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Properties

Label 20.4.401...000.1
Degree $20$
Signature $[4, 8]$
Discriminant $4.014\times 10^{33}$
Root discriminant \(47.88\)
Ramified primes $2,5,19,1699$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.D_5\wr C_2$ (as 20T760)

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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^20 + 26*x^18 + 250*x^16 + 1067*x^14 + 1588*x^12 - 2479*x^10 - 11776*x^8 - 14727*x^6 - 6438*x^4 + 38*x^2 + 361)
 
Copy content gp:K = bnfinit(y^20 + 26*y^18 + 250*y^16 + 1067*y^14 + 1588*y^12 - 2479*y^10 - 11776*y^8 - 14727*y^6 - 6438*y^4 + 38*y^2 + 361, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 26*x^18 + 250*x^16 + 1067*x^14 + 1588*x^12 - 2479*x^10 - 11776*x^8 - 14727*x^6 - 6438*x^4 + 38*x^2 + 361);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 26*x^18 + 250*x^16 + 1067*x^14 + 1588*x^12 - 2479*x^10 - 11776*x^8 - 14727*x^6 - 6438*x^4 + 38*x^2 + 361)
 

\( x^{20} + 26 x^{18} + 250 x^{16} + 1067 x^{14} + 1588 x^{12} - 2479 x^{10} - 11776 x^{8} - 14727 x^{6} + \cdots + 361 \) 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:  $20$
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:  $[4, 8]$
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:   \(4014163686424571853629440000000000\) \(\medspace = 2^{20}\cdot 5^{10}\cdot 19^{6}\cdot 1699^{4}\) 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:  \(47.88\)
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:   \(2\), \(5\), \(19\), \(1699\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{396336765148673}a^{18}-\frac{2636541828650}{396336765148673}a^{16}-\frac{83105492813617}{396336765148673}a^{14}-\frac{24005712035949}{396336765148673}a^{12}+\frac{157716436612237}{396336765148673}a^{10}-\frac{68680251329648}{396336765148673}a^{8}+\frac{72639693679119}{396336765148673}a^{6}+\frac{79908139679898}{396336765148673}a^{4}+\frac{158374222590161}{396336765148673}a^{2}+\frac{8893071598967}{20859829744667}$, $\frac{1}{396336765148673}a^{19}-\frac{2636541828650}{396336765148673}a^{17}-\frac{83105492813617}{396336765148673}a^{15}-\frac{24005712035949}{396336765148673}a^{13}+\frac{157716436612237}{396336765148673}a^{11}-\frac{68680251329648}{396336765148673}a^{9}+\frac{72639693679119}{396336765148673}a^{7}+\frac{79908139679898}{396336765148673}a^{5}+\frac{158374222590161}{396336765148673}a^{3}+\frac{8893071598967}{20859829744667}a$ 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:  $C_{2}\times C_{2}$, which has order $4$ (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:  $11$
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:   $\frac{2829904}{245541883}a^{18}+\frac{69795953}{245541883}a^{16}+\frac{614868062}{245541883}a^{14}+\frac{2214317531}{245541883}a^{12}+\frac{1680273198}{245541883}a^{10}-\frac{8754796095}{245541883}a^{8}-\frac{21398354434}{245541883}a^{6}-\frac{15436714210}{245541883}a^{4}-\frac{1830561005}{245541883}a^{2}+\frac{37360538}{12923257}$, $\frac{8008289538745}{396336765148673}a^{18}+\frac{193484355401218}{396336765148673}a^{16}+\frac{16\cdots 47}{396336765148673}a^{14}+\frac{54\cdots 73}{396336765148673}a^{12}+\frac{22\cdots 46}{396336765148673}a^{10}-\frac{25\cdots 21}{396336765148673}a^{8}-\frac{48\cdots 63}{396336765148673}a^{6}-\frac{23\cdots 66}{396336765148673}a^{4}+\frac{21\cdots 50}{396336765148673}a^{2}+\frac{80192588580589}{20859829744667}$, $\frac{2474023273527}{396336765148673}a^{18}+\frac{59170240446863}{396336765148673}a^{16}+\frac{495531711699290}{396336765148673}a^{14}+\frac{16\cdots 42}{396336765148673}a^{12}+\frac{606830466765102}{396336765148673}a^{10}-\frac{73\cdots 82}{396336765148673}a^{8}-\frac{14\cdots 84}{396336765148673}a^{6}-\frac{77\cdots 78}{396336765148673}a^{4}+\frac{266026269252461}{396336765148673}a^{2}+\frac{49538125591440}{20859829744667}$, $\frac{2800984729529}{396336765148673}a^{18}+\frac{70624218262711}{396336765148673}a^{16}+\frac{644993371848368}{396336765148673}a^{14}+\frac{24\cdots 44}{396336765148673}a^{12}+\frac{25\cdots 46}{396336765148673}a^{10}-\frac{88\cdots 10}{396336765148673}a^{8}-\frac{26\cdots 00}{396336765148673}a^{6}-\frac{21\cdots 40}{396336765148673}a^{4}-\frac{20\cdots 92}{396336765148673}a^{2}+\frac{102261076217997}{20859829744667}$, $\frac{3729541911129}{396336765148673}a^{18}+\frac{93474401554976}{396336765148673}a^{16}+\frac{845214112108156}{396336765148673}a^{14}+\frac{31\cdots 79}{396336765148673}a^{12}+\frac{30\cdots 62}{396336765148673}a^{10}-\frac{11\cdots 80}{396336765148673}a^{8}-\frac{32\cdots 80}{396336765148673}a^{6}-\frac{25\cdots 69}{396336765148673}a^{4}-\frac{34\cdots 41}{396336765148673}a^{2}+\frac{49916737117560}{20859829744667}$, $\frac{2637504001528}{396336765148673}a^{18}+\frac{64897229354787}{396336765148673}a^{16}+\frac{570262541773829}{396336765148673}a^{14}+\frac{20\cdots 93}{396336765148673}a^{12}+\frac{15\cdots 24}{396336765148673}a^{10}-\frac{80\cdots 96}{396336765148673}a^{8}-\frac{20\cdots 42}{396336765148673}a^{6}-\frac{14\cdots 09}{396336765148673}a^{4}-\frac{678233680818929}{396336765148673}a^{2}+\frac{107189345521719}{20859829744667}$, $\frac{4485725705619}{396336765148673}a^{18}+\frac{111794386843450}{396336765148673}a^{16}+\frac{10\cdots 51}{396336765148673}a^{14}+\frac{37\cdots 16}{396336765148673}a^{12}+\frac{31\cdots 67}{396336765148673}a^{10}-\frac{14\cdots 65}{396336765148673}a^{8}-\frac{37\cdots 01}{396336765148673}a^{6}-\frac{26\cdots 51}{396336765148673}a^{4}-\frac{488095885003577}{396336765148673}a^{2}+\frac{105475753390074}{20859829744667}$, $\frac{3370881320007}{396336765148673}a^{18}+\frac{85495004598814}{396336765148673}a^{16}+\frac{786956446154242}{396336765148673}a^{14}+\frac{30\cdots 15}{396336765148673}a^{12}+\frac{32\cdots 00}{396336765148673}a^{10}-\frac{10\cdots 76}{396336765148673}a^{8}-\frac{31\cdots 85}{396336765148673}a^{6}-\frac{27\cdots 32}{396336765148673}a^{4}-\frac{52\cdots 75}{396336765148673}a^{2}+\frac{90005092394930}{20859829744667}$, $\frac{2838198079505}{396336765148673}a^{18}+\frac{62277531304484}{396336765148673}a^{16}+\frac{434288317423642}{396336765148673}a^{14}+\frac{731621706228947}{396336765148673}a^{12}-\frac{28\cdots 02}{396336765148673}a^{10}-\frac{89\cdots 14}{396336765148673}a^{8}+\frac{19\cdots 90}{396336765148673}a^{6}+\frac{19\cdots 34}{396336765148673}a^{4}+\frac{85\cdots 40}{396336765148673}a^{2}-\frac{106613783891142}{20859829744667}$, $\frac{31\cdots 24}{396336765148673}a^{19}+\frac{45\cdots 88}{396336765148673}a^{18}+\frac{87\cdots 96}{396336765148673}a^{17}+\frac{12\cdots 00}{396336765148673}a^{16}+\frac{96\cdots 02}{396336765148673}a^{15}+\frac{14\cdots 88}{396336765148673}a^{14}+\frac{54\cdots 98}{396336765148673}a^{13}+\frac{79\cdots 70}{396336765148673}a^{12}+\frac{16\cdots 60}{396336765148673}a^{11}+\frac{24\cdots 84}{396336765148673}a^{10}+\frac{28\cdots 48}{396336765148673}a^{9}+\frac{42\cdots 20}{396336765148673}a^{8}+\frac{25\cdots 44}{396336765148673}a^{7}+\frac{37\cdots 88}{396336765148673}a^{6}+\frac{91\cdots 50}{396336765148673}a^{5}+\frac{13\cdots 40}{396336765148673}a^{4}-\frac{29\cdots 18}{396336765148673}a^{3}-\frac{43\cdots 08}{396336765148673}a^{2}-\frac{27\cdots 92}{20859829744667}a-\frac{40\cdots 47}{20859829744667}$, $\frac{34\cdots 65}{396336765148673}a^{19}-\frac{15\cdots 09}{396336765148673}a^{18}+\frac{91\cdots 30}{396336765148673}a^{17}-\frac{40\cdots 92}{396336765148673}a^{16}+\frac{88\cdots 77}{396336765148673}a^{15}-\frac{39\cdots 50}{396336765148673}a^{14}+\frac{38\cdots 53}{396336765148673}a^{13}-\frac{17\cdots 08}{396336765148673}a^{12}+\frac{62\cdots 63}{396336765148673}a^{11}-\frac{27\cdots 06}{396336765148673}a^{10}-\frac{74\cdots 24}{396336765148673}a^{9}+\frac{32\cdots 94}{396336765148673}a^{8}-\frac{42\cdots 23}{396336765148673}a^{7}+\frac{18\cdots 94}{396336765148673}a^{6}-\frac{59\cdots 46}{396336765148673}a^{5}+\frac{26\cdots 09}{396336765148673}a^{4}-\frac{34\cdots 04}{396336765148673}a^{3}+\frac{14\cdots 05}{396336765148673}a^{2}-\frac{34\cdots 20}{20859829744667}a+\frac{15\cdots 12}{20859829744667}$ 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:  \( 531384738.658 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 531384738.658 \cdot 1}{2\cdot\sqrt{4014163686424571853629440000000000}}\cr\approx \mathstrut & 0.162982311266 \end{aligned}\] (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^20 + 26*x^18 + 250*x^16 + 1067*x^14 + 1588*x^12 - 2479*x^10 - 11776*x^8 - 14727*x^6 - 6438*x^4 + 38*x^2 + 361) 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^20 + 26*x^18 + 250*x^16 + 1067*x^14 + 1588*x^12 - 2479*x^10 - 11776*x^8 - 14727*x^6 - 6438*x^4 + 38*x^2 + 361, 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^20 + 26*x^18 + 250*x^16 + 1067*x^14 + 1588*x^12 - 2479*x^10 - 11776*x^8 - 14727*x^6 - 6438*x^4 + 38*x^2 + 361); 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^20 + 26*x^18 + 250*x^16 + 1067*x^14 + 1588*x^12 - 2479*x^10 - 11776*x^8 - 14727*x^6 - 6438*x^4 + 38*x^2 + 361); 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

$C_2^9.D_5\wr C_2$ (as 20T760):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
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 solvable group of order 102400
The 130 conjugacy class representatives for $C_2^9.D_5\wr C_2$
Character table for $C_2^9.D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.4.358950742275544609437706240000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ R ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ R ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{7}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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 2.10.2.20a18.2$x^{20} + 2 x^{17} + 2 x^{16} + 2 x^{15} + 4 x^{13} + 5 x^{12} + 4 x^{11} + 5 x^{10} + 4 x^{9} + 6 x^{8} + 6 x^{7} + 5 x^{6} + 4 x^{5} + 3 x^{4} + 4 x^{3} + 7 x^{2} + 2 x + 3$$2$$10$$20$20T340not computed
\(5\) Copy content Toggle raw display 5.5.2.5a1.2$x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 24 x + 14$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
5.5.2.5a1.2$x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 24 x + 14$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(19\) Copy content Toggle raw display 19.1.2.1a1.1$x^{2} + 19$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
19.4.1.0a1.1$x^{4} + 2 x^{2} + 11 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(1699\) Copy content Toggle raw display $\Q_{1699}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1699}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1699}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1699}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $4$$2$$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)