Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 11*x^13 - 22*x^12 + 31*x^11 - 39*x^10 - 30*x^9 + 189*x^8 - 211*x^7 - 4*x^6 + 166*x^5 - 112*x^4 + 3*x^3 + 28*x^2 - 11*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 + 2*y^14 + 11*y^13 - 22*y^12 + 31*y^11 - 39*y^10 - 30*y^9 + 189*y^8 - 211*y^7 - 4*y^6 + 166*y^5 - 112*y^4 + 3*y^3 + 28*y^2 - 11*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 2*x^14 + 11*x^13 - 22*x^12 + 31*x^11 - 39*x^10 - 30*x^9 + 189*x^8 - 211*x^7 - 4*x^6 + 166*x^5 - 112*x^4 + 3*x^3 + 28*x^2 - 11*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 2*x^14 + 11*x^13 - 22*x^12 + 31*x^11 - 39*x^10 - 30*x^9 + 189*x^8 - 211*x^7 - 4*x^6 + 166*x^5 - 112*x^4 + 3*x^3 + 28*x^2 - 11*x + 1)
\( x^{16} - 4 x^{15} + 2 x^{14} + 11 x^{13} - 22 x^{12} + 31 x^{11} - 39 x^{10} - 30 x^{9} + 189 x^{8} + \cdots + 1 \)
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: | $[10, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-17156774645460546875\)
\(\medspace = -\,5^{8}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}\cdot 179\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.93\) | 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: | $5^{1/2}19^{1/2}29^{1/2}31^{1/2}179^{1/2}\approx 3909.922633505681$ | ||
| Ramified primes: |
\(5\), \(19\), \(29\), \(31\), \(179\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-179}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}-\frac{2}{7}a^{12}-\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{2}{7}a^{9}-\frac{1}{7}a^{8}-\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{49}a^{14}+\frac{2}{49}a^{13}+\frac{5}{49}a^{12}+\frac{23}{49}a^{11}+\frac{22}{49}a^{10}+\frac{5}{49}a^{9}-\frac{11}{49}a^{8}+\frac{6}{49}a^{7}-\frac{19}{49}a^{6}+\frac{13}{49}a^{5}-\frac{2}{7}a^{2}+\frac{17}{49}a+\frac{11}{49}$, $\frac{1}{676739}a^{15}+\frac{2337}{676739}a^{14}-\frac{39670}{676739}a^{13}+\frac{329169}{676739}a^{12}-\frac{27760}{676739}a^{11}-\frac{212539}{676739}a^{10}-\frac{247350}{676739}a^{9}-\frac{241181}{676739}a^{8}+\frac{182502}{676739}a^{7}+\frac{44477}{96677}a^{6}-\frac{196662}{676739}a^{5}-\frac{2171}{13811}a^{4}+\frac{943}{96677}a^{3}-\frac{112053}{676739}a^{2}+\frac{258687}{676739}a+\frac{291559}{676739}$
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
Trivial group, which has order $1$ (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: | $12$ | 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: |
$\frac{776714}{676739}a^{15}-\frac{2706068}{676739}a^{14}+\frac{220919}{676739}a^{13}+\frac{8465661}{676739}a^{12}-\frac{12802158}{676739}a^{11}+\frac{18157526}{676739}a^{10}-\frac{21613989}{676739}a^{9}-\frac{33420635}{676739}a^{8}+\frac{128340028}{676739}a^{7}-\frac{14439899}{96677}a^{6}-\frac{46321188}{676739}a^{5}+\frac{296804}{1973}a^{4}-\frac{5922380}{96677}a^{3}-\frac{13876111}{676739}a^{2}+\frac{15526387}{676739}a-\frac{2874359}{676739}$, $\frac{367124}{676739}a^{15}-\frac{1683096}{676739}a^{14}+\frac{774324}{676739}a^{13}+\frac{5293899}{676739}a^{12}-\frac{8373830}{676739}a^{11}+\frac{10279980}{676739}a^{10}-\frac{15463682}{676739}a^{9}-\frac{14464358}{676739}a^{8}+\frac{82398326}{676739}a^{7}-\frac{10695134}{96677}a^{6}-\frac{29862911}{676739}a^{5}+\frac{1399544}{13811}a^{4}-\frac{3620887}{96677}a^{3}-\frac{8726201}{676739}a^{2}+\frac{9029584}{676739}a-\frac{727898}{676739}$, $\frac{1295583}{676739}a^{15}-\frac{4579500}{676739}a^{14}+\frac{450168}{676739}a^{13}+\frac{14398267}{676739}a^{12}-\frac{21621274}{676739}a^{11}+\frac{30267234}{676739}a^{10}-\frac{36910611}{676739}a^{9}-\frac{55722988}{676739}a^{8}+\frac{217868864}{676739}a^{7}-\frac{24431733}{96677}a^{6}-\frac{81222968}{676739}a^{5}+\frac{498053}{1973}a^{4}-\frac{9336338}{96677}a^{3}-\frac{22147329}{676739}a^{2}+\frac{23277503}{676739}a-\frac{3719264}{676739}$, $\frac{112485}{676739}a^{15}-\frac{28751}{676739}a^{14}-\frac{912620}{676739}a^{13}+\frac{623632}{676739}a^{12}+\frac{1450789}{676739}a^{11}-\frac{1105611}{676739}a^{10}+\frac{2782287}{676739}a^{9}-\frac{9633721}{676739}a^{8}-\frac{775401}{676739}a^{7}+\frac{4299222}{96677}a^{6}-\frac{23828393}{676739}a^{5}-\frac{326351}{13811}a^{4}+\frac{3871955}{96677}a^{3}-\frac{7848667}{676739}a^{2}-\frac{5844569}{676739}a+\frac{3300014}{676739}$, $\frac{800753}{676739}a^{15}-\frac{2475286}{676739}a^{14}-\frac{502455}{676739}a^{13}+\frac{8098204}{676739}a^{12}-\frac{10774683}{676739}a^{11}+\frac{16116006}{676739}a^{10}-\frac{17246533}{676739}a^{9}-\frac{38298665}{676739}a^{8}+\frac{116674484}{676739}a^{7}-\frac{10312291}{96677}a^{6}-\frac{56455736}{676739}a^{5}+\frac{1753210}{13811}a^{4}-\frac{3859615}{96677}a^{3}-\frac{14773697}{676739}a^{2}+\frac{12344357}{676739}a-\frac{1706092}{676739}$, $\frac{251275}{676739}a^{15}-\frac{1243224}{676739}a^{14}+\frac{962959}{676739}a^{13}+\frac{3510174}{676739}a^{12}-\frac{7205871}{676739}a^{11}+\frac{8788660}{676739}a^{10}-\frac{12762730}{676739}a^{9}-\frac{6095560}{676739}a^{8}+\frac{60941672}{676739}a^{7}-\frac{10275086}{96677}a^{6}-\frac{6562890}{676739}a^{5}+\frac{1210140}{13811}a^{4}-\frac{4325845}{96677}a^{3}-\frac{5225330}{676739}a^{2}+\frac{9495905}{676739}a-\frac{1696353}{676739}$, $\frac{637415}{676739}a^{15}-\frac{2211554}{676739}a^{14}+\frac{334472}{676739}a^{13}+\frac{6567731}{676739}a^{12}-\frac{10828835}{676739}a^{11}+\frac{15984392}{676739}a^{10}-\frac{18871695}{676739}a^{9}-\frac{24740911}{676739}a^{8}+\frac{103097074}{676739}a^{7}-\frac{12923496}{96677}a^{6}-\frac{22831600}{676739}a^{5}+\frac{1620527}{13811}a^{4}-\frac{5800617}{96677}a^{3}-\frac{5772754}{676739}a^{2}+\frac{11748111}{676739}a-\frac{2578068}{676739}$, $\frac{1570811}{676739}a^{15}-\frac{4991886}{676739}a^{14}-\frac{773910}{676739}a^{13}+\frac{16343443}{676739}a^{12}-\frac{21843464}{676739}a^{11}+\frac{32021193}{676739}a^{10}-\frac{35496915}{676739}a^{9}-\frac{74681892}{676739}a^{8}+\frac{235953878}{676739}a^{7}-\frac{3049278}{13811}a^{6}-\frac{115548785}{676739}a^{5}+\frac{3568353}{13811}a^{4}-\frac{7440539}{96677}a^{3}-\frac{31491728}{676739}a^{2}+\frac{22798805}{676739}a-\frac{2624761}{676739}$, $\frac{87246}{676739}a^{15}+\frac{29731}{676739}a^{14}-\frac{1117100}{676739}a^{13}+\frac{913834}{676739}a^{12}+\frac{2282059}{676739}a^{11}-\frac{2918297}{676739}a^{10}+\frac{4623769}{676739}a^{9}-\frac{12716943}{676739}a^{8}-\frac{740092}{676739}a^{7}+\frac{6062215}{96677}a^{6}-\frac{40757562}{676739}a^{5}-\frac{291124}{13811}a^{4}+\frac{4378938}{96677}a^{3}-\frac{8608697}{676739}a^{2}-\frac{3623862}{676739}a+\frac{2425041}{676739}$, $\frac{918669}{676739}a^{15}-\frac{2904618}{676739}a^{14}-\frac{679262}{676739}a^{13}+\frac{9833904}{676739}a^{12}-\frac{11965479}{676739}a^{11}+\frac{17409958}{676739}a^{10}-\frac{19922904}{676739}a^{9}-\frac{45810452}{676739}a^{8}+\frac{137245214}{676739}a^{7}-\frac{10651395}{96677}a^{6}-\frac{80903261}{676739}a^{5}+\frac{1906245}{13811}a^{4}-\frac{1992703}{96677}a^{3}-\frac{22310461}{676739}a^{2}+\frac{9004701}{676739}a+\frac{1025828}{676739}$, $\frac{1129190}{676739}a^{15}-\frac{4024985}{676739}a^{14}+\frac{444949}{676739}a^{13}+\frac{12531567}{676739}a^{12}-\frac{18878344}{676739}a^{11}+\frac{26955381}{676739}a^{10}-\frac{33067421}{676739}a^{9}-\frac{48071426}{676739}a^{8}+\frac{190086395}{676739}a^{7}-\frac{21609967}{96677}a^{6}-\frac{65322057}{676739}a^{5}+\frac{2958267}{13811}a^{4}-\frac{8702860}{96677}a^{3}-\frac{16928131}{676739}a^{2}+\frac{20461943}{676739}a-\frac{3707258}{676739}$, $\frac{6656}{676739}a^{15}-\frac{189468}{676739}a^{14}+\frac{492374}{676739}a^{13}+\frac{220422}{676739}a^{12}-\frac{1733377}{676739}a^{11}+\frac{2252339}{676739}a^{10}-\frac{3363607}{676739}a^{9}+\frac{2962592}{676739}a^{8}+\frac{9060634}{676739}a^{7}-\frac{3407244}{96677}a^{6}+\frac{12771861}{676739}a^{5}+\frac{262485}{13811}a^{4}-\frac{2341456}{96677}a^{3}+\frac{2841920}{676739}a^{2}+\frac{4395200}{676739}a-\frac{1278951}{676739}$
| 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: | \( 4615.91677539 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{3}\cdot 4615.91677539 \cdot 1}{2\cdot\sqrt{17156774645460546875}}\cr\approx \mathstrut & 0.141530392980 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 11*x^13 - 22*x^12 + 31*x^11 - 39*x^10 - 30*x^9 + 189*x^8 - 211*x^7 - 4*x^6 + 166*x^5 - 112*x^4 + 3*x^3 + 28*x^2 - 11*x + 1)
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 - 4*x^15 + 2*x^14 + 11*x^13 - 22*x^12 + 31*x^11 - 39*x^10 - 30*x^9 + 189*x^8 - 211*x^7 - 4*x^6 + 166*x^5 - 112*x^4 + 3*x^3 + 28*x^2 - 11*x + 1, 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 - 4*x^15 + 2*x^14 + 11*x^13 - 22*x^12 + 31*x^11 - 39*x^10 - 30*x^9 + 189*x^8 - 211*x^7 - 4*x^6 + 166*x^5 - 112*x^4 + 3*x^3 + 28*x^2 - 11*x + 1);
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 - 4*x^15 + 2*x^14 + 11*x^13 - 22*x^12 + 31*x^11 - 39*x^10 - 30*x^9 + 189*x^8 - 211*x^7 - 4*x^6 + 166*x^5 - 112*x^4 + 3*x^3 + 28*x^2 - 11*x + 1);
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
$C_4^4.C_2\wr D_4$ (as 16T1823):
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 32768 |
| The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$ |
| Character table for $C_4^4.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.309593125.1 |
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: | 16.4.5214028287839453125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $16$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | R | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(19\)
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.8.0.1 | $x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(179\)
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |