317,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{527}a^{13}+\frac{257}{527}a^{12}-\frac{57}{527}a^{11}-\frac{142}{527}a^{10}-\frac{241}{527}a^{9}+\frac{66}{527}a^{8}-\frac{226}{527}a^{7}+\frac{159}{527}a^{6}+\frac{183}{527}a^{5}-\frac{230}{527}a^{4}+\frac{98}{527}a^{3}-\frac{2}{17}a^{2}+\frac{162}{527}a+\frac{16}{527}$, $\frac{1}{10\!\cdots\!01}a^{14}-\frac{7}{10\!\cdots\!01}a^{13}-\frac{12\!\cdots\!98}{10\!\cdots\!01}a^{12}-\frac{31\!\cdots\!22}{10\!\cdots\!01}a^{11}-\frac{14\!\cdots\!77}{62\!\cdots\!53}a^{10}-\frac{25\!\cdots\!41}{10\!\cdots\!01}a^{9}+\frac{36\!\cdots\!72}{10\!\cdots\!01}a^{8}-\frac{68\!\cdots\!15}{32\!\cdots\!77}a^{7}+\frac{36\!\cdots\!04}{10\!\cdots\!01}a^{6}-\frac{60\!\cdots\!72}{10\!\cdots\!01}a^{5}+\frac{48\!\cdots\!26}{10\!\cdots\!01}a^{4}+\frac{28\!\cdots\!89}{10\!\cdots\!01}a^{3}+\frac{34\!\cdots\!17}{10\!\cdots\!01}a^{2}+\frac{42\!\cdots\!36}{10\!\cdots\!01}a-\frac{21\!\cdots\!91}{10\!\cdots\!01}$, $\frac{1}{11\!\cdots\!91}a^{15}+\frac{53888}{11\!\cdots\!91}a^{14}+\frac{93\!\cdots\!76}{11\!\cdots\!91}a^{13}+\frac{19\!\cdots\!23}{44\!\cdots\!63}a^{12}+\frac{13\!\cdots\!03}{11\!\cdots\!03}a^{11}+\frac{19\!\cdots\!74}{11\!\cdots\!91}a^{10}-\frac{44\!\cdots\!32}{11\!\cdots\!91}a^{9}-\frac{33\!\cdots\!65}{11\!\cdots\!91}a^{8}-\frac{16\!\cdots\!44}{11\!\cdots\!91}a^{7}-\frac{14\!\cdots\!23}{11\!\cdots\!91}a^{6}-\frac{25\!\cdots\!76}{11\!\cdots\!91}a^{5}+\frac{30\!\cdots\!85}{11\!\cdots\!91}a^{4}-\frac{49\!\cdots\!41}{11\!\cdots\!91}a^{3}-\frac{37\!\cdots\!79}{11\!\cdots\!91}a^{2}+\frac{35\!\cdots\!25}{11\!\cdots\!91}a+\frac{14\!\cdots\!92}{66\!\cdots\!23}$
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
$C_{4}\times C_{432}\times C_{2160}$, which has order $3732480$ (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: |
$\frac{24469776}{22\!\cdots\!69}a^{14}-\frac{171288432}{22\!\cdots\!69}a^{13}+\frac{4125472414}{22\!\cdots\!69}a^{12}-\frac{22526084868}{22\!\cdots\!69}a^{11}+\frac{17935750246}{131011398757457}a^{10}-\frac{1322132033916}{22\!\cdots\!69}a^{9}+\frac{13026116000027}{22\!\cdots\!69}a^{8}-\frac{1383005648}{69354896113}a^{7}+\frac{346380131708704}{22\!\cdots\!69}a^{6}-\frac{889009299698960}{22\!\cdots\!69}a^{5}+\frac{57\!\cdots\!25}{22\!\cdots\!69}a^{4}-\frac{10\!\cdots\!60}{22\!\cdots\!69}a^{3}+\frac{54\!\cdots\!88}{22\!\cdots\!69}a^{2}-\frac{49\!\cdots\!56}{22\!\cdots\!69}a+\frac{23\!\cdots\!61}{22\!\cdots\!69}$, $\frac{11\!\cdots\!78}{10\!\cdots\!01}a^{14}-\frac{82\!\cdots\!46}{10\!\cdots\!01}a^{13}+\frac{18\!\cdots\!03}{10\!\cdots\!01}a^{12}-\frac{10\!\cdots\!20}{10\!\cdots\!01}a^{11}+\frac{71\!\cdots\!31}{62\!\cdots\!53}a^{10}-\frac{52\!\cdots\!48}{10\!\cdots\!01}a^{9}+\frac{45\!\cdots\!26}{10\!\cdots\!01}a^{8}-\frac{46\!\cdots\!56}{32\!\cdots\!77}a^{7}+\frac{10\!\cdots\!19}{10\!\cdots\!01}a^{6}-\frac{25\!\cdots\!64}{10\!\cdots\!01}a^{5}+\frac{13\!\cdots\!52}{10\!\cdots\!01}a^{4}-\frac{23\!\cdots\!64}{10\!\cdots\!01}a^{3}+\frac{10\!\cdots\!49}{10\!\cdots\!01}a^{2}-\frac{96\!\cdots\!84}{10\!\cdots\!01}a+\frac{36\!\cdots\!17}{10\!\cdots\!01}$, $\frac{37811930988036}{10\!\cdots\!01}a^{14}-\frac{264683516916252}{10\!\cdots\!01}a^{13}+\frac{64\!\cdots\!62}{10\!\cdots\!01}a^{12}-\frac{35\!\cdots\!96}{10\!\cdots\!01}a^{11}+\frac{28\!\cdots\!72}{62\!\cdots\!53}a^{10}-\frac{21\!\cdots\!46}{10\!\cdots\!01}a^{9}+\frac{21\!\cdots\!97}{10\!\cdots\!01}a^{8}-\frac{23\!\cdots\!44}{32\!\cdots\!77}a^{7}+\frac{64\!\cdots\!72}{10\!\cdots\!01}a^{6}-\frac{16\!\cdots\!52}{10\!\cdots\!01}a^{5}+\frac{13\!\cdots\!76}{10\!\cdots\!01}a^{4}-\frac{24\!\cdots\!92}{10\!\cdots\!01}a^{3}+\frac{18\!\cdots\!07}{10\!\cdots\!01}a^{2}-\frac{17\!\cdots\!64}{10\!\cdots\!01}a+\frac{12\!\cdots\!51}{10\!\cdots\!01}$, $\frac{10\!\cdots\!24}{11\!\cdots\!91}a^{15}-\frac{81\!\cdots\!80}{11\!\cdots\!91}a^{14}+\frac{59\!\cdots\!96}{36\!\cdots\!61}a^{13}-\frac{10\!\cdots\!14}{11\!\cdots\!91}a^{12}+\frac{13\!\cdots\!24}{11\!\cdots\!91}a^{11}-\frac{63\!\cdots\!84}{11\!\cdots\!91}a^{10}+\frac{34\!\cdots\!32}{66\!\cdots\!23}a^{9}-\frac{21\!\cdots\!79}{11\!\cdots\!91}a^{8}+\frac{15\!\cdots\!24}{11\!\cdots\!91}a^{7}-\frac{46\!\cdots\!80}{11\!\cdots\!91}a^{6}+\frac{28\!\cdots\!28}{11\!\cdots\!91}a^{5}-\frac{61\!\cdots\!32}{11\!\cdots\!91}a^{4}+\frac{35\!\cdots\!44}{11\!\cdots\!91}a^{3}-\frac{46\!\cdots\!40}{11\!\cdots\!91}a^{2}+\frac{22\!\cdots\!93}{11\!\cdots\!91}a-\frac{21\!\cdots\!65}{11\!\cdots\!91}$, $\frac{18\!\cdots\!32}{11\!\cdots\!91}a^{15}-\frac{13\!\cdots\!90}{11\!\cdots\!91}a^{14}+\frac{31\!\cdots\!42}{11\!\cdots\!91}a^{13}-\frac{18\!\cdots\!08}{11\!\cdots\!91}a^{12}+\frac{22\!\cdots\!68}{11\!\cdots\!91}a^{11}-\frac{10\!\cdots\!19}{11\!\cdots\!91}a^{10}+\frac{95\!\cdots\!71}{11\!\cdots\!91}a^{9}-\frac{35\!\cdots\!74}{11\!\cdots\!91}a^{8}+\frac{25\!\cdots\!04}{11\!\cdots\!91}a^{7}-\frac{72\!\cdots\!08}{11\!\cdots\!91}a^{6}+\frac{42\!\cdots\!64}{11\!\cdots\!91}a^{5}-\frac{88\!\cdots\!08}{11\!\cdots\!91}a^{4}+\frac{41\!\cdots\!71}{11\!\cdots\!91}a^{3}-\frac{54\!\cdots\!78}{11\!\cdots\!91}a^{2}+\frac{18\!\cdots\!65}{11\!\cdots\!91}a-\frac{74\!\cdots\!25}{11\!\cdots\!91}$, $\frac{17\!\cdots\!06}{11\!\cdots\!91}a^{15}-\frac{85\!\cdots\!27}{66\!\cdots\!23}a^{14}+\frac{86\!\cdots\!05}{44\!\cdots\!63}a^{13}-\frac{13\!\cdots\!79}{11\!\cdots\!91}a^{12}+\frac{11\!\cdots\!67}{11\!\cdots\!91}a^{11}-\frac{34\!\cdots\!32}{66\!\cdots\!23}a^{10}+\frac{31\!\cdots\!86}{11\!\cdots\!91}a^{9}-\frac{14\!\cdots\!89}{11\!\cdots\!91}a^{8}+\frac{52\!\cdots\!55}{11\!\cdots\!91}a^{7}-\frac{25\!\cdots\!90}{11\!\cdots\!03}a^{6}+\frac{30\!\cdots\!72}{66\!\cdots\!23}a^{5}-\frac{29\!\cdots\!61}{11\!\cdots\!91}a^{4}+\frac{16\!\cdots\!93}{66\!\cdots\!23}a^{3}-\frac{24\!\cdots\!36}{11\!\cdots\!91}a^{2}+\frac{77\!\cdots\!58}{11\!\cdots\!91}a-\frac{10\!\cdots\!29}{11\!\cdots\!91}$, $\frac{16\!\cdots\!04}{11\!\cdots\!91}a^{15}-\frac{12\!\cdots\!30}{11\!\cdots\!91}a^{14}+\frac{26\!\cdots\!96}{11\!\cdots\!91}a^{13}-\frac{15\!\cdots\!69}{11\!\cdots\!91}a^{12}+\frac{18\!\cdots\!74}{11\!\cdots\!91}a^{11}-\frac{50\!\cdots\!97}{66\!\cdots\!23}a^{10}+\frac{73\!\cdots\!34}{11\!\cdots\!91}a^{9}-\frac{26\!\cdots\!29}{11\!\cdots\!91}a^{8}+\frac{17\!\cdots\!44}{11\!\cdots\!91}a^{7}-\frac{49\!\cdots\!80}{11\!\cdots\!91}a^{6}+\frac{25\!\cdots\!77}{11\!\cdots\!91}a^{5}-\frac{30\!\cdots\!86}{66\!\cdots\!23}a^{4}+\frac{21\!\cdots\!99}{11\!\cdots\!91}a^{3}-\frac{26\!\cdots\!10}{11\!\cdots\!91}a^{2}+\frac{77\!\cdots\!49}{11\!\cdots\!91}a-\frac{35\!\cdots\!65}{11\!\cdots\!91}$
| 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: | \( 15753.94986242651 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 15753.94986242651 \cdot 3732480}{2\cdot\sqrt{6996429487079101204569997541638144}}\cr\approx \mathstrut & 0.853802984819478 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278)
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 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278, 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 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278);
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 - 8*x^15 + 172*x^14 - 1064*x^13 + 12790*x^12 - 63272*x^11 + 545756*x^10 - 2173192*x^9 + 14717225*x^8 - 46483984*x^7 + 257684608*x^6 - 618835120*x^5 + 2865222448*x^4 - 4749263824*x^3 + 18507232360*x^2 - 16228594896*x + 53156361278);
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
$C_2\times C_8$ (as 16T5):
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)
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_8\times C_2$ |
| Character table for $C_8\times C_2$ |
Intermediate fields
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.31.6 | $x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 2$ | $8$ | $1$ | $31$ | $C_8$ | $[3, 4, 5]$ |
| 2.8.31.6 | $x^{8} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 2$ | $8$ | $1$ | $31$ | $C_8$ | $[3, 4, 5]$ | |
|
\(79\)
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |