Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 21*x^16 + 7*x^15 - 171*x^14 + 414*x^13 - 353*x^12 - 237*x^11 + 757*x^10 - 490*x^9 - 127*x^8 + 306*x^7 - 159*x^6 + 12*x^5 + 57*x^4 - 27*x^3 - 9*x^2 + 4*x + 1)
gp: K = bnfinit(y^18 - 8*y^17 + 21*y^16 + 7*y^15 - 171*y^14 + 414*y^13 - 353*y^12 - 237*y^11 + 757*y^10 - 490*y^9 - 127*y^8 + 306*y^7 - 159*y^6 + 12*y^5 + 57*y^4 - 27*y^3 - 9*y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 8*x^17 + 21*x^16 + 7*x^15 - 171*x^14 + 414*x^13 - 353*x^12 - 237*x^11 + 757*x^10 - 490*x^9 - 127*x^8 + 306*x^7 - 159*x^6 + 12*x^5 + 57*x^4 - 27*x^3 - 9*x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 8*x^17 + 21*x^16 + 7*x^15 - 171*x^14 + 414*x^13 - 353*x^12 - 237*x^11 + 757*x^10 - 490*x^9 - 127*x^8 + 306*x^7 - 159*x^6 + 12*x^5 + 57*x^4 - 27*x^3 - 9*x^2 + 4*x + 1)
\( x^{18} - 8 x^{17} + 21 x^{16} + 7 x^{15} - 171 x^{14} + 414 x^{13} - 353 x^{12} - 237 x^{11} + 757 x^{10} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(394876295174073378555289\)
\(\medspace = 19^{16}\cdot 37^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.46\) | 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: | $19^{8/9}37^{1/2}\approx 83.32411881951172$ | ||
| Ramified primes: |
\(19\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{47336503631}a^{17}-\frac{5466586411}{47336503631}a^{16}+\frac{14124400574}{47336503631}a^{15}-\frac{18309942608}{47336503631}a^{14}-\frac{20192058552}{47336503631}a^{13}+\frac{15212203268}{47336503631}a^{12}-\frac{16580939398}{47336503631}a^{11}+\frac{19020404842}{47336503631}a^{10}-\frac{12959835560}{47336503631}a^{9}-\frac{2068262192}{47336503631}a^{8}+\frac{1686551846}{47336503631}a^{7}+\frac{3601929604}{47336503631}a^{6}-\frac{12898478986}{47336503631}a^{5}+\frac{17797757282}{47336503631}a^{4}+\frac{20617878008}{47336503631}a^{3}-\frac{1705251726}{47336503631}a^{2}+\frac{1130556165}{47336503631}a+\frac{19259764943}{47336503631}$
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: | $13$ | 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: |
$\frac{167447353824}{47336503631}a^{17}-\frac{1489116332103}{47336503631}a^{16}+\frac{4714444110867}{47336503631}a^{15}-\frac{2023034964122}{47336503631}a^{14}-\frac{29326952285119}{47336503631}a^{13}+\frac{94119184143051}{47336503631}a^{12}-\frac{121728418943700}{47336503631}a^{11}+\frac{20521686997031}{47336503631}a^{10}+\frac{146275005212095}{47336503631}a^{9}-\frac{183555727556215}{47336503631}a^{8}+\frac{62452737520407}{47336503631}a^{7}+\frac{43738057216735}{47336503631}a^{6}-\frac{57151456922132}{47336503631}a^{5}+\frac{29490624760658}{47336503631}a^{4}-\frac{349286936374}{47336503631}a^{3}-\frac{8148375477668}{47336503631}a^{2}+\frac{1599329642442}{47336503631}a+\frac{723506633311}{47336503631}$, $\frac{284118854686}{47336503631}a^{17}-\frac{2245456893852}{47336503631}a^{16}+\frac{5851616263669}{47336503631}a^{15}+\frac{1766822226659}{47336503631}a^{14}-\frac{46470416372209}{47336503631}a^{13}+\frac{114195206236417}{47336503631}a^{12}-\frac{105848901271960}{47336503631}a^{11}-\frac{39869819525222}{47336503631}a^{10}+\frac{181453032263735}{47336503631}a^{9}-\frac{143633450405660}{47336503631}a^{8}+\frac{12187360199272}{47336503631}a^{7}+\frac{49788843150746}{47336503631}a^{6}-\frac{46221740071806}{47336503631}a^{5}+\frac{17349251924696}{47336503631}a^{4}+\frac{5058376259779}{47336503631}a^{3}-\frac{4193710938255}{47336503631}a^{2}-\frac{91867746350}{47336503631}a+\frac{60145689870}{47336503631}$, $\frac{737015615}{47336503631}a^{17}-\frac{43594916543}{47336503631}a^{16}+\frac{293821053531}{47336503631}a^{15}-\frac{624548615564}{47336503631}a^{14}-\frac{704123768017}{47336503631}a^{13}+\frac{6213125655084}{47336503631}a^{12}-\frac{12322503691489}{47336503631}a^{11}+\frac{7235798407793}{47336503631}a^{10}+\frac{10549928401069}{47336503631}a^{9}-\frac{19867390312398}{47336503631}a^{8}+\frac{8069314820578}{47336503631}a^{7}+\frac{4683135522191}{47336503631}a^{6}-\frac{5400617128392}{47336503631}a^{5}+\frac{2859444609818}{47336503631}a^{4}+\frac{54279910188}{47336503631}a^{3}-\frac{1144038743740}{47336503631}a^{2}+\frac{150303100407}{47336503631}a+\frac{147740449281}{47336503631}$, $\frac{169256806743}{47336503631}a^{17}-\frac{951004432085}{47336503631}a^{16}+\frac{496090713466}{47336503631}a^{15}+\frac{8464965956512}{47336503631}a^{14}-\frac{23717291568503}{47336503631}a^{13}+\frac{4801022776774}{47336503631}a^{12}+\frac{80948725633346}{47336503631}a^{11}-\frac{137038832053069}{47336503631}a^{10}+\frac{21802133695965}{47336503631}a^{9}+\frac{155012047965260}{47336503631}a^{8}-\frac{138104432195677}{47336503631}a^{7}+\frac{1828438923690}{47336503631}a^{6}+\frac{45712619084312}{47336503631}a^{5}-\frac{38185567371761}{47336503631}a^{4}+\frac{13366103928518}{47336503631}a^{3}+\frac{10074955273563}{47336503631}a^{2}-\frac{4257379432506}{47336503631}a-\frac{1381494163462}{47336503631}$, $\frac{115599063558}{47336503631}a^{17}-\frac{1338047378310}{47336503631}a^{16}+\frac{5649346603734}{47336503631}a^{15}-\frac{7322692345417}{47336503631}a^{14}-\frac{23457248571723}{47336503631}a^{13}+\frac{115607309114727}{47336503631}a^{12}-\frac{199120130596795}{47336503631}a^{11}+\frac{104404810935640}{47336503631}a^{10}+\frac{170200826968839}{47336503631}a^{9}-\frac{318512888683318}{47336503631}a^{8}+\frac{158361107215527}{47336503631}a^{7}+\frac{52643539749247}{47336503631}a^{6}-\frac{97334976284510}{47336503631}a^{5}+\frac{58394263906275}{47336503631}a^{4}-\frac{8253447758551}{47336503631}a^{3}-\frac{15412704955558}{47336503631}a^{2}+\frac{4315814786563}{47336503631}a+\frac{1589380302613}{47336503631}$, $\frac{429778386373}{47336503631}a^{17}-\frac{3293304574912}{47336503631}a^{16}+\frac{8021093349316}{47336503631}a^{15}+\frac{4884104736727}{47336503631}a^{14}-\frac{69750272944151}{47336503631}a^{13}+\frac{155352519154497}{47336503631}a^{12}-\frac{116767583126673}{47336503631}a^{11}-\frac{100453655733625}{47336503631}a^{10}+\frac{257522603868165}{47336503631}a^{9}-\frac{145071766119409}{47336503631}a^{8}-\frac{36152780470722}{47336503631}a^{7}+\frac{75102730739566}{47336503631}a^{6}-\frac{47704011366678}{47336503631}a^{5}+\frac{9075686469996}{47336503631}a^{4}+\frac{13364357958743}{47336503631}a^{3}-\frac{3352314643295}{47336503631}a^{2}-\frac{1882570438657}{47336503631}a-\frac{221892247222}{47336503631}$, $\frac{233793215116}{47336503631}a^{17}-\frac{2107297796052}{47336503631}a^{16}+\frac{6784077595845}{47336503631}a^{15}-\frac{3239343571962}{47336503631}a^{14}-\frac{41554003646325}{47336503631}a^{13}+\frac{135895355636851}{47336503631}a^{12}-\frac{177873694488324}{47336503631}a^{11}+\frac{30713563044583}{47336503631}a^{10}+\frac{216145749573365}{47336503631}a^{9}-\frac{271620865704576}{47336503631}a^{8}+\frac{88140288256009}{47336503631}a^{7}+\frac{70796012432759}{47336503631}a^{6}-\frac{85929268136989}{47336503631}a^{5}+\frac{41904033315928}{47336503631}a^{4}+\frac{1278785127106}{47336503631}a^{3}-\frac{12809023445382}{47336503631}a^{2}+\frac{2198858042072}{47336503631}a+\frac{1212237356719}{47336503631}$, $\frac{45369048100}{47336503631}a^{17}-\frac{492580954924}{47336503631}a^{16}+\frac{1959777895564}{47336503631}a^{15}-\frac{2196457465364}{47336503631}a^{14}-\frac{9051565625915}{47336503631}a^{13}+\frac{40155256547114}{47336503631}a^{12}-\frac{64981276460872}{47336503631}a^{11}+\frac{27882931066651}{47336503631}a^{10}+\frac{64354651175546}{47336503631}a^{9}-\frac{105993931695907}{47336503631}a^{8}+\frac{44934245975698}{47336503631}a^{7}+\frac{24916459918284}{47336503631}a^{6}-\frac{34636046745875}{47336503631}a^{5}+\frac{18003263506674}{47336503631}a^{4}-\frac{741777310128}{47336503631}a^{3}-\frac{6002676062342}{47336503631}a^{2}+\frac{1361547606601}{47336503631}a+\frac{687747056972}{47336503631}$, $a$, $\frac{89347595081}{47336503631}a^{17}-\frac{607181072485}{47336503631}a^{16}+\frac{1063169323780}{47336503631}a^{15}+\frac{2543034641502}{47336503631}a^{14}-\frac{13821454052341}{47336503631}a^{13}+\frac{19619368467367}{47336503631}a^{12}+\frac{5424471323942}{47336503631}a^{11}-\frac{45839324067336}{47336503631}a^{10}+\frac{38698031901556}{47336503631}a^{9}+\frac{18107701889012}{47336503631}a^{8}-\frac{39804606959306}{47336503631}a^{7}+\frac{13336138976073}{47336503631}a^{6}+\frac{3643357839588}{47336503631}a^{5}-\frac{8321226713428}{47336503631}a^{4}+\frac{5714728475532}{47336503631}a^{3}+\frac{1245885993023}{47336503631}a^{2}-\frac{1165817858895}{47336503631}a-\frac{259747232712}{47336503631}$, $\frac{26728364514}{47336503631}a^{17}-\frac{235002346775}{47336503631}a^{16}+\frac{740925040005}{47336503631}a^{15}-\frac{341517647823}{47336503631}a^{14}-\frac{4478977729323}{47336503631}a^{13}+\frac{14656140740872}{47336503631}a^{12}-\frac{19842614271695}{47336503631}a^{11}+\frac{5871613274861}{47336503631}a^{10}+\frac{19574720598835}{47336503631}a^{9}-\frac{28462948380093}{47336503631}a^{8}+\frac{13811363468946}{47336503631}a^{7}+\frac{2478279383117}{47336503631}a^{6}-\frac{7487362314001}{47336503631}a^{5}+\frac{5357220525829}{47336503631}a^{4}-\frac{1280531096881}{47336503631}a^{3}-\frac{618246471476}{47336503631}a^{2}+\frac{175950951777}{47336503631}a-\frac{5298936848}{47336503631}$, $\frac{83415568329}{47336503631}a^{17}-\frac{722454951669}{47336503631}a^{16}+\frac{2180775763848}{47336503631}a^{15}-\frac{488159070577}{47336503631}a^{14}-\frac{14824363570028}{47336503631}a^{13}+\frac{43697940542830}{47336503631}a^{12}-\frac{50377642902076}{47336503631}a^{11}-\frac{3580770504549}{47336503631}a^{10}+\frac{76978894880036}{47336503631}a^{9}-\frac{77817582793514}{47336503631}a^{8}+\frac{10856163511178}{47336503631}a^{7}+\frac{32084054795947}{47336503631}a^{6}-\frac{26548428360829}{47336503631}a^{5}+\frac{8646919399992}{47336503631}a^{4}+\frac{3731141744794}{47336503631}a^{3}-\frac{4856078241806}{47336503631}a^{2}+\frac{356166080534}{47336503631}a+\frac{624770041447}{47336503631}$, $\frac{78355051148}{47336503631}a^{17}-\frac{902344882796}{47336503631}a^{16}+\frac{3785312170956}{47336503631}a^{15}-\frac{4813057604860}{47336503631}a^{14}-\frac{16004847335524}{47336503631}a^{13}+\frac{77546866634082}{47336503631}a^{12}-\frac{132044710695333}{47336503631}a^{11}+\frac{66489653465998}{47336503631}a^{10}+\frac{117312563516866}{47336503631}a^{9}-\frac{212984496336728}{47336503631}a^{8}+\frac{102149165612340}{47336503631}a^{7}+\frac{39143794509745}{47336503631}a^{6}-\frac{66935348267363}{47336503631}a^{5}+\frac{38801148646485}{47336503631}a^{4}-\frac{4520179803622}{47336503631}a^{3}-\frac{10804461440929}{47336503631}a^{2}+\frac{3002089338829}{47336503631}a+\frac{1187867839028}{47336503631}$
| 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: | \( 139723.055343 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 139723.055343 \cdot 1}{2\cdot\sqrt{394876295174073378555289}}\cr\approx \mathstrut & 0.177429968957 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 21*x^16 + 7*x^15 - 171*x^14 + 414*x^13 - 353*x^12 - 237*x^11 + 757*x^10 - 490*x^9 - 127*x^8 + 306*x^7 - 159*x^6 + 12*x^5 + 57*x^4 - 27*x^3 - 9*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^18 - 8*x^17 + 21*x^16 + 7*x^15 - 171*x^14 + 414*x^13 - 353*x^12 - 237*x^11 + 757*x^10 - 490*x^9 - 127*x^8 + 306*x^7 - 159*x^6 + 12*x^5 + 57*x^4 - 27*x^3 - 9*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^18 - 8*x^17 + 21*x^16 + 7*x^15 - 171*x^14 + 414*x^13 - 353*x^12 - 237*x^11 + 757*x^10 - 490*x^9 - 127*x^8 + 306*x^7 - 159*x^6 + 12*x^5 + 57*x^4 - 27*x^3 - 9*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^18 - 8*x^17 + 21*x^16 + 7*x^15 - 171*x^14 + 414*x^13 - 353*x^12 - 237*x^11 + 757*x^10 - 490*x^9 - 127*x^8 + 306*x^7 - 159*x^6 + 12*x^5 + 57*x^4 - 27*x^3 - 9*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
$C_2^8:C_9$ (as 18T368):
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 2304 |
| The 40 conjugacy class representatives for $C_2^8:C_9$ |
| Character table for $C_2^8:C_9$ |
Intermediate fields
| 3.3.361.1, \(\Q(\zeta_{19})^+\) |
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 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.6.394876295174073378555289.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |