Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729)
gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 - 32*y^13 + 41*y^12 - 113*y^11 - 320*y^10 + 501*y^9 + 2370*y^8 + 5301*y^7 + 11097*y^6 + 14526*y^5 + 13716*y^4 + 13284*y^3 + 9315*y^2 + 3159*y + 729, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729)
\( x^{16} - 2 x^{15} + 2 x^{14} - 32 x^{13} + 41 x^{12} - 113 x^{11} - 320 x^{10} + 501 x^{9} + 2370 x^{8} + \cdots + 729 \)
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: |
\(17415183620366462784744081\)
\(\medspace = 3^{8}\cdot 61^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.81\) | 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: | $3^{1/2}61^{3/4}\approx 37.80576266511387$ | ||
| Ramified primes: |
\(3\), \(61\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{4}{9}a^{5}+\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{27}a^{11}+\frac{1}{27}a^{10}-\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{1}{27}a^{6}+\frac{7}{27}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{405}a^{12}+\frac{1}{405}a^{11}-\frac{2}{81}a^{10}+\frac{4}{405}a^{9}-\frac{4}{405}a^{8}-\frac{23}{405}a^{7}-\frac{1}{81}a^{6}-\frac{13}{27}a^{5}+\frac{8}{135}a^{4}-\frac{2}{45}a^{3}+\frac{7}{45}a^{2}+\frac{7}{15}a-\frac{2}{5}$, $\frac{1}{405}a^{13}+\frac{4}{405}a^{11}-\frac{16}{405}a^{10}+\frac{22}{405}a^{9}-\frac{4}{405}a^{8}+\frac{1}{135}a^{7}-\frac{8}{81}a^{6}+\frac{7}{15}a^{5}+\frac{1}{135}a^{4}-\frac{11}{45}a^{3}-\frac{1}{45}a^{2}+\frac{7}{15}a+\frac{2}{5}$, $\frac{1}{397305}a^{14}-\frac{56}{397305}a^{13}-\frac{217}{397305}a^{12}+\frac{1999}{397305}a^{11}-\frac{10432}{397305}a^{10}-\frac{1411}{79461}a^{9}-\frac{15464}{397305}a^{8}+\frac{2762}{26487}a^{7}-\frac{2183}{14715}a^{6}+\frac{4147}{14715}a^{5}+\frac{43}{1635}a^{4}-\frac{4856}{14715}a^{3}-\frac{496}{2943}a^{2}+\frac{664}{4905}a+\frac{155}{327}$, $\frac{1}{38\!\cdots\!75}a^{15}+\frac{2345760614}{38\!\cdots\!75}a^{14}+\frac{25797844289}{428568977905875}a^{13}-\frac{1223363400472}{12\!\cdots\!25}a^{12}-\frac{301692671461}{257141386743525}a^{11}-\frac{17810200289326}{12\!\cdots\!25}a^{10}-\frac{69475657740793}{38\!\cdots\!75}a^{9}+\frac{137323076254888}{38\!\cdots\!75}a^{8}+\frac{88340107619776}{12\!\cdots\!25}a^{7}-\frac{4311660772289}{428568977905875}a^{6}-\frac{38387285922847}{142856325968625}a^{5}+\frac{4163869430237}{47618775322875}a^{4}-\frac{61105709244691}{142856325968625}a^{3}-\frac{41100184322789}{142856325968625}a^{2}-\frac{4999551551843}{47618775322875}a+\frac{6539745510517}{15872925107625}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{3}\times C_{6}$, which has order $18$ (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: |
\( -\frac{34529650088}{35386429368375} a^{15} + \frac{88405322068}{35386429368375} a^{14} - \frac{124215397663}{35386429368375} a^{13} + \frac{1191386976583}{35386429368375} a^{12} - \frac{422481201821}{7077285873675} a^{11} + \frac{5297549546164}{35386429368375} a^{10} + \frac{7636268460259}{35386429368375} a^{9} - \frac{6815990014598}{11795476456125} a^{8} - \frac{7750305281671}{3931825485375} a^{7} - \frac{1816696502927}{436869498375} a^{6} - \frac{11515100244364}{1310608495125} a^{5} - \frac{12892658031593}{1310608495125} a^{4} - \frac{12039532228142}{1310608495125} a^{3} - \frac{3971055818381}{436869498375} a^{2} - \frac{833390383222}{145623166125} a - \frac{44362097007}{48541055375} \)
(order $6$)
| 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: |
$\frac{2365571637928}{38\!\cdots\!75}a^{15}-\frac{10126679204108}{38\!\cdots\!75}a^{14}+\frac{22383924845453}{38\!\cdots\!75}a^{13}-\frac{106969940148698}{38\!\cdots\!75}a^{12}+\frac{60917844613201}{771424160230575}a^{11}-\frac{745941808978109}{38\!\cdots\!75}a^{10}+\frac{393707355395821}{38\!\cdots\!75}a^{9}+\frac{521568546294013}{12\!\cdots\!25}a^{8}+\frac{26131959173464}{47618775322875}a^{7}+\frac{417396013107533}{428568977905875}a^{6}+\frac{364589026570834}{142856325968625}a^{5}-\frac{8102768162842}{142856325968625}a^{4}+\frac{236534567671177}{142856325968625}a^{3}+\frac{37502789019512}{15872925107625}a^{2}+\frac{13948290651907}{15872925107625}a-\frac{436291732083}{5290975035875}$, $\frac{1280802081344}{38\!\cdots\!75}a^{15}-\frac{2096845127878}{12\!\cdots\!25}a^{14}+\frac{15024441628844}{38\!\cdots\!75}a^{13}-\frac{62578003157579}{38\!\cdots\!75}a^{12}+\frac{39052723056808}{771424160230575}a^{11}-\frac{482489931290357}{38\!\cdots\!75}a^{10}+\frac{124790694963461}{12\!\cdots\!25}a^{9}+\frac{910801637051572}{38\!\cdots\!75}a^{8}+\frac{152933270236744}{12\!\cdots\!25}a^{7}+\frac{88316902490234}{428568977905875}a^{6}+\frac{134070229827307}{142856325968625}a^{5}-\frac{29230264433947}{47618775322875}a^{4}+\frac{84518234750782}{47618775322875}a^{3}+\frac{259071497017384}{142856325968625}a^{2}+\frac{75927948157408}{47618775322875}a+\frac{8698776067423}{15872925107625}$, $\frac{3539953107058}{38\!\cdots\!75}a^{15}-\frac{1510008024157}{428568977905875}a^{14}+\frac{27705594364108}{38\!\cdots\!75}a^{13}-\frac{146917308516853}{38\!\cdots\!75}a^{12}+\frac{75948493143716}{771424160230575}a^{11}-\frac{911281617783274}{38\!\cdots\!75}a^{10}+\frac{4333453056109}{428568977905875}a^{9}+\frac{29\!\cdots\!29}{38\!\cdots\!75}a^{8}+\frac{917224722735383}{12\!\cdots\!25}a^{7}+\frac{11\!\cdots\!13}{428568977905875}a^{6}+\frac{577550139857024}{142856325968625}a^{5}+\frac{45889946876032}{15872925107625}a^{4}+\frac{168297606725074}{47618775322875}a^{3}+\frac{383916055467188}{142856325968625}a^{2}+\frac{28426608850556}{47618775322875}a+\frac{945560480111}{15872925107625}$, $\frac{5706770422849}{38\!\cdots\!75}a^{15}-\frac{21718102191764}{38\!\cdots\!75}a^{14}+\frac{43551946289324}{38\!\cdots\!75}a^{13}-\frac{242263448323409}{38\!\cdots\!75}a^{12}+\frac{127254071089498}{771424160230575}a^{11}-\frac{15\!\cdots\!72}{38\!\cdots\!75}a^{10}+\frac{316744428730243}{38\!\cdots\!75}a^{9}+\frac{12\!\cdots\!79}{12\!\cdots\!25}a^{8}+\frac{83561831415937}{47618775322875}a^{7}+\frac{18\!\cdots\!39}{428568977905875}a^{6}+\frac{207078469875824}{47618775322875}a^{5}+\frac{546190649051114}{142856325968625}a^{4}+\frac{129760128895891}{142856325968625}a^{3}-\frac{27622400986954}{15872925107625}a^{2}+\frac{53490369473381}{15872925107625}a+\frac{19638687092861}{5290975035875}$, $\frac{306713520140}{30856966409223}a^{15}-\frac{231916924183}{10285655469741}a^{14}+\frac{2113158761437}{154284832046115}a^{13}-\frac{9429728232872}{30856966409223}a^{12}+\frac{75751905394153}{154284832046115}a^{11}-\frac{136564243326667}{154284832046115}a^{10}-\frac{163920101948447}{51428277348705}a^{9}+\frac{10\!\cdots\!27}{154284832046115}a^{8}+\frac{13\!\cdots\!32}{51428277348705}a^{7}+\frac{142613898305917}{3428551823247}a^{6}+\frac{353880268815469}{5714253038745}a^{5}+\frac{247352407626278}{5714253038745}a^{4}-\frac{48700781929343}{1904751012915}a^{3}-\frac{247994376050299}{5714253038745}a^{2}-\frac{29533812201442}{1904751012915}a-\frac{2846652338722}{634917004305}$, $\frac{579343486954}{38\!\cdots\!75}a^{15}+\frac{188640615406}{38\!\cdots\!75}a^{14}-\frac{8727838509871}{38\!\cdots\!75}a^{13}-\frac{630229179464}{38\!\cdots\!75}a^{12}-\frac{5597271117182}{771424160230575}a^{11}+\frac{197428069086013}{38\!\cdots\!75}a^{10}-\frac{603576771431747}{38\!\cdots\!75}a^{9}+\frac{125383439844059}{12\!\cdots\!25}a^{8}+\frac{553514656357718}{428568977905875}a^{7}+\frac{31235517670819}{428568977905875}a^{6}-\frac{88438393230638}{142856325968625}a^{5}-\frac{41543055891284}{15872925107625}a^{4}-\frac{12\!\cdots\!64}{142856325968625}a^{3}-\frac{47752592749153}{5290975035875}a^{2}-\frac{18465732369708}{5290975035875}a-\frac{4609419928544}{5290975035875}$, $\frac{14044562645236}{38\!\cdots\!75}a^{15}-\frac{13208303788496}{38\!\cdots\!75}a^{14}+\frac{5915503077236}{38\!\cdots\!75}a^{13}-\frac{434967650829251}{38\!\cdots\!75}a^{12}+\frac{19376036759842}{771424160230575}a^{11}-\frac{11\!\cdots\!83}{38\!\cdots\!75}a^{10}-\frac{58\!\cdots\!48}{38\!\cdots\!75}a^{9}+\frac{217582817694602}{428568977905875}a^{8}+\frac{42\!\cdots\!12}{428568977905875}a^{7}+\frac{12\!\cdots\!71}{428568977905875}a^{6}+\frac{96\!\cdots\!08}{142856325968625}a^{5}+\frac{49\!\cdots\!82}{47618775322875}a^{4}+\frac{16\!\cdots\!74}{142856325968625}a^{3}+\frac{47\!\cdots\!32}{47618775322875}a^{2}+\frac{340056712877953}{5290975035875}a+\frac{102536550046629}{5290975035875}$
| 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: | \( 2820654.85259 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2820654.85259 \cdot 18}{6\cdot\sqrt{17415183620366462784744081}}\cr\approx \mathstrut & 4.92545232683 \end{aligned}\] (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 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729)
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 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729, 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 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729);
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 + 2*x^14 - 32*x^13 + 41*x^12 - 113*x^11 - 320*x^10 + 501*x^9 + 2370*x^8 + 5301*x^7 + 11097*x^6 + 14526*x^5 + 13716*x^4 + 13284*x^3 + 9315*x^2 + 3159*x + 729);
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
| \(\Q(\sqrt{-183}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{61})\), 4.2.11163.1 x2, 4.0.549.1 x2, 8.0.1121513121.2, 8.2.1391050107747.2 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(61\)
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |