Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336)
gp: K = bnfinit(y^16 - 6*y^15 + 40*y^14 - 92*y^13 + 116*y^12 - 70*y^11 + 545*y^10 - 13523*y^9 + 54152*y^8 - 97086*y^7 + 176183*y^6 - 418415*y^5 + 678907*y^4 - 641024*y^3 + 340920*y^2 - 82952*y + 16336, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336)
\( x^{16} - 6 x^{15} + 40 x^{14} - 92 x^{13} + 116 x^{12} - 70 x^{11} + 545 x^{10} - 13523 x^{9} + \cdots + 16336 \)
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: |
\(1800057439498232157527332231681\)
\(\medspace = 41^{14}\cdot 83^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(77.80\) | 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: | $41^{7/8}83^{1/2}\approx 234.8145050561505$ | ||
| Ramified primes: |
\(41\), \(83\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{664}a^{14}-\frac{23}{664}a^{13}+\frac{43}{664}a^{12}+\frac{133}{664}a^{11}+\frac{95}{664}a^{10}-\frac{169}{664}a^{9}+\frac{45}{332}a^{8}+\frac{55}{664}a^{7}+\frac{37}{664}a^{6}-\frac{199}{664}a^{5}-\frac{63}{332}a^{4}-\frac{89}{664}a^{3}+\frac{59}{166}a^{2}+\frac{11}{166}a-\frac{8}{83}$, $\frac{1}{19\!\cdots\!44}a^{15}-\frac{86\!\cdots\!45}{19\!\cdots\!44}a^{14}-\frac{25\!\cdots\!07}{19\!\cdots\!44}a^{13}+\frac{14\!\cdots\!85}{19\!\cdots\!44}a^{12}-\frac{32\!\cdots\!31}{19\!\cdots\!44}a^{11}-\frac{16\!\cdots\!93}{19\!\cdots\!44}a^{10}+\frac{95\!\cdots\!90}{23\!\cdots\!93}a^{9}+\frac{89\!\cdots\!97}{19\!\cdots\!44}a^{8}+\frac{46\!\cdots\!47}{19\!\cdots\!44}a^{7}-\frac{87\!\cdots\!41}{19\!\cdots\!44}a^{6}+\frac{32\!\cdots\!13}{95\!\cdots\!72}a^{5}-\frac{70\!\cdots\!01}{19\!\cdots\!44}a^{4}+\frac{22\!\cdots\!82}{23\!\cdots\!93}a^{3}+\frac{35\!\cdots\!75}{95\!\cdots\!72}a^{2}-\frac{16\!\cdots\!77}{47\!\cdots\!86}a-\frac{63\!\cdots\!53}{23\!\cdots\!33}$
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
$C_{2}$, which has order $2$ (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: |
$\frac{19\!\cdots\!93}{38\!\cdots\!06}a^{15}-\frac{74\!\cdots\!97}{38\!\cdots\!06}a^{14}+\frac{70\!\cdots\!71}{38\!\cdots\!06}a^{13}-\frac{34\!\cdots\!23}{19\!\cdots\!03}a^{12}+\frac{35\!\cdots\!11}{38\!\cdots\!06}a^{11}+\frac{19\!\cdots\!01}{19\!\cdots\!03}a^{10}+\frac{88\!\cdots\!74}{19\!\cdots\!03}a^{9}-\frac{10\!\cdots\!99}{19\!\cdots\!03}a^{8}+\frac{63\!\cdots\!61}{38\!\cdots\!06}a^{7}-\frac{17\!\cdots\!99}{38\!\cdots\!06}a^{6}+\frac{25\!\cdots\!89}{38\!\cdots\!06}a^{5}-\frac{40\!\cdots\!23}{38\!\cdots\!06}a^{4}+\frac{90\!\cdots\!09}{38\!\cdots\!06}a^{3}-\frac{62\!\cdots\!63}{38\!\cdots\!06}a^{2}+\frac{82\!\cdots\!69}{19\!\cdots\!03}a+\frac{13\!\cdots\!69}{18\!\cdots\!43}$, $\frac{59\!\cdots\!23}{77\!\cdots\!12}a^{15}-\frac{11\!\cdots\!11}{77\!\cdots\!12}a^{14}+\frac{45\!\cdots\!81}{77\!\cdots\!12}a^{13}-\frac{14\!\cdots\!89}{38\!\cdots\!06}a^{12}-\frac{14\!\cdots\!39}{77\!\cdots\!12}a^{11}-\frac{14\!\cdots\!21}{19\!\cdots\!03}a^{10}-\frac{41\!\cdots\!83}{38\!\cdots\!06}a^{9}-\frac{73\!\cdots\!85}{38\!\cdots\!06}a^{8}+\frac{12\!\cdots\!87}{77\!\cdots\!12}a^{7}-\frac{17\!\cdots\!95}{77\!\cdots\!12}a^{6}+\frac{23\!\cdots\!61}{77\!\cdots\!12}a^{5}-\frac{85\!\cdots\!69}{77\!\cdots\!12}a^{4}+\frac{13\!\cdots\!67}{77\!\cdots\!12}a^{3}-\frac{10\!\cdots\!41}{77\!\cdots\!12}a^{2}+\frac{12\!\cdots\!43}{38\!\cdots\!06}a+\frac{10\!\cdots\!25}{18\!\cdots\!43}$, $\frac{66\!\cdots\!05}{77\!\cdots\!12}a^{15}-\frac{43\!\cdots\!29}{77\!\cdots\!12}a^{14}+\frac{28\!\cdots\!19}{77\!\cdots\!12}a^{13}-\frac{36\!\cdots\!35}{38\!\cdots\!06}a^{12}+\frac{96\!\cdots\!39}{77\!\cdots\!12}a^{11}-\frac{20\!\cdots\!35}{19\!\cdots\!03}a^{10}+\frac{19\!\cdots\!13}{38\!\cdots\!06}a^{9}-\frac{45\!\cdots\!75}{38\!\cdots\!06}a^{8}+\frac{41\!\cdots\!13}{77\!\cdots\!12}a^{7}-\frac{80\!\cdots\!41}{77\!\cdots\!12}a^{6}+\frac{11\!\cdots\!71}{77\!\cdots\!12}a^{5}-\frac{29\!\cdots\!55}{77\!\cdots\!12}a^{4}+\frac{55\!\cdots\!45}{77\!\cdots\!12}a^{3}-\frac{41\!\cdots\!19}{77\!\cdots\!12}a^{2}+\frac{53\!\cdots\!89}{38\!\cdots\!06}a-\frac{61\!\cdots\!21}{18\!\cdots\!43}$, $\frac{96\!\cdots\!53}{23\!\cdots\!93}a^{15}-\frac{21\!\cdots\!29}{95\!\cdots\!72}a^{14}+\frac{72\!\cdots\!37}{47\!\cdots\!86}a^{13}-\frac{72\!\cdots\!51}{23\!\cdots\!93}a^{12}+\frac{15\!\cdots\!41}{47\!\cdots\!86}a^{11}-\frac{23\!\cdots\!58}{23\!\cdots\!93}a^{10}+\frac{49\!\cdots\!22}{23\!\cdots\!93}a^{9}-\frac{51\!\cdots\!85}{95\!\cdots\!72}a^{8}+\frac{18\!\cdots\!81}{95\!\cdots\!72}a^{7}-\frac{72\!\cdots\!36}{23\!\cdots\!93}a^{6}+\frac{26\!\cdots\!05}{47\!\cdots\!86}a^{5}-\frac{13\!\cdots\!21}{95\!\cdots\!72}a^{4}+\frac{19\!\cdots\!43}{95\!\cdots\!72}a^{3}-\frac{14\!\cdots\!53}{95\!\cdots\!72}a^{2}+\frac{17\!\cdots\!51}{47\!\cdots\!86}a+\frac{33\!\cdots\!36}{23\!\cdots\!33}$, $\frac{86\!\cdots\!13}{19\!\cdots\!44}a^{15}-\frac{36\!\cdots\!53}{19\!\cdots\!44}a^{14}+\frac{28\!\cdots\!25}{19\!\cdots\!44}a^{13}-\frac{28\!\cdots\!23}{19\!\cdots\!44}a^{12}+\frac{48\!\cdots\!53}{19\!\cdots\!44}a^{11}+\frac{28\!\cdots\!87}{19\!\cdots\!44}a^{10}+\frac{64\!\cdots\!09}{23\!\cdots\!93}a^{9}-\frac{10\!\cdots\!59}{19\!\cdots\!44}a^{8}+\frac{27\!\cdots\!43}{19\!\cdots\!44}a^{7}-\frac{33\!\cdots\!73}{19\!\cdots\!44}a^{6}+\frac{44\!\cdots\!17}{95\!\cdots\!72}a^{5}-\frac{19\!\cdots\!93}{19\!\cdots\!44}a^{4}+\frac{27\!\cdots\!71}{23\!\cdots\!93}a^{3}-\frac{69\!\cdots\!09}{95\!\cdots\!72}a^{2}+\frac{42\!\cdots\!51}{23\!\cdots\!93}a-\frac{11\!\cdots\!99}{28\!\cdots\!51}$, $\frac{31\!\cdots\!05}{19\!\cdots\!44}a^{15}-\frac{50\!\cdots\!11}{47\!\cdots\!86}a^{14}+\frac{16\!\cdots\!99}{23\!\cdots\!93}a^{13}-\frac{15\!\cdots\!07}{95\!\cdots\!72}a^{12}+\frac{60\!\cdots\!53}{47\!\cdots\!86}a^{11}+\frac{81\!\cdots\!99}{47\!\cdots\!86}a^{10}+\frac{80\!\cdots\!33}{19\!\cdots\!44}a^{9}-\frac{43\!\cdots\!31}{19\!\cdots\!44}a^{8}+\frac{94\!\cdots\!81}{95\!\cdots\!72}a^{7}-\frac{15\!\cdots\!49}{95\!\cdots\!72}a^{6}+\frac{32\!\cdots\!33}{19\!\cdots\!44}a^{5}-\frac{68\!\cdots\!01}{19\!\cdots\!44}a^{4}+\frac{14\!\cdots\!91}{19\!\cdots\!44}a^{3}-\frac{42\!\cdots\!19}{47\!\cdots\!86}a^{2}+\frac{26\!\cdots\!31}{47\!\cdots\!86}a-\frac{30\!\cdots\!68}{23\!\cdots\!33}$, $\frac{49\!\cdots\!49}{19\!\cdots\!44}a^{15}-\frac{20\!\cdots\!29}{19\!\cdots\!44}a^{14}+\frac{16\!\cdots\!83}{19\!\cdots\!44}a^{13}-\frac{16\!\cdots\!97}{19\!\cdots\!44}a^{12}+\frac{26\!\cdots\!23}{19\!\cdots\!44}a^{11}+\frac{12\!\cdots\!93}{19\!\cdots\!44}a^{10}+\frac{14\!\cdots\!57}{95\!\cdots\!72}a^{9}-\frac{62\!\cdots\!77}{19\!\cdots\!44}a^{8}+\frac{15\!\cdots\!27}{19\!\cdots\!44}a^{7}-\frac{19\!\cdots\!07}{19\!\cdots\!44}a^{6}+\frac{12\!\cdots\!67}{47\!\cdots\!86}a^{5}-\frac{11\!\cdots\!31}{19\!\cdots\!44}a^{4}+\frac{31\!\cdots\!41}{47\!\cdots\!86}a^{3}-\frac{19\!\cdots\!63}{47\!\cdots\!86}a^{2}+\frac{48\!\cdots\!97}{47\!\cdots\!86}a-\frac{52\!\cdots\!31}{23\!\cdots\!33}$
| 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: | \( 93875598.4999 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 93875598.4999 \cdot 2}{2\cdot\sqrt{1800057439498232157527332231681}}\cr\approx \mathstrut & 0.169960689361 \end{aligned}\] (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 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336)
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 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336, 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 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336);
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 + 40*x^14 - 92*x^13 + 116*x^12 - 70*x^11 + 545*x^10 - 13523*x^9 + 54152*x^8 - 97086*x^7 + 176183*x^6 - 418415*x^5 + 678907*x^4 - 641024*x^3 + 340920*x^2 - 82952*x + 16336);
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
$C_2^2:C_8$ (as 16T24):
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 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.68921.1, 4.2.139523.2, 4.2.5720443.1, 8.4.1341662192766209.3, 8.0.194754273881.1, 8.4.32723468116249.1 |
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 16 sibling: | 16.8.85427703782145180664454699324763267601.2 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(41\)
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 41.8.7.3 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
|
\(83\)
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |