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Properties

Label 16.0.233...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.337\times 10^{23}$
Root discriminant \(28.88\)
Ramified primes $3,5,7,17$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $D_8:C_2$ (as 16T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 16*x^14 - 11*x^13 + 59*x^12 - 349*x^11 + 586*x^10 + 206*x^9 - 1354*x^8 + 280*x^7 + 1715*x^6 - 713*x^5 - 377*x^4 - 596*x^3 + 153*x^2 + 209*x + 361)
 
gp: K = bnfinit(y^16 - 7*y^15 + 16*y^14 - 11*y^13 + 59*y^12 - 349*y^11 + 586*y^10 + 206*y^9 - 1354*y^8 + 280*y^7 + 1715*y^6 - 713*y^5 - 377*y^4 - 596*y^3 + 153*y^2 + 209*y + 361, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 16*x^14 - 11*x^13 + 59*x^12 - 349*x^11 + 586*x^10 + 206*x^9 - 1354*x^8 + 280*x^7 + 1715*x^6 - 713*x^5 - 377*x^4 - 596*x^3 + 153*x^2 + 209*x + 361);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 16*x^14 - 11*x^13 + 59*x^12 - 349*x^11 + 586*x^10 + 206*x^9 - 1354*x^8 + 280*x^7 + 1715*x^6 - 713*x^5 - 377*x^4 - 596*x^3 + 153*x^2 + 209*x + 361)
 

\( x^{16} - 7 x^{15} + 16 x^{14} - 11 x^{13} + 59 x^{12} - 349 x^{11} + 586 x^{10} + 206 x^{9} - 1354 x^{8} + \cdots + 361 \) Copy content Toggle raw display

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:   \(233704553418537641015625\) \(\medspace = 3^{6}\cdot 5^{8}\cdot 7^{6}\cdot 17^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(28.88\)
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:  $3^{1/2}5^{1/2}7^{1/2}17^{1/2}\approx 42.24926034855522$
Ramified primes:   \(3\), \(5\), \(7\), \(17\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{53\!\cdots\!09}a^{15}+\frac{16\!\cdots\!88}{53\!\cdots\!09}a^{14}-\frac{17\!\cdots\!45}{53\!\cdots\!09}a^{13}+\frac{12\!\cdots\!41}{53\!\cdots\!09}a^{12}+\frac{23\!\cdots\!49}{53\!\cdots\!09}a^{11}+\frac{26\!\cdots\!51}{53\!\cdots\!09}a^{10}+\frac{47\!\cdots\!47}{53\!\cdots\!09}a^{9}+\frac{19\!\cdots\!11}{53\!\cdots\!09}a^{8}+\frac{25\!\cdots\!31}{53\!\cdots\!09}a^{7}+\frac{25\!\cdots\!54}{53\!\cdots\!09}a^{6}+\frac{63\!\cdots\!60}{53\!\cdots\!09}a^{5}-\frac{18\!\cdots\!06}{53\!\cdots\!09}a^{4}+\frac{19\!\cdots\!56}{53\!\cdots\!09}a^{3}+\frac{20\!\cdots\!95}{53\!\cdots\!09}a^{2}-\frac{13\!\cdots\!71}{53\!\cdots\!09}a+\frac{10\!\cdots\!51}{28\!\cdots\!11}$ Copy content Toggle raw display

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

$C_{2}$, which has order $2$ (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 $2$) Copy content Toggle raw display
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:   $\frac{113782483739885}{13\!\cdots\!11}a^{15}-\frac{710499445958061}{13\!\cdots\!11}a^{14}+\frac{11\!\cdots\!56}{13\!\cdots\!11}a^{13}+\frac{386698195060702}{13\!\cdots\!11}a^{12}+\frac{50\!\cdots\!94}{13\!\cdots\!11}a^{11}-\frac{33\!\cdots\!42}{13\!\cdots\!11}a^{10}+\frac{33\!\cdots\!36}{13\!\cdots\!11}a^{9}+\frac{88\!\cdots\!64}{13\!\cdots\!11}a^{8}-\frac{16\!\cdots\!79}{13\!\cdots\!11}a^{7}-\frac{59\!\cdots\!14}{13\!\cdots\!11}a^{6}+\frac{21\!\cdots\!65}{13\!\cdots\!11}a^{5}+\frac{51\!\cdots\!53}{13\!\cdots\!11}a^{4}-\frac{52\!\cdots\!37}{13\!\cdots\!11}a^{3}-\frac{15\!\cdots\!57}{13\!\cdots\!11}a^{2}-\frac{12\!\cdots\!18}{13\!\cdots\!11}a+\frac{12\!\cdots\!20}{13\!\cdots\!11}$, $\frac{42\!\cdots\!79}{53\!\cdots\!09}a^{15}-\frac{36\!\cdots\!33}{53\!\cdots\!09}a^{14}+\frac{11\!\cdots\!91}{53\!\cdots\!09}a^{13}-\frac{12\!\cdots\!46}{53\!\cdots\!09}a^{12}+\frac{22\!\cdots\!15}{53\!\cdots\!09}a^{11}-\frac{17\!\cdots\!81}{53\!\cdots\!09}a^{10}+\frac{46\!\cdots\!68}{53\!\cdots\!09}a^{9}-\frac{16\!\cdots\!49}{53\!\cdots\!09}a^{8}-\frac{11\!\cdots\!41}{53\!\cdots\!09}a^{7}+\frac{13\!\cdots\!49}{53\!\cdots\!09}a^{6}+\frac{10\!\cdots\!66}{53\!\cdots\!09}a^{5}-\frac{23\!\cdots\!08}{53\!\cdots\!09}a^{4}+\frac{66\!\cdots\!77}{53\!\cdots\!09}a^{3}+\frac{10\!\cdots\!16}{53\!\cdots\!09}a^{2}+\frac{33\!\cdots\!35}{53\!\cdots\!09}a-\frac{15\!\cdots\!75}{28\!\cdots\!11}$, $\frac{49\!\cdots\!29}{53\!\cdots\!09}a^{15}-\frac{39\!\cdots\!52}{53\!\cdots\!09}a^{14}+\frac{10\!\cdots\!34}{53\!\cdots\!09}a^{13}-\frac{76\!\cdots\!67}{53\!\cdots\!09}a^{12}+\frac{24\!\cdots\!27}{53\!\cdots\!09}a^{11}-\frac{18\!\cdots\!47}{53\!\cdots\!09}a^{10}+\frac{37\!\cdots\!22}{53\!\cdots\!09}a^{9}+\frac{67\!\cdots\!23}{53\!\cdots\!09}a^{8}-\frac{11\!\cdots\!39}{53\!\cdots\!09}a^{7}+\frac{91\!\cdots\!77}{53\!\cdots\!09}a^{6}+\frac{10\!\cdots\!60}{53\!\cdots\!09}a^{5}-\frac{17\!\cdots\!85}{53\!\cdots\!09}a^{4}+\frac{78\!\cdots\!23}{53\!\cdots\!09}a^{3}+\frac{71\!\cdots\!35}{53\!\cdots\!09}a^{2}+\frac{15\!\cdots\!68}{53\!\cdots\!09}a-\frac{10\!\cdots\!15}{28\!\cdots\!11}$, $\frac{14\!\cdots\!88}{53\!\cdots\!09}a^{15}-\frac{10\!\cdots\!96}{53\!\cdots\!09}a^{14}+\frac{25\!\cdots\!67}{53\!\cdots\!09}a^{13}-\frac{22\!\cdots\!46}{53\!\cdots\!09}a^{12}+\frac{86\!\cdots\!75}{53\!\cdots\!09}a^{11}-\frac{50\!\cdots\!72}{53\!\cdots\!09}a^{10}+\frac{96\!\cdots\!50}{53\!\cdots\!09}a^{9}+\frac{31\!\cdots\!23}{53\!\cdots\!09}a^{8}-\frac{22\!\cdots\!71}{53\!\cdots\!09}a^{7}+\frac{16\!\cdots\!85}{53\!\cdots\!09}a^{6}+\frac{24\!\cdots\!63}{53\!\cdots\!09}a^{5}-\frac{31\!\cdots\!51}{53\!\cdots\!09}a^{4}-\frac{14\!\cdots\!88}{53\!\cdots\!09}a^{3}+\frac{73\!\cdots\!09}{53\!\cdots\!09}a^{2}-\frac{82\!\cdots\!21}{53\!\cdots\!09}a+\frac{12\!\cdots\!43}{28\!\cdots\!11}$, $\frac{10\!\cdots\!24}{53\!\cdots\!09}a^{15}-\frac{17\!\cdots\!02}{53\!\cdots\!09}a^{14}-\frac{19\!\cdots\!61}{53\!\cdots\!09}a^{13}+\frac{54\!\cdots\!06}{53\!\cdots\!09}a^{12}+\frac{35\!\cdots\!36}{53\!\cdots\!09}a^{11}+\frac{10\!\cdots\!14}{53\!\cdots\!09}a^{10}-\frac{11\!\cdots\!14}{53\!\cdots\!09}a^{9}+\frac{23\!\cdots\!04}{53\!\cdots\!09}a^{8}+\frac{34\!\cdots\!73}{53\!\cdots\!09}a^{7}-\frac{39\!\cdots\!17}{53\!\cdots\!09}a^{6}+\frac{12\!\cdots\!39}{53\!\cdots\!09}a^{5}+\frac{55\!\cdots\!88}{53\!\cdots\!09}a^{4}-\frac{53\!\cdots\!73}{53\!\cdots\!09}a^{3}-\frac{38\!\cdots\!70}{53\!\cdots\!09}a^{2}-\frac{20\!\cdots\!19}{53\!\cdots\!09}a-\frac{45\!\cdots\!36}{28\!\cdots\!11}$, $\frac{67\!\cdots\!76}{53\!\cdots\!09}a^{15}-\frac{47\!\cdots\!32}{53\!\cdots\!09}a^{14}+\frac{10\!\cdots\!29}{53\!\cdots\!09}a^{13}-\frac{69\!\cdots\!10}{53\!\cdots\!09}a^{12}+\frac{46\!\cdots\!14}{53\!\cdots\!09}a^{11}-\frac{25\!\cdots\!72}{53\!\cdots\!09}a^{10}+\frac{37\!\cdots\!20}{53\!\cdots\!09}a^{9}+\frac{12\!\cdots\!17}{53\!\cdots\!09}a^{8}-\frac{52\!\cdots\!05}{53\!\cdots\!09}a^{7}-\frac{51\!\cdots\!55}{53\!\cdots\!09}a^{6}+\frac{97\!\cdots\!06}{53\!\cdots\!09}a^{5}+\frac{62\!\cdots\!15}{53\!\cdots\!09}a^{4}-\frac{27\!\cdots\!60}{53\!\cdots\!09}a^{3}-\frac{10\!\cdots\!54}{53\!\cdots\!09}a^{2}-\frac{66\!\cdots\!63}{53\!\cdots\!09}a+\frac{58\!\cdots\!40}{28\!\cdots\!11}$, $\frac{20\!\cdots\!59}{53\!\cdots\!09}a^{15}-\frac{14\!\cdots\!56}{53\!\cdots\!09}a^{14}+\frac{32\!\cdots\!95}{53\!\cdots\!09}a^{13}-\frac{16\!\cdots\!55}{53\!\cdots\!09}a^{12}+\frac{96\!\cdots\!59}{53\!\cdots\!09}a^{11}-\frac{67\!\cdots\!88}{53\!\cdots\!09}a^{10}+\frac{11\!\cdots\!64}{53\!\cdots\!09}a^{9}+\frac{75\!\cdots\!18}{53\!\cdots\!09}a^{8}-\frac{38\!\cdots\!13}{53\!\cdots\!09}a^{7}+\frac{17\!\cdots\!04}{53\!\cdots\!09}a^{6}+\frac{41\!\cdots\!78}{53\!\cdots\!09}a^{5}-\frac{36\!\cdots\!98}{53\!\cdots\!09}a^{4}-\frac{38\!\cdots\!45}{53\!\cdots\!09}a^{3}+\frac{24\!\cdots\!73}{53\!\cdots\!09}a^{2}-\frac{67\!\cdots\!13}{53\!\cdots\!09}a+\frac{73\!\cdots\!17}{28\!\cdots\!11}$ Copy content Toggle raw display (assuming GRH)
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:  \( 42426.8341092 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 42426.8341092 \cdot 2}{2\cdot\sqrt{233704553418537641015625}}\cr\approx \mathstrut & 0.213179768238 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 16*x^14 - 11*x^13 + 59*x^12 - 349*x^11 + 586*x^10 + 206*x^9 - 1354*x^8 + 280*x^7 + 1715*x^6 - 713*x^5 - 377*x^4 - 596*x^3 + 153*x^2 + 209*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 - 7*x^15 + 16*x^14 - 11*x^13 + 59*x^12 - 349*x^11 + 586*x^10 + 206*x^9 - 1354*x^8 + 280*x^7 + 1715*x^6 - 713*x^5 - 377*x^4 - 596*x^3 + 153*x^2 + 209*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 - 7*x^15 + 16*x^14 - 11*x^13 + 59*x^12 - 349*x^11 + 586*x^10 + 206*x^9 - 1354*x^8 + 280*x^7 + 1715*x^6 - 713*x^5 - 377*x^4 - 596*x^3 + 153*x^2 + 209*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 - 7*x^15 + 16*x^14 - 11*x^13 + 59*x^12 - 349*x^11 + 586*x^10 + 206*x^9 - 1354*x^8 + 280*x^7 + 1715*x^6 - 713*x^5 - 377*x^4 - 596*x^3 + 153*x^2 + 209*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

$D_8:C_2$ (as 16T45):

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 11 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), 4.4.151725.2, 4.4.151725.1, \(\Q(\sqrt{5}, \sqrt{17})\), 8.8.23020475625.1

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 8 siblings: 8.0.119435644125.2, 8.0.119435644125.1
Degree 16 siblings: deg 16, deg 16, 16.0.356621827188841175390625.1
Minimal sibling: 8.0.119435644125.2

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 R R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ R ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{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}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.4.2.2$x^{4} - 42 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(17\) Copy content Toggle raw display 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$