Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153)
gp: K = bnfinit(y^16 - 6*y^15 + 30*y^14 - 108*y^13 + 411*y^12 - 1060*y^11 + 796*y^10 + 5577*y^9 + 18564*y^8 - 64314*y^7 - 50075*y^6 - 50668*y^5 + 836728*y^4 + 2574663*y^3 + 5459082*y^2 + 3924023*y + 4936153, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153)
\( x^{16} - 6 x^{15} + 30 x^{14} - 108 x^{13} + 411 x^{12} - 1060 x^{11} + 796 x^{10} + 5577 x^{9} + \cdots + 4936153 \)
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: |
\(24722989956581301387479701814209\)
\(\medspace = 17^{14}\cdot 59^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(91.64\) | 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: | $17^{7/8}59^{1/2}\approx 91.6365672680861$ | ||
| Ramified primes: |
\(17\), \(59\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{26}a^{14}-\frac{2}{13}a^{13}-\frac{1}{13}a^{12}-\frac{3}{26}a^{11}-\frac{1}{13}a^{10}-\frac{2}{13}a^{9}-\frac{9}{26}a^{8}+\frac{4}{13}a^{6}-\frac{1}{2}a^{5}+\frac{2}{13}a^{4}-\frac{6}{13}a^{3}-\frac{7}{26}a^{2}-\frac{6}{13}a+\frac{3}{13}$, $\frac{1}{13\!\cdots\!46}a^{15}+\frac{23\!\cdots\!59}{13\!\cdots\!46}a^{14}-\frac{92\!\cdots\!08}{67\!\cdots\!73}a^{13}-\frac{13\!\cdots\!12}{67\!\cdots\!73}a^{12}+\frac{44\!\cdots\!85}{10\!\cdots\!42}a^{11}+\frac{28\!\cdots\!00}{67\!\cdots\!73}a^{10}-\frac{72\!\cdots\!60}{67\!\cdots\!73}a^{9}+\frac{24\!\cdots\!01}{13\!\cdots\!46}a^{8}+\frac{78\!\cdots\!23}{67\!\cdots\!73}a^{7}+\frac{48\!\cdots\!34}{67\!\cdots\!73}a^{6}+\frac{54\!\cdots\!43}{13\!\cdots\!46}a^{5}+\frac{32\!\cdots\!40}{67\!\cdots\!73}a^{4}-\frac{37\!\cdots\!38}{75\!\cdots\!57}a^{3}-\frac{66\!\cdots\!47}{13\!\cdots\!46}a^{2}-\frac{27\!\cdots\!07}{67\!\cdots\!73}a+\frac{34\!\cdots\!63}{13\!\cdots\!46}$
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
$C_{2}\times C_{626}$, which has order $1252$ (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: |
$\frac{87\!\cdots\!54}{24\!\cdots\!51}a^{15}-\frac{63\!\cdots\!64}{24\!\cdots\!51}a^{14}+\frac{32\!\cdots\!34}{24\!\cdots\!51}a^{13}-\frac{12\!\cdots\!57}{24\!\cdots\!51}a^{12}+\frac{44\!\cdots\!34}{24\!\cdots\!51}a^{11}-\frac{11\!\cdots\!38}{24\!\cdots\!51}a^{10}+\frac{79\!\cdots\!55}{24\!\cdots\!51}a^{9}+\frac{84\!\cdots\!40}{24\!\cdots\!51}a^{8}-\frac{50\!\cdots\!04}{24\!\cdots\!51}a^{7}-\frac{42\!\cdots\!85}{24\!\cdots\!51}a^{6}-\frac{13\!\cdots\!52}{24\!\cdots\!51}a^{5}+\frac{94\!\cdots\!13}{24\!\cdots\!51}a^{4}+\frac{51\!\cdots\!11}{24\!\cdots\!51}a^{3}+\frac{13\!\cdots\!46}{18\!\cdots\!27}a^{2}+\frac{94\!\cdots\!89}{24\!\cdots\!51}a+\frac{11\!\cdots\!81}{24\!\cdots\!51}$, $\frac{63\!\cdots\!06}{24\!\cdots\!51}a^{15}-\frac{46\!\cdots\!21}{24\!\cdots\!51}a^{14}+\frac{24\!\cdots\!87}{24\!\cdots\!51}a^{13}-\frac{91\!\cdots\!57}{24\!\cdots\!51}a^{12}+\frac{32\!\cdots\!87}{24\!\cdots\!51}a^{11}-\frac{85\!\cdots\!04}{24\!\cdots\!51}a^{10}+\frac{60\!\cdots\!74}{24\!\cdots\!51}a^{9}+\frac{62\!\cdots\!43}{24\!\cdots\!51}a^{8}-\frac{43\!\cdots\!23}{24\!\cdots\!51}a^{7}-\frac{28\!\cdots\!82}{24\!\cdots\!51}a^{6}-\frac{97\!\cdots\!20}{24\!\cdots\!51}a^{5}+\frac{55\!\cdots\!22}{24\!\cdots\!51}a^{4}+\frac{36\!\cdots\!29}{24\!\cdots\!51}a^{3}+\frac{10\!\cdots\!41}{18\!\cdots\!27}a^{2}+\frac{70\!\cdots\!58}{24\!\cdots\!51}a+\frac{95\!\cdots\!90}{24\!\cdots\!51}$, $\frac{23\!\cdots\!48}{24\!\cdots\!51}a^{15}-\frac{16\!\cdots\!43}{24\!\cdots\!51}a^{14}+\frac{85\!\cdots\!47}{24\!\cdots\!51}a^{13}-\frac{31\!\cdots\!00}{24\!\cdots\!51}a^{12}+\frac{11\!\cdots\!47}{24\!\cdots\!51}a^{11}-\frac{29\!\cdots\!34}{24\!\cdots\!51}a^{10}+\frac{18\!\cdots\!81}{24\!\cdots\!51}a^{9}+\frac{22\!\cdots\!97}{24\!\cdots\!51}a^{8}-\frac{68\!\cdots\!81}{24\!\cdots\!51}a^{7}-\frac{14\!\cdots\!03}{24\!\cdots\!51}a^{6}-\frac{33\!\cdots\!32}{24\!\cdots\!51}a^{5}+\frac{38\!\cdots\!91}{24\!\cdots\!51}a^{4}+\frac{14\!\cdots\!82}{24\!\cdots\!51}a^{3}+\frac{29\!\cdots\!05}{18\!\cdots\!27}a^{2}+\frac{23\!\cdots\!31}{24\!\cdots\!51}a+\frac{39\!\cdots\!42}{24\!\cdots\!51}$, $\frac{68\!\cdots\!05}{13\!\cdots\!46}a^{15}-\frac{25\!\cdots\!22}{67\!\cdots\!73}a^{14}+\frac{25\!\cdots\!65}{13\!\cdots\!46}a^{13}-\frac{49\!\cdots\!62}{67\!\cdots\!73}a^{12}+\frac{17\!\cdots\!54}{67\!\cdots\!73}a^{11}-\frac{88\!\cdots\!93}{13\!\cdots\!46}a^{10}+\frac{25\!\cdots\!49}{67\!\cdots\!73}a^{9}+\frac{36\!\cdots\!15}{67\!\cdots\!73}a^{8}-\frac{66\!\cdots\!57}{13\!\cdots\!46}a^{7}-\frac{12\!\cdots\!23}{67\!\cdots\!73}a^{6}-\frac{10\!\cdots\!88}{67\!\cdots\!73}a^{5}+\frac{61\!\cdots\!61}{13\!\cdots\!46}a^{4}+\frac{21\!\cdots\!35}{75\!\cdots\!57}a^{3}+\frac{59\!\cdots\!12}{51\!\cdots\!21}a^{2}+\frac{79\!\cdots\!59}{13\!\cdots\!46}a+\frac{17\!\cdots\!83}{13\!\cdots\!46}$, $\frac{10\!\cdots\!73}{13\!\cdots\!46}a^{15}-\frac{75\!\cdots\!63}{13\!\cdots\!46}a^{14}+\frac{38\!\cdots\!33}{13\!\cdots\!46}a^{13}-\frac{72\!\cdots\!31}{67\!\cdots\!73}a^{12}+\frac{53\!\cdots\!63}{13\!\cdots\!46}a^{11}-\frac{15\!\cdots\!21}{13\!\cdots\!46}a^{10}+\frac{87\!\cdots\!89}{67\!\cdots\!73}a^{9}+\frac{62\!\cdots\!39}{13\!\cdots\!46}a^{8}+\frac{91\!\cdots\!19}{13\!\cdots\!46}a^{7}-\frac{49\!\cdots\!66}{67\!\cdots\!73}a^{6}+\frac{41\!\cdots\!19}{13\!\cdots\!46}a^{5}+\frac{13\!\cdots\!35}{13\!\cdots\!46}a^{4}+\frac{46\!\cdots\!91}{75\!\cdots\!57}a^{3}+\frac{93\!\cdots\!29}{10\!\cdots\!42}a^{2}+\frac{10\!\cdots\!43}{13\!\cdots\!46}a-\frac{46\!\cdots\!91}{13\!\cdots\!46}$, $\frac{17\!\cdots\!73}{13\!\cdots\!46}a^{15}-\frac{10\!\cdots\!49}{13\!\cdots\!46}a^{14}+\frac{42\!\cdots\!69}{13\!\cdots\!46}a^{13}-\frac{55\!\cdots\!21}{67\!\cdots\!73}a^{12}+\frac{26\!\cdots\!19}{13\!\cdots\!46}a^{11}+\frac{14\!\cdots\!45}{13\!\cdots\!46}a^{10}-\frac{32\!\cdots\!13}{67\!\cdots\!73}a^{9}+\frac{35\!\cdots\!27}{13\!\cdots\!46}a^{8}-\frac{27\!\cdots\!75}{13\!\cdots\!46}a^{7}-\frac{25\!\cdots\!31}{67\!\cdots\!73}a^{6}-\frac{18\!\cdots\!93}{13\!\cdots\!46}a^{5}+\frac{38\!\cdots\!45}{13\!\cdots\!46}a^{4}+\frac{25\!\cdots\!07}{58\!\cdots\!89}a^{3}+\frac{56\!\cdots\!27}{13\!\cdots\!46}a^{2}+\frac{20\!\cdots\!33}{10\!\cdots\!42}a+\frac{55\!\cdots\!13}{13\!\cdots\!46}$, $\frac{16\!\cdots\!67}{13\!\cdots\!46}a^{15}-\frac{12\!\cdots\!27}{13\!\cdots\!46}a^{14}+\frac{69\!\cdots\!45}{13\!\cdots\!46}a^{13}-\frac{15\!\cdots\!77}{67\!\cdots\!73}a^{12}+\frac{11\!\cdots\!97}{13\!\cdots\!46}a^{11}-\frac{39\!\cdots\!31}{13\!\cdots\!46}a^{10}+\frac{33\!\cdots\!93}{67\!\cdots\!73}a^{9}-\frac{19\!\cdots\!21}{13\!\cdots\!46}a^{8}+\frac{18\!\cdots\!81}{13\!\cdots\!46}a^{7}-\frac{30\!\cdots\!15}{67\!\cdots\!73}a^{6}+\frac{51\!\cdots\!83}{13\!\cdots\!46}a^{5}-\frac{33\!\cdots\!85}{10\!\cdots\!42}a^{4}+\frac{11\!\cdots\!05}{75\!\cdots\!57}a^{3}-\frac{39\!\cdots\!93}{13\!\cdots\!46}a^{2}-\frac{13\!\cdots\!25}{13\!\cdots\!46}a+\frac{48\!\cdots\!43}{13\!\cdots\!46}$
| 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: | \( 1288346.89197 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1288346.89197 \cdot 1252}{2\cdot\sqrt{24722989956581301387479701814209}}\cr\approx \mathstrut & 0.393999434016 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153)
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 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153, 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 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153);
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 - 6*x^15 + 30*x^14 - 108*x^13 + 411*x^12 - 1060*x^11 + 796*x^10 + 5577*x^9 + 18564*x^8 - 64314*x^7 - 50075*x^6 - 50668*x^5 + 836728*x^4 + 2574663*x^3 + 5459082*x^2 + 3924023*x + 4936153);
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
$C_2^2:C_8$ (as 16T24):
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 $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.289867.1, 4.2.17051.1, 8.4.1428388920713.1, 8.0.4972221833001953.2, 8.4.84022877689.1 |
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 sibling: | 16.8.2040294908815649000428369.1 |
| Minimal sibling: | 16.8.2040294908815649000428369.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.14.1 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
|
\(59\)
| 59.4.2.2 | $x^{4} - 13924 x^{3} - 56674102 x^{2} - 11222744 x + 6962$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 59.4.2.2 | $x^{4} - 13924 x^{3} - 56674102 x^{2} - 11222744 x + 6962$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 59.8.4.1 | $x^{8} + 240 x^{6} + 80 x^{5} + 21130 x^{4} - 9280 x^{3} + 808256 x^{2} - 825840 x + 11417625$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |