Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 17*x^13 + 108*x^12 - 47*x^11 + 453*x^10 - 4466*x^9 + 10451*x^8 - 27969*x^7 + 64250*x^6 - 133292*x^5 + 359613*x^4 - 301048*x^3 + 604461*x^2 - 1094445*x + 543169)
gp: K = bnfinit(y^16 - 2*y^15 - 15*y^14 + 17*y^13 + 108*y^12 - 47*y^11 + 453*y^10 - 4466*y^9 + 10451*y^8 - 27969*y^7 + 64250*y^6 - 133292*y^5 + 359613*y^4 - 301048*y^3 + 604461*y^2 - 1094445*y + 543169, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 15*x^14 + 17*x^13 + 108*x^12 - 47*x^11 + 453*x^10 - 4466*x^9 + 10451*x^8 - 27969*x^7 + 64250*x^6 - 133292*x^5 + 359613*x^4 - 301048*x^3 + 604461*x^2 - 1094445*x + 543169);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 15*x^14 + 17*x^13 + 108*x^12 - 47*x^11 + 453*x^10 - 4466*x^9 + 10451*x^8 - 27969*x^7 + 64250*x^6 - 133292*x^5 + 359613*x^4 - 301048*x^3 + 604461*x^2 - 1094445*x + 543169)
\( x^{16} - 2 x^{15} - 15 x^{14} + 17 x^{13} + 108 x^{12} - 47 x^{11} + 453 x^{10} - 4466 x^{9} + \cdots + 543169 \)
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: |
\(1411114225648575427835680561\)
\(\medspace = 11^{8}\cdot 37^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(49.76\) | 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: | $11^{1/2}37^{3/4}\approx 49.75624935454214$ | ||
| Ramified primes: |
\(11\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{1}{7}a^{11}+\frac{3}{7}a^{10}+\frac{2}{7}a^{9}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{2}{7}a^{3}+\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{13}+\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{9}-\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{2}{7}a^{4}-\frac{3}{7}a^{2}-\frac{2}{7}$, $\frac{1}{301763}a^{14}+\frac{1766}{43109}a^{13}-\frac{6406}{301763}a^{12}-\frac{121995}{301763}a^{11}-\frac{7859}{43109}a^{10}-\frac{16359}{43109}a^{9}+\frac{122091}{301763}a^{8}-\frac{5821}{27433}a^{7}+\frac{47862}{301763}a^{6}-\frac{14367}{43109}a^{5}-\frac{4760}{43109}a^{4}-\frac{13535}{301763}a^{3}-\frac{126433}{301763}a^{2}+\frac{7747}{27433}a+\frac{10225}{27433}$, $\frac{1}{43\!\cdots\!47}a^{15}+\frac{33\!\cdots\!75}{43\!\cdots\!47}a^{14}+\frac{12\!\cdots\!01}{62\!\cdots\!21}a^{13}-\frac{23\!\cdots\!62}{43\!\cdots\!47}a^{12}-\frac{16\!\cdots\!70}{43\!\cdots\!47}a^{11}-\frac{19\!\cdots\!54}{43\!\cdots\!47}a^{10}-\frac{16\!\cdots\!19}{43\!\cdots\!47}a^{9}-\frac{23\!\cdots\!76}{62\!\cdots\!21}a^{8}+\frac{28\!\cdots\!26}{62\!\cdots\!21}a^{7}-\frac{64\!\cdots\!43}{43\!\cdots\!47}a^{6}-\frac{71\!\cdots\!40}{43\!\cdots\!47}a^{5}-\frac{14\!\cdots\!65}{43\!\cdots\!47}a^{4}+\frac{90\!\cdots\!03}{43\!\cdots\!47}a^{3}+\frac{65\!\cdots\!53}{43\!\cdots\!47}a^{2}-\frac{43\!\cdots\!03}{39\!\cdots\!77}a-\frac{50\!\cdots\!51}{59\!\cdots\!31}$
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
$C_{4}$, which has order $4$ (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: |
$\frac{42\!\cdots\!13}{43\!\cdots\!47}a^{15}-\frac{74\!\cdots\!84}{43\!\cdots\!47}a^{14}-\frac{43\!\cdots\!70}{43\!\cdots\!47}a^{13}-\frac{67\!\cdots\!06}{43\!\cdots\!47}a^{12}-\frac{59\!\cdots\!34}{43\!\cdots\!47}a^{11}+\frac{22\!\cdots\!08}{62\!\cdots\!21}a^{10}+\frac{10\!\cdots\!01}{43\!\cdots\!47}a^{9}-\frac{26\!\cdots\!66}{39\!\cdots\!77}a^{8}+\frac{58\!\cdots\!19}{43\!\cdots\!47}a^{7}-\frac{33\!\cdots\!74}{43\!\cdots\!47}a^{6}+\frac{75\!\cdots\!75}{43\!\cdots\!47}a^{5}-\frac{14\!\cdots\!15}{43\!\cdots\!47}a^{4}+\frac{30\!\cdots\!61}{43\!\cdots\!47}a^{3}-\frac{24\!\cdots\!38}{39\!\cdots\!77}a^{2}+\frac{15\!\cdots\!53}{56\!\cdots\!11}a-\frac{11\!\cdots\!16}{59\!\cdots\!31}$, $\frac{11\!\cdots\!02}{55\!\cdots\!27}a^{15}-\frac{34\!\cdots\!97}{38\!\cdots\!89}a^{14}-\frac{12\!\cdots\!66}{38\!\cdots\!89}a^{13}-\frac{19\!\cdots\!96}{55\!\cdots\!27}a^{12}+\frac{85\!\cdots\!78}{38\!\cdots\!89}a^{11}+\frac{21\!\cdots\!08}{38\!\cdots\!89}a^{10}+\frac{36\!\cdots\!41}{38\!\cdots\!89}a^{9}-\frac{32\!\cdots\!81}{35\!\cdots\!99}a^{8}+\frac{43\!\cdots\!15}{38\!\cdots\!89}a^{7}-\frac{21\!\cdots\!63}{38\!\cdots\!89}a^{6}+\frac{60\!\cdots\!98}{38\!\cdots\!89}a^{5}-\frac{12\!\cdots\!04}{38\!\cdots\!89}a^{4}+\frac{32\!\cdots\!44}{38\!\cdots\!89}a^{3}-\frac{18\!\cdots\!05}{35\!\cdots\!99}a^{2}+\frac{77\!\cdots\!93}{35\!\cdots\!99}a-\frac{17\!\cdots\!08}{74\!\cdots\!71}$, $\frac{43\!\cdots\!15}{46\!\cdots\!59}a^{15}-\frac{34\!\cdots\!78}{46\!\cdots\!59}a^{14}+\frac{12\!\cdots\!20}{46\!\cdots\!59}a^{13}+\frac{81\!\cdots\!15}{66\!\cdots\!37}a^{12}-\frac{10\!\cdots\!19}{46\!\cdots\!59}a^{11}-\frac{44\!\cdots\!26}{46\!\cdots\!59}a^{10}+\frac{14\!\cdots\!96}{46\!\cdots\!59}a^{9}-\frac{20\!\cdots\!90}{42\!\cdots\!69}a^{8}+\frac{13\!\cdots\!67}{46\!\cdots\!59}a^{7}-\frac{35\!\cdots\!67}{46\!\cdots\!59}a^{6}+\frac{19\!\cdots\!03}{46\!\cdots\!59}a^{5}+\frac{36\!\cdots\!52}{46\!\cdots\!59}a^{4}+\frac{24\!\cdots\!63}{46\!\cdots\!59}a^{3}+\frac{21\!\cdots\!30}{60\!\cdots\!67}a^{2}-\frac{43\!\cdots\!26}{60\!\cdots\!67}a+\frac{78\!\cdots\!44}{42\!\cdots\!69}$, $\frac{86\!\cdots\!60}{65\!\cdots\!41}a^{15}-\frac{10\!\cdots\!39}{65\!\cdots\!41}a^{14}-\frac{18\!\cdots\!89}{65\!\cdots\!41}a^{13}+\frac{16\!\cdots\!32}{59\!\cdots\!31}a^{12}+\frac{26\!\cdots\!77}{93\!\cdots\!63}a^{11}-\frac{14\!\cdots\!80}{65\!\cdots\!41}a^{10}+\frac{59\!\cdots\!75}{93\!\cdots\!63}a^{9}-\frac{28\!\cdots\!74}{65\!\cdots\!41}a^{8}+\frac{20\!\cdots\!11}{65\!\cdots\!41}a^{7}-\frac{35\!\cdots\!99}{59\!\cdots\!31}a^{6}+\frac{93\!\cdots\!12}{65\!\cdots\!41}a^{5}+\frac{57\!\cdots\!30}{65\!\cdots\!41}a^{4}+\frac{24\!\cdots\!54}{65\!\cdots\!41}a^{3}-\frac{33\!\cdots\!49}{93\!\cdots\!63}a^{2}-\frac{48\!\cdots\!44}{59\!\cdots\!31}a+\frac{40\!\cdots\!94}{59\!\cdots\!31}$, $\frac{48\!\cdots\!66}{39\!\cdots\!77}a^{15}-\frac{56\!\cdots\!35}{43\!\cdots\!47}a^{14}+\frac{10\!\cdots\!89}{43\!\cdots\!47}a^{13}+\frac{54\!\cdots\!05}{43\!\cdots\!47}a^{12}-\frac{19\!\cdots\!66}{43\!\cdots\!47}a^{11}-\frac{19\!\cdots\!71}{43\!\cdots\!47}a^{10}+\frac{18\!\cdots\!27}{43\!\cdots\!47}a^{9}-\frac{65\!\cdots\!08}{43\!\cdots\!47}a^{8}+\frac{35\!\cdots\!54}{56\!\cdots\!11}a^{7}-\frac{95\!\cdots\!66}{43\!\cdots\!47}a^{6}+\frac{23\!\cdots\!17}{43\!\cdots\!47}a^{5}-\frac{46\!\cdots\!71}{43\!\cdots\!47}a^{4}+\frac{12\!\cdots\!76}{62\!\cdots\!21}a^{3}-\frac{23\!\cdots\!76}{62\!\cdots\!21}a^{2}+\frac{22\!\cdots\!81}{39\!\cdots\!77}a-\frac{17\!\cdots\!64}{59\!\cdots\!31}$, $\frac{18\!\cdots\!33}{43\!\cdots\!47}a^{15}+\frac{36\!\cdots\!39}{43\!\cdots\!47}a^{14}-\frac{87\!\cdots\!66}{43\!\cdots\!47}a^{13}-\frac{85\!\cdots\!04}{62\!\cdots\!21}a^{12}-\frac{58\!\cdots\!89}{43\!\cdots\!47}a^{11}+\frac{40\!\cdots\!48}{43\!\cdots\!47}a^{10}+\frac{20\!\cdots\!13}{43\!\cdots\!47}a^{9}-\frac{39\!\cdots\!50}{43\!\cdots\!47}a^{8}+\frac{12\!\cdots\!71}{43\!\cdots\!47}a^{7}-\frac{51\!\cdots\!68}{43\!\cdots\!47}a^{6}+\frac{13\!\cdots\!07}{62\!\cdots\!21}a^{5}-\frac{50\!\cdots\!45}{62\!\cdots\!21}a^{4}+\frac{30\!\cdots\!35}{43\!\cdots\!47}a^{3}-\frac{64\!\cdots\!93}{43\!\cdots\!47}a^{2}+\frac{15\!\cdots\!66}{56\!\cdots\!11}a-\frac{81\!\cdots\!07}{59\!\cdots\!31}$, $\frac{81\!\cdots\!67}{43\!\cdots\!47}a^{15}-\frac{38\!\cdots\!31}{43\!\cdots\!47}a^{14}-\frac{21\!\cdots\!11}{43\!\cdots\!47}a^{13}+\frac{19\!\cdots\!25}{43\!\cdots\!47}a^{12}+\frac{18\!\cdots\!85}{43\!\cdots\!47}a^{11}-\frac{35\!\cdots\!05}{62\!\cdots\!21}a^{10}+\frac{11\!\cdots\!94}{62\!\cdots\!21}a^{9}-\frac{96\!\cdots\!91}{43\!\cdots\!47}a^{8}-\frac{39\!\cdots\!06}{43\!\cdots\!47}a^{7}+\frac{25\!\cdots\!08}{43\!\cdots\!47}a^{6}-\frac{85\!\cdots\!11}{43\!\cdots\!47}a^{5}+\frac{19\!\cdots\!48}{43\!\cdots\!47}a^{4}-\frac{23\!\cdots\!18}{43\!\cdots\!47}a^{3}+\frac{85\!\cdots\!40}{62\!\cdots\!21}a^{2}-\frac{71\!\cdots\!68}{39\!\cdots\!77}a+\frac{42\!\cdots\!86}{59\!\cdots\!31}$
| 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: | \( 1290119.85307 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1290119.85307 \cdot 4}{2\cdot\sqrt{1411114225648575427835680561}}\cr\approx \mathstrut & 0.166846813185 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 17*x^13 + 108*x^12 - 47*x^11 + 453*x^10 - 4466*x^9 + 10451*x^8 - 27969*x^7 + 64250*x^6 - 133292*x^5 + 359613*x^4 - 301048*x^3 + 604461*x^2 - 1094445*x + 543169)
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^15 - 15*x^14 + 17*x^13 + 108*x^12 - 47*x^11 + 453*x^10 - 4466*x^9 + 10451*x^8 - 27969*x^7 + 64250*x^6 - 133292*x^5 + 359613*x^4 - 301048*x^3 + 604461*x^2 - 1094445*x + 543169, 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^15 - 15*x^14 + 17*x^13 + 108*x^12 - 47*x^11 + 453*x^10 - 4466*x^9 + 10451*x^8 - 27969*x^7 + 64250*x^6 - 133292*x^5 + 359613*x^4 - 301048*x^3 + 604461*x^2 - 1094445*x + 543169);
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^15 - 15*x^14 + 17*x^13 + 108*x^12 - 47*x^11 + 453*x^10 - 4466*x^9 + 10451*x^8 - 27969*x^7 + 64250*x^6 - 133292*x^5 + 359613*x^4 - 301048*x^3 + 604461*x^2 - 1094445*x + 543169);
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
$\SD_{16}$ (as 16T12):
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 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{-407}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{37})\), 4.2.15059.1 x2, 4.0.4477.1 x2, 8.0.27439591201.1, 8.2.3414981850379.1 x4 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(37\)
| 37.4.3.2 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 37.4.3.2 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 37.4.3.2 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 37.4.3.2 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |