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Properties

Label 21.9.834...152.1
Degree $21$
Signature $[9, 6]$
Discriminant $8.345\times 10^{74}$
Root discriminant \(3695.62\)
Ramified primes $2,3,37,59,109,10859$
Class number not computed
Class group not computed
Galois group $C_3^6.(C_2\times S_7)$ (as 21T138)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752)
 
gp: K = bnfinit(y^21 - 26208*y^19 - 195536*y^18 + 291973068*y^17 + 4409577342*y^16 - 1795665131924*y^15 - 41002370088498*y^14 + 6572542654341105*y^13 + 203858863109865284*y^12 - 14213279062376569485*y^11 - 582361703119054694040*y^10 + 16275661558142313035035*y^9 + 945745018043095393911180*y^8 - 5646302360560715152043103*y^7 - 792530253662409376623342416*y^6 - 4726551871709798292359565120*y^5 + 266740346394447654507510010344*y^4 + 2440298711745375561122234921808*y^3 - 44488899123058840085730493001568*y^2 - 315068960394756661920290849696640*y + 3724414958441731059478927509146752, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752)
 

\( x^{21} - 26208 x^{19} - 195536 x^{18} + 291973068 x^{17} + 4409577342 x^{16} - 1795665131924 x^{15} + \cdots + 37\!\cdots\!52 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[9, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(834507966212271820403374346664412679514757637054312831716289049278398513152\) \(\medspace = 2^{14}\cdot 3^{21}\cdot 37^{12}\cdot 59^{3}\cdot 109^{12}\cdot 10859^{3}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(3695.62\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(37\), \(59\), \(109\), \(10859\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{1922043}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4033}a^{6}-\frac{2010}{4033}a^{4}-\frac{1952}{4033}a^{3}$, $\frac{1}{4033}a^{7}-\frac{2010}{4033}a^{5}-\frac{1952}{4033}a^{4}$, $\frac{1}{4033}a^{8}-\frac{1952}{4033}a^{5}+\frac{966}{4033}a^{4}+\frac{589}{4033}a^{3}$, $\frac{1}{16265089}a^{9}-\frac{2010}{16265089}a^{7}-\frac{1952}{16265089}a^{6}-\frac{159}{4033}a^{5}+\frac{1130}{4033}a^{4}-\frac{765}{4033}a^{3}$, $\frac{1}{16265089}a^{10}-\frac{2010}{16265089}a^{8}-\frac{1952}{16265089}a^{7}+\frac{1130}{4033}a^{5}-\frac{1748}{4033}a^{4}+\frac{173}{4033}a^{3}$, $\frac{1}{16265089}a^{11}-\frac{1952}{16265089}a^{8}+\frac{966}{16265089}a^{7}+\frac{589}{16265089}a^{6}-\frac{251}{4033}a^{5}+\frac{1991}{4033}a^{4}-\frac{1121}{4033}a^{3}$, $\frac{1}{65597103937}a^{12}-\frac{2010}{65597103937}a^{10}-\frac{1952}{65597103937}a^{9}-\frac{159}{16265089}a^{8}+\frac{1130}{16265089}a^{7}-\frac{765}{16265089}a^{6}-\frac{1646}{4033}a^{5}+\frac{1794}{4033}a^{4}+\frac{1138}{4033}a^{3}$, $\frac{1}{65597103937}a^{13}-\frac{2010}{65597103937}a^{11}-\frac{1952}{65597103937}a^{10}+\frac{1130}{16265089}a^{8}-\frac{1748}{16265089}a^{7}+\frac{173}{16265089}a^{6}-\frac{792}{4033}a^{5}-\frac{514}{4033}a^{4}-\frac{419}{4033}a^{3}$, $\frac{1}{65597103937}a^{14}-\frac{1952}{65597103937}a^{11}+\frac{966}{65597103937}a^{10}+\frac{589}{65597103937}a^{9}-\frac{251}{16265089}a^{8}+\frac{1991}{16265089}a^{7}-\frac{1121}{16265089}a^{6}-\frac{3}{109}a^{5}+\frac{732}{4033}a^{4}-\frac{1884}{4033}a^{3}$, $\frac{1}{529106240355842}a^{15}-\frac{1005}{264553120177921}a^{13}-\frac{976}{264553120177921}a^{12}+\frac{1937}{65597103937}a^{11}+\frac{565}{65597103937}a^{10}+\frac{1634}{65597103937}a^{9}-\frac{1799}{16265089}a^{8}-\frac{3283}{32530178}a^{7}+\frac{1904}{16265089}a^{6}-\frac{1511}{8066}a^{5}-\frac{1222}{4033}a^{4}+\frac{1087}{8066}a^{3}-\frac{1}{2}a$, $\frac{1}{10\!\cdots\!84}a^{16}+\frac{1514}{264553120177921}a^{14}-\frac{488}{264553120177921}a^{13}-\frac{1387}{131194207874}a^{11}+\frac{46}{65597103937}a^{10}-\frac{1341}{131194207874}a^{9}-\frac{5327}{65060356}a^{8}-\frac{647}{16265089}a^{7}-\frac{709}{65060356}a^{6}-\frac{2011}{4033}a^{5}+\frac{2403}{16132}a^{4}+\frac{182}{4033}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{21\!\cdots\!68}a^{17}-\frac{244}{264553120177921}a^{14}-\frac{1775}{529106240355842}a^{13}+\frac{2311}{10\!\cdots\!84}a^{12}+\frac{3967}{131194207874}a^{11}+\frac{107}{7091578804}a^{10}+\frac{4993}{524776831496}a^{9}-\frac{2599}{32530178}a^{8}-\frac{1749}{130120712}a^{7}+\frac{1375}{16265089}a^{6}+\frac{9051}{32264}a^{5}+\frac{3691}{8066}a^{4}+\frac{11569}{32264}a^{3}$, $\frac{1}{17\!\cdots\!88}a^{18}+\frac{757}{21\!\cdots\!86}a^{16}-\frac{122}{10\!\cdots\!93}a^{15}-\frac{2053}{10\!\cdots\!84}a^{14}+\frac{6679}{21\!\cdots\!68}a^{13}-\frac{7249}{10\!\cdots\!84}a^{12}-\frac{2737}{524776831496}a^{11}+\frac{31825}{1049553662992}a^{10}-\frac{7451}{262388415748}a^{9}-\frac{29013}{260241424}a^{8}-\frac{2881}{32530178}a^{7}+\frac{16595}{260241424}a^{6}+\frac{11}{16132}a^{5}+\frac{19177}{64528}a^{4}+\frac{706}{4033}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{34\!\cdots\!76}a^{19}+\frac{757}{42\!\cdots\!72}a^{17}-\frac{61}{10\!\cdots\!93}a^{16}-\frac{1}{21\!\cdots\!68}a^{15}-\frac{25585}{42\!\cdots\!36}a^{14}+\frac{14155}{21\!\cdots\!68}a^{13}-\frac{13569}{42\!\cdots\!36}a^{12}-\frac{21519}{2099107325984}a^{11}+\frac{9621}{524776831496}a^{10}+\frac{43163}{2099107325984}a^{9}-\frac{5029}{65060356}a^{8}+\frac{35331}{520482848}a^{7}+\frac{435}{3516776}a^{6}-\frac{14583}{129056}a^{5}+\frac{585}{4033}a^{4}-\frac{5869}{16132}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{98\!\cdots\!36}a^{20}-\frac{22\!\cdots\!25}{24\!\cdots\!84}a^{19}-\frac{13\!\cdots\!09}{24\!\cdots\!84}a^{18}-\frac{13\!\cdots\!89}{61\!\cdots\!96}a^{17}+\frac{32\!\cdots\!55}{24\!\cdots\!84}a^{16}+\frac{15\!\cdots\!55}{49\!\cdots\!68}a^{15}-\frac{34\!\cdots\!59}{61\!\cdots\!48}a^{14}-\frac{19\!\cdots\!37}{12\!\cdots\!96}a^{13}+\frac{23\!\cdots\!65}{24\!\cdots\!92}a^{12}+\frac{44\!\cdots\!89}{20\!\cdots\!44}a^{11}-\frac{16\!\cdots\!93}{60\!\cdots\!24}a^{10}-\frac{32\!\cdots\!09}{15\!\cdots\!56}a^{9}+\frac{12\!\cdots\!27}{10\!\cdots\!72}a^{8}-\frac{11\!\cdots\!27}{47\!\cdots\!54}a^{7}+\frac{88\!\cdots\!73}{15\!\cdots\!28}a^{6}-\frac{23\!\cdots\!01}{93\!\cdots\!04}a^{5}+\frac{34\!\cdots\!87}{93\!\cdots\!04}a^{4}-\frac{17\!\cdots\!19}{46\!\cdots\!52}a^{3}+\frac{41\!\cdots\!63}{14\!\cdots\!93}a^{2}-\frac{38\!\cdots\!34}{14\!\cdots\!93}a+\frac{44\!\cdots\!08}{14\!\cdots\!93}$ Copy content Toggle raw display

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

not computed

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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{9}\cdot(2\pi)^{6}\cdot R \cdot h}{2\cdot\sqrt{834507966212271820403374346664412679514757637054312831716289049278398513152}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752)
 
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^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752, 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^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752);
 
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^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752);
 
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

$C_3^6.(C_2\times S_7)$ (as 21T138):

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 non-solvable group of order 7348320
The 118 conjugacy class representatives for $C_3^6.(C_2\times S_7)$
Character table for $C_3^6.(C_2\times S_7)$

Intermediate fields

7.3.640681.1

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 42 siblings: data not computed
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 R R ${\href{/padicField/5.14.0.1}{14} }{,}\,{\href{/padicField/5.7.0.1}{7} }$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.5.0.1}{5} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }$ ${\href{/padicField/23.7.0.1}{7} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ R $18{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }$ 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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.7.0.1$x^{7} + x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
2.14.14.15$x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$$2$$7$$14$$C_{14}$$[2]^{7}$
\(3\) Copy content Toggle raw display Deg $21$$3$$7$$21$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
37.3.2.3$x^{3} + 111$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.3$x^{3} + 111$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.4.2$x^{6} - 2294 x^{3} - 4603947$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.6.4.1$x^{6} + 99 x^{5} + 3273 x^{4} + 36407 x^{3} + 10209 x^{2} + 120831 x + 1323720$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(59\) Copy content Toggle raw display 59.6.3.1$x^{6} + 17405 x^{2} - 11706603$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.15.0.1$x^{15} + 57 x^{6} + 24 x^{5} + 23 x^{4} + 13 x^{3} + 39 x^{2} + 58 x + 57$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(109\) Copy content Toggle raw display 109.3.0.1$x^{3} + x + 103$$1$$3$$0$$C_3$$[\ ]^{3}$
109.9.6.1$x^{9} + 3 x^{7} + 636 x^{6} + 3 x^{5} + 291 x^{4} - 168296 x^{3} + 636 x^{2} - 73140 x + 9528237$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
109.9.6.3$x^{9} - 218 x^{6} + 11881 x^{3} + 13738962661$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
\(10859\) Copy content Toggle raw display Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $6$$2$$3$$3$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$