Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1)
gp: K = bnfinit(y^20 - 4*y^19 + 6*y^18 - 4*y^17 - 2*y^16 + 10*y^15 - 10*y^14 - 12*y^13 + 43*y^12 - 38*y^11 + 21*y^10 - 38*y^9 + 43*y^8 - 12*y^7 - 10*y^6 + 10*y^5 - 2*y^4 - 4*y^3 + 6*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1)
\( x^{20} - 4 x^{19} + 6 x^{18} - 4 x^{17} - 2 x^{16} + 10 x^{15} - 10 x^{14} - 12 x^{13} + 43 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(287532152115567788032\)
\(\medspace = 2^{30}\cdot 193^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.54\) | 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: | $2^{3/2}193^{1/2}\approx 39.293765408777$ | ||
| Ramified primes: |
\(2\), \(193\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{193}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3092351}a^{18}+\frac{814921}{3092351}a^{17}-\frac{1435426}{3092351}a^{16}-\frac{88748}{3092351}a^{15}-\frac{715639}{3092351}a^{14}+\frac{636475}{3092351}a^{13}+\frac{4222}{181903}a^{12}+\frac{1063649}{3092351}a^{11}-\frac{1172759}{3092351}a^{10}+\frac{1031245}{3092351}a^{9}-\frac{1172759}{3092351}a^{8}+\frac{1063649}{3092351}a^{7}+\frac{4222}{181903}a^{6}+\frac{636475}{3092351}a^{5}-\frac{715639}{3092351}a^{4}-\frac{88748}{3092351}a^{3}-\frac{1435426}{3092351}a^{2}+\frac{814921}{3092351}a+\frac{1}{3092351}$, $\frac{1}{3092351}a^{19}+\frac{167338}{3092351}a^{17}-\frac{371927}{3092351}a^{16}+\frac{1080432}{3092351}a^{15}+\frac{318553}{3092351}a^{14}+\frac{169178}{3092351}a^{13}-\frac{349391}{3092351}a^{12}+\frac{1090514}{3092351}a^{11}+\frac{429979}{3092351}a^{10}-\frac{886942}{3092351}a^{9}+\frac{27199}{181903}a^{8}-\frac{757304}{3092351}a^{7}-\frac{776565}{3092351}a^{6}-\frac{618235}{3092351}a^{5}-\frac{406670}{3092351}a^{4}+\frac{360645}{3092351}a^{3}+\frac{531742}{3092351}a^{2}-\frac{1489586}{3092351}a-\frac{814921}{3092351}$
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 $1$
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: | $9$ | 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: |
$\frac{13983957}{3092351}a^{19}-\frac{48093261}{3092351}a^{18}+\frac{56956252}{3092351}a^{17}-\frac{23728783}{3092351}a^{16}-\frac{41951534}{3092351}a^{15}+\frac{116454852}{3092351}a^{14}-\frac{74046976}{3092351}a^{13}-\frac{210439531}{3092351}a^{12}+\frac{484757959}{3092351}a^{11}-\frac{258982479}{3092351}a^{10}+\frac{143153370}{3092351}a^{9}-\frac{446034696}{3092351}a^{8}+\frac{356073431}{3092351}a^{7}+\frac{32021208}{3092351}a^{6}-\frac{128572265}{3092351}a^{5}+\frac{65436014}{3092351}a^{4}+\frac{15879883}{3092351}a^{3}-\frac{47652050}{3092351}a^{2}+\frac{55099029}{3092351}a-\frac{22996286}{3092351}$, $\frac{597373}{3092351}a^{19}-\frac{1725800}{3092351}a^{18}+\frac{1160595}{3092351}a^{17}+\frac{630385}{3092351}a^{16}-\frac{1980459}{3092351}a^{15}+\frac{2862794}{3092351}a^{14}+\frac{505371}{3092351}a^{13}-\frac{11246046}{3092351}a^{12}+\frac{13990972}{3092351}a^{11}+\frac{4739105}{3092351}a^{10}-\frac{1464155}{3092351}a^{9}-\frac{23934622}{3092351}a^{8}+\frac{10652414}{3092351}a^{7}+\frac{15175731}{3092351}a^{6}+\frac{1842283}{3092351}a^{5}-\frac{9047742}{3092351}a^{4}+\frac{1921838}{3092351}a^{3}-\frac{529797}{3092351}a^{2}-\frac{791977}{3092351}a+\frac{2920193}{3092351}$, $\frac{9912023}{3092351}a^{19}-\frac{33667745}{3092351}a^{18}+\frac{39480510}{3092351}a^{17}-\frac{16862401}{3092351}a^{16}-\frac{29098098}{3092351}a^{15}+\frac{81915764}{3092351}a^{14}-\frac{51522395}{3092351}a^{13}-\frac{147216621}{3092351}a^{12}+\frac{336631463}{3092351}a^{11}-\frac{180161191}{3092351}a^{10}+\frac{110583846}{3092351}a^{9}-\frac{310975607}{3092351}a^{8}+\frac{234681893}{3092351}a^{7}+\frac{16478816}{3092351}a^{6}-\frac{85491782}{3092351}a^{5}+\frac{56685460}{3092351}a^{4}+\frac{8151822}{3092351}a^{3}-\frac{33901382}{3092351}a^{2}+\frac{37567205}{3092351}a-\frac{16223477}{3092351}$, $\frac{813378}{181903}a^{19}-\frac{46372892}{3092351}a^{18}+\frac{53607127}{3092351}a^{17}-\frac{22785245}{3092351}a^{16}-\frac{39946474}{3092351}a^{15}+\frac{111747923}{3092351}a^{14}-\frac{68368494}{3092351}a^{13}-\frac{12050754}{181903}a^{12}+\frac{459533158}{3092351}a^{11}-\frac{239008636}{3092351}a^{10}+\frac{158302973}{3092351}a^{9}-\frac{433020439}{3092351}a^{8}+\frac{316227799}{3092351}a^{7}+\frac{1306111}{181903}a^{6}-\frac{107813883}{3092351}a^{5}+\frac{69682341}{3092351}a^{4}+\frac{10870453}{3092351}a^{3}-\frac{48601309}{3092351}a^{2}+\frac{56242178}{3092351}a-\frac{23447967}{3092351}$, $\frac{420587}{181903}a^{19}-\frac{22758229}{3092351}a^{18}+\frac{23229970}{3092351}a^{17}-\frac{6142781}{3092351}a^{16}-\frac{22609901}{3092351}a^{15}+\frac{52945981}{3092351}a^{14}-\frac{23659533}{3092351}a^{13}-\frac{6690265}{181903}a^{12}+\frac{217389973}{3092351}a^{11}-\frac{74246345}{3092351}a^{10}+\frac{54526109}{3092351}a^{9}-\frac{219881299}{3092351}a^{8}+\frac{130275955}{3092351}a^{7}+\frac{2893526}{181903}a^{6}-\frac{48012101}{3092351}a^{5}+\frac{14990676}{3092351}a^{4}+\frac{10084992}{3092351}a^{3}-\frac{21073303}{3092351}a^{2}+\frac{21278030}{3092351}a-\frac{5731913}{3092351}$, $\frac{9471583}{3092351}a^{19}-\frac{31844330}{3092351}a^{18}+\frac{36713517}{3092351}a^{17}-\frac{15488030}{3092351}a^{16}-\frac{26748782}{3092351}a^{15}+\frac{75600407}{3092351}a^{14}-\frac{46074492}{3092351}a^{13}-\frac{140613504}{3092351}a^{12}+\frac{313427917}{3092351}a^{11}-\frac{160494017}{3092351}a^{10}+\frac{6351541}{181903}a^{9}-\frac{307588083}{3092351}a^{8}+\frac{222594438}{3092351}a^{7}+\frac{16594249}{3092351}a^{6}-\frac{66657405}{3092351}a^{5}+\frac{40099327}{3092351}a^{4}+\frac{8755249}{3092351}a^{3}-\frac{28004161}{3092351}a^{2}+\frac{34124878}{3092351}a-\frac{16683041}{3092351}$, $\frac{6203593}{3092351}a^{19}-\frac{21648320}{3092351}a^{18}+\frac{25684762}{3092351}a^{17}-\frac{600795}{181903}a^{16}-\frac{1134072}{181903}a^{15}+\frac{53071820}{3092351}a^{14}-\frac{33855338}{3092351}a^{13}-\frac{94782746}{3092351}a^{12}+\frac{219833437}{3092351}a^{11}-\frac{116731915}{3092351}a^{10}+\frac{57031601}{3092351}a^{9}-\frac{200366926}{3092351}a^{8}+\frac{157314667}{3092351}a^{7}+\frac{20926317}{3092351}a^{6}-\frac{65589521}{3092351}a^{5}+\frac{33444951}{3092351}a^{4}+\frac{5030714}{3092351}a^{3}-\frac{1244080}{181903}a^{2}+\frac{25510300}{3092351}a-\frac{13320600}{3092351}$, $\frac{492432}{3092351}a^{19}-\frac{3851597}{3092351}a^{18}+\frac{8249188}{3092351}a^{17}-\frac{6267325}{3092351}a^{16}-\frac{463208}{3092351}a^{15}+\frac{10702160}{3092351}a^{14}-\frac{17236230}{3092351}a^{13}-\frac{2889407}{3092351}a^{12}+\frac{50562106}{3092351}a^{11}-\frac{64852990}{3092351}a^{10}+\frac{16571875}{3092351}a^{9}-\frac{36786463}{3092351}a^{8}+\frac{71776937}{3092351}a^{7}-\frac{15321555}{3092351}a^{6}-\frac{1228674}{181903}a^{5}+\frac{10937410}{3092351}a^{4}-\frac{10620625}{3092351}a^{3}-\frac{43217}{3092351}a^{2}+\frac{8846072}{3092351}a-\frac{5732550}{3092351}$, $\frac{9491484}{3092351}a^{19}-\frac{31482324}{3092351}a^{18}+\frac{35543505}{3092351}a^{17}-\frac{14466181}{3092351}a^{16}-\frac{27062002}{3092351}a^{15}+\frac{4411156}{181903}a^{14}-\frac{43952811}{3092351}a^{13}-\frac{141553205}{3092351}a^{12}+\frac{307585789}{3092351}a^{11}-\frac{150758217}{3092351}a^{10}+\frac{107070660}{3092351}a^{9}-\frac{302373989}{3092351}a^{8}+\frac{205194980}{3092351}a^{7}+\frac{21522375}{3092351}a^{6}-\frac{62806969}{3092351}a^{5}+\frac{48104819}{3092351}a^{4}+\frac{4638423}{3092351}a^{3}-\frac{31271310}{3092351}a^{2}+\frac{35341406}{3092351}a-\frac{14682861}{3092351}$
| 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: | \( 52.0455573525 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 52.0455573525 \cdot 1}{2\cdot\sqrt{287532152115567788032}}\cr\approx \mathstrut & 0.147166561949 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*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^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*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^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*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^20 - 4*x^19 + 6*x^18 - 4*x^17 - 2*x^16 + 10*x^15 - 10*x^14 - 12*x^13 + 43*x^12 - 38*x^11 + 21*x^10 - 38*x^9 + 43*x^8 - 12*x^7 - 10*x^6 + 10*x^5 - 2*x^4 - 4*x^3 + 6*x^2 - 4*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
$D_5\wr C_2$ (as 20T48):
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 200 |
| The 14 conjugacy class representatives for $D_5\wr C_2$ |
| Character table for $D_5\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.12352.2, 10.2.1220575232.1 x5 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.8774784915636224.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{5}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{10}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(193\)
| 193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 193.2.0.1 | $x^{2} + 192 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |