Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113)
gp: K = bnfinit(y^16 - 2*y^15 - 2*y^14 - 84*y^13 - 146*y^12 + 108*y^11 + 3511*y^10 + 8153*y^9 + 24141*y^8 - 27890*y^7 + 107105*y^6 + 473927*y^5 + 371280*y^4 + 129812*y^3 + 81805*y^2 - 11559*y + 12113, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113)
\( x^{16} - 2 x^{15} - 2 x^{14} - 84 x^{13} - 146 x^{12} + 108 x^{11} + 3511 x^{10} + 8153 x^{9} + \cdots + 12113 \)
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: |
\(459602138371809190522512417121\)
\(\medspace = 41^{14}\cdot 59^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(71.43\) | 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: | $41^{7/8}59^{1/2}\approx 197.97569691730635$ | ||
| Ramified primes: |
\(41\), \(59\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{12}+\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}+\frac{3}{8}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}+\frac{3}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{13}+\frac{1}{8}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{3}{8}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}+\frac{3}{8}a^{2}+\frac{3}{8}a$, $\frac{1}{1328}a^{14}-\frac{1}{1328}a^{13}-\frac{31}{664}a^{12}+\frac{245}{1328}a^{11}-\frac{227}{1328}a^{10}+\frac{183}{664}a^{9}-\frac{329}{664}a^{8}-\frac{155}{1328}a^{7}-\frac{17}{83}a^{6}+\frac{239}{1328}a^{5}+\frac{55}{332}a^{4}+\frac{113}{664}a^{3}+\frac{149}{664}a^{2}+\frac{36}{83}a-\frac{605}{1328}$, $\frac{1}{55\!\cdots\!36}a^{15}+\frac{18\!\cdots\!77}{55\!\cdots\!36}a^{14}-\frac{94\!\cdots\!31}{27\!\cdots\!68}a^{13}+\frac{12\!\cdots\!87}{55\!\cdots\!36}a^{12}-\frac{14\!\cdots\!27}{55\!\cdots\!36}a^{11}-\frac{22\!\cdots\!19}{27\!\cdots\!68}a^{10}+\frac{95\!\cdots\!77}{27\!\cdots\!68}a^{9}-\frac{11\!\cdots\!83}{55\!\cdots\!36}a^{8}-\frac{75\!\cdots\!75}{27\!\cdots\!68}a^{7}-\frac{24\!\cdots\!91}{55\!\cdots\!36}a^{6}+\frac{10\!\cdots\!83}{27\!\cdots\!68}a^{5}-\frac{65\!\cdots\!49}{27\!\cdots\!68}a^{4}-\frac{66\!\cdots\!49}{34\!\cdots\!71}a^{3}-\frac{46\!\cdots\!03}{33\!\cdots\!96}a^{2}-\frac{11\!\cdots\!65}{55\!\cdots\!36}a+\frac{19\!\cdots\!51}{68\!\cdots\!42}$
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: |
$\frac{17\!\cdots\!00}{13\!\cdots\!27}a^{15}-\frac{23\!\cdots\!56}{11\!\cdots\!41}a^{14}-\frac{45\!\cdots\!94}{11\!\cdots\!41}a^{13}-\frac{11\!\cdots\!72}{11\!\cdots\!41}a^{12}-\frac{24\!\cdots\!02}{11\!\cdots\!41}a^{11}+\frac{15\!\cdots\!29}{11\!\cdots\!41}a^{10}+\frac{50\!\cdots\!07}{11\!\cdots\!41}a^{9}+\frac{12\!\cdots\!45}{11\!\cdots\!41}a^{8}+\frac{34\!\cdots\!92}{11\!\cdots\!41}a^{7}-\frac{33\!\cdots\!71}{11\!\cdots\!41}a^{6}+\frac{12\!\cdots\!69}{11\!\cdots\!41}a^{5}+\frac{75\!\cdots\!89}{11\!\cdots\!41}a^{4}+\frac{66\!\cdots\!96}{11\!\cdots\!41}a^{3}-\frac{13\!\cdots\!77}{11\!\cdots\!41}a^{2}+\frac{22\!\cdots\!41}{11\!\cdots\!41}a+\frac{42\!\cdots\!20}{11\!\cdots\!41}$, $\frac{65\!\cdots\!15}{13\!\cdots\!27}a^{15}-\frac{28\!\cdots\!79}{11\!\cdots\!41}a^{14}+\frac{41\!\cdots\!31}{11\!\cdots\!41}a^{13}-\frac{46\!\cdots\!63}{11\!\cdots\!41}a^{12}+\frac{67\!\cdots\!71}{11\!\cdots\!41}a^{11}+\frac{17\!\cdots\!14}{11\!\cdots\!41}a^{10}+\frac{15\!\cdots\!15}{11\!\cdots\!41}a^{9}-\frac{14\!\cdots\!30}{11\!\cdots\!41}a^{8}+\frac{45\!\cdots\!45}{11\!\cdots\!41}a^{7}-\frac{46\!\cdots\!30}{11\!\cdots\!41}a^{6}+\frac{14\!\cdots\!39}{11\!\cdots\!41}a^{5}+\frac{52\!\cdots\!51}{11\!\cdots\!41}a^{4}-\frac{42\!\cdots\!62}{11\!\cdots\!41}a^{3}+\frac{17\!\cdots\!37}{11\!\cdots\!41}a^{2}-\frac{53\!\cdots\!18}{11\!\cdots\!41}a+\frac{81\!\cdots\!60}{11\!\cdots\!41}$, $\frac{57\!\cdots\!15}{13\!\cdots\!27}a^{15}-\frac{99\!\cdots\!47}{11\!\cdots\!41}a^{14}-\frac{93\!\cdots\!51}{11\!\cdots\!41}a^{13}-\frac{39\!\cdots\!79}{11\!\cdots\!41}a^{12}-\frac{66\!\cdots\!35}{11\!\cdots\!41}a^{11}+\frac{63\!\cdots\!01}{11\!\cdots\!41}a^{10}+\frac{16\!\cdots\!36}{11\!\cdots\!41}a^{9}+\frac{37\!\cdots\!05}{11\!\cdots\!41}a^{8}+\frac{10\!\cdots\!21}{11\!\cdots\!41}a^{7}-\frac{14\!\cdots\!43}{11\!\cdots\!41}a^{6}+\frac{51\!\cdots\!46}{11\!\cdots\!41}a^{5}+\frac{22\!\cdots\!18}{11\!\cdots\!41}a^{4}+\frac{15\!\cdots\!26}{11\!\cdots\!41}a^{3}+\frac{13\!\cdots\!06}{11\!\cdots\!41}a^{2}+\frac{15\!\cdots\!05}{11\!\cdots\!41}a-\frac{23\!\cdots\!44}{11\!\cdots\!41}$, $\frac{66\!\cdots\!01}{34\!\cdots\!71}a^{15}-\frac{20\!\cdots\!31}{34\!\cdots\!71}a^{14}+\frac{21\!\cdots\!16}{34\!\cdots\!71}a^{13}-\frac{55\!\cdots\!45}{34\!\cdots\!71}a^{12}-\frac{40\!\cdots\!54}{34\!\cdots\!71}a^{11}+\frac{15\!\cdots\!83}{34\!\cdots\!71}a^{10}+\frac{22\!\cdots\!55}{34\!\cdots\!71}a^{9}+\frac{31\!\cdots\!96}{34\!\cdots\!71}a^{8}+\frac{11\!\cdots\!73}{34\!\cdots\!71}a^{7}-\frac{34\!\cdots\!00}{34\!\cdots\!71}a^{6}+\frac{92\!\cdots\!87}{34\!\cdots\!71}a^{5}+\frac{22\!\cdots\!00}{34\!\cdots\!71}a^{4}-\frac{60\!\cdots\!59}{34\!\cdots\!71}a^{3}-\frac{10\!\cdots\!17}{34\!\cdots\!71}a^{2}+\frac{44\!\cdots\!84}{34\!\cdots\!71}a-\frac{14\!\cdots\!02}{34\!\cdots\!71}$, $\frac{17\!\cdots\!89}{13\!\cdots\!84}a^{15}-\frac{18\!\cdots\!39}{68\!\cdots\!42}a^{14}-\frac{70\!\cdots\!43}{13\!\cdots\!84}a^{13}-\frac{12\!\cdots\!47}{13\!\cdots\!84}a^{12}-\frac{10\!\cdots\!71}{68\!\cdots\!42}a^{11}+\frac{33\!\cdots\!69}{13\!\cdots\!84}a^{10}+\frac{29\!\cdots\!13}{68\!\cdots\!42}a^{9}+\frac{10\!\cdots\!77}{13\!\cdots\!84}a^{8}+\frac{29\!\cdots\!11}{13\!\cdots\!84}a^{7}-\frac{22\!\cdots\!49}{13\!\cdots\!84}a^{6}+\frac{16\!\cdots\!35}{13\!\cdots\!84}a^{5}+\frac{32\!\cdots\!15}{68\!\cdots\!42}a^{4}+\frac{24\!\cdots\!61}{68\!\cdots\!42}a^{3}+\frac{62\!\cdots\!17}{68\!\cdots\!42}a^{2}+\frac{45\!\cdots\!15}{13\!\cdots\!84}a-\frac{45\!\cdots\!71}{13\!\cdots\!84}$, $\frac{28\!\cdots\!39}{13\!\cdots\!84}a^{15}-\frac{10\!\cdots\!45}{27\!\cdots\!68}a^{14}-\frac{12\!\cdots\!33}{27\!\cdots\!68}a^{13}-\frac{11\!\cdots\!85}{68\!\cdots\!42}a^{12}-\frac{87\!\cdots\!45}{27\!\cdots\!68}a^{11}+\frac{54\!\cdots\!05}{27\!\cdots\!68}a^{10}+\frac{99\!\cdots\!09}{13\!\cdots\!84}a^{9}+\frac{59\!\cdots\!75}{34\!\cdots\!71}a^{8}+\frac{13\!\cdots\!21}{27\!\cdots\!68}a^{7}-\frac{72\!\cdots\!61}{13\!\cdots\!84}a^{6}+\frac{57\!\cdots\!75}{27\!\cdots\!68}a^{5}+\frac{34\!\cdots\!52}{34\!\cdots\!71}a^{4}+\frac{11\!\cdots\!15}{13\!\cdots\!84}a^{3}+\frac{37\!\cdots\!41}{13\!\cdots\!84}a^{2}+\frac{15\!\cdots\!35}{13\!\cdots\!84}a-\frac{14\!\cdots\!73}{27\!\cdots\!68}$, $\frac{78\!\cdots\!33}{13\!\cdots\!84}a^{15}-\frac{11\!\cdots\!55}{27\!\cdots\!68}a^{14}+\frac{33\!\cdots\!73}{27\!\cdots\!68}a^{13}-\frac{19\!\cdots\!80}{34\!\cdots\!71}a^{12}+\frac{27\!\cdots\!97}{27\!\cdots\!68}a^{11}-\frac{74\!\cdots\!69}{27\!\cdots\!68}a^{10}+\frac{12\!\cdots\!83}{13\!\cdots\!84}a^{9}+\frac{19\!\cdots\!54}{34\!\cdots\!71}a^{8}+\frac{21\!\cdots\!31}{27\!\cdots\!68}a^{7}-\frac{66\!\cdots\!61}{16\!\cdots\!48}a^{6}+\frac{29\!\cdots\!77}{27\!\cdots\!68}a^{5}-\frac{31\!\cdots\!25}{68\!\cdots\!42}a^{4}+\frac{52\!\cdots\!79}{13\!\cdots\!84}a^{3}-\frac{24\!\cdots\!01}{13\!\cdots\!84}a^{2}+\frac{72\!\cdots\!95}{13\!\cdots\!84}a-\frac{20\!\cdots\!47}{27\!\cdots\!68}$
| 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: | \( 67708958.4741 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 67708958.4741 \cdot 2}{2\cdot\sqrt{459602138371809190522512417121}}\cr\approx \mathstrut & 0.242601822950 \end{aligned}\] (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 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113)
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 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113, 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 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113);
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 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113);
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
| \(\Q(\sqrt{41}) \), 4.2.99179.1, 4.4.68921.1, 4.2.4066339.1, 8.0.194754273881.1, 8.4.677939627379761.1, 8.4.16535112862921.1 |
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: | 16.8.5569165027023164184679061585237737681.1 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | R |
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 | $x^{8} + 41$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |