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Properties

Label 20.8.206...544.1
Degree $20$
Signature $[8, 6]$
Discriminant $2.061\times 10^{24}$
Root discriminant \(16.43\)
Ramified primes $2,3,17,4153$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.C_2^4:S_5$ (as 20T964)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 4*x^16 + 14*x^14 - 33*x^12 - 3*x^10 + 65*x^8 - 14*x^6 - 43*x^4 + 4*x^2 + 1)
 
gp: K = bnfinit(y^20 - 5*y^18 + 4*y^16 + 14*y^14 - 33*y^12 - 3*y^10 + 65*y^8 - 14*y^6 - 43*y^4 + 4*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^18 + 4*x^16 + 14*x^14 - 33*x^12 - 3*x^10 + 65*x^8 - 14*x^6 - 43*x^4 + 4*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^18 + 4*x^16 + 14*x^14 - 33*x^12 - 3*x^10 + 65*x^8 - 14*x^6 - 43*x^4 + 4*x^2 + 1)
 

\( x^{20} - 5x^{18} + 4x^{16} + 14x^{14} - 33x^{12} - 3x^{10} + 65x^{8} - 14x^{6} - 43x^{4} + 4x^{2} + 1 \) 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:  $[8, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2060764600365471574508544\) \(\medspace = 2^{10}\cdot 3^{4}\cdot 17^{4}\cdot 4153^{4}\) 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:  \(16.43\)
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:  $2^{15/8}3^{1/2}17^{1/2}4153^{1/2}\approx 1688.0966012168024$
Ramified primes:   \(2\), \(3\), \(17\), \(4153\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{6}a^{16}-\frac{1}{2}a^{11}+\frac{1}{3}a^{10}-\frac{1}{2}a^{9}+\frac{1}{6}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{6}a^{17}-\frac{1}{2}a^{12}+\frac{1}{3}a^{11}-\frac{1}{2}a^{10}+\frac{1}{6}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{35538}a^{18}+\frac{1207}{17769}a^{16}+\frac{1873}{5923}a^{14}-\frac{1}{2}a^{13}-\frac{917}{17769}a^{12}-\frac{1}{2}a^{11}+\frac{5771}{35538}a^{10}-\frac{1}{2}a^{9}+\frac{11381}{35538}a^{8}-\frac{1}{2}a^{7}+\frac{6523}{35538}a^{6}-\frac{8759}{17769}a^{4}-\frac{1}{2}a^{3}+\frac{1490}{17769}a^{2}-\frac{2795}{17769}$, $\frac{1}{35538}a^{19}+\frac{1207}{17769}a^{17}-\frac{2177}{11846}a^{15}-\frac{1}{2}a^{14}-\frac{917}{17769}a^{13}-\frac{1}{2}a^{12}+\frac{5771}{35538}a^{11}+\frac{11381}{35538}a^{9}-\frac{5623}{17769}a^{7}-\frac{1}{2}a^{6}+\frac{251}{35538}a^{5}-\frac{14789}{35538}a^{3}-\frac{2795}{17769}a-\frac{1}{2}$ 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 $1$ (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:  $13$
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:   $\frac{7277}{17769}a^{19}+\frac{109}{17769}a^{18}-\frac{36478}{17769}a^{17}-\frac{3409}{17769}a^{16}+\frac{21761}{11846}a^{15}+\frac{5550}{5923}a^{14}+\frac{87346}{17769}a^{13}-\frac{22216}{17769}a^{12}-\frac{76500}{5923}a^{11}-\frac{39059}{35538}a^{10}+\frac{4151}{17769}a^{9}+\frac{188857}{35538}a^{8}+\frac{813349}{35538}a^{7}-\frac{123889}{35538}a^{6}-\frac{226511}{35538}a^{5}-\frac{176279}{35538}a^{4}-\frac{465203}{35538}a^{3}+\frac{76054}{17769}a^{2}+\frac{36272}{17769}a+\frac{7441}{35538}$, $\frac{7277}{17769}a^{19}-\frac{109}{17769}a^{18}-\frac{36478}{17769}a^{17}+\frac{3409}{17769}a^{16}+\frac{21761}{11846}a^{15}-\frac{5550}{5923}a^{14}+\frac{87346}{17769}a^{13}+\frac{22216}{17769}a^{12}-\frac{76500}{5923}a^{11}+\frac{39059}{35538}a^{10}+\frac{4151}{17769}a^{9}-\frac{188857}{35538}a^{8}+\frac{813349}{35538}a^{7}+\frac{123889}{35538}a^{6}-\frac{226511}{35538}a^{5}+\frac{176279}{35538}a^{4}-\frac{465203}{35538}a^{3}-\frac{76054}{17769}a^{2}+\frac{36272}{17769}a-\frac{7441}{35538}$, $\frac{317}{5923}a^{18}-\frac{2410}{17769}a^{16}-\frac{3200}{5923}a^{14}+\frac{10922}{5923}a^{12}-\frac{14246}{17769}a^{10}-\frac{92758}{17769}a^{8}+\frac{42125}{5923}a^{6}+\frac{72857}{17769}a^{4}-\frac{44481}{5923}a^{2}-\frac{9082}{17769}$, $\frac{64}{17769}a^{18}-\frac{5425}{17769}a^{16}+\frac{8747}{5923}a^{14}-\frac{28531}{17769}a^{12}-\frac{39343}{17769}a^{10}+\frac{142007}{17769}a^{8}-\frac{44522}{17769}a^{6}-\frac{161626}{17769}a^{4}+\frac{66337}{17769}a^{2}+\frac{15389}{17769}$, $\frac{3174}{5923}a^{19}-\frac{48439}{17769}a^{17}+\frac{12952}{5923}a^{15}+\frac{48577}{5923}a^{13}-\frac{339743}{17769}a^{11}-\frac{26941}{17769}a^{9}+\frac{234114}{5923}a^{7}-\frac{192406}{17769}a^{5}-\frac{160432}{5923}a^{3}+\frac{73097}{17769}a$, $a$, $\frac{1577}{35538}a^{19}-\frac{7435}{35538}a^{18}-\frac{537}{11846}a^{17}+\frac{17069}{17769}a^{16}-\frac{9635}{11846}a^{15}-\frac{7487}{11846}a^{14}+\frac{75205}{35538}a^{13}-\frac{40939}{17769}a^{12}-\frac{13238}{17769}a^{11}+\frac{91220}{17769}a^{10}-\frac{66745}{11846}a^{9}+\frac{25775}{17769}a^{8}+\frac{123643}{17769}a^{7}-\frac{308977}{35538}a^{6}+\frac{23523}{5923}a^{5}-\frac{17989}{17769}a^{4}-\frac{120161}{17769}a^{3}+\frac{143795}{35538}a^{2}+\frac{1640}{5923}a-\frac{41}{35538}$, $\frac{7277}{17769}a^{19}+\frac{15}{5923}a^{18}-\frac{36478}{17769}a^{17}+\frac{672}{5923}a^{16}+\frac{21761}{11846}a^{15}-\frac{3197}{5923}a^{14}+\frac{87346}{17769}a^{13}+\frac{2105}{5923}a^{12}-\frac{76500}{5923}a^{11}+\frac{13209}{11846}a^{10}+\frac{4151}{17769}a^{9}-\frac{31719}{11846}a^{8}+\frac{813349}{35538}a^{7}-\frac{11615}{11846}a^{6}-\frac{226511}{35538}a^{5}+\frac{48991}{11846}a^{4}-\frac{465203}{35538}a^{3}+\frac{3239}{5923}a^{2}+\frac{54041}{17769}a-\frac{19625}{11846}$, $\frac{7277}{17769}a^{19}-\frac{15}{5923}a^{18}-\frac{36478}{17769}a^{17}-\frac{672}{5923}a^{16}+\frac{21761}{11846}a^{15}+\frac{3197}{5923}a^{14}+\frac{87346}{17769}a^{13}-\frac{2105}{5923}a^{12}-\frac{76500}{5923}a^{11}-\frac{13209}{11846}a^{10}+\frac{4151}{17769}a^{9}+\frac{31719}{11846}a^{8}+\frac{813349}{35538}a^{7}+\frac{11615}{11846}a^{6}-\frac{226511}{35538}a^{5}-\frac{48991}{11846}a^{4}-\frac{465203}{35538}a^{3}-\frac{3239}{5923}a^{2}+\frac{54041}{17769}a+\frac{19625}{11846}$, $\frac{1573}{11846}a^{19}-\frac{1705}{35538}a^{18}-\frac{2672}{5923}a^{17}+\frac{12461}{35538}a^{16}-\frac{4352}{5923}a^{15}-\frac{7859}{11846}a^{14}+\frac{47003}{11846}a^{13}-\frac{18143}{35538}a^{12}-\frac{43637}{11846}a^{11}+\frac{17527}{5923}a^{10}-\frac{91761}{11846}a^{9}-\frac{83779}{35538}a^{8}+\frac{179733}{11846}a^{7}-\frac{79121}{17769}a^{6}+\frac{51289}{11846}a^{5}+\frac{91057}{17769}a^{4}-\frac{163397}{11846}a^{3}+\frac{53824}{17769}a^{2}+\frac{2585}{11846}a-\frac{20309}{17769}$, $\frac{1573}{11846}a^{19}+\frac{1705}{35538}a^{18}-\frac{2672}{5923}a^{17}-\frac{12461}{35538}a^{16}-\frac{4352}{5923}a^{15}+\frac{7859}{11846}a^{14}+\frac{47003}{11846}a^{13}+\frac{18143}{35538}a^{12}-\frac{43637}{11846}a^{11}-\frac{17527}{5923}a^{10}-\frac{91761}{11846}a^{9}+\frac{83779}{35538}a^{8}+\frac{179733}{11846}a^{7}+\frac{79121}{17769}a^{6}+\frac{51289}{11846}a^{5}-\frac{91057}{17769}a^{4}-\frac{163397}{11846}a^{3}-\frac{53824}{17769}a^{2}+\frac{2585}{11846}a+\frac{20309}{17769}$, $\frac{16639}{35538}a^{19}-\frac{7403}{35538}a^{18}-\frac{80239}{35538}a^{17}+\frac{34387}{35538}a^{16}+\frac{9867}{5923}a^{15}-\frac{6075}{11846}a^{14}+\frac{206675}{35538}a^{13}-\frac{52514}{17769}a^{12}-\frac{462001}{35538}a^{11}+\frac{65115}{11846}a^{10}-\frac{51194}{17769}a^{9}+\frac{53873}{17769}a^{8}+\frac{436913}{17769}a^{7}-\frac{455621}{35538}a^{6}+\frac{18106}{17769}a^{5}-\frac{25819}{17769}a^{4}-\frac{542129}{35538}a^{3}+\frac{310231}{35538}a^{2}-\frac{133447}{35538}a+\frac{14192}{17769}$, $\frac{479}{11846}a^{19}+\frac{325}{35538}a^{18}-\frac{980}{17769}a^{17}-\frac{3209}{35538}a^{16}-\frac{3464}{5923}a^{15}+\frac{3235}{11846}a^{14}+\frac{15887}{11846}a^{13}-\frac{9673}{35538}a^{12}+\frac{727}{35538}a^{11}-\frac{336}{5923}a^{10}-\frac{158827}{35538}a^{9}+\frac{14719}{35538}a^{8}+\frac{44557}{11846}a^{7}+\frac{2732}{17769}a^{6}+\frac{194843}{35538}a^{5}+\frac{2288}{17769}a^{4}-\frac{47407}{11846}a^{3}-\frac{13282}{17769}a^{2}-\frac{42703}{35538}a+\frac{3767}{17769}$ 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:  \( 30273.1886548 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 30273.1886548 \cdot 1}{2\cdot\sqrt{2060764600365471574508544}}\cr\approx \mathstrut & 0.166086049203 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 + 4*x^16 + 14*x^14 - 33*x^12 - 3*x^10 + 65*x^8 - 14*x^6 - 43*x^4 + 4*x^2 + 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 - 5*x^18 + 4*x^16 + 14*x^14 - 33*x^12 - 3*x^10 + 65*x^8 - 14*x^6 - 43*x^4 + 4*x^2 + 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 - 5*x^18 + 4*x^16 + 14*x^14 - 33*x^12 - 3*x^10 + 65*x^8 - 14*x^6 - 43*x^4 + 4*x^2 + 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 - 5*x^18 + 4*x^16 + 14*x^14 - 33*x^12 - 3*x^10 + 65*x^8 - 14*x^6 - 43*x^4 + 4*x^2 + 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

$C_2^9.C_2^4:S_5$ (as 20T964):

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 non-solvable group of order 983040
The 155 conjugacy class representatives for $C_2^9.C_2^4:S_5$
Character table for $C_2^9.C_2^4:S_5$

Intermediate fields

5.5.70601.1, 10.6.44860510809.1

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 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.5.0.1}{5} }^{4}$ $16{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ R ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ $16{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.10.0.1}{10} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(2\) Copy content Toggle raw display 2.10.0.1$x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.5$x^{10} + 34 x^{8} - 24 x^{7} + 368 x^{6} - 496 x^{5} + 1568 x^{4} - 1760 x^{3} + 2992 x^{2} - 1856 x + 352$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
\(3\) 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$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.6.0.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(17\) Copy content Toggle raw display 17.3.0.1$x^{3} + x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} + x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} + x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} + x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
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}$
\(4153\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$