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Properties

Label 20.4.128...401.1
Degree $20$
Signature $[4, 8]$
Discriminant $1.286\times 10^{22}$
Root discriminant \(12.75\)
Ramified primes $11,23$
Class number $1$
Class group trivial
Galois group $C_2\wr C_5$ (as 20T41)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 9*x^16 + 24*x^15 - 39*x^14 + 48*x^13 - 58*x^12 + 60*x^11 - 59*x^10 + 60*x^9 - 58*x^8 + 48*x^7 - 39*x^6 + 24*x^5 - 9*x^4 + 2*x^3 + 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^20 - 2*y^19 + 2*y^18 + 2*y^17 - 9*y^16 + 24*y^15 - 39*y^14 + 48*y^13 - 58*y^12 + 60*y^11 - 59*y^10 + 60*y^9 - 58*y^8 + 48*y^7 - 39*y^6 + 24*y^5 - 9*y^4 + 2*y^3 + 2*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 9*x^16 + 24*x^15 - 39*x^14 + 48*x^13 - 58*x^12 + 60*x^11 - 59*x^10 + 60*x^9 - 58*x^8 + 48*x^7 - 39*x^6 + 24*x^5 - 9*x^4 + 2*x^3 + 2*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 9*x^16 + 24*x^15 - 39*x^14 + 48*x^13 - 58*x^12 + 60*x^11 - 59*x^10 + 60*x^9 - 58*x^8 + 48*x^7 - 39*x^6 + 24*x^5 - 9*x^4 + 2*x^3 + 2*x^2 - 2*x + 1)
 

\( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} - 9 x^{16} + 24 x^{15} - 39 x^{14} + 48 x^{13} - 58 x^{12} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(12858618354751897106401\) \(\medspace = 11^{16}\cdot 23^{4}\) 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:  \(12.75\)
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:  $11^{4/5}23^{1/2}\approx 32.65713384043754$
Ramified primes:   \(11\), \(23\) 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 not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{706}a^{18}-\frac{51}{706}a^{17}+\frac{29}{706}a^{16}+\frac{22}{353}a^{15}-\frac{38}{353}a^{14}+\frac{87}{353}a^{13}-\frac{17}{706}a^{12}+\frac{1}{706}a^{11}-\frac{45}{353}a^{10}+\frac{233}{706}a^{9}-\frac{45}{353}a^{8}+\frac{1}{706}a^{7}+\frac{168}{353}a^{6}+\frac{87}{353}a^{5}-\frac{38}{353}a^{4}-\frac{309}{706}a^{3}+\frac{29}{706}a^{2}+\frac{151}{353}a-\frac{176}{353}$, $\frac{1}{16238}a^{19}-\frac{3}{16238}a^{18}-\frac{1007}{16238}a^{17}+\frac{3907}{16238}a^{16}-\frac{394}{8119}a^{15}-\frac{3121}{16238}a^{14}-\frac{26}{353}a^{13}+\frac{1887}{8119}a^{12}-\frac{1433}{8119}a^{11}+\frac{957}{8119}a^{10}+\frac{7917}{16238}a^{9}-\frac{83}{16238}a^{8}-\frac{6323}{16238}a^{7}-\frac{60}{353}a^{6}+\frac{3687}{16238}a^{5}-\frac{1133}{16238}a^{4}-\frac{1577}{8119}a^{3}-\frac{777}{16238}a^{2}+\frac{3907}{16238}a+\frac{3554}{8119}$ 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

Trivial group, which has order $1$

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:  $11$
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:   $\frac{702}{8119}a^{19}-\frac{2221}{8119}a^{18}+\frac{5304}{8119}a^{17}-\frac{1567}{16238}a^{16}-\frac{20407}{16238}a^{15}+\frac{34285}{8119}a^{14}-\frac{2427}{353}a^{13}+\frac{85699}{8119}a^{12}-\frac{167569}{16238}a^{11}+\frac{150471}{16238}a^{10}-\frac{166699}{16238}a^{9}+\frac{139619}{16238}a^{8}-\frac{70839}{8119}a^{7}+\frac{5719}{706}a^{6}-\frac{100227}{16238}a^{5}+\frac{9036}{8119}a^{4}+\frac{2757}{16238}a^{3}-\frac{21054}{8119}a^{2}+\frac{33071}{16238}a-\frac{3484}{8119}$, $a$, $\frac{7167}{16238}a^{19}-\frac{3287}{8119}a^{18}-\frac{2794}{8119}a^{17}+\frac{17045}{8119}a^{16}-\frac{46337}{16238}a^{15}+\frac{41518}{8119}a^{14}-\frac{1387}{353}a^{13}-\frac{14439}{16238}a^{12}+\frac{3628}{8119}a^{11}-\frac{23505}{16238}a^{10}+\frac{28670}{8119}a^{9}-\frac{22207}{16238}a^{8}+\frac{26363}{16238}a^{7}-\frac{4811}{706}a^{6}+\frac{47015}{8119}a^{5}-\frac{80193}{16238}a^{4}+\frac{43518}{8119}a^{3}-\frac{12787}{16238}a^{2}-\frac{3586}{8119}a+\frac{3119}{16238}$, $\frac{3573}{16238}a^{19}-\frac{1887}{16238}a^{18}+\frac{1469}{8119}a^{17}+\frac{7589}{16238}a^{16}-\frac{15573}{16238}a^{15}+\frac{27766}{8119}a^{14}-\frac{2843}{706}a^{13}+\frac{92269}{16238}a^{12}-\frac{139469}{16238}a^{11}+\frac{92615}{16238}a^{10}-\frac{62655}{8119}a^{9}+\frac{110173}{16238}a^{8}-\frac{77379}{16238}a^{7}+\frac{3491}{706}a^{6}-\frac{82399}{16238}a^{5}+\frac{22055}{16238}a^{4}-\frac{587}{8119}a^{3}+\frac{2458}{8119}a^{2}+\frac{31739}{16238}a+\frac{1365}{16238}$, $\frac{1539}{16238}a^{19}-\frac{4367}{8119}a^{18}+\frac{16067}{16238}a^{17}-\frac{9033}{16238}a^{16}-\frac{14944}{8119}a^{15}+\frac{48454}{8119}a^{14}-\frac{4402}{353}a^{13}+\frac{150211}{8119}a^{12}-\frac{355391}{16238}a^{11}+\frac{188549}{8119}a^{10}-\frac{393287}{16238}a^{9}+\frac{388935}{16238}a^{8}-\frac{357769}{16238}a^{7}+\frac{7845}{353}a^{6}-\frac{139410}{8119}a^{5}+\frac{92443}{8119}a^{4}-\frac{90683}{16238}a^{3}+\frac{8160}{8119}a^{2}+\frac{28043}{16238}a-\frac{8718}{8119}$, $\frac{2938}{8119}a^{19}-\frac{10286}{8119}a^{18}+\frac{21859}{16238}a^{17}+\frac{17171}{16238}a^{16}-\frac{91399}{16238}a^{15}+\frac{100606}{8119}a^{14}-\frac{14713}{706}a^{13}+\frac{201100}{8119}a^{12}-\frac{189114}{8119}a^{11}+\frac{429189}{16238}a^{10}-\frac{205742}{8119}a^{9}+\frac{369939}{16238}a^{8}-\frac{402179}{16238}a^{7}+\frac{7176}{353}a^{6}-\frac{100207}{8119}a^{5}+\frac{55075}{8119}a^{4}-\frac{10608}{8119}a^{3}-\frac{27837}{8119}a^{2}+\frac{9121}{16238}a+\frac{7647}{16238}$, $\frac{1231}{16238}a^{19}+\frac{1845}{8119}a^{18}-\frac{4294}{8119}a^{17}+\frac{3041}{8119}a^{16}+\frac{12463}{16238}a^{15}-\frac{13459}{8119}a^{14}+\frac{2785}{706}a^{13}-\frac{49714}{8119}a^{12}+\frac{67935}{16238}a^{11}-\frac{47258}{8119}a^{10}+\frac{107585}{16238}a^{9}-\frac{84647}{16238}a^{8}+\frac{107319}{16238}a^{7}-\frac{2282}{353}a^{6}+\frac{42611}{16238}a^{5}-\frac{23933}{8119}a^{4}+\frac{15438}{8119}a^{3}+\frac{12687}{16238}a^{2}+\frac{8124}{8119}a-\frac{1535}{8119}$, $\frac{1544}{8119}a^{19}-\frac{4595}{16238}a^{18}+\frac{2706}{8119}a^{17}+\frac{5479}{16238}a^{16}-\frac{9770}{8119}a^{15}+\frac{66949}{16238}a^{14}-\frac{2264}{353}a^{13}+\frac{143187}{16238}a^{12}-\frac{174447}{16238}a^{11}+\frac{188515}{16238}a^{10}-\frac{185419}{16238}a^{9}+\frac{192219}{16238}a^{8}-\frac{197535}{16238}a^{7}+\frac{3791}{353}a^{6}-\frac{71509}{8119}a^{5}+\frac{84167}{16238}a^{4}-\frac{59243}{16238}a^{3}+\frac{9345}{16238}a^{2}-\frac{10805}{16238}a+\frac{4258}{8119}$, $\frac{9791}{16238}a^{19}-\frac{4717}{16238}a^{18}-\frac{1041}{8119}a^{17}+\frac{29737}{16238}a^{16}-\frac{18908}{8119}a^{15}+\frac{58752}{8119}a^{14}-\frac{2452}{353}a^{13}+\frac{55124}{8119}a^{12}-\frac{110328}{8119}a^{11}+\frac{84634}{8119}a^{10}-\frac{186887}{16238}a^{9}+\frac{232145}{16238}a^{8}-\frac{93778}{8119}a^{7}+\frac{2469}{353}a^{6}-\frac{82422}{8119}a^{5}+\frac{40087}{8119}a^{4}-\frac{39815}{16238}a^{3}+\frac{32700}{8119}a^{2}-\frac{9270}{8119}a-\frac{1341}{16238}$, $\frac{379}{706}a^{19}-\frac{278}{353}a^{18}+\frac{157}{353}a^{17}+\frac{441}{353}a^{16}-\frac{1345}{353}a^{15}+\frac{7073}{706}a^{14}-\frac{10839}{706}a^{13}+\frac{5823}{353}a^{12}-\frac{8198}{353}a^{11}+\frac{7915}{353}a^{10}-\frac{14609}{706}a^{9}+\frac{8429}{353}a^{8}-\frac{15557}{706}a^{7}+\frac{11555}{706}a^{6}-\frac{10961}{706}a^{5}+\frac{3435}{353}a^{4}-\frac{686}{353}a^{3}+\frac{2295}{706}a^{2}+\frac{645}{706}a-\frac{321}{353}$, $\frac{2397}{16238}a^{19}-\frac{4619}{8119}a^{18}+\frac{4533}{16238}a^{17}+\frac{9461}{16238}a^{16}-\frac{38463}{16238}a^{15}+\frac{35472}{8119}a^{14}-\frac{5637}{706}a^{13}+\frac{117701}{16238}a^{12}-\frac{54361}{8119}a^{11}+\frac{72129}{8119}a^{10}-\frac{38087}{8119}a^{9}+\frac{45413}{8119}a^{8}-\frac{97533}{16238}a^{7}+\frac{1921}{706}a^{6}-\frac{9516}{8119}a^{5}+\frac{21613}{16238}a^{4}+\frac{38487}{16238}a^{3}-\frac{30105}{16238}a^{2}+\frac{1361}{8119}a-\frac{30309}{16238}$ Copy content Toggle raw display
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:  \( 845.372840783 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 845.372840783 \cdot 1}{2\cdot\sqrt{12858618354751897106401}}\cr\approx \mathstrut & 0.144870537596 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 9*x^16 + 24*x^15 - 39*x^14 + 48*x^13 - 58*x^12 + 60*x^11 - 59*x^10 + 60*x^9 - 58*x^8 + 48*x^7 - 39*x^6 + 24*x^5 - 9*x^4 + 2*x^3 + 2*x^2 - 2*x + 1)
 
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^20 - 2*x^19 + 2*x^18 + 2*x^17 - 9*x^16 + 24*x^15 - 39*x^14 + 48*x^13 - 58*x^12 + 60*x^11 - 59*x^10 + 60*x^9 - 58*x^8 + 48*x^7 - 39*x^6 + 24*x^5 - 9*x^4 + 2*x^3 + 2*x^2 - 2*x + 1, 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^20 - 2*x^19 + 2*x^18 + 2*x^17 - 9*x^16 + 24*x^15 - 39*x^14 + 48*x^13 - 58*x^12 + 60*x^11 - 59*x^10 + 60*x^9 - 58*x^8 + 48*x^7 - 39*x^6 + 24*x^5 - 9*x^4 + 2*x^3 + 2*x^2 - 2*x + 1);
 
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^20 - 2*x^19 + 2*x^18 + 2*x^17 - 9*x^16 + 24*x^15 - 39*x^14 + 48*x^13 - 58*x^12 + 60*x^11 - 59*x^10 + 60*x^9 - 58*x^8 + 48*x^7 - 39*x^6 + 24*x^5 - 9*x^4 + 2*x^3 + 2*x^2 - 2*x + 1);
 
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

$C_2\wr C_5$ (as 20T41):

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 160
The 16 conjugacy class representatives for $C_2\wr C_5$
Character table for $C_2\wr C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.4930254263.1 x2, 10.6.113395848049.1

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

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.4.4930254263.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{4}$ ${\href{/padicField/3.5.0.1}{5} }^{4}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ R ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ R ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{10}$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(11\) Copy content Toggle raw display 11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$