Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 10*x^14 - 4*x^13 + 28*x^12 + 43*x^11 + 14*x^10 - 28*x^9 - 114*x^8 - 156*x^7 + 37*x^6 + 162*x^5 + 25*x^4 - 51*x^3 - 11*x^2 + 5*x + 1)
gp: K = bnfinit(y^16 - y^15 - 10*y^14 - 4*y^13 + 28*y^12 + 43*y^11 + 14*y^10 - 28*y^9 - 114*y^8 - 156*y^7 + 37*y^6 + 162*y^5 + 25*y^4 - 51*y^3 - 11*y^2 + 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 10*x^14 - 4*x^13 + 28*x^12 + 43*x^11 + 14*x^10 - 28*x^9 - 114*x^8 - 156*x^7 + 37*x^6 + 162*x^5 + 25*x^4 - 51*x^3 - 11*x^2 + 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 10*x^14 - 4*x^13 + 28*x^12 + 43*x^11 + 14*x^10 - 28*x^9 - 114*x^8 - 156*x^7 + 37*x^6 + 162*x^5 + 25*x^4 - 51*x^3 - 11*x^2 + 5*x + 1)
\( x^{16} - x^{15} - 10 x^{14} - 4 x^{13} + 28 x^{12} + 43 x^{11} + 14 x^{10} - 28 x^{9} - 114 x^{8} + \cdots + 1 \)
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: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(302231793843017578125\)
\(\medspace = 3^{8}\cdot 5^{12}\cdot 181\cdot 1021^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.06\) | 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: | $3^{1/2}5^{3/4}181^{1/2}1021^{1/2}\approx 2489.661417672909$ | ||
| Ramified primes: |
\(3\), \(5\), \(181\), \(1021\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{181}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{30060775669}a^{15}+\frac{414839405}{30060775669}a^{14}+\frac{14231131903}{30060775669}a^{13}+\frac{14630491726}{30060775669}a^{12}+\frac{11036475821}{30060775669}a^{11}-\frac{722722087}{30060775669}a^{10}+\frac{9844216050}{30060775669}a^{9}+\frac{9105216313}{30060775669}a^{8}-\frac{9608340498}{30060775669}a^{7}-\frac{5374468257}{30060775669}a^{6}-\frac{12426560802}{30060775669}a^{5}-\frac{4967399849}{30060775669}a^{4}+\frac{3607806877}{30060775669}a^{3}+\frac{3809619100}{30060775669}a^{2}-\frac{14566545688}{30060775669}a+\frac{13529734576}{30060775669}$
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: | $13$ | 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: |
$\frac{981065}{68475571}a^{15}+\frac{1578967}{68475571}a^{14}-\frac{5609742}{68475571}a^{13}-\frac{36484796}{68475571}a^{12}-\frac{43707312}{68475571}a^{11}+\frac{77096229}{68475571}a^{10}+\frac{263252953}{68475571}a^{9}+\frac{300991555}{68475571}a^{8}+\frac{38460829}{68475571}a^{7}-\frac{467259369}{68475571}a^{6}-\frac{1056265187}{68475571}a^{5}-\frac{865057206}{68475571}a^{4}+\frac{101471661}{68475571}a^{3}+\frac{470137472}{68475571}a^{2}+\frac{197798909}{68475571}a-\frac{88625030}{68475571}$, $\frac{35242811855}{30060775669}a^{15}-\frac{55045017188}{30060775669}a^{14}-\frac{328935035991}{30060775669}a^{13}+\frac{49818921992}{30060775669}a^{12}+\frac{1032210445027}{30060775669}a^{11}+\frac{983089440463}{30060775669}a^{10}-\frac{240702126309}{30060775669}a^{9}-\frac{1204657214499}{30060775669}a^{8}-\frac{3559809189102}{30060775669}a^{7}-\frac{3400651139165}{30060775669}a^{6}+\frac{4052626721445}{30060775669}a^{5}+\frac{4788751925925}{30060775669}a^{4}-\frac{1613428928801}{30060775669}a^{3}-\frac{1824379232753}{30060775669}a^{2}+\frac{241699089264}{30060775669}a+\frac{185242339953}{30060775669}$, $\frac{6601421457}{30060775669}a^{15}-\frac{30152158714}{30060775669}a^{14}-\frac{40592842299}{30060775669}a^{13}+\frac{203116082529}{30060775669}a^{12}+\frac{264150220988}{30060775669}a^{11}-\frac{341028708774}{30060775669}a^{10}-\frac{851036672587}{30060775669}a^{9}-\frac{549272137265}{30060775669}a^{8}-\frac{241427445242}{30060775669}a^{7}+\frac{1541720884210}{30060775669}a^{6}+\frac{3807066101994}{30060775669}a^{5}+\frac{358423586015}{30060775669}a^{4}-\frac{2919392528718}{30060775669}a^{3}-\frac{770422142400}{30060775669}a^{2}+\frac{540651823572}{30060775669}a+\frac{147006353690}{30060775669}$, $\frac{32161308805}{30060775669}a^{15}-\frac{22731976203}{30060775669}a^{14}-\frac{329988687930}{30060775669}a^{13}-\frac{219093738414}{30060775669}a^{12}+\frac{844928961763}{30060775669}a^{11}+\frac{1598402309483}{30060775669}a^{10}+\frac{886551929969}{30060775669}a^{9}-\frac{596984655206}{30060775669}a^{8}-\frac{3769484371331}{30060775669}a^{7}-\frac{6094964757963}{30060775669}a^{6}-\frac{512592723783}{30060775669}a^{5}+\frac{4889523376448}{30060775669}a^{4}+\frac{1818333448030}{30060775669}a^{3}-\frac{896301544558}{30060775669}a^{2}-\frac{324612530843}{30060775669}a-\frac{10601432071}{30060775669}$, $\frac{46516343069}{30060775669}a^{15}-\frac{68320350504}{30060775669}a^{14}-\frac{431840808637}{30060775669}a^{13}+\frac{9335765345}{30060775669}a^{12}+\frac{1293563492182}{30060775669}a^{11}+\frac{1438305213238}{30060775669}a^{10}+\frac{16085917665}{30060775669}a^{9}-\frac{1383555311040}{30060775669}a^{8}-\frac{4803984790354}{30060775669}a^{7}-\frac{5114690526725}{30060775669}a^{6}+\frac{4033264080615}{30060775669}a^{5}+\frac{5969532846319}{30060775669}a^{4}-\frac{948971298413}{30060775669}a^{3}-\frac{1915486081994}{30060775669}a^{2}+\frac{105300981238}{30060775669}a+\frac{182448640304}{30060775669}$, $\frac{81354331884}{30060775669}a^{15}-\frac{105001644900}{30060775669}a^{14}-\frac{779871621552}{30060775669}a^{13}-\frac{99794811269}{30060775669}a^{12}+\frac{2271346316563}{30060775669}a^{11}+\frac{2809032800086}{30060775669}a^{10}+\frac{416399581394}{30060775669}a^{9}-\frac{2215651447986}{30060775669}a^{8}-\frac{8538372528053}{30060775669}a^{7}-\frac{10290975748067}{30060775669}a^{6}+\frac{5601097732694}{30060775669}a^{5}+\frac{10853177338698}{30060775669}a^{4}-\frac{1179835669924}{30060775669}a^{3}-\frac{3162336555979}{30060775669}a^{2}+\frac{212810056248}{30060775669}a+\frac{174282438020}{30060775669}$, $\frac{28412776881}{30060775669}a^{15}-\frac{20803686491}{30060775669}a^{14}-\frac{291257088985}{30060775669}a^{13}-\frac{185195031309}{30060775669}a^{12}+\frac{752390599640}{30060775669}a^{11}+\frac{1389047228777}{30060775669}a^{10}+\frac{739978076059}{30060775669}a^{9}-\frac{543721416718}{30060775669}a^{8}-\frac{3295169250044}{30060775669}a^{7}-\frac{5280793932688}{30060775669}a^{6}-\frac{303015819217}{30060775669}a^{5}+\frac{4299594536481}{30060775669}a^{4}+\frac{1411579628085}{30060775669}a^{3}-\frac{912404104628}{30060775669}a^{2}-\frac{249702340785}{30060775669}a+\frac{48100733996}{30060775669}$, $\frac{29733390092}{30060775669}a^{15}-\frac{17871567853}{30060775669}a^{14}-\frac{314845873781}{30060775669}a^{13}-\frac{216455708509}{30060775669}a^{12}+\frac{815236239192}{30060775669}a^{11}+\frac{1504360364283}{30060775669}a^{10}+\frac{787447705540}{30060775669}a^{9}-\frac{550195662326}{30060775669}a^{8}-\frac{3401573302548}{30060775669}a^{7}-\frac{5743956616628}{30060775669}a^{6}-\frac{371069283407}{30060775669}a^{5}+\frac{4678031264123}{30060775669}a^{4}+\frac{995256016733}{30060775669}a^{3}-\frac{1211774873689}{30060775669}a^{2}-\frac{79125810120}{30060775669}a+\frac{93193275465}{30060775669}$, $\frac{2813655040}{30060775669}a^{15}+\frac{4301206355}{30060775669}a^{14}-\frac{24814289682}{30060775669}a^{13}-\frac{98441708246}{30060775669}a^{12}-\frac{38850827064}{30060775669}a^{11}+\frac{317751324499}{30060775669}a^{10}+\frac{579890399572}{30060775669}a^{9}+\frac{330511148690}{30060775669}a^{8}-\frac{390086336750}{30060775669}a^{7}-\frac{1391930457522}{30060775669}a^{6}-\frac{2011375463730}{30060775669}a^{5}-\frac{445011803475}{30060775669}a^{4}+\frac{1611237796582}{30060775669}a^{3}+\frac{701930847501}{30060775669}a^{2}-\frac{329577020582}{30060775669}a-\frac{106057415086}{30060775669}$, $\frac{89297301688}{30060775669}a^{15}-\frac{121254503341}{30060775669}a^{14}-\frac{826482026375}{30060775669}a^{13}-\frac{91758337379}{30060775669}a^{12}+\frac{2322199041838}{30060775669}a^{11}+\frac{2968717809770}{30060775669}a^{10}+\frac{752467681028}{30060775669}a^{9}-\frac{1973684247839}{30060775669}a^{8}-\frac{9150935625696}{30060775669}a^{7}-\frac{11041241949733}{30060775669}a^{6}+\frac{4905082662531}{30060775669}a^{5}+\frac{9684036975083}{30060775669}a^{4}-\frac{483141902082}{30060775669}a^{3}-\frac{2229355288635}{30060775669}a^{2}-\frac{1912034238}{30060775669}a+\frac{112187380726}{30060775669}$, $\frac{33285162853}{30060775669}a^{15}-\frac{20887777591}{30060775669}a^{14}-\frac{344282763234}{30060775669}a^{13}-\frac{250340499362}{30060775669}a^{12}+\frac{860254642289}{30060775669}a^{11}+\frac{1707940730360}{30060775669}a^{10}+\frac{1030746473594}{30060775669}a^{9}-\frac{531001972285}{30060775669}a^{8}-\frac{3907423978862}{30060775669}a^{7}-\frac{6574600613851}{30060775669}a^{6}-\frac{962124217887}{30060775669}a^{5}+\frac{4923302247252}{30060775669}a^{4}+\frac{2046688862523}{30060775669}a^{3}-\frac{804730260622}{30060775669}a^{2}-\frac{365672356688}{30060775669}a-\frac{11032119606}{30060775669}$, $\frac{6093849701}{30060775669}a^{15}-\frac{32110942056}{30060775669}a^{14}-\frac{30927789327}{30060775669}a^{13}+\frac{224503631621}{30060775669}a^{12}+\frac{242760191259}{30060775669}a^{11}-\frac{411243929804}{30060775669}a^{10}-\frac{899806326079}{30060775669}a^{9}-\frac{556569377685}{30060775669}a^{8}-\frac{188068358706}{30060775669}a^{7}+\frac{1797144977616}{30060775669}a^{6}+\frac{3964733553484}{30060775669}a^{5}+\frac{165073340948}{30060775669}a^{4}-\frac{2832267653389}{30060775669}a^{3}-\frac{550118123653}{30060775669}a^{2}+\frac{507111928763}{30060775669}a+\frac{103828741226}{30060775669}$, $a$
| 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: | \( 32380.1670441 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 32380.1670441 \cdot 1}{2\cdot\sqrt{302231793843017578125}}\cr\approx \mathstrut & 0.150590889820 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 10*x^14 - 4*x^13 + 28*x^12 + 43*x^11 + 14*x^10 - 28*x^9 - 114*x^8 - 156*x^7 + 37*x^6 + 162*x^5 + 25*x^4 - 51*x^3 - 11*x^2 + 5*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 - x^15 - 10*x^14 - 4*x^13 + 28*x^12 + 43*x^11 + 14*x^10 - 28*x^9 - 114*x^8 - 156*x^7 + 37*x^6 + 162*x^5 + 25*x^4 - 51*x^3 - 11*x^2 + 5*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 - x^15 - 10*x^14 - 4*x^13 + 28*x^12 + 43*x^11 + 14*x^10 - 28*x^9 - 114*x^8 - 156*x^7 + 37*x^6 + 162*x^5 + 25*x^4 - 51*x^3 - 11*x^2 + 5*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 - x^15 - 10*x^14 - 4*x^13 + 28*x^12 + 43*x^11 + 14*x^10 - 28*x^9 - 114*x^8 - 156*x^7 + 37*x^6 + 162*x^5 + 25*x^4 - 51*x^3 - 11*x^2 + 5*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
$C_4^4.C_2\wr C_4$ (as 16T1771):
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 16384 |
| The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$ |
| Character table for $C_4^4.C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.8.1292203125.1 |
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: | 16.8.53578799887939453125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $16$ | R | R | $16$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | $16$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(181\)
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.1 | $x^{2} + 181$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.4.0.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(1021\)
| $\Q_{1021}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1021}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |