Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361)
gp: K = bnfinit(y^16 - 4*y^15 + 14*y^14 - 30*y^13 + 41*y^12 - 62*y^11 + 45*y^10 - 128*y^9 + 267*y^8 - 454*y^7 + 1110*y^6 - 1124*y^5 + 1666*y^4 - 1540*y^3 + 1071*y^2 - 798*y + 361, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361)
\( x^{16} - 4 x^{15} + 14 x^{14} - 30 x^{13} + 41 x^{12} - 62 x^{11} + 45 x^{10} - 128 x^{9} + 267 x^{8} + \cdots + 361 \)
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: |
\(26873856000000000000\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.38\) | 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: | $2^{3/2}3^{1/2}5^{3/4}\approx 16.3807251762544$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{27474}a^{14}-\frac{239}{4579}a^{13}-\frac{2069}{27474}a^{12}+\frac{1595}{13737}a^{11}+\frac{4937}{13737}a^{10}-\frac{2144}{13737}a^{9}-\frac{2845}{9158}a^{8}-\frac{2068}{13737}a^{7}+\frac{3073}{13737}a^{6}+\frac{6182}{13737}a^{5}-\frac{1999}{4579}a^{4}-\frac{1890}{4579}a^{3}-\frac{5131}{13737}a^{2}-\frac{6503}{13737}a-\frac{475}{1446}$, $\frac{1}{12\!\cdots\!06}a^{15}-\frac{15755822554}{21\!\cdots\!01}a^{14}-\frac{17\!\cdots\!73}{12\!\cdots\!06}a^{13}-\frac{592277838126173}{63\!\cdots\!03}a^{12}-\frac{751200758188061}{21\!\cdots\!01}a^{11}-\frac{396058893816431}{63\!\cdots\!03}a^{10}-\frac{21\!\cdots\!85}{12\!\cdots\!06}a^{9}-\frac{483584240210977}{63\!\cdots\!03}a^{8}-\frac{17\!\cdots\!54}{63\!\cdots\!03}a^{7}-\frac{22\!\cdots\!11}{63\!\cdots\!03}a^{6}+\frac{534470531442427}{21\!\cdots\!01}a^{5}-\frac{111388409689270}{63\!\cdots\!03}a^{4}+\frac{31\!\cdots\!23}{63\!\cdots\!03}a^{3}-\frac{554444734176579}{21\!\cdots\!01}a^{2}+\frac{32165052804613}{411433377169026}a-\frac{18045844430611}{335643018216837}$
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
$C_{2}$, which has order $2$
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: |
\( \frac{39588464}{6140979987} a^{15} - \frac{425424005}{12281959974} a^{14} + \frac{718735085}{6140979987} a^{13} - \frac{1102067725}{4093986658} a^{12} + \frac{2207046845}{6140979987} a^{11} - \frac{2167136359}{6140979987} a^{10} + \frac{473539990}{2046993329} a^{9} - \frac{2229001775}{4093986658} a^{8} + \frac{14455901725}{6140979987} a^{7} - \frac{26757015230}{6140979987} a^{6} + \frac{48270245006}{6140979987} a^{5} - \frac{21114357805}{2046993329} a^{4} + \frac{48719748340}{6140979987} a^{3} - \frac{46854375710}{6140979987} a^{2} + \frac{20790331760}{6140979987} a + \frac{216648805}{646418946} \)
(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: |
$\frac{8242459537271}{42\!\cdots\!02}a^{15}+\frac{31620525091439}{12\!\cdots\!06}a^{14}-\frac{131348031153487}{12\!\cdots\!06}a^{13}+\frac{808509240571111}{12\!\cdots\!06}a^{12}-\frac{311161903015820}{21\!\cdots\!01}a^{11}+\frac{567642272705203}{63\!\cdots\!03}a^{10}-\frac{822266268243807}{42\!\cdots\!02}a^{9}-\frac{970423874117541}{42\!\cdots\!02}a^{8}-\frac{25\!\cdots\!93}{63\!\cdots\!03}a^{7}+\frac{28\!\cdots\!76}{21\!\cdots\!01}a^{6}-\frac{61\!\cdots\!25}{63\!\cdots\!03}a^{5}+\frac{18\!\cdots\!46}{335643018216837}a^{4}-\frac{77\!\cdots\!97}{63\!\cdots\!03}a^{3}+\frac{22\!\cdots\!74}{63\!\cdots\!03}a^{2}-\frac{705067519420591}{411433377169026}a-\frac{130701120145553}{223762012144558}$, $\frac{6439245098887}{42\!\cdots\!02}a^{15}-\frac{30123269557867}{42\!\cdots\!02}a^{14}+\frac{365452437852121}{12\!\cdots\!06}a^{13}-\frac{946505656741711}{12\!\cdots\!06}a^{12}+\frac{882515594126132}{63\!\cdots\!03}a^{11}-\frac{14\!\cdots\!27}{63\!\cdots\!03}a^{10}+\frac{25\!\cdots\!59}{12\!\cdots\!06}a^{9}-\frac{34\!\cdots\!89}{12\!\cdots\!06}a^{8}+\frac{29\!\cdots\!59}{63\!\cdots\!03}a^{7}-\frac{24\!\cdots\!96}{21\!\cdots\!01}a^{6}+\frac{65\!\cdots\!23}{21\!\cdots\!01}a^{5}-\frac{25\!\cdots\!22}{63\!\cdots\!03}a^{4}+\frac{37\!\cdots\!37}{63\!\cdots\!03}a^{3}-\frac{11\!\cdots\!99}{21\!\cdots\!01}a^{2}+\frac{69729554481959}{21654388272054}a-\frac{488057903682219}{223762012144558}$, $\frac{3163380798634}{335643018216837}a^{15}-\frac{299755183407812}{63\!\cdots\!03}a^{14}+\frac{10\!\cdots\!09}{63\!\cdots\!03}a^{13}-\frac{23\!\cdots\!34}{63\!\cdots\!03}a^{12}+\frac{10\!\cdots\!33}{21\!\cdots\!01}a^{11}-\frac{32\!\cdots\!04}{63\!\cdots\!03}a^{10}+\frac{19\!\cdots\!62}{63\!\cdots\!03}a^{9}-\frac{19\!\cdots\!41}{21\!\cdots\!01}a^{8}+\frac{19\!\cdots\!98}{63\!\cdots\!03}a^{7}-\frac{12\!\cdots\!29}{21\!\cdots\!01}a^{6}+\frac{70\!\cdots\!83}{63\!\cdots\!03}a^{5}-\frac{84\!\cdots\!49}{63\!\cdots\!03}a^{4}+\frac{27\!\cdots\!28}{21\!\cdots\!01}a^{3}-\frac{21\!\cdots\!19}{21\!\cdots\!01}a^{2}+\frac{443697345273162}{68572229528171}a-\frac{337582622592988}{335643018216837}$, $\frac{25402557979457}{63\!\cdots\!03}a^{15}-\frac{43201355472493}{21\!\cdots\!01}a^{14}+\frac{126931874122251}{21\!\cdots\!01}a^{13}-\frac{801885251719129}{63\!\cdots\!03}a^{12}+\frac{759746007607832}{63\!\cdots\!03}a^{11}-\frac{193951751655812}{21\!\cdots\!01}a^{10}+\frac{281572481783572}{21\!\cdots\!01}a^{9}-\frac{758637544992577}{21\!\cdots\!01}a^{8}+\frac{95\!\cdots\!50}{63\!\cdots\!03}a^{7}-\frac{10\!\cdots\!31}{63\!\cdots\!03}a^{6}+\frac{62\!\cdots\!82}{21\!\cdots\!01}a^{5}-\frac{29\!\cdots\!27}{63\!\cdots\!03}a^{4}+\frac{97\!\cdots\!25}{63\!\cdots\!03}a^{3}-\frac{573721979283844}{111881006072279}a^{2}+\frac{43501007904041}{68572229528171}a-\frac{563062717619704}{335643018216837}$, $\frac{52110132127891}{63\!\cdots\!03}a^{15}-\frac{381519815315645}{12\!\cdots\!06}a^{14}+\frac{637322093721749}{63\!\cdots\!03}a^{13}-\frac{24\!\cdots\!83}{12\!\cdots\!06}a^{12}+\frac{12\!\cdots\!62}{63\!\cdots\!03}a^{11}-\frac{17\!\cdots\!11}{63\!\cdots\!03}a^{10}+\frac{80507046975277}{21\!\cdots\!01}a^{9}-\frac{32\!\cdots\!25}{42\!\cdots\!02}a^{8}+\frac{194447422789753}{111881006072279}a^{7}-\frac{17\!\cdots\!78}{63\!\cdots\!03}a^{6}+\frac{42\!\cdots\!19}{63\!\cdots\!03}a^{5}-\frac{26\!\cdots\!56}{63\!\cdots\!03}a^{4}+\frac{13\!\cdots\!24}{21\!\cdots\!01}a^{3}-\frac{21\!\cdots\!91}{63\!\cdots\!03}a^{2}+\frac{384226597718336}{205716688584513}a-\frac{178448845376515}{671286036433674}$, $\frac{43435572134437}{12\!\cdots\!06}a^{15}-\frac{37294516041890}{63\!\cdots\!03}a^{14}+\frac{70894210540129}{12\!\cdots\!06}a^{13}+\frac{238919809747858}{63\!\cdots\!03}a^{12}-\frac{10\!\cdots\!46}{63\!\cdots\!03}a^{11}+\frac{386096919958401}{21\!\cdots\!01}a^{10}-\frac{813034413867915}{42\!\cdots\!02}a^{9}-\frac{13\!\cdots\!76}{63\!\cdots\!03}a^{8}+\frac{10\!\cdots\!23}{63\!\cdots\!03}a^{7}+\frac{28\!\cdots\!16}{21\!\cdots\!01}a^{6}-\frac{84\!\cdots\!24}{63\!\cdots\!03}a^{5}+\frac{98\!\cdots\!38}{21\!\cdots\!01}a^{4}-\frac{12\!\cdots\!37}{21\!\cdots\!01}a^{3}+\frac{29\!\cdots\!84}{63\!\cdots\!03}a^{2}-\frac{547814913110103}{137144459056342}a+\frac{771424367444275}{335643018216837}$, $\frac{41772409409607}{42\!\cdots\!02}a^{15}-\frac{211590769552828}{63\!\cdots\!03}a^{14}+\frac{12\!\cdots\!31}{12\!\cdots\!06}a^{13}-\frac{267052561205829}{21\!\cdots\!01}a^{12}-\frac{371466824465453}{63\!\cdots\!03}a^{11}+\frac{13\!\cdots\!75}{63\!\cdots\!03}a^{10}-\frac{72\!\cdots\!35}{12\!\cdots\!06}a^{9}-\frac{26\!\cdots\!45}{63\!\cdots\!03}a^{8}+\frac{39\!\cdots\!41}{21\!\cdots\!01}a^{7}-\frac{92\!\cdots\!38}{63\!\cdots\!03}a^{6}+\frac{18\!\cdots\!67}{63\!\cdots\!03}a^{5}+\frac{30\!\cdots\!18}{63\!\cdots\!03}a^{4}-\frac{47\!\cdots\!92}{63\!\cdots\!03}a^{3}+\frac{61\!\cdots\!27}{63\!\cdots\!03}a^{2}-\frac{247944061317629}{21654388272054}a+\frac{16\!\cdots\!36}{335643018216837}$
| 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: | \( 2522.8409077 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2522.8409077 \cdot 2}{6\cdot\sqrt{26873856000000000000}}\cr\approx \mathstrut & 0.39404204447 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361)
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 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361, 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 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361);
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 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361);
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
$C_4\times D_4$ (as 16T19):
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 $C_4 \times D_4$ |
| Character table for $C_4 \times D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.0.8000.2, 4.4.72000.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.9000.1, 4.0.9000.2, 8.0.81000000.1, 8.0.5184000000.3, 8.0.3240000.1 |
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 siblings: | 16.0.6561000000000000.1, 16.0.331776000000000000.1, 16.8.26873856000000000000.4 |
| Minimal sibling: | 16.0.6561000000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ |
| 2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |