Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561)
gp: K = bnfinit(y^32 - 3*y^24 + 117486*y^16 + 35572*y^8 + 6561, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561)
\( x^{32} - 3x^{24} + 117486x^{16} + 35572x^{8} + 6561 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(43005107648088506255732033296018914516707307660871991296\)
\(\medspace = 2^{96}\cdot 13^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.77\) | 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: | $2^{3}13^{3/4}\approx 54.770600336184366$ | ||
| Ramified primes: |
\(2\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(208=2^{4}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{208}(1,·)$, $\chi_{208}(131,·)$, $\chi_{208}(5,·)$, $\chi_{208}(129,·)$, $\chi_{208}(21,·)$, $\chi_{208}(151,·)$, $\chi_{208}(25,·)$, $\chi_{208}(155,·)$, $\chi_{208}(157,·)$, $\chi_{208}(31,·)$, $\chi_{208}(161,·)$, $\chi_{208}(27,·)$, $\chi_{208}(135,·)$, $\chi_{208}(47,·)$, $\chi_{208}(177,·)$, $\chi_{208}(51,·)$, $\chi_{208}(181,·)$, $\chi_{208}(183,·)$, $\chi_{208}(57,·)$, $\chi_{208}(187,·)$, $\chi_{208}(53,·)$, $\chi_{208}(73,·)$, $\chi_{208}(203,·)$, $\chi_{208}(77,·)$, $\chi_{208}(79,·)$, $\chi_{208}(83,·)$, $\chi_{208}(207,·)$, $\chi_{208}(99,·)$, $\chi_{208}(103,·)$, $\chi_{208}(105,·)$, $\chi_{208}(109,·)$, $\chi_{208}(125,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{7}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{8}$, $\frac{1}{3}a^{17}-\frac{1}{3}a$, $\frac{1}{9}a^{18}-\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{19}+\frac{2}{27}a^{11}-\frac{8}{27}a^{3}$, $\frac{1}{27}a^{20}+\frac{2}{27}a^{12}-\frac{8}{27}a^{4}$, $\frac{1}{27}a^{21}+\frac{2}{27}a^{13}-\frac{8}{27}a^{5}$, $\frac{1}{27}a^{22}+\frac{2}{27}a^{14}-\frac{8}{27}a^{6}$, $\frac{1}{27}a^{23}+\frac{2}{27}a^{15}-\frac{8}{27}a^{7}$, $\frac{1}{24421218093}a^{24}-\frac{2831412994}{24421218093}a^{16}+\frac{12033653563}{24421218093}a^{8}+\frac{169209443}{904489559}$, $\frac{1}{24421218093}a^{25}-\frac{2831412994}{24421218093}a^{17}+\frac{3893247532}{24421218093}a^{9}-\frac{396861230}{2713468677}a$, $\frac{1}{24421218093}a^{26}-\frac{117944317}{24421218093}a^{18}+\frac{1179778855}{24421218093}a^{10}-\frac{2999562808}{8140406031}a^{2}$, $\frac{1}{24421218093}a^{27}-\frac{117944317}{24421218093}a^{19}+\frac{1179778855}{24421218093}a^{11}-\frac{2999562808}{8140406031}a^{3}$, $\frac{1}{73263654279}a^{28}-\frac{113603764}{8140406031}a^{20}-\frac{2923202098}{24421218093}a^{12}-\frac{9903177983}{73263654279}a^{4}$, $\frac{1}{219790962837}a^{29}+\frac{563678267}{73263654279}a^{21}-\frac{371407660}{24421218093}a^{13}-\frac{31610927399}{219790962837}a^{5}$, $\frac{1}{659372888511}a^{30}-\frac{2149790410}{219790962837}a^{22}+\frac{5960019253}{73263654279}a^{14}+\frac{326566937965}{659372888511}a^{6}$, $\frac{1}{1978118665533}a^{31}-\frac{2149790410}{659372888511}a^{23}-\frac{18461198840}{219790962837}a^{15}-\frac{552596913383}{1978118665533}a^{7}$
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
$C_{5}\times C_{5}\times C_{15}$, which has order $375$ (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: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{327797006}{1978118665533} a^{31} - \frac{343926968}{659372888511} a^{23} + \frac{4279075670711}{219790962837} a^{15} + \frac{5965382536868}{1978118665533} a^{7} \)
(order $16$)
| 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: |
$\frac{72278381}{659372888511}a^{30}+\frac{534637}{8140406031}a^{26}-\frac{79316876}{219790962837}a^{22}-\frac{570895}{2713468677}a^{18}+\frac{943528268096}{73263654279}a^{14}+\frac{6978621676}{904489559}a^{10}+\frac{97979779304}{659372888511}a^{6}+\frac{6973200247}{8140406031}a^{2}+1$, $\frac{5993117}{219790962837}a^{29}-\frac{534637}{8140406031}a^{26}-\frac{7785128}{73263654279}a^{21}+\frac{570895}{2713468677}a^{18}+\frac{8693100391}{2713468677}a^{13}-\frac{6978621676}{904489559}a^{10}-\frac{419585889307}{219790962837}a^{5}-\frac{6973200247}{8140406031}a^{2}-1$, $\frac{65394}{904489559}a^{27}-\frac{534637}{8140406031}a^{26}+\frac{858068}{24421218093}a^{25}-\frac{5383763}{24421218093}a^{19}+\frac{570895}{2713468677}a^{18}-\frac{2953487}{24421218093}a^{17}+\frac{207439477526}{24421218093}a^{11}-\frac{6978621676}{904489559}a^{10}+\frac{100840752737}{24421218093}a^{9}+\frac{49916865853}{24421218093}a^{3}-\frac{6973200247}{8140406031}a^{2}-\frac{838856612}{2713468677}a$, $\frac{54299030}{659372888511}a^{30}-\frac{372910}{24421218093}a^{26}-\frac{55961492}{219790962837}a^{22}+\frac{1466977}{24421218093}a^{18}+\frac{708814557539}{73263654279}a^{14}-\frac{43790902810}{24421218093}a^{10}+\frac{1356737447225}{659372888511}a^{6}+\frac{4627886278}{8140406031}a^{2}$, $\frac{54299030}{659372888511}a^{30}+\frac{372910}{24421218093}a^{26}-\frac{55961492}{219790962837}a^{22}-\frac{1466977}{24421218093}a^{18}+\frac{708814557539}{73263654279}a^{14}+\frac{43790902810}{24421218093}a^{10}+\frac{1356737447225}{659372888511}a^{6}-\frac{4627886278}{8140406031}a^{2}$, $\frac{65394}{904489559}a^{28}-\frac{597383}{24421218093}a^{24}-\frac{5383763}{24421218093}a^{20}+\frac{2255429}{24421218093}a^{16}+\frac{207439477526}{24421218093}a^{12}-\frac{70308570149}{24421218093}a^{8}+\frac{49916865853}{24421218093}a^{4}+\frac{194957494}{904489559}$, $\frac{9574}{234069183}a^{28}+\frac{2089069}{24421218093}a^{26}-\frac{8845}{78023061}a^{20}-\frac{6624565}{24421218093}a^{18}+\frac{124981039}{26007687}a^{12}+\frac{245472635179}{24421218093}a^{10}+\frac{606387187}{234069183}a^{4}+\frac{9084516689}{8140406031}a^{2}-1$, $\frac{366482351}{1978118665533}a^{31}+\frac{35685785}{219790962837}a^{29}+\frac{65394}{904489559}a^{27}+\frac{858068}{24421218093}a^{25}-\frac{350353010}{659372888511}a^{23}-\frac{35131769}{73263654279}a^{21}-\frac{5383763}{24421218093}a^{19}-\frac{2953487}{24421218093}a^{17}+\frac{4784040311093}{219790962837}a^{15}+\frac{465842398645}{24421218093}a^{13}+\frac{207439477526}{24421218093}a^{11}+\frac{100840752737}{24421218093}a^{9}+\frac{18731504371841}{1978118665533}a^{7}+\frac{1461006542555}{219790962837}a^{5}+\frac{49916865853}{24421218093}a^{3}-\frac{838856612}{2713468677}a$, $\frac{72278381}{659372888511}a^{30}-\frac{1392682}{73263654279}a^{28}-\frac{858068}{24421218093}a^{26}-\frac{79316876}{219790962837}a^{22}+\frac{1805186}{24421218093}a^{20}+\frac{2953487}{24421218093}a^{18}+\frac{943528268096}{73263654279}a^{14}-\frac{54549373642}{24421218093}a^{12}-\frac{100840752737}{24421218093}a^{10}+\frac{97979779304}{659372888511}a^{6}+\frac{97515167333}{73263654279}a^{4}+\frac{838856612}{2713468677}a^{2}-1$, $\frac{327797006}{1978118665533}a^{31}+\frac{534637}{8140406031}a^{26}-\frac{2834912}{24421218093}a^{25}+\frac{658948}{8140406031}a^{24}-\frac{343926968}{659372888511}a^{23}-\frac{570895}{2713468677}a^{18}+\frac{8809133}{24421218093}a^{17}-\frac{1951882}{8140406031}a^{16}+\frac{4279075670711}{219790962837}a^{15}+\frac{6978621676}{904489559}a^{10}-\frac{333054667694}{24421218093}a^{9}+\frac{77404638319}{8140406031}a^{8}+\frac{5965382536868}{1978118665533}a^{7}+\frac{6973200247}{8140406031}a^{2}-\frac{6191428924}{2713468677}a+\frac{2343428512}{904489559}$, $\frac{72278381}{659372888511}a^{30}-\frac{5993117}{219790962837}a^{29}-\frac{534637}{8140406031}a^{27}-\frac{597383}{24421218093}a^{24}-\frac{79316876}{219790962837}a^{22}+\frac{7785128}{73263654279}a^{21}+\frac{570895}{2713468677}a^{19}+\frac{2255429}{24421218093}a^{16}+\frac{943528268096}{73263654279}a^{14}-\frac{8693100391}{2713468677}a^{13}-\frac{6978621676}{904489559}a^{11}-\frac{70308570149}{24421218093}a^{8}+\frac{97979779304}{659372888511}a^{6}+\frac{419585889307}{219790962837}a^{5}-\frac{6973200247}{8140406031}a^{3}+\frac{194957494}{904489559}$, $\frac{179335736}{659372888511}a^{30}-\frac{9574}{234069183}a^{28}+\frac{65394}{904489559}a^{27}-\frac{858091}{24421218093}a^{25}-\frac{184712183}{219790962837}a^{22}+\frac{8845}{78023061}a^{20}-\frac{5383763}{24421218093}a^{19}+\frac{2204101}{24421218093}a^{17}+\frac{2341055464031}{73263654279}a^{14}-\frac{124981039}{26007687}a^{12}+\frac{207439477526}{24421218093}a^{11}-\frac{100840979632}{24421218093}a^{9}+\frac{4480999406969}{659372888511}a^{6}-\frac{606387187}{234069183}a^{4}+\frac{49916865853}{24421218093}a^{3}-\frac{8123126278}{2713468677}a$, $\frac{327797006}{1978118665533}a^{31}-\frac{72278381}{659372888511}a^{30}-\frac{7597097}{73263654279}a^{28}-\frac{858091}{24421218093}a^{25}-\frac{343926968}{659372888511}a^{23}+\frac{79316876}{219790962837}a^{22}+\frac{8748427}{24421218093}a^{20}+\frac{2204101}{24421218093}a^{17}+\frac{4279075670711}{219790962837}a^{15}-\frac{943528268096}{73263654279}a^{14}-\frac{297521532536}{24421218093}a^{12}-\frac{100840979632}{24421218093}a^{9}+\frac{5965382536868}{1978118665533}a^{7}-\frac{97979779304}{659372888511}a^{6}+\frac{132271532443}{73263654279}a^{4}-\frac{8123126278}{2713468677}a$, $\frac{72278381}{659372888511}a^{31}+\frac{72278381}{659372888511}a^{30}+\frac{858068}{24421218093}a^{25}+\frac{597383}{24421218093}a^{24}-\frac{79316876}{219790962837}a^{23}-\frac{79316876}{219790962837}a^{22}-\frac{2953487}{24421218093}a^{17}-\frac{2255429}{24421218093}a^{16}+\frac{943528268096}{73263654279}a^{15}+\frac{943528268096}{73263654279}a^{14}+\frac{100840752737}{24421218093}a^{9}+\frac{70308570149}{24421218093}a^{8}+\frac{97979779304}{659372888511}a^{7}+\frac{97979779304}{659372888511}a^{6}-\frac{838856612}{2713468677}a-\frac{194957494}{904489559}$, $\frac{9139574}{86005159371}a^{31}-\frac{5993117}{219790962837}a^{30}-\frac{25788160}{219790962837}a^{29}+\frac{1392682}{73263654279}a^{28}+\frac{534637}{8140406031}a^{27}+\frac{1231001}{24421218093}a^{26}-\frac{9655}{904489559}a^{25}-\frac{597406}{24421218093}a^{24}-\frac{9139547}{28668386457}a^{23}+\frac{7785128}{73263654279}a^{22}+\frac{26765608}{73263654279}a^{21}-\frac{1805186}{24421218093}a^{20}-\frac{570895}{2713468677}a^{19}-\frac{3671078}{24421218093}a^{18}+\frac{25854}{904489559}a^{17}+\frac{1506043}{24421218093}a^{16}+\frac{119308292234}{9556128819}a^{15}-\frac{8693100391}{2713468677}a^{14}-\frac{336640824638}{24421218093}a^{13}+\frac{54549373642}{24421218093}a^{12}+\frac{6978621676}{904489559}a^{11}+\frac{144631882442}{24421218093}a^{10}-\frac{3392464732}{2713468677}a^{9}-\frac{70308797044}{24421218093}a^{8}+\frac{325113725393}{86005159371}a^{7}+\frac{419585889307}{219790962837}a^{6}-\frac{592168860571}{219790962837}a^{5}-\frac{97515167333}{73263654279}a^{4}+\frac{6973200247}{8140406031}a^{3}+\frac{11601086525}{8140406031}a^{2}+\frac{253984130}{2713468677}a-\frac{983391018}{904489559}$
| 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: | \( 6740125694616.533 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 6740125694616.533 \cdot 375}{16\cdot\sqrt{43005107648088506255732033296018914516707307660871991296}}\cr\approx \mathstrut & 0.142133730510525 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561)
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^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561, 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^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561);
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^32 - 3*x^24 + 117486*x^16 + 35572*x^8 + 6561);
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
$C_2\times C_4^2$ (as 32T36):
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 32 |
| The 32 conjugacy class representatives for $C_2\times C_4^2$ |
| Character table for $C_2\times C_4^2$ is not computed |
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 | ${\href{/padicField/3.4.0.1}{4} }^{8}$ | ${\href{/padicField/5.4.0.1}{4} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{16}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $32$ | $8$ | $4$ | $96$ | |||
|
\(13\)
| 13.16.12.1 | $x^{16} + 12 x^{14} + 48 x^{13} + 114 x^{12} + 432 x^{11} + 888 x^{10} - 5280 x^{9} + 4933 x^{8} + 13680 x^{7} + 64788 x^{6} - 10416 x^{5} + 182568 x^{4} + 90432 x^{3} + 720840 x^{2} + 400992 x + 573316$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
| 13.16.12.1 | $x^{16} + 12 x^{14} + 48 x^{13} + 114 x^{12} + 432 x^{11} + 888 x^{10} - 5280 x^{9} + 4933 x^{8} + 13680 x^{7} + 64788 x^{6} - 10416 x^{5} + 182568 x^{4} + 90432 x^{3} + 720840 x^{2} + 400992 x + 573316$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |