Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 2*x^13 + 57*x^12 + 12*x^11 + 171*x^10 + 128*x^9 + 1436*x^8 + 684*x^7 + 5805*x^6 + 1084*x^5 + 18353*x^4 + 1488*x^3 + 40675*x^2 + 2430*x + 42947)
gp: K = bnfinit(y^16 - 2*y^14 - 2*y^13 + 57*y^12 + 12*y^11 + 171*y^10 + 128*y^9 + 1436*y^8 + 684*y^7 + 5805*y^6 + 1084*y^5 + 18353*y^4 + 1488*y^3 + 40675*y^2 + 2430*y + 42947, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^14 - 2*x^13 + 57*x^12 + 12*x^11 + 171*x^10 + 128*x^9 + 1436*x^8 + 684*x^7 + 5805*x^6 + 1084*x^5 + 18353*x^4 + 1488*x^3 + 40675*x^2 + 2430*x + 42947);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^14 - 2*x^13 + 57*x^12 + 12*x^11 + 171*x^10 + 128*x^9 + 1436*x^8 + 684*x^7 + 5805*x^6 + 1084*x^5 + 18353*x^4 + 1488*x^3 + 40675*x^2 + 2430*x + 42947)
\( x^{16} - 2 x^{14} - 2 x^{13} + 57 x^{12} + 12 x^{11} + 171 x^{10} + 128 x^{9} + 1436 x^{8} + \cdots + 42947 \)
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: |
\(33378638618235270076563456\)
\(\medspace = 2^{24}\cdot 3^{4}\cdot 89^{2}\cdot 1327^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.37\) | 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: | $2^{3/2}3^{1/2}89^{1/2}1327^{1/2}\approx 1683.5890234852448$ | ||
| Ramified primes: |
\(2\), \(3\), \(89\), \(1327\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{73\!\cdots\!68}a^{15}+\frac{22\!\cdots\!72}{18\!\cdots\!17}a^{14}-\frac{18\!\cdots\!41}{36\!\cdots\!34}a^{13}-\frac{19\!\cdots\!49}{18\!\cdots\!17}a^{12}+\frac{15\!\cdots\!37}{36\!\cdots\!34}a^{11}+\frac{76\!\cdots\!23}{18\!\cdots\!17}a^{10}-\frac{37\!\cdots\!35}{36\!\cdots\!34}a^{9}-\frac{27\!\cdots\!27}{36\!\cdots\!34}a^{8}-\frac{10\!\cdots\!19}{73\!\cdots\!68}a^{7}-\frac{83\!\cdots\!08}{18\!\cdots\!17}a^{6}+\frac{32\!\cdots\!43}{18\!\cdots\!17}a^{5}+\frac{12\!\cdots\!77}{36\!\cdots\!34}a^{4}+\frac{28\!\cdots\!17}{73\!\cdots\!68}a^{3}+\frac{77\!\cdots\!25}{36\!\cdots\!34}a^{2}+\frac{34\!\cdots\!51}{73\!\cdots\!68}a-\frac{27\!\cdots\!42}{18\!\cdots\!17}$
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
$C_{140}$, which has order $140$ (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 $2$)
| 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: |
$\frac{51\!\cdots\!89}{31\!\cdots\!12}a^{15}-\frac{18\!\cdots\!83}{78\!\cdots\!53}a^{14}-\frac{13\!\cdots\!63}{78\!\cdots\!53}a^{13}+\frac{10\!\cdots\!29}{31\!\cdots\!12}a^{12}-\frac{31\!\cdots\!81}{15\!\cdots\!06}a^{11}-\frac{26\!\cdots\!45}{31\!\cdots\!12}a^{10}-\frac{12\!\cdots\!32}{78\!\cdots\!53}a^{9}-\frac{59\!\cdots\!97}{31\!\cdots\!12}a^{8}-\frac{82\!\cdots\!25}{31\!\cdots\!12}a^{7}+\frac{84\!\cdots\!71}{31\!\cdots\!12}a^{6}-\frac{38\!\cdots\!25}{15\!\cdots\!06}a^{5}+\frac{10\!\cdots\!23}{15\!\cdots\!06}a^{4}-\frac{17\!\cdots\!27}{31\!\cdots\!12}a^{3}+\frac{72\!\cdots\!41}{15\!\cdots\!06}a^{2}-\frac{14\!\cdots\!37}{31\!\cdots\!12}a+\frac{29\!\cdots\!57}{31\!\cdots\!12}$, $\frac{92\!\cdots\!99}{73\!\cdots\!68}a^{15}+\frac{20\!\cdots\!21}{18\!\cdots\!17}a^{14}-\frac{43\!\cdots\!52}{18\!\cdots\!17}a^{13}-\frac{47\!\cdots\!79}{73\!\cdots\!68}a^{12}+\frac{21\!\cdots\!97}{36\!\cdots\!34}a^{11}+\frac{54\!\cdots\!75}{73\!\cdots\!68}a^{10}+\frac{47\!\cdots\!93}{18\!\cdots\!17}a^{9}+\frac{23\!\cdots\!43}{73\!\cdots\!68}a^{8}+\frac{11\!\cdots\!47}{73\!\cdots\!68}a^{7}+\frac{11\!\cdots\!59}{73\!\cdots\!68}a^{6}+\frac{21\!\cdots\!55}{36\!\cdots\!34}a^{5}+\frac{13\!\cdots\!41}{36\!\cdots\!34}a^{4}+\frac{12\!\cdots\!97}{73\!\cdots\!68}a^{3}+\frac{11\!\cdots\!23}{36\!\cdots\!34}a^{2}+\frac{12\!\cdots\!87}{73\!\cdots\!68}a-\frac{34\!\cdots\!39}{73\!\cdots\!68}$, $\frac{35\!\cdots\!69}{18\!\cdots\!17}a^{15}-\frac{22\!\cdots\!73}{36\!\cdots\!34}a^{14}-\frac{10\!\cdots\!71}{36\!\cdots\!34}a^{13}-\frac{14\!\cdots\!35}{36\!\cdots\!34}a^{12}+\frac{40\!\cdots\!01}{36\!\cdots\!34}a^{11}+\frac{64\!\cdots\!37}{36\!\cdots\!34}a^{10}+\frac{14\!\cdots\!29}{36\!\cdots\!34}a^{9}+\frac{88\!\cdots\!99}{36\!\cdots\!34}a^{8}+\frac{11\!\cdots\!05}{36\!\cdots\!34}a^{7}+\frac{24\!\cdots\!12}{18\!\cdots\!17}a^{6}+\frac{24\!\cdots\!61}{18\!\cdots\!17}a^{5}+\frac{45\!\cdots\!46}{18\!\cdots\!17}a^{4}+\frac{87\!\cdots\!55}{18\!\cdots\!17}a^{3}+\frac{12\!\cdots\!69}{36\!\cdots\!34}a^{2}+\frac{20\!\cdots\!45}{36\!\cdots\!34}a+\frac{84\!\cdots\!01}{18\!\cdots\!17}$, $\frac{78\!\cdots\!15}{73\!\cdots\!68}a^{15}+\frac{30\!\cdots\!23}{73\!\cdots\!68}a^{14}-\frac{37\!\cdots\!19}{73\!\cdots\!68}a^{13}-\frac{93\!\cdots\!73}{36\!\cdots\!34}a^{12}+\frac{49\!\cdots\!71}{73\!\cdots\!68}a^{11}+\frac{75\!\cdots\!40}{18\!\cdots\!17}a^{10}+\frac{40\!\cdots\!25}{73\!\cdots\!68}a^{9}+\frac{33\!\cdots\!08}{18\!\cdots\!17}a^{8}+\frac{48\!\cdots\!81}{36\!\cdots\!34}a^{7}+\frac{64\!\cdots\!87}{73\!\cdots\!68}a^{6}+\frac{40\!\cdots\!83}{18\!\cdots\!17}a^{5}+\frac{30\!\cdots\!59}{18\!\cdots\!17}a^{4}+\frac{64\!\cdots\!45}{73\!\cdots\!68}a^{3}+\frac{30\!\cdots\!69}{73\!\cdots\!68}a^{2}+\frac{40\!\cdots\!95}{36\!\cdots\!34}a+\frac{78\!\cdots\!67}{73\!\cdots\!68}$, $\frac{28\!\cdots\!81}{36\!\cdots\!34}a^{15}+\frac{45\!\cdots\!54}{18\!\cdots\!17}a^{14}-\frac{42\!\cdots\!12}{18\!\cdots\!17}a^{13}-\frac{69\!\cdots\!41}{36\!\cdots\!34}a^{12}+\frac{38\!\cdots\!11}{18\!\cdots\!17}a^{11}+\frac{70\!\cdots\!81}{36\!\cdots\!34}a^{10}-\frac{26\!\cdots\!38}{18\!\cdots\!17}a^{9}+\frac{11\!\cdots\!51}{36\!\cdots\!34}a^{8}+\frac{14\!\cdots\!41}{36\!\cdots\!34}a^{7}+\frac{16\!\cdots\!97}{36\!\cdots\!34}a^{6}-\frac{93\!\cdots\!51}{18\!\cdots\!17}a^{5}+\frac{58\!\cdots\!81}{18\!\cdots\!17}a^{4}+\frac{40\!\cdots\!23}{36\!\cdots\!34}a^{3}+\frac{19\!\cdots\!11}{18\!\cdots\!17}a^{2}+\frac{11\!\cdots\!33}{36\!\cdots\!34}a+\frac{79\!\cdots\!03}{36\!\cdots\!34}$, $\frac{51\!\cdots\!89}{31\!\cdots\!12}a^{15}+\frac{18\!\cdots\!83}{78\!\cdots\!53}a^{14}+\frac{13\!\cdots\!63}{78\!\cdots\!53}a^{13}-\frac{10\!\cdots\!29}{31\!\cdots\!12}a^{12}+\frac{31\!\cdots\!81}{15\!\cdots\!06}a^{11}+\frac{26\!\cdots\!45}{31\!\cdots\!12}a^{10}+\frac{12\!\cdots\!32}{78\!\cdots\!53}a^{9}+\frac{59\!\cdots\!97}{31\!\cdots\!12}a^{8}+\frac{82\!\cdots\!25}{31\!\cdots\!12}a^{7}-\frac{84\!\cdots\!71}{31\!\cdots\!12}a^{6}+\frac{38\!\cdots\!25}{15\!\cdots\!06}a^{5}-\frac{10\!\cdots\!23}{15\!\cdots\!06}a^{4}+\frac{17\!\cdots\!27}{31\!\cdots\!12}a^{3}-\frac{72\!\cdots\!41}{15\!\cdots\!06}a^{2}+\frac{14\!\cdots\!37}{31\!\cdots\!12}a+\frac{22\!\cdots\!55}{31\!\cdots\!12}$, $\frac{10\!\cdots\!20}{18\!\cdots\!17}a^{15}+\frac{70\!\cdots\!23}{73\!\cdots\!68}a^{14}+\frac{14\!\cdots\!49}{73\!\cdots\!68}a^{13}-\frac{18\!\cdots\!43}{73\!\cdots\!68}a^{12}-\frac{25\!\cdots\!03}{73\!\cdots\!68}a^{11}+\frac{30\!\cdots\!41}{73\!\cdots\!68}a^{10}-\frac{27\!\cdots\!67}{73\!\cdots\!68}a^{9}+\frac{41\!\cdots\!85}{73\!\cdots\!68}a^{8}-\frac{39\!\cdots\!09}{73\!\cdots\!68}a^{7}+\frac{90\!\cdots\!42}{18\!\cdots\!17}a^{6}-\frac{49\!\cdots\!61}{36\!\cdots\!34}a^{5}+\frac{15\!\cdots\!15}{36\!\cdots\!34}a^{4}-\frac{12\!\cdots\!18}{18\!\cdots\!17}a^{3}+\frac{62\!\cdots\!15}{73\!\cdots\!68}a^{2}-\frac{65\!\cdots\!81}{73\!\cdots\!68}a+\frac{89\!\cdots\!99}{36\!\cdots\!34}$
| 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: | \( 7729.95963966061 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7729.95963966061 \cdot 140}{2\cdot\sqrt{33378638618235270076563456}}\cr\approx \mathstrut & 0.227499215529261 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 - 2*x^13 + 57*x^12 + 12*x^11 + 171*x^10 + 128*x^9 + 1436*x^8 + 684*x^7 + 5805*x^6 + 1084*x^5 + 18353*x^4 + 1488*x^3 + 40675*x^2 + 2430*x + 42947)
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 - 2*x^14 - 2*x^13 + 57*x^12 + 12*x^11 + 171*x^10 + 128*x^9 + 1436*x^8 + 684*x^7 + 5805*x^6 + 1084*x^5 + 18353*x^4 + 1488*x^3 + 40675*x^2 + 2430*x + 42947, 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 - 2*x^14 - 2*x^13 + 57*x^12 + 12*x^11 + 171*x^10 + 128*x^9 + 1436*x^8 + 684*x^7 + 5805*x^6 + 1084*x^5 + 18353*x^4 + 1488*x^3 + 40675*x^2 + 2430*x + 42947);
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 - 2*x^14 - 2*x^13 + 57*x^12 + 12*x^11 + 171*x^10 + 128*x^9 + 1436*x^8 + 684*x^7 + 5805*x^6 + 1084*x^5 + 18353*x^4 + 1488*x^3 + 40675*x^2 + 2430*x + 42947);
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
$C_2^5:S_4$ (as 16T1045):
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 768 |
| The 40 conjugacy class representatives for $C_2^5:S_4$ |
| Character table for $C_2^5:S_4$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.4.3981.1, 8.0.5777424912384.1, 8.8.1410504129.1, 8.0.64914886656.1 |
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 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.2 | $x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(89\)
| 89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(1327\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |