Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 7*y^14 - 21*y^13 + 48*y^12 - 25*y^11 + 66*y^10 - 183*y^9 + 463*y^8 - 626*y^7 + 649*y^6 - 393*y^5 + 298*y^4 - 137*y^3 + 43*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*x + 1)
\( x^{16} - 2 x^{15} + 7 x^{14} - 21 x^{13} + 48 x^{12} - 25 x^{11} + 66 x^{10} - 183 x^{9} + 463 x^{8} + \cdots + 1 \)
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: |
\(6157543352187744140625\)
\(\medspace = 3^{12}\cdot 5^{12}\cdot 83^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.01\) | 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: | $3^{7/6}5^{3/4}83^{1/2}\approx 109.75088053305629$ | ||
| Ramified primes: |
\(3\), \(5\), \(83\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{57\!\cdots\!81}a^{15}+\frac{19\!\cdots\!09}{57\!\cdots\!81}a^{14}-\frac{46\!\cdots\!69}{57\!\cdots\!81}a^{13}+\frac{27\!\cdots\!97}{57\!\cdots\!81}a^{12}+\frac{15\!\cdots\!32}{57\!\cdots\!81}a^{11}+\frac{19\!\cdots\!78}{57\!\cdots\!81}a^{10}+\frac{21\!\cdots\!95}{57\!\cdots\!81}a^{9}+\frac{27\!\cdots\!06}{57\!\cdots\!81}a^{8}-\frac{41\!\cdots\!87}{57\!\cdots\!81}a^{7}+\frac{39\!\cdots\!60}{57\!\cdots\!81}a^{6}+\frac{23\!\cdots\!08}{57\!\cdots\!81}a^{5}+\frac{16\!\cdots\!65}{57\!\cdots\!81}a^{4}+\frac{81\!\cdots\!22}{57\!\cdots\!81}a^{3}+\frac{27\!\cdots\!99}{57\!\cdots\!81}a^{2}+\frac{21\!\cdots\!01}{57\!\cdots\!81}a-\frac{27\!\cdots\!19}{57\!\cdots\!81}$
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
$C_{2}\times C_{2}$, which has order $4$
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: |
\( \frac{73029584819897249}{579933626724499081} a^{15} - \frac{162775257629643180}{579933626724499081} a^{14} + \frac{517094340249579661}{579933626724499081} a^{13} - \frac{1606345805009401631}{579933626724499081} a^{12} + \frac{3685058594269131720}{579933626724499081} a^{11} - \frac{2124085387859260981}{579933626724499081} a^{10} + \frac{4139394661666996272}{579933626724499081} a^{9} - \frac{14288086806047614416}{579933626724499081} a^{8} + \frac{35293955087986134994}{579933626724499081} a^{7} - \frac{49083804118573453558}{579933626724499081} a^{6} + \frac{47041820676990360675}{579933626724499081} a^{5} - \frac{27206203005309694641}{579933626724499081} a^{4} + \frac{16934801109233966399}{579933626724499081} a^{3} - \frac{10941782442828618849}{579933626724499081} a^{2} + \frac{643279930590324427}{579933626724499081} a + \frac{52169438270252402}{579933626724499081} \)
(order $10$)
| 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: |
$\frac{13\!\cdots\!01}{57\!\cdots\!81}a^{15}-\frac{34\!\cdots\!69}{57\!\cdots\!81}a^{14}+\frac{10\!\cdots\!56}{57\!\cdots\!81}a^{13}-\frac{33\!\cdots\!08}{57\!\cdots\!81}a^{12}+\frac{78\!\cdots\!36}{57\!\cdots\!81}a^{11}-\frac{62\!\cdots\!31}{57\!\cdots\!81}a^{10}+\frac{95\!\cdots\!01}{57\!\cdots\!81}a^{9}-\frac{29\!\cdots\!95}{57\!\cdots\!81}a^{8}+\frac{74\!\cdots\!97}{57\!\cdots\!81}a^{7}-\frac{11\!\cdots\!41}{57\!\cdots\!81}a^{6}+\frac{12\!\cdots\!02}{57\!\cdots\!81}a^{5}-\frac{84\!\cdots\!85}{57\!\cdots\!81}a^{4}+\frac{53\!\cdots\!51}{57\!\cdots\!81}a^{3}-\frac{31\!\cdots\!96}{57\!\cdots\!81}a^{2}+\frac{81\!\cdots\!34}{57\!\cdots\!81}a-\frac{86\!\cdots\!47}{57\!\cdots\!81}$, $\frac{409070270401848}{964947798210481}a^{15}-\frac{630666209682373}{964947798210481}a^{14}+\frac{25\!\cdots\!43}{964947798210481}a^{13}-\frac{73\!\cdots\!35}{964947798210481}a^{12}+\frac{15\!\cdots\!50}{964947798210481}a^{11}-\frac{17\!\cdots\!29}{964947798210481}a^{10}+\frac{23\!\cdots\!40}{964947798210481}a^{9}-\frac{63\!\cdots\!79}{964947798210481}a^{8}+\frac{15\!\cdots\!95}{964947798210481}a^{7}-\frac{17\!\cdots\!83}{964947798210481}a^{6}+\frac{15\!\cdots\!10}{964947798210481}a^{5}-\frac{54\!\cdots\!25}{964947798210481}a^{4}+\frac{64\!\cdots\!12}{964947798210481}a^{3}-\frac{11\!\cdots\!51}{964947798210481}a^{2}-\frac{916459979407734}{964947798210481}a+\frac{630390918892568}{964947798210481}$, $\frac{16\!\cdots\!82}{57\!\cdots\!81}a^{15}-\frac{58\!\cdots\!18}{57\!\cdots\!81}a^{14}+\frac{72\!\cdots\!02}{57\!\cdots\!81}a^{13}-\frac{17\!\cdots\!68}{57\!\cdots\!81}a^{12}+\frac{29\!\cdots\!56}{57\!\cdots\!81}a^{11}+\frac{68\!\cdots\!62}{57\!\cdots\!81}a^{10}+\frac{92\!\cdots\!49}{57\!\cdots\!81}a^{9}-\frac{14\!\cdots\!07}{57\!\cdots\!81}a^{8}+\frac{33\!\cdots\!84}{57\!\cdots\!81}a^{7}+\frac{35\!\cdots\!26}{57\!\cdots\!81}a^{6}-\frac{14\!\cdots\!16}{57\!\cdots\!81}a^{5}+\frac{48\!\cdots\!03}{57\!\cdots\!81}a^{4}+\frac{93\!\cdots\!36}{57\!\cdots\!81}a^{3}+\frac{24\!\cdots\!80}{57\!\cdots\!81}a^{2}-\frac{63\!\cdots\!94}{57\!\cdots\!81}a+\frac{73\!\cdots\!49}{57\!\cdots\!81}$, $\frac{22\!\cdots\!37}{57\!\cdots\!81}a^{15}-\frac{33\!\cdots\!33}{57\!\cdots\!81}a^{14}+\frac{13\!\cdots\!24}{57\!\cdots\!81}a^{13}-\frac{39\!\cdots\!00}{57\!\cdots\!81}a^{12}+\frac{84\!\cdots\!04}{57\!\cdots\!81}a^{11}-\frac{79\!\cdots\!01}{57\!\cdots\!81}a^{10}+\frac{13\!\cdots\!79}{57\!\cdots\!81}a^{9}-\frac{33\!\cdots\!98}{57\!\cdots\!81}a^{8}+\frac{84\!\cdots\!37}{57\!\cdots\!81}a^{7}-\frac{92\!\cdots\!85}{57\!\cdots\!81}a^{6}+\frac{86\!\cdots\!60}{57\!\cdots\!81}a^{5}-\frac{30\!\cdots\!69}{57\!\cdots\!81}a^{4}+\frac{37\!\cdots\!91}{57\!\cdots\!81}a^{3}-\frac{57\!\cdots\!10}{57\!\cdots\!81}a^{2}+\frac{15\!\cdots\!92}{57\!\cdots\!81}a-\frac{12\!\cdots\!03}{57\!\cdots\!81}$, $\frac{33\!\cdots\!07}{57\!\cdots\!81}a^{15}-\frac{46\!\cdots\!74}{57\!\cdots\!81}a^{14}+\frac{19\!\cdots\!01}{57\!\cdots\!81}a^{13}-\frac{57\!\cdots\!01}{57\!\cdots\!81}a^{12}+\frac{12\!\cdots\!04}{57\!\cdots\!81}a^{11}-\frac{50\!\cdots\!67}{57\!\cdots\!81}a^{10}+\frac{21\!\cdots\!87}{57\!\cdots\!81}a^{9}-\frac{53\!\cdots\!82}{57\!\cdots\!81}a^{8}+\frac{12\!\cdots\!04}{57\!\cdots\!81}a^{7}-\frac{12\!\cdots\!94}{57\!\cdots\!81}a^{6}+\frac{13\!\cdots\!62}{57\!\cdots\!81}a^{5}-\frac{83\!\cdots\!70}{57\!\cdots\!81}a^{4}+\frac{11\!\cdots\!13}{57\!\cdots\!81}a^{3}-\frac{56\!\cdots\!79}{57\!\cdots\!81}a^{2}+\frac{12\!\cdots\!15}{57\!\cdots\!81}a+\frac{50\!\cdots\!27}{57\!\cdots\!81}$, $\frac{17\!\cdots\!97}{57\!\cdots\!81}a^{15}-\frac{21\!\cdots\!15}{57\!\cdots\!81}a^{14}+\frac{95\!\cdots\!26}{57\!\cdots\!81}a^{13}-\frac{27\!\cdots\!83}{57\!\cdots\!81}a^{12}+\frac{55\!\cdots\!12}{57\!\cdots\!81}a^{11}+\frac{21\!\cdots\!55}{57\!\cdots\!81}a^{10}+\frac{86\!\cdots\!00}{57\!\cdots\!81}a^{9}-\frac{23\!\cdots\!91}{57\!\cdots\!81}a^{8}+\frac{56\!\cdots\!94}{57\!\cdots\!81}a^{7}-\frac{46\!\cdots\!02}{57\!\cdots\!81}a^{6}+\frac{31\!\cdots\!89}{57\!\cdots\!81}a^{5}+\frac{13\!\cdots\!14}{57\!\cdots\!81}a^{4}+\frac{81\!\cdots\!99}{57\!\cdots\!81}a^{3}+\frac{97\!\cdots\!00}{57\!\cdots\!81}a^{2}-\frac{65\!\cdots\!02}{57\!\cdots\!81}a+\frac{15\!\cdots\!57}{57\!\cdots\!81}$, $\frac{21\!\cdots\!22}{57\!\cdots\!81}a^{15}-\frac{42\!\cdots\!51}{57\!\cdots\!81}a^{14}+\frac{14\!\cdots\!68}{57\!\cdots\!81}a^{13}-\frac{44\!\cdots\!36}{57\!\cdots\!81}a^{12}+\frac{10\!\cdots\!22}{57\!\cdots\!81}a^{11}-\frac{44\!\cdots\!83}{57\!\cdots\!81}a^{10}+\frac{13\!\cdots\!98}{57\!\cdots\!81}a^{9}-\frac{38\!\cdots\!47}{57\!\cdots\!81}a^{8}+\frac{97\!\cdots\!26}{57\!\cdots\!81}a^{7}-\frac{12\!\cdots\!36}{57\!\cdots\!81}a^{6}+\frac{12\!\cdots\!57}{57\!\cdots\!81}a^{5}-\frac{68\!\cdots\!47}{57\!\cdots\!81}a^{4}+\frac{53\!\cdots\!07}{57\!\cdots\!81}a^{3}-\frac{22\!\cdots\!41}{57\!\cdots\!81}a^{2}+\frac{47\!\cdots\!65}{57\!\cdots\!81}a-\frac{10\!\cdots\!15}{57\!\cdots\!81}$
| 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: | \( 14789.2708979 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 14789.2708979 \cdot 4}{10\cdot\sqrt{6157543352187744140625}}\cr\approx \mathstrut & 0.183122607292 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*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^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*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(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*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^16 - 2*x^15 + 7*x^14 - 21*x^13 + 48*x^12 - 25*x^11 + 66*x^10 - 183*x^9 + 463*x^8 - 626*x^7 + 649*x^6 - 393*x^5 + 298*x^4 - 137*x^3 + 43*x^2 - 8*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
$C_4\times S_4$ (as 16T181):
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 96 |
| The 20 conjugacy class representatives for $C_4\times S_4$ |
| Character table for $C_4\times S_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.56025.1, 8.8.3138800625.1 |
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
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Minimal sibling: | 12.0.49260346817501953125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | R | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.12.12.23 | $x^{12} + 6 x^{11} + 12 x^{10} + 12 x^{9} + 189 x^{8} + 108 x^{7} + 324 x^{6} + 648 x^{5} + 891 x^{4} + 918 x^{3} + 648 x^{2} + 324 x + 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(83\)
| 83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 83.8.4.1 | $x^{8} + 340 x^{6} + 84 x^{5} + 42018 x^{4} - 13608 x^{3} + 2234480 x^{2} - 1707972 x + 43166893$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |