Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25)
gp: K = bnfinit(y^20 - 3*y^19 + 2*y^18 + 6*y^17 - 14*y^16 + 7*y^15 + 10*y^14 - 12*y^13 - 11*y^12 + 14*y^11 + 30*y^10 - 56*y^9 + 16*y^8 + 38*y^7 + 8*y^6 - 106*y^5 + 111*y^4 - 25*y^3 - 40*y^2 + 25, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25)
\( x^{20} - 3 x^{19} + 2 x^{18} + 6 x^{17} - 14 x^{16} + 7 x^{15} + 10 x^{14} - 12 x^{13} - 11 x^{12} + \cdots + 25 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(428717762000000000000000\)
\(\medspace = 2^{16}\cdot 5^{15}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.19\) | 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: | $2^{4/5}5^{3/4}11^{4/5}\approx 39.642921082121845$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{7}-\frac{2}{5}a^{2}$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{8}-\frac{2}{5}a^{3}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{9}-\frac{2}{5}a^{4}$, $\frac{1}{5}a^{15}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{17}+\frac{2}{25}a^{16}-\frac{1}{25}a^{14}-\frac{1}{25}a^{12}-\frac{2}{25}a^{11}+\frac{6}{25}a^{9}-\frac{1}{5}a^{8}-\frac{12}{25}a^{7}+\frac{11}{25}a^{6}-\frac{2}{5}a^{5}+\frac{12}{25}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{475}a^{18}-\frac{6}{475}a^{17}-\frac{31}{475}a^{16}+\frac{44}{475}a^{15}-\frac{47}{475}a^{14}-\frac{6}{475}a^{13}+\frac{31}{475}a^{12}+\frac{11}{475}a^{11}-\frac{29}{475}a^{10}+\frac{72}{475}a^{9}-\frac{227}{475}a^{8}+\frac{12}{475}a^{7}+\frac{62}{475}a^{6}-\frac{2}{25}a^{5}-\frac{9}{25}a^{4}-\frac{5}{19}a^{3}+\frac{5}{19}a^{2}-\frac{8}{19}a+\frac{3}{19}$, $\frac{1}{13775}a^{19}-\frac{7}{13775}a^{18}+\frac{146}{13775}a^{17}-\frac{723}{13775}a^{16}+\frac{384}{13775}a^{15}+\frac{164}{2755}a^{14}-\frac{1008}{13775}a^{13}+\frac{569}{13775}a^{12}+\frac{758}{13775}a^{11}-\frac{89}{13775}a^{10}+\frac{5287}{13775}a^{9}-\frac{6791}{13775}a^{8}-\frac{4662}{13775}a^{7}+\frac{2446}{13775}a^{6}+\frac{328}{725}a^{5}+\frac{3618}{13775}a^{4}-\frac{1109}{2755}a^{3}+\frac{196}{551}a^{2}+\frac{125}{551}a+\frac{225}{551}$
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: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{3354}{2755} a^{19} + \frac{1233}{551} a^{18} + \frac{89}{551} a^{17} - \frac{19816}{2755} a^{16} + \frac{24256}{2755} a^{15} + \frac{1031}{551} a^{14} - \frac{29017}{2755} a^{13} + \frac{359}{145} a^{12} + \frac{9404}{551} a^{11} + \frac{6592}{2755} a^{10} - \frac{95262}{2755} a^{9} + \frac{4181}{145} a^{8} + \frac{42468}{2755} a^{7} - \frac{83304}{2755} a^{6} - \frac{6696}{145} a^{5} + \frac{213689}{2755} a^{4} - \frac{120294}{2755} a^{3} - \frac{67493}{2755} a^{2} + \frac{11789}{551} a + \frac{14701}{551} \)
(order $10$)
| 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: |
$\frac{12}{19}a^{19}-\frac{109}{95}a^{18}-\frac{3}{95}a^{17}+\frac{70}{19}a^{16}-\frac{88}{19}a^{15}-\frac{73}{95}a^{14}+\frac{105}{19}a^{13}-\frac{147}{95}a^{12}-\frac{172}{19}a^{11}-\frac{18}{19}a^{10}+\frac{1678}{95}a^{9}-\frac{1518}{95}a^{8}-\frac{754}{95}a^{7}+\frac{308}{19}a^{6}+24a^{5}-\frac{3909}{95}a^{4}+\frac{2288}{95}a^{3}+\frac{265}{19}a^{2}-\frac{191}{19}a-\frac{263}{19}$, $\frac{5438}{13775}a^{19}-\frac{8428}{13775}a^{18}-\frac{1551}{13775}a^{17}+\frac{30306}{13775}a^{16}-\frac{32201}{13775}a^{15}-\frac{1571}{2755}a^{14}+\frac{40763}{13775}a^{13}-\frac{8749}{13775}a^{12}-\frac{63566}{13775}a^{11}-\frac{21084}{13775}a^{10}+\frac{124516}{13775}a^{9}-\frac{111754}{13775}a^{8}-\frac{67958}{13775}a^{7}+\frac{101563}{13775}a^{6}+\frac{9183}{725}a^{5}-\frac{288046}{13775}a^{4}+\frac{37566}{2755}a^{3}+\frac{19659}{2755}a^{2}-\frac{3113}{551}a-\frac{2773}{551}$, $\frac{3901}{13775}a^{19}-\frac{7442}{13775}a^{18}+\frac{189}{13775}a^{17}+\frac{21838}{13775}a^{16}-\frac{30316}{13775}a^{15}-\frac{1098}{13775}a^{14}+\frac{31512}{13775}a^{13}-\frac{781}{725}a^{12}-\frac{51183}{13775}a^{11}+\frac{2406}{13775}a^{10}+\frac{20936}{2755}a^{9}-\frac{5459}{725}a^{8}-\frac{19093}{13775}a^{7}+\frac{107769}{13775}a^{6}+\frac{6718}{725}a^{5}-\frac{263976}{13775}a^{4}+\frac{38088}{2755}a^{3}+\frac{9132}{2755}a^{2}-\frac{3548}{551}a-\frac{3788}{551}$, $\frac{81}{13775}a^{19}-\frac{1234}{13775}a^{18}+\frac{951}{13775}a^{17}+\frac{13}{145}a^{16}-\frac{6509}{13775}a^{15}+\frac{2446}{13775}a^{14}+\frac{5004}{13775}a^{13}-\frac{6546}{13775}a^{12}-\frac{284}{551}a^{11}+\frac{12134}{13775}a^{10}+\frac{18216}{13775}a^{9}-\frac{21227}{13775}a^{8}+\frac{10543}{13775}a^{7}+\frac{1378}{551}a^{6}+\frac{207}{725}a^{5}-\frac{49664}{13775}a^{4}+\frac{1238}{551}a^{3}-\frac{132}{551}a^{2}-\frac{1069}{551}a-\frac{857}{551}$, $\frac{59}{13775}a^{19}-\frac{1341}{13775}a^{18}+\frac{958}{13775}a^{17}+\frac{197}{725}a^{16}-\frac{7156}{13775}a^{15}-\frac{89}{551}a^{14}+\frac{12216}{13775}a^{13}-\frac{1258}{13775}a^{12}-\frac{13423}{13775}a^{11}+\frac{5131}{13775}a^{10}+\frac{28023}{13775}a^{9}-\frac{19203}{13775}a^{8}-\frac{28326}{13775}a^{7}+\frac{29474}{13775}a^{6}+\frac{1343}{725}a^{5}-\frac{51018}{13775}a^{4}-\frac{2008}{2755}a^{3}+\frac{1414}{551}a^{2}-\frac{78}{551}a+\frac{22}{551}$, $\frac{468}{725}a^{19}-\frac{14133}{13775}a^{18}-\frac{414}{2755}a^{17}+\frac{48756}{13775}a^{16}-\frac{55728}{13775}a^{15}-\frac{10876}{13775}a^{14}+\frac{68363}{13775}a^{13}-\frac{3827}{2755}a^{12}-\frac{112941}{13775}a^{11}-\frac{23932}{13775}a^{10}+\frac{43196}{2755}a^{9}-\frac{200219}{13775}a^{8}-\frac{19863}{2755}a^{7}+\frac{201643}{13775}a^{6}+\frac{15909}{725}a^{5}-\frac{27143}{725}a^{4}+\frac{63993}{2755}a^{3}+\frac{38056}{2755}a^{2}-\frac{5667}{551}a-\frac{7173}{551}$, $\frac{1481}{13775}a^{19}-\frac{1986}{13775}a^{18}-\frac{3217}{13775}a^{17}+\frac{8907}{13775}a^{16}-\frac{1987}{13775}a^{15}-\frac{526}{551}a^{14}+\frac{5176}{13775}a^{13}+\frac{12622}{13775}a^{12}-\frac{14492}{13775}a^{11}-\frac{24973}{13775}a^{10}+\frac{35137}{13775}a^{9}+\frac{18782}{13775}a^{8}-\frac{29856}{13775}a^{7}-\frac{10739}{13775}a^{6}+\frac{3701}{725}a^{5}-\frac{19502}{13775}a^{4}-\frac{3349}{2755}a^{3}+\frac{2946}{2755}a^{2}-\frac{939}{551}a-\frac{884}{551}$, $\frac{11383}{13775}a^{19}-\frac{19216}{13775}a^{18}-\frac{26}{2755}a^{17}+\frac{12766}{2755}a^{16}-\frac{79123}{13775}a^{15}-\frac{418}{725}a^{14}+\frac{95051}{13775}a^{13}-\frac{6094}{2755}a^{12}-\frac{30583}{2755}a^{11}-\frac{19837}{13775}a^{10}+\frac{292658}{13775}a^{9}-\frac{294338}{13775}a^{8}-\frac{26833}{2755}a^{7}+\frac{55333}{2755}a^{6}+\frac{20609}{725}a^{5}-\frac{714897}{13775}a^{4}+\frac{89244}{2755}a^{3}+\frac{49616}{2755}a^{2}-\frac{6361}{551}a-\frac{9124}{551}$, $\frac{1152}{2755}a^{19}-\frac{10769}{13775}a^{18}-\frac{1403}{13775}a^{17}+\frac{1441}{551}a^{16}-\frac{41866}{13775}a^{15}-\frac{587}{551}a^{14}+\frac{55574}{13775}a^{13}-\frac{6752}{13775}a^{12}-\frac{17867}{2755}a^{11}-\frac{16914}{13775}a^{10}+\frac{35627}{2755}a^{9}-\frac{127052}{13775}a^{8}-\frac{106159}{13775}a^{7}+\frac{28978}{2755}a^{6}+\frac{13038}{725}a^{5}-\frac{74764}{2755}a^{4}+\frac{1712}{145}a^{3}+\frac{31647}{2755}a^{2}-\frac{3492}{551}a-\frac{4969}{551}$
| 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: | \( 33038.0055622 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 33038.0055622 \cdot 1}{10\cdot\sqrt{428717762000000000000000}}\cr\approx \mathstrut & 0.483867850184 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25)
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 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25, 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 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25);
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 - 3*x^19 + 2*x^18 + 6*x^17 - 14*x^16 + 7*x^15 + 10*x^14 - 12*x^13 - 11*x^12 + 14*x^11 + 30*x^10 - 56*x^9 + 16*x^8 + 38*x^7 + 8*x^6 - 106*x^5 + 111*x^4 - 25*x^3 - 40*x^2 + 25);
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
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 100 |
| The 10 conjugacy class representatives for $C_5:F_5$ |
| Character table for $C_5:F_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.292820000000.1 x5 |
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 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Minimal sibling: | 10.2.292820000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 $20$ | $5$ | $4$ | $16$ | |||
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(11\)
| 11.5.4.5 | $x^{5} + 22$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 11.5.4.5 | $x^{5} + 22$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |