Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848)
gp: K = bnfinit(y^16 - 2*y^15 - 10*y^14 + 42*y^13 + 122*y^12 - 708*y^11 + 749*y^10 + 2585*y^9 - 6198*y^8 + 2352*y^7 + 10883*y^6 - 13571*y^5 + 429*y^4 + 4438*y^3 + 1024*y^2 - 2584*y + 848, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848)
\( x^{16} - 2 x^{15} - 10 x^{14} + 42 x^{13} + 122 x^{12} - 708 x^{11} + 749 x^{10} + 2585 x^{9} + \cdots + 848 \)
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: |
\(5823129643646193092027401\)
\(\medspace = 7^{12}\cdot 29^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(35.30\) | 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: | $7^{3/4}29^{3/4}\approx 53.78015238265253$ | ||
| Ramified primes: |
\(7\), \(29\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{80}a^{14}-\frac{9}{80}a^{13}-\frac{9}{80}a^{12}-\frac{17}{80}a^{11}-\frac{1}{80}a^{10}-\frac{7}{80}a^{9}-\frac{1}{4}a^{8}-\frac{1}{80}a^{7}+\frac{9}{80}a^{6}+\frac{31}{80}a^{5}-\frac{3}{20}a^{4}+\frac{31}{80}a^{3}+\frac{9}{20}a^{2}-\frac{1}{5}a+\frac{3}{10}$, $\frac{1}{50\!\cdots\!60}a^{15}+\frac{46\!\cdots\!91}{12\!\cdots\!40}a^{14}+\frac{25\!\cdots\!37}{25\!\cdots\!80}a^{13}+\frac{30\!\cdots\!13}{25\!\cdots\!80}a^{12}+\frac{44\!\cdots\!29}{25\!\cdots\!80}a^{11}-\frac{60\!\cdots\!45}{25\!\cdots\!68}a^{10}+\frac{84\!\cdots\!29}{50\!\cdots\!60}a^{9}+\frac{74\!\cdots\!19}{50\!\cdots\!60}a^{8}-\frac{49\!\cdots\!71}{12\!\cdots\!40}a^{7}-\frac{40\!\cdots\!33}{12\!\cdots\!40}a^{6}+\frac{68\!\cdots\!91}{50\!\cdots\!60}a^{5}+\frac{45\!\cdots\!51}{10\!\cdots\!72}a^{4}-\frac{76\!\cdots\!81}{50\!\cdots\!60}a^{3}-\frac{56\!\cdots\!57}{12\!\cdots\!40}a^{2}-\frac{16\!\cdots\!83}{63\!\cdots\!20}a+\frac{29\!\cdots\!83}{11\!\cdots\!40}$
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
$C_{5}$, which has order $5$ (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: |
$\frac{22\!\cdots\!51}{50\!\cdots\!60}a^{15}-\frac{36\!\cdots\!09}{25\!\cdots\!80}a^{14}-\frac{77\!\cdots\!87}{12\!\cdots\!40}a^{13}+\frac{34\!\cdots\!61}{12\!\cdots\!40}a^{12}+\frac{76\!\cdots\!53}{12\!\cdots\!40}a^{11}-\frac{11\!\cdots\!59}{25\!\cdots\!80}a^{10}+\frac{16\!\cdots\!93}{50\!\cdots\!60}a^{9}+\frac{11\!\cdots\!89}{50\!\cdots\!60}a^{8}-\frac{12\!\cdots\!51}{25\!\cdots\!80}a^{7}-\frac{76\!\cdots\!57}{50\!\cdots\!36}a^{6}+\frac{54\!\cdots\!99}{50\!\cdots\!60}a^{5}-\frac{57\!\cdots\!31}{50\!\cdots\!60}a^{4}-\frac{21\!\cdots\!13}{50\!\cdots\!60}a^{3}+\frac{15\!\cdots\!53}{25\!\cdots\!68}a^{2}+\frac{17\!\cdots\!41}{63\!\cdots\!20}a+\frac{79\!\cdots\!11}{11\!\cdots\!40}$, $\frac{19\!\cdots\!89}{80\!\cdots\!70}a^{15}-\frac{17\!\cdots\!41}{64\!\cdots\!60}a^{14}-\frac{17\!\cdots\!31}{64\!\cdots\!60}a^{13}+\frac{50\!\cdots\!33}{64\!\cdots\!60}a^{12}+\frac{23\!\cdots\!89}{64\!\cdots\!60}a^{11}-\frac{91\!\cdots\!51}{64\!\cdots\!60}a^{10}+\frac{35\!\cdots\!11}{64\!\cdots\!60}a^{9}+\frac{11\!\cdots\!27}{16\!\cdots\!40}a^{8}-\frac{58\!\cdots\!59}{64\!\cdots\!60}a^{7}-\frac{36\!\cdots\!21}{12\!\cdots\!32}a^{6}+\frac{15\!\cdots\!13}{64\!\cdots\!60}a^{5}-\frac{86\!\cdots\!09}{80\!\cdots\!70}a^{4}-\frac{69\!\cdots\!91}{64\!\cdots\!60}a^{3}+\frac{21\!\cdots\!95}{32\!\cdots\!08}a^{2}+\frac{50\!\cdots\!57}{80\!\cdots\!70}a-\frac{45\!\cdots\!83}{15\!\cdots\!90}$, $\frac{18\!\cdots\!89}{14\!\cdots\!16}a^{15}-\frac{47\!\cdots\!37}{29\!\cdots\!32}a^{14}-\frac{39\!\cdots\!19}{29\!\cdots\!32}a^{13}+\frac{12\!\cdots\!53}{29\!\cdots\!32}a^{12}+\frac{53\!\cdots\!13}{29\!\cdots\!32}a^{11}-\frac{21\!\cdots\!03}{29\!\cdots\!32}a^{10}+\frac{11\!\cdots\!25}{29\!\cdots\!32}a^{9}+\frac{50\!\cdots\!49}{14\!\cdots\!16}a^{8}-\frac{15\!\cdots\!55}{29\!\cdots\!32}a^{7}-\frac{29\!\cdots\!17}{29\!\cdots\!32}a^{6}+\frac{37\!\cdots\!71}{29\!\cdots\!32}a^{5}-\frac{99\!\cdots\!89}{14\!\cdots\!16}a^{4}-\frac{18\!\cdots\!37}{29\!\cdots\!32}a^{3}+\frac{11\!\cdots\!71}{73\!\cdots\!08}a^{2}+\frac{12\!\cdots\!33}{36\!\cdots\!54}a-\frac{95\!\cdots\!69}{69\!\cdots\!18}$, $\frac{98\!\cdots\!59}{50\!\cdots\!60}a^{15}-\frac{75\!\cdots\!33}{25\!\cdots\!80}a^{14}-\frac{26\!\cdots\!79}{12\!\cdots\!40}a^{13}+\frac{91\!\cdots\!83}{12\!\cdots\!40}a^{12}+\frac{34\!\cdots\!49}{12\!\cdots\!40}a^{11}-\frac{31\!\cdots\!79}{25\!\cdots\!80}a^{10}+\frac{85\!\cdots\!53}{10\!\cdots\!72}a^{9}+\frac{28\!\cdots\!41}{50\!\cdots\!60}a^{8}-\frac{24\!\cdots\!27}{25\!\cdots\!80}a^{7}-\frac{74\!\cdots\!73}{25\!\cdots\!80}a^{6}+\frac{11\!\cdots\!07}{50\!\cdots\!60}a^{5}-\frac{91\!\cdots\!71}{50\!\cdots\!60}a^{4}-\frac{42\!\cdots\!81}{50\!\cdots\!60}a^{3}+\frac{10\!\cdots\!19}{12\!\cdots\!40}a^{2}+\frac{14\!\cdots\!07}{63\!\cdots\!20}a-\frac{38\!\cdots\!85}{23\!\cdots\!28}$, $\frac{61\!\cdots\!39}{50\!\cdots\!60}a^{15}-\frac{10\!\cdots\!41}{25\!\cdots\!80}a^{14}-\frac{15\!\cdots\!63}{12\!\cdots\!40}a^{13}+\frac{94\!\cdots\!69}{12\!\cdots\!40}a^{12}+\frac{14\!\cdots\!17}{12\!\cdots\!40}a^{11}-\frac{30\!\cdots\!51}{25\!\cdots\!80}a^{10}+\frac{83\!\cdots\!77}{50\!\cdots\!60}a^{9}+\frac{20\!\cdots\!01}{50\!\cdots\!60}a^{8}-\frac{34\!\cdots\!19}{25\!\cdots\!80}a^{7}+\frac{23\!\cdots\!91}{50\!\cdots\!36}a^{6}+\frac{11\!\cdots\!11}{50\!\cdots\!60}a^{5}-\frac{18\!\cdots\!19}{50\!\cdots\!60}a^{4}-\frac{49\!\cdots\!57}{50\!\cdots\!60}a^{3}+\frac{55\!\cdots\!09}{25\!\cdots\!68}a^{2}-\frac{21\!\cdots\!71}{63\!\cdots\!20}a-\frac{52\!\cdots\!01}{11\!\cdots\!40}$, $\frac{41\!\cdots\!61}{50\!\cdots\!60}a^{15}-\frac{77\!\cdots\!49}{12\!\cdots\!40}a^{14}-\frac{23\!\cdots\!43}{25\!\cdots\!80}a^{13}+\frac{58\!\cdots\!73}{25\!\cdots\!80}a^{12}+\frac{33\!\cdots\!29}{25\!\cdots\!80}a^{11}-\frac{10\!\cdots\!03}{25\!\cdots\!68}a^{10}+\frac{11\!\cdots\!49}{50\!\cdots\!60}a^{9}+\frac{12\!\cdots\!59}{50\!\cdots\!60}a^{8}-\frac{28\!\cdots\!51}{12\!\cdots\!40}a^{7}-\frac{25\!\cdots\!23}{12\!\cdots\!40}a^{6}+\frac{41\!\cdots\!51}{50\!\cdots\!60}a^{5}-\frac{75\!\cdots\!53}{10\!\cdots\!72}a^{4}-\frac{24\!\cdots\!21}{50\!\cdots\!60}a^{3}+\frac{37\!\cdots\!53}{12\!\cdots\!40}a^{2}+\frac{12\!\cdots\!67}{63\!\cdots\!20}a-\frac{91\!\cdots\!37}{11\!\cdots\!40}$, $\frac{76\!\cdots\!49}{25\!\cdots\!80}a^{15}+\frac{24\!\cdots\!01}{31\!\cdots\!10}a^{14}-\frac{45\!\cdots\!71}{12\!\cdots\!40}a^{13}-\frac{33\!\cdots\!57}{12\!\cdots\!40}a^{12}+\frac{89\!\cdots\!69}{12\!\cdots\!40}a^{11}+\frac{22\!\cdots\!43}{15\!\cdots\!05}a^{10}-\frac{10\!\cdots\!23}{25\!\cdots\!80}a^{9}+\frac{17\!\cdots\!71}{25\!\cdots\!80}a^{8}+\frac{58\!\cdots\!19}{31\!\cdots\!10}a^{7}-\frac{23\!\cdots\!09}{12\!\cdots\!84}a^{6}+\frac{18\!\cdots\!51}{25\!\cdots\!80}a^{5}+\frac{19\!\cdots\!51}{25\!\cdots\!80}a^{4}+\frac{32\!\cdots\!43}{25\!\cdots\!80}a^{3}-\frac{26\!\cdots\!61}{12\!\cdots\!84}a^{2}-\frac{73\!\cdots\!33}{15\!\cdots\!05}a+\frac{55\!\cdots\!19}{59\!\cdots\!70}$
| 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: | \( 343368.044177 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 343368.044177 \cdot 5}{2\cdot\sqrt{5823129643646193092027401}}\cr\approx \mathstrut & 0.864093226339 \end{aligned}\] (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 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848)
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 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848, 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 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848);
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 - 10*x^14 + 42*x^13 + 122*x^12 - 708*x^11 + 749*x^10 + 2585*x^9 - 6198*x^8 + 2352*x^7 + 10883*x^6 - 13571*x^5 + 429*x^4 + 4438*x^3 + 1024*x^2 - 2584*x + 848);
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
$C_4\wr C_2$ (as 16T42):
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 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-203}) \), 4.2.5887.1 x2, 4.0.1421.1 x2, \(\Q(\sqrt{-7}, \sqrt{29})\), 8.0.2869341461.1, 8.0.2413116168701.1, 8.0.1698181681.1 |
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
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.0.2413116168701.1, 8.0.2869341461.1 |
| Degree 16 sibling: | 16.4.4897252030306448390395044241.1 |
| Minimal sibling: | 8.0.2869341461.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{16}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
|
\(29\)
| 29.8.6.2 | $x^{8} - 1914 x^{4} - 2069701$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 29.8.4.1 | $x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |