Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367)
gp: K = bnfinit(y^16 - 6*y^15 - 6*y^14 + 44*y^13 + 162*y^12 - 479*y^11 - 287*y^10 + 2522*y^9 + 3180*y^8 - 9028*y^7 - 10699*y^6 + 401*y^5 + 17424*y^4 + 6594*y^3 + 99876*y^2 - 192542*y + 220367, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367)
\( x^{16} - 6 x^{15} - 6 x^{14} + 44 x^{13} + 162 x^{12} - 479 x^{11} - 287 x^{10} + 2522 x^{9} + \cdots + 220367 \)
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: |
\(2420667705185442556922185849\)
\(\medspace = 7^{12}\cdot 53^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(51.46\) | 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: | $7^{3/4}53^{3/4}\approx 84.53380782538902$ | ||
| Ramified primes: |
\(7\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}-\frac{3}{7}a^{9}+\frac{3}{7}a^{8}+\frac{3}{7}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}$, $\frac{1}{14}a^{11}-\frac{1}{14}a^{10}+\frac{2}{7}a^{9}+\frac{1}{7}a^{8}+\frac{2}{7}a^{7}-\frac{3}{14}a^{6}+\frac{3}{14}a^{5}+\frac{1}{7}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{154}a^{12}-\frac{2}{77}a^{11}-\frac{5}{154}a^{10}+\frac{13}{77}a^{9}+\frac{2}{77}a^{8}+\frac{3}{14}a^{7}-\frac{3}{77}a^{6}+\frac{23}{154}a^{5}-\frac{16}{77}a^{4}+\frac{1}{22}a^{2}-\frac{2}{11}a-\frac{5}{22}$, $\frac{1}{308}a^{13}-\frac{5}{154}a^{11}+\frac{17}{308}a^{10}-\frac{17}{77}a^{9}-\frac{17}{308}a^{8}-\frac{18}{77}a^{7}-\frac{39}{154}a^{6}+\frac{115}{308}a^{5}+\frac{5}{22}a^{4}+\frac{1}{44}a^{3}-\frac{1}{2}a^{2}-\frac{5}{22}a-\frac{9}{44}$, $\frac{1}{303688}a^{14}-\frac{171}{303688}a^{13}-\frac{201}{75922}a^{12}-\frac{1091}{43384}a^{11}+\frac{3767}{303688}a^{10}-\frac{114897}{303688}a^{9}+\frac{11573}{27608}a^{8}+\frac{30003}{75922}a^{7}+\frac{60897}{303688}a^{6}-\frac{2949}{17864}a^{5}-\frac{150455}{303688}a^{4}+\frac{18727}{43384}a^{3}-\frac{7915}{21692}a^{2}-\frac{1019}{43384}a-\frac{5975}{43384}$, $\frac{1}{27\!\cdots\!72}a^{15}-\frac{19\!\cdots\!29}{69\!\cdots\!18}a^{14}-\frac{92\!\cdots\!69}{57\!\cdots\!28}a^{13}+\frac{78\!\cdots\!63}{27\!\cdots\!72}a^{12}-\frac{18\!\cdots\!31}{69\!\cdots\!18}a^{11}+\frac{36\!\cdots\!61}{16\!\cdots\!02}a^{10}-\frac{14\!\cdots\!96}{49\!\cdots\!37}a^{9}-\frac{93\!\cdots\!75}{27\!\cdots\!72}a^{8}+\frac{88\!\cdots\!47}{25\!\cdots\!52}a^{7}+\frac{72\!\cdots\!61}{28\!\cdots\!64}a^{6}+\frac{48\!\cdots\!47}{12\!\cdots\!76}a^{5}+\frac{54\!\cdots\!04}{34\!\cdots\!59}a^{4}-\frac{55\!\cdots\!25}{39\!\cdots\!96}a^{3}+\frac{46\!\cdots\!39}{39\!\cdots\!96}a^{2}-\frac{20\!\cdots\!09}{90\!\cdots\!34}a+\frac{12\!\cdots\!47}{39\!\cdots\!96}$
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
$C_{4}$, 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: |
$\frac{15\!\cdots\!63}{10\!\cdots\!04}a^{15}-\frac{22\!\cdots\!37}{26\!\cdots\!26}a^{14}-\frac{26\!\cdots\!49}{10\!\cdots\!04}a^{13}+\frac{15\!\cdots\!03}{15\!\cdots\!72}a^{12}+\frac{36\!\cdots\!87}{13\!\cdots\!63}a^{11}-\frac{45\!\cdots\!47}{18\!\cdots\!88}a^{10}-\frac{83\!\cdots\!30}{18\!\cdots\!09}a^{9}-\frac{12\!\cdots\!55}{10\!\cdots\!04}a^{8}+\frac{14\!\cdots\!67}{10\!\cdots\!04}a^{7}+\frac{46\!\cdots\!73}{53\!\cdots\!52}a^{6}-\frac{10\!\cdots\!50}{13\!\cdots\!63}a^{5}-\frac{17\!\cdots\!89}{26\!\cdots\!26}a^{4}+\frac{41\!\cdots\!11}{15\!\cdots\!72}a^{3}+\frac{84\!\cdots\!37}{15\!\cdots\!72}a^{2}-\frac{95\!\cdots\!18}{18\!\cdots\!09}a+\frac{69\!\cdots\!67}{15\!\cdots\!72}$, $\frac{72\!\cdots\!55}{23\!\cdots\!88}a^{15}-\frac{22\!\cdots\!29}{16\!\cdots\!92}a^{14}-\frac{15\!\cdots\!37}{33\!\cdots\!84}a^{13}+\frac{24\!\cdots\!89}{23\!\cdots\!88}a^{12}+\frac{78\!\cdots\!33}{11\!\cdots\!44}a^{11}-\frac{53\!\cdots\!01}{10\!\cdots\!09}a^{10}-\frac{23\!\cdots\!79}{11\!\cdots\!44}a^{9}+\frac{92\!\cdots\!71}{23\!\cdots\!88}a^{8}+\frac{27\!\cdots\!35}{23\!\cdots\!88}a^{7}-\frac{36\!\cdots\!36}{29\!\cdots\!61}a^{6}-\frac{37\!\cdots\!73}{11\!\cdots\!44}a^{5}-\frac{13\!\cdots\!99}{11\!\cdots\!44}a^{4}+\frac{19\!\cdots\!37}{33\!\cdots\!84}a^{3}+\frac{81\!\cdots\!97}{33\!\cdots\!84}a^{2}+\frac{37\!\cdots\!35}{16\!\cdots\!92}a-\frac{75\!\cdots\!27}{33\!\cdots\!84}$, $\frac{25\!\cdots\!75}{23\!\cdots\!88}a^{15}-\frac{42\!\cdots\!51}{83\!\cdots\!46}a^{14}-\frac{44\!\cdots\!95}{33\!\cdots\!84}a^{13}+\frac{68\!\cdots\!81}{23\!\cdots\!88}a^{12}+\frac{67\!\cdots\!90}{29\!\cdots\!61}a^{11}-\frac{88\!\cdots\!23}{36\!\cdots\!76}a^{10}-\frac{21\!\cdots\!02}{29\!\cdots\!61}a^{9}+\frac{40\!\cdots\!09}{23\!\cdots\!88}a^{8}+\frac{16\!\cdots\!67}{23\!\cdots\!88}a^{7}-\frac{23\!\cdots\!83}{11\!\cdots\!44}a^{6}-\frac{97\!\cdots\!25}{58\!\cdots\!22}a^{5}-\frac{13\!\cdots\!39}{58\!\cdots\!22}a^{4}+\frac{58\!\cdots\!19}{33\!\cdots\!84}a^{3}+\frac{13\!\cdots\!89}{33\!\cdots\!84}a^{2}+\frac{69\!\cdots\!69}{41\!\cdots\!23}a-\frac{22\!\cdots\!89}{33\!\cdots\!84}$, $\frac{36\!\cdots\!95}{13\!\cdots\!36}a^{15}-\frac{74\!\cdots\!57}{69\!\cdots\!18}a^{14}-\frac{46\!\cdots\!81}{11\!\cdots\!44}a^{13}+\frac{98\!\cdots\!01}{13\!\cdots\!36}a^{12}+\frac{33\!\cdots\!79}{63\!\cdots\!38}a^{11}-\frac{10\!\cdots\!86}{24\!\cdots\!79}a^{10}-\frac{19\!\cdots\!61}{99\!\cdots\!74}a^{9}+\frac{86\!\cdots\!87}{13\!\cdots\!36}a^{8}+\frac{23\!\cdots\!07}{13\!\cdots\!36}a^{7}+\frac{34\!\cdots\!45}{49\!\cdots\!37}a^{6}-\frac{26\!\cdots\!85}{69\!\cdots\!18}a^{5}+\frac{91\!\cdots\!81}{69\!\cdots\!18}a^{4}+\frac{27\!\cdots\!13}{19\!\cdots\!48}a^{3}-\frac{10\!\cdots\!47}{19\!\cdots\!48}a^{2}-\frac{16\!\cdots\!41}{99\!\cdots\!74}a-\frac{11\!\cdots\!95}{19\!\cdots\!48}$, $\frac{28\!\cdots\!95}{35\!\cdots\!48}a^{15}-\frac{48\!\cdots\!63}{17\!\cdots\!74}a^{14}-\frac{13\!\cdots\!47}{73\!\cdots\!52}a^{13}+\frac{12\!\cdots\!85}{35\!\cdots\!48}a^{12}+\frac{35\!\cdots\!21}{17\!\cdots\!74}a^{11}-\frac{67\!\cdots\!01}{44\!\cdots\!79}a^{10}-\frac{23\!\cdots\!29}{23\!\cdots\!62}a^{9}+\frac{10\!\cdots\!63}{35\!\cdots\!48}a^{8}+\frac{19\!\cdots\!91}{35\!\cdots\!48}a^{7}-\frac{10\!\cdots\!40}{12\!\cdots\!91}a^{6}-\frac{23\!\cdots\!67}{17\!\cdots\!74}a^{5}-\frac{35\!\cdots\!15}{17\!\cdots\!74}a^{4}-\frac{23\!\cdots\!83}{51\!\cdots\!64}a^{3}+\frac{14\!\cdots\!45}{51\!\cdots\!64}a^{2}-\frac{70\!\cdots\!99}{15\!\cdots\!46}a-\frac{43\!\cdots\!67}{51\!\cdots\!64}$, $\frac{22\!\cdots\!87}{69\!\cdots\!18}a^{15}-\frac{31\!\cdots\!95}{13\!\cdots\!36}a^{14}+\frac{28\!\cdots\!35}{14\!\cdots\!82}a^{13}+\frac{36\!\cdots\!37}{69\!\cdots\!18}a^{12}+\frac{45\!\cdots\!51}{13\!\cdots\!36}a^{11}-\frac{23\!\cdots\!43}{17\!\cdots\!53}a^{10}+\frac{46\!\cdots\!25}{19\!\cdots\!48}a^{9}+\frac{10\!\cdots\!39}{69\!\cdots\!18}a^{8}+\frac{48\!\cdots\!49}{69\!\cdots\!18}a^{7}-\frac{24\!\cdots\!17}{19\!\cdots\!48}a^{6}+\frac{16\!\cdots\!63}{69\!\cdots\!18}a^{5}-\frac{10\!\cdots\!73}{13\!\cdots\!36}a^{4}+\frac{41\!\cdots\!66}{49\!\cdots\!37}a^{3}-\frac{10\!\cdots\!79}{99\!\cdots\!74}a^{2}+\frac{89\!\cdots\!81}{19\!\cdots\!48}a-\frac{67\!\cdots\!47}{99\!\cdots\!74}$, $\frac{12\!\cdots\!17}{27\!\cdots\!72}a^{15}-\frac{22\!\cdots\!19}{12\!\cdots\!76}a^{14}-\frac{26\!\cdots\!71}{39\!\cdots\!96}a^{13}+\frac{10\!\cdots\!59}{16\!\cdots\!16}a^{12}+\frac{12\!\cdots\!73}{13\!\cdots\!36}a^{11}-\frac{62\!\cdots\!43}{34\!\cdots\!06}a^{10}-\frac{48\!\cdots\!47}{19\!\cdots\!48}a^{9}+\frac{89\!\cdots\!93}{16\!\cdots\!16}a^{8}+\frac{81\!\cdots\!89}{27\!\cdots\!72}a^{7}+\frac{28\!\cdots\!99}{14\!\cdots\!82}a^{6}-\frac{56\!\cdots\!33}{13\!\cdots\!36}a^{5}-\frac{16\!\cdots\!61}{13\!\cdots\!36}a^{4}-\frac{39\!\cdots\!53}{39\!\cdots\!96}a^{3}-\frac{20\!\cdots\!45}{23\!\cdots\!88}a^{2}+\frac{30\!\cdots\!91}{19\!\cdots\!48}a-\frac{23\!\cdots\!05}{39\!\cdots\!96}$
| 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: | \( 13170551.5373 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 13170551.5373 \cdot 4}{2\cdot\sqrt{2420667705185442556922185849}}\cr\approx \mathstrut & 1.30048490777 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367)
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 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367, 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 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367);
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 - 6*x^15 - 6*x^14 + 44*x^13 + 162*x^12 - 479*x^11 - 287*x^10 + 2522*x^9 + 3180*x^8 - 9028*x^7 - 10699*x^6 + 401*x^5 + 17424*x^4 + 6594*x^3 + 99876*x^2 - 192542*x + 220367);
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
$C_2^3:C_4$ (as 16T33):
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 32 |
| The 11 conjugacy class representatives for $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{-371}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-7}) \), 4.0.2597.2 x2, \(\Q(\sqrt{-7}, \sqrt{53})\), 4.2.19663.1 x2, 8.0.928307199169.1 x2, 8.0.49200281555957.1 x2, 8.0.18945044881.1 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
|
\(53\)
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 53.4.3.2 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 53.4.3.2 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |