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Properties

Label 21.5.127...000.1
Degree $21$
Signature $(5, 8)$
Discriminant $1.280\times 10^{39}$
Root discriminant \(72.82\)
Ramified primes $2,3,5,113$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3:S_7$ (as 21T58)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - 18*x^18 - 351*x^15 + 351*x^12 + 19359*x^9 + 99630*x^6 + 40095*x^3 + 2187)
 
Copy content gp:K = bnfinit(y^21 - 18*y^18 - 351*y^15 + 351*y^12 + 19359*y^9 + 99630*y^6 + 40095*y^3 + 2187, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 18*x^18 - 351*x^15 + 351*x^12 + 19359*x^9 + 99630*x^6 + 40095*x^3 + 2187);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 18*x^18 - 351*x^15 + 351*x^12 + 19359*x^9 + 99630*x^6 + 40095*x^3 + 2187)
 

\( x^{21} - 18x^{18} - 351x^{15} + 351x^{12} + 19359x^{9} + 99630x^{6} + 40095x^{3} + 2187 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $21$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(5, 8)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(1279958175800508388164740942598144000000\) \(\medspace = 2^{18}\cdot 3^{36}\cdot 5^{6}\cdot 113^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(72.82\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(3\), \(5\), \(113\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{9}a^{6}$, $\frac{1}{9}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{27}a^{9}$, $\frac{1}{27}a^{10}$, $\frac{1}{27}a^{11}$, $\frac{1}{81}a^{12}$, $\frac{1}{81}a^{13}$, $\frac{1}{81}a^{14}$, $\frac{1}{243}a^{15}$, $\frac{1}{243}a^{16}$, $\frac{1}{243}a^{17}$, $\frac{1}{24173092521}a^{18}+\frac{10084394}{8057697507}a^{15}-\frac{9967820}{2685899169}a^{12}-\frac{671267}{99477747}a^{9}-\frac{5155024}{298433241}a^{6}-\frac{403081}{3684361}a^{3}+\frac{5038957}{33159249}$, $\frac{1}{24173092521}a^{19}+\frac{10084394}{8057697507}a^{16}-\frac{9967820}{2685899169}a^{13}-\frac{671267}{99477747}a^{10}-\frac{5155024}{298433241}a^{7}-\frac{403081}{3684361}a^{4}+\frac{5038957}{33159249}a$, $\frac{1}{72519277563}a^{20}+\frac{10084394}{24173092521}a^{17}-\frac{43127069}{8057697507}a^{14}+\frac{3013094}{298433241}a^{11}-\frac{5155024}{895299723}a^{8}-\frac{4893604}{33159249}a^{5}+\frac{38198206}{99477747}a^{2}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $12$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{273257}{24173092521}a^{18}+\frac{1818389}{8057697507}a^{15}+\frac{9558382}{2685899169}a^{12}-\frac{396709}{33159249}a^{9}-\frac{57982099}{298433241}a^{6}-\frac{2951639}{3684361}a^{3}+\frac{7541776}{33159249}$, $\frac{1468054}{24173092521}a^{18}-\frac{9155509}{8057697507}a^{15}-\frac{55120082}{2685899169}a^{12}+\frac{3516613}{99477747}a^{9}+\frac{347069966}{298433241}a^{6}+\frac{60587284}{11053083}a^{3}-\frac{2720483}{33159249}$, $\frac{5305220}{24173092521}a^{18}+\frac{32581145}{8057697507}a^{15}+\frac{201970417}{2685899169}a^{12}-\frac{3513880}{33159249}a^{9}-\frac{1245309169}{298433241}a^{6}-\frac{226079786}{11053083}a^{3}-\frac{54185732}{33159249}$, $\frac{2578973}{24173092521}a^{18}+\frac{16009820}{8057697507}a^{15}+\frac{97180108}{2685899169}a^{12}-\frac{5971723}{99477747}a^{9}-\frac{599494696}{298433241}a^{6}-\frac{103376953}{11053083}a^{3}+\frac{7746682}{33159249}$, $\frac{323668}{8057697507}a^{18}-\frac{1878874}{2685899169}a^{15}-\frac{13126139}{895299723}a^{12}+\frac{1046803}{99477747}a^{9}+\frac{79207814}{99477747}a^{6}+\frac{40798172}{11053083}a^{3}+\frac{11472211}{11053083}$, $\frac{7541776}{24173092521}a^{20}-\frac{44977399}{8057697507}a^{17}-\frac{295947653}{2685899169}a^{14}+\frac{9831634}{99477747}a^{11}+\frac{1813195607}{298433241}a^{8}+\frac{350012251}{11053083}a^{5}+\frac{494491933}{33159249}a^{2}-2$, $\frac{34498714}{72519277563}a^{20}-\frac{2262950}{2685899169}a^{19}+\frac{20033320}{24173092521}a^{18}-\frac{224659675}{24173092521}a^{17}+\frac{14192741}{895299723}a^{16}-\frac{125735131}{8057697507}a^{15}-\frac{1234457912}{8057697507}a^{14}+\frac{28133102}{99477747}a^{13}-\frac{744907484}{2685899169}a^{12}+\frac{121758920}{298433241}a^{11}-\frac{51228659}{99477747}a^{10}+\frac{48921676}{99477747}a^{9}+\frac{7900213115}{895299723}a^{8}-\frac{177597034}{11053083}a^{7}+\frac{4689395246}{298433241}a^{6}+\frac{1148050262}{33159249}a^{5}-\frac{274898451}{3684361}a^{4}+\frac{833230756}{11053083}a^{3}-\frac{4369298315}{99477747}a^{2}+\frac{53230493}{3684361}a-\frac{94753697}{33159249}$, $\frac{28000729}{8057697507}a^{20}-\frac{33131237}{24173092521}a^{19}+\frac{997114}{895299723}a^{18}+\frac{163562518}{2685899169}a^{17}+\frac{232517240}{8057697507}a^{16}-\frac{8652508}{298433241}a^{15}+\frac{1135978058}{895299723}a^{14}+\frac{977080210}{2685899169}a^{13}-\frac{510793}{3684361}a^{12}-\frac{117945244}{99477747}a^{11}-\frac{75924241}{99477747}a^{10}+\frac{94496390}{99477747}a^{9}-\frac{6866518730}{99477747}a^{8}-\frac{6725330440}{298433241}a^{7}+\frac{427372765}{33159249}a^{6}-\frac{1323701523}{3684361}a^{5}-\frac{1071448933}{11053083}a^{4}+\frac{101923472}{3684361}a^{3}-\frac{1367515918}{11053083}a^{2}-\frac{1035596228}{33159249}a+\frac{29154920}{3684361}$, $\frac{72250897}{72519277563}a^{20}+\frac{23065078}{24173092521}a^{19}+\frac{21822208}{24173092521}a^{18}+\frac{439183369}{24173092521}a^{17}-\frac{141154210}{8057697507}a^{16}-\frac{132669472}{8057697507}a^{15}+\frac{2783079902}{8057697507}a^{14}-\frac{882403400}{2685899169}a^{13}-\frac{838138139}{2685899169}a^{12}-\frac{128438147}{298433241}a^{11}+\frac{45572416}{99477747}a^{10}+\frac{38062061}{99477747}a^{9}-\frac{17132047391}{895299723}a^{8}+\frac{5449840706}{298433241}a^{7}+\frac{5179153814}{298433241}a^{6}-\frac{3139568246}{33159249}a^{5}+\frac{323959684}{3684361}a^{4}+\frac{957081176}{11053083}a^{3}-\frac{2129527030}{99477747}a^{2}+\frac{337428409}{33159249}a+\frac{323141200}{33159249}$, $\frac{12080744}{24173092521}a^{20}+\frac{18684520}{24173092521}a^{19}-\frac{6064028}{24173092521}a^{18}-\frac{65357615}{8057697507}a^{17}-\frac{131900278}{8057697507}a^{16}+\frac{73630523}{8057697507}a^{15}-\frac{537322090}{2685899169}a^{14}-\frac{566798534}{2685899169}a^{13}-\frac{59601164}{2685899169}a^{12}+\frac{6415348}{99477747}a^{11}+\frac{77400941}{99477747}a^{10}-\frac{113842262}{99477747}a^{9}+\frac{3640443430}{298433241}a^{8}+\frac{2898562688}{298433241}a^{7}+\frac{1516392356}{298433241}a^{6}+\frac{182617378}{3684361}a^{5}+\frac{217856246}{3684361}a^{4}+\frac{63592341}{3684361}a^{3}+\frac{169159571}{33159249}a^{2}+\frac{243426976}{33159249}a+\frac{70570198}{33159249}$, $\frac{788534923}{72519277563}a^{20}+\frac{143764544}{24173092521}a^{19}+\frac{2718685}{24173092521}a^{18}-\frac{4783563127}{24173092521}a^{17}-\frac{863147201}{8057697507}a^{16}+\frac{24673196}{8057697507}a^{15}-\frac{30405675602}{8057697507}a^{14}-\frac{5643105931}{2685899169}a^{13}-\frac{430700936}{2685899169}a^{12}+\frac{1346378822}{298433241}a^{11}+\frac{246680719}{99477747}a^{10}-\frac{117942400}{99477747}a^{9}+\frac{186070191275}{895299723}a^{8}+\frac{35153514370}{298433241}a^{7}+\frac{4147709531}{298433241}a^{6}+\frac{34704626297}{33159249}a^{5}+\frac{2067988923}{3684361}a^{4}+\frac{792968099}{11053083}a^{3}+\frac{36919174558}{99477747}a^{2}+\frac{6240863216}{33159249}a+\frac{554705416}{33159249}$, $\frac{74193719762}{72519277563}a^{20}+\frac{6363045290}{24173092521}a^{19}-\frac{8560512442}{24173092521}a^{18}-\frac{454419212117}{24173092521}a^{17}-\frac{38944683869}{8057697507}a^{16}+\frac{52409297653}{8057697507}a^{15}-\frac{2836872356014}{8057697507}a^{14}-\frac{243466646608}{2685899169}a^{13}+\frac{327455054015}{2685899169}a^{12}+\frac{146505583126}{298433241}a^{11}+\frac{12448681207}{99477747}a^{10}-\frac{622667826}{3684361}a^{9}+\frac{17568297608473}{895299723}a^{8}+\frac{1507207483342}{298433241}a^{7}-\frac{2027478730235}{298433241}a^{6}+\frac{3136351382707}{33159249}a^{5}+\frac{269699778685}{11053083}a^{4}-\frac{362447366866}{11053083}a^{3}+\frac{555722005835}{99477747}a^{2}+\frac{58003180769}{33159249}a-\frac{72089458804}{33159249}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 763087169026.0542 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{8}\cdot 763087169026.0542 \cdot 1}{2\cdot\sqrt{1279958175800508388164740942598144000000}}\cr\approx \mathstrut & 0.828963085187727 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 - 18*x^18 - 351*x^15 + 351*x^12 + 19359*x^9 + 99630*x^6 + 40095*x^3 + 2187) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 - 18*x^18 - 351*x^15 + 351*x^12 + 19359*x^9 + 99630*x^6 + 40095*x^3 + 2187, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 18*x^18 - 351*x^15 + 351*x^12 + 19359*x^9 + 99630*x^6 + 40095*x^3 + 2187); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - 18*x^18 - 351*x^15 + 351*x^12 + 19359*x^9 + 99630*x^6 + 40095*x^3 + 2187); 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

$C_3:S_7$ (as 21T58):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 15120
The 24 conjugacy class representatives for $C_3:S_7$
Character table for $C_3:S_7$

Intermediate fields

3.1.243.1, 7.5.61291200.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 42 sibling: 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 R $21$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ $21$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ $21$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ $21$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$$[2, 2]^{2}$$
2.6.0.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.8.12.14$x^{8} + 4 x^{7} + 12 x^{6} + 22 x^{5} + 31 x^{4} + 30 x^{3} + 22 x^{2} + 10 x + 9$$4$$2$$12$$D_4$$$[2, 2]^{2}$$
\(3\) Copy content Toggle raw display 3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$$[\frac{5}{2}]_{2}$$
3.15.25.41$x^{15} + 6 x^{11} + 3 x^{10} + 12 x^{7} + 12 x^{6} + 3 x^{5} + 8 x^{3} + 12 x^{2} + 6 x + 4$$3$$5$$25$$S_3 \times C_5$$$[\frac{5}{2}]_{2}^{5}$$
\(5\) Copy content Toggle raw display 5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
5.4.2.2$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.4.2.2$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.4.2.2$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.6.0.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(113\) Copy content Toggle raw display $\Q_{113}$$x + 110$$1$$1$$0$Trivial$$[\ ]$$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$$[\ ]_{2}$$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
113.2.0.1$x^{2} + 101 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$$[\ ]_{2}$$
113.4.2.1$x^{4} + 202 x^{3} + 10207 x^{2} + 606 x + 122$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
113.4.2.1$x^{4} + 202 x^{3} + 10207 x^{2} + 606 x + 122$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)