Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16)
gp: K = bnfinit(y^20 - 2*y^19 + 6*y^18 - 8*y^17 + 16*y^16 - 14*y^15 + 32*y^14 - 12*y^13 + 32*y^12 + 8*y^11 + 48*y^10 - 16*y^9 + 48*y^8 + 8*y^7 + 24*y^6 + 8*y^5 + 32*y^4 + 16*y^3 + 32*y^2 + 16*y + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16)
\( x^{20} - 2 x^{19} + 6 x^{18} - 8 x^{17} + 16 x^{16} - 14 x^{15} + 32 x^{14} - 12 x^{13} + 32 x^{12} + \cdots + 16 \)
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: |
\(15233950180173557331984384\)
\(\medspace = 2^{16}\cdot 3^{10}\cdot 89^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.16\) | 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: | $2^{4/5}3^{1/2}89^{1/2}\approx 28.44982682753188$ | ||
| Ramified primes: |
\(2\), \(3\), \(89\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{4}a^{12}$, $\frac{1}{4}a^{13}$, $\frac{1}{12}a^{14}+\frac{1}{12}a^{11}+\frac{1}{12}a^{10}-\frac{1}{6}a^{9}-\frac{1}{6}a^{7}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{24}a^{15}-\frac{1}{12}a^{12}-\frac{1}{12}a^{11}-\frac{1}{12}a^{10}+\frac{1}{6}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{24}a^{16}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}-\frac{1}{12}a^{11}+\frac{1}{6}a^{9}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{24}a^{17}-\frac{1}{12}a^{13}-\frac{1}{12}a^{12}+\frac{1}{12}a^{11}-\frac{1}{6}a^{9}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{2136}a^{18}-\frac{13}{712}a^{17}-\frac{7}{534}a^{16}-\frac{5}{534}a^{15}-\frac{37}{1068}a^{14}-\frac{11}{178}a^{13}+\frac{29}{534}a^{12}-\frac{13}{178}a^{11}+\frac{1}{178}a^{10}+\frac{1}{6}a^{9}+\frac{20}{89}a^{8}+\frac{95}{534}a^{7}-\frac{73}{534}a^{6}-\frac{3}{89}a^{5}+\frac{1}{267}a^{4}-\frac{4}{267}a^{3}+\frac{10}{267}a^{2}+\frac{37}{89}a+\frac{28}{89}$, $\frac{1}{98256}a^{19}-\frac{1}{24564}a^{18}+\frac{475}{24564}a^{17}+\frac{53}{12282}a^{16}-\frac{209}{49128}a^{15}-\frac{13}{24564}a^{14}+\frac{663}{8188}a^{13}-\frac{1381}{12282}a^{12}+\frac{863}{24564}a^{11}-\frac{793}{12282}a^{10}-\frac{860}{6141}a^{9}+\frac{1391}{12282}a^{8}-\frac{77}{6141}a^{7}+\frac{1873}{12282}a^{6}-\frac{937}{12282}a^{5}+\frac{5371}{12282}a^{4}-\frac{332}{6141}a^{3}+\frac{329}{2047}a^{2}-\frac{2777}{6141}a-\frac{2357}{6141}$
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
$C_{2}$, which has order $2$
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: |
\( -\frac{2533}{98256} a^{19} + \frac{1153}{24564} a^{18} - \frac{3035}{24564} a^{17} + \frac{875}{6141} a^{16} - \frac{13127}{49128} a^{15} + \frac{1347}{8188} a^{14} - \frac{12775}{24564} a^{13} + \frac{1133}{24564} a^{12} - \frac{9673}{24564} a^{11} - \frac{1577}{12282} a^{10} - \frac{7816}{6141} a^{9} + \frac{2030}{2047} a^{8} - \frac{13361}{12282} a^{7} - \frac{268}{6141} a^{6} - \frac{1797}{2047} a^{5} - \frac{1807}{12282} a^{4} - \frac{1899}{2047} a^{3} + \frac{374}{6141} a^{2} - \frac{2283}{2047} a + \frac{2747}{6141} \)
(order $6$)
| 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: |
$\frac{4039}{98256}a^{19}-\frac{549}{8188}a^{18}+\frac{8861}{49128}a^{17}-\frac{2821}{16376}a^{16}+\frac{8059}{24564}a^{15}-\frac{251}{24564}a^{14}+\frac{906}{2047}a^{13}+\frac{17935}{24564}a^{12}-\frac{4585}{12282}a^{11}+\frac{14763}{8188}a^{10}+\frac{2266}{6141}a^{9}+\frac{6218}{6141}a^{8}-\frac{1883}{6141}a^{7}+\frac{12352}{6141}a^{6}-\frac{1825}{12282}a^{5}+\frac{7071}{4094}a^{4}-\frac{292}{2047}a^{3}+\frac{2628}{2047}a^{2}+\frac{6202}{6141}a+\frac{3365}{6141}$, $\frac{2905}{98256}a^{19}-\frac{1653}{16376}a^{18}+\frac{12239}{49128}a^{17}-\frac{1839}{4094}a^{16}+\frac{4627}{6141}a^{15}-\frac{12109}{12282}a^{14}+\frac{11771}{8188}a^{13}-\frac{38033}{24564}a^{12}+\frac{15659}{12282}a^{11}-\frac{1412}{2047}a^{10}+\frac{7228}{6141}a^{9}-\frac{11335}{6141}a^{8}+\frac{18311}{12282}a^{7}-\frac{287}{6141}a^{6}+\frac{1583}{12282}a^{5}-\frac{3353}{4094}a^{4}+\frac{1487}{2047}a^{3}+\frac{1268}{2047}a^{2}-\frac{1706}{6141}a-\frac{31}{6141}$, $\frac{88}{2047}a^{19}-\frac{4607}{49128}a^{18}+\frac{2099}{8188}a^{17}-\frac{9289}{24564}a^{16}+\frac{17603}{24564}a^{15}-\frac{4735}{6141}a^{14}+\frac{9454}{6141}a^{13}-\frac{6436}{6141}a^{12}+\frac{18881}{12282}a^{11}-\frac{4411}{8188}a^{10}+\frac{13067}{6141}a^{9}-\frac{6477}{4094}a^{8}+\frac{86}{69}a^{7}+\frac{4037}{12282}a^{6}+\frac{4829}{6141}a^{5}-\frac{3121}{2047}a^{4}+\frac{11074}{6141}a^{3}-\frac{3181}{6141}a^{2}+\frac{628}{2047}a-\frac{1412}{2047}$, $\frac{569}{98256}a^{19}+\frac{395}{16376}a^{18}+\frac{415}{12282}a^{17}-\frac{433}{12282}a^{16}+\frac{1059}{4094}a^{15}-\frac{1229}{4094}a^{14}+\frac{11237}{12282}a^{13}-\frac{10085}{24564}a^{12}+\frac{47837}{24564}a^{11}-\frac{22615}{24564}a^{10}+\frac{18209}{12282}a^{9}+\frac{3923}{6141}a^{8}+\frac{2730}{2047}a^{7}-\frac{18851}{12282}a^{6}+\frac{22757}{12282}a^{5}+\frac{10715}{12282}a^{4}+\frac{2849}{2047}a^{3}+\frac{3485}{6141}a^{2}+\frac{4196}{6141}a+\frac{2366}{6141}$, $\frac{999}{32752}a^{19}-\frac{189}{4094}a^{18}+\frac{1979}{16376}a^{17}-\frac{1077}{16376}a^{16}+\frac{1177}{6141}a^{15}+\frac{211}{4094}a^{14}+\frac{844}{2047}a^{13}+\frac{9715}{24564}a^{12}+\frac{7237}{24564}a^{11}+\frac{24295}{24564}a^{10}+\frac{7341}{4094}a^{9}-\frac{2066}{6141}a^{8}+\frac{495}{2047}a^{7}+\frac{4903}{2047}a^{6}+\frac{3671}{4094}a^{5}-\frac{13739}{12282}a^{4}+\frac{7451}{6141}a^{3}+\frac{14867}{6141}a^{2}+\frac{8713}{6141}a+\frac{1385}{2047}$, $\frac{3323}{98256}a^{19}-\frac{3725}{49128}a^{18}+\frac{1205}{8188}a^{17}-\frac{649}{4094}a^{16}+\frac{1085}{8188}a^{15}+\frac{358}{2047}a^{14}-\frac{2275}{8188}a^{13}+\frac{13375}{12282}a^{12}-\frac{49847}{24564}a^{11}+\frac{63307}{24564}a^{10}-\frac{4150}{2047}a^{9}+\frac{4993}{4094}a^{8}-\frac{25453}{12282}a^{7}+\frac{18800}{6141}a^{6}-\frac{34319}{12282}a^{5}+\frac{28399}{12282}a^{4}-\frac{7493}{6141}a^{3}+\frac{353}{2047}a^{2}-\frac{1128}{2047}a+\frac{3329}{6141}$, $\frac{4199}{98256}a^{19}-\frac{95}{2047}a^{18}+\frac{6485}{49128}a^{17}-\frac{3707}{49128}a^{16}+\frac{2305}{12282}a^{15}+\frac{895}{6141}a^{14}+\frac{9757}{24564}a^{13}+\frac{1650}{2047}a^{12}-\frac{950}{6141}a^{11}+\frac{5414}{6141}a^{10}+\frac{13865}{12282}a^{9}+\frac{5}{12282}a^{8}-\frac{1979}{12282}a^{7}+\frac{1670}{2047}a^{6}+\frac{5260}{6141}a^{5}+\frac{3031}{12282}a^{4}+\frac{1375}{2047}a^{3}+\frac{1055}{6141}a^{2}-\frac{1417}{6141}a-\frac{1730}{6141}$, $\frac{275}{4272}a^{19}-\frac{185}{2136}a^{18}+\frac{287}{1068}a^{17}-\frac{257}{1068}a^{16}+\frac{421}{712}a^{15}-\frac{259}{1068}a^{14}+\frac{239}{178}a^{13}+\frac{80}{267}a^{12}+\frac{1307}{1068}a^{11}+\frac{149}{178}a^{10}+\frac{1637}{534}a^{9}-\frac{42}{89}a^{8}+\frac{335}{534}a^{7}+\frac{148}{89}a^{6}+\frac{443}{267}a^{5}-\frac{179}{534}a^{4}+\frac{512}{267}a^{3}+\frac{421}{267}a^{2}+\frac{196}{89}a+\frac{56}{267}$, $\frac{333}{32752}a^{19}-\frac{20}{2047}a^{18}+\frac{725}{24564}a^{17}+\frac{715}{24564}a^{16}-\frac{1873}{24564}a^{15}+\frac{97}{276}a^{14}-\frac{1791}{4094}a^{13}+\frac{28799}{24564}a^{12}-\frac{10011}{8188}a^{11}+\frac{27053}{12282}a^{10}-\frac{9635}{6141}a^{9}+\frac{13142}{6141}a^{8}-\frac{5259}{4094}a^{7}+\frac{32645}{12282}a^{6}-\frac{3969}{2047}a^{5}+\frac{30089}{12282}a^{4}-\frac{2032}{6141}a^{3}+\frac{10445}{6141}a^{2}-\frac{3988}{6141}a+\frac{684}{2047}$
| 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: | \( 64033.5089322 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 64033.5089322 \cdot 2}{6\cdot\sqrt{15233950180173557331984384}}\cr\approx \mathstrut & 0.524419056076 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16)
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 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16, 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 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16);
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 + 6*x^18 - 8*x^17 + 16*x^16 - 14*x^15 + 32*x^14 - 12*x^13 + 32*x^12 + 8*x^11 + 48*x^10 - 16*x^9 + 48*x^8 + 8*x^7 + 24*x^6 + 8*x^5 + 32*x^4 + 16*x^3 + 32*x^2 + 16*x + 16);
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 non-solvable group of order 120 |
| The 7 conjugacy class representatives for $S_5$ |
| Character table for $S_5$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 10.4.433674369792.1 |
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 5 sibling: | 5.3.380208.1 |
| Degree 6 sibling: | 6.0.3421872.4 |
| Degree 10 siblings: | 10.0.3903069328128.1, 10.4.433674369792.1 |
| Degree 12 sibling: | 12.0.11709207984384.1 |
| Degree 15 sibling: | 15.3.1483978183108890624.1 |
| Degree 20 siblings: | deg 20, deg 20 |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | 5.3.380208.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(89\)
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.4.2.2 | $x^{4} - 7298 x^{2} + 23763$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.2 | $x^{4} - 7298 x^{2} + 23763$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.2 | $x^{4} - 7298 x^{2} + 23763$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 89.4.2.2 | $x^{4} - 7298 x^{2} + 23763$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |