Properties

Label 20.0.148...000.1
Degree $20$
Signature $(0, 10)$
Discriminant $1.482\times 10^{38}$
Root discriminant \(81.01\)
Ramified primes $2,3,5,173$
Class number $1968$ (GRH)
Class group [2, 2, 492] (GRH)
Galois group $D_5^2.D_4$ (as 20T157)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 + 78*x^18 + 2331*x^16 + 34488*x^14 + 274536*x^12 + 1203552*x^10 + 2835648*x^8 + 3302208*x^6 + 1586304*x^4 + 279936*x^2 + 15552)
 
Copy content gp:K = bnfinit(y^20 + 78*y^18 + 2331*y^16 + 34488*y^14 + 274536*y^12 + 1203552*y^10 + 2835648*y^8 + 3302208*y^6 + 1586304*y^4 + 279936*y^2 + 15552, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 78*x^18 + 2331*x^16 + 34488*x^14 + 274536*x^12 + 1203552*x^10 + 2835648*x^8 + 3302208*x^6 + 1586304*x^4 + 279936*x^2 + 15552);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 + 78*x^18 + 2331*x^16 + 34488*x^14 + 274536*x^12 + 1203552*x^10 + 2835648*x^8 + 3302208*x^6 + 1586304*x^4 + 279936*x^2 + 15552)
 

\( x^{20} + 78 x^{18} + 2331 x^{16} + 34488 x^{14} + 274536 x^{12} + 1203552 x^{10} + 2835648 x^{8} + \cdots + 15552 \) 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:  $(0, 10)$
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:   \(148184566209122179801351864839045120000\) \(\medspace = 2^{64}\cdot 3^{15}\cdot 5^{4}\cdot 173^{4}\) 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:  \(81.01\)
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:  $2^{27/8}3^{3/4}5^{1/2}173^{1/2}\approx 695.5449954322967$
Ramified primes:   \(2\), \(3\), \(5\), \(173\) 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:  \(\Q(\sqrt{3}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
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 a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{6}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{18}a^{8}-\frac{1}{6}a^{4}$, $\frac{1}{18}a^{9}-\frac{1}{6}a^{5}$, $\frac{1}{36}a^{10}+\frac{1}{12}a^{6}-\frac{1}{6}a^{4}$, $\frac{1}{36}a^{11}-\frac{1}{12}a^{7}-\frac{1}{6}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{864}a^{12}-\frac{1}{144}a^{10}+\frac{5}{288}a^{8}+\frac{1}{24}a^{6}+\frac{1}{24}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{1728}a^{13}+\frac{1}{96}a^{11}-\frac{11}{576}a^{9}-\frac{1}{36}a^{8}+\frac{1}{16}a^{7}+\frac{1}{48}a^{5}-\frac{1}{12}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{8}a$, $\frac{1}{1728}a^{14}-\frac{7}{576}a^{10}-\frac{1}{36}a^{9}+\frac{5}{288}a^{8}+\frac{7}{48}a^{6}-\frac{1}{12}a^{5}+\frac{1}{24}a^{4}-\frac{1}{2}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}$, $\frac{1}{1728}a^{15}-\frac{7}{576}a^{11}+\frac{5}{288}a^{9}-\frac{1}{48}a^{7}+\frac{1}{24}a^{5}-\frac{1}{8}a^{3}+\frac{1}{4}a$, $\frac{1}{51840}a^{16}-\frac{1}{8640}a^{14}-\frac{1}{5760}a^{12}-\frac{1}{72}a^{11}+\frac{13}{1440}a^{10}-\frac{5}{288}a^{8}-\frac{1}{24}a^{7}+\frac{1}{8}a^{6}-\frac{1}{12}a^{5}-\frac{37}{240}a^{4}-\frac{1}{2}a^{3}-\frac{1}{10}a^{2}-\frac{3}{10}$, $\frac{1}{51840}a^{17}-\frac{1}{8640}a^{15}-\frac{1}{5760}a^{13}+\frac{13}{1440}a^{11}-\frac{5}{288}a^{9}-\frac{1}{24}a^{7}-\frac{37}{240}a^{5}+\frac{2}{5}a^{3}-\frac{3}{10}a$, $\frac{1}{12621585265920}a^{18}-\frac{10556197}{1577698158240}a^{16}-\frac{73008181}{1402398362880}a^{14}+\frac{374646151}{701199181440}a^{12}-\frac{649574467}{70119918144}a^{10}-\frac{180849059}{8764989768}a^{8}+\frac{1279182043}{58433265120}a^{6}-\frac{1}{6}a^{5}-\frac{3865135997}{29216632560}a^{4}-\frac{730280243}{2434719380}a^{2}-\frac{1}{2}a-\frac{52967555}{486943876}$, $\frac{1}{12621585265920}a^{19}-\frac{10556197}{1577698158240}a^{17}-\frac{73008181}{1402398362880}a^{15}-\frac{93421237}{2103597544320}a^{13}+\frac{63087247}{7791102016}a^{11}-\frac{107696813}{70119918144}a^{9}-\frac{1}{36}a^{8}+\frac{2496541733}{58433265120}a^{7}-\frac{1}{6}a^{6}+\frac{65937153}{4869438760}a^{5}-\frac{1}{12}a^{4}-\frac{60800199}{1217359690}a^{3}+\frac{15800859}{973887752}a$ 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:  $C_{2}\times C_{2}\times C_{492}$, which has order $1968$ (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:  $C_{2}\times C_{2}\times C_{492}$, which has order $1968$ (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);
 
Relative class number:   $1968$ (assuming GRH)

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:  $9$
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:   $\frac{5175623}{87649897680}a^{18}+\frac{7272657271}{1577698158240}a^{16}+\frac{36279943849}{262949693040}a^{14}+\frac{538375006787}{262949693040}a^{12}+\frac{71830329317}{4382494884}a^{10}+\frac{2544089074759}{35059959072}a^{8}+\frac{844170875883}{4869438760}a^{6}+\frac{2975673860093}{14608316280}a^{4}+\frac{57163696256}{608679845}a^{2}+\frac{6486237445}{486943876}$, $\frac{372104101}{6310792632960}a^{18}+\frac{3570447007}{788849079120}a^{16}+\frac{92603562751}{701199181440}a^{14}+\frac{396810431323}{210359754432}a^{12}+\frac{2504959485061}{175299795360}a^{10}+\frac{2058804198323}{35059959072}a^{8}+\frac{1241699182291}{9738877520}a^{6}+\frac{321408022871}{2434719380}a^{4}+\frac{12147584677}{243471938}a^{2}+\frac{11424282723}{2434719380}$, $\frac{1076405}{26854436736}a^{18}+\frac{207923363}{67136091840}a^{16}+\frac{1361610431}{14919131520}a^{14}+\frac{7401798949}{5594674320}a^{12}+\frac{38166318463}{3729782880}a^{10}+\frac{4022088665}{93244572}a^{8}+\frac{3974954885}{41442032}a^{6}+\frac{15630911167}{155407620}a^{4}+\frac{945361349}{25901270}a^{2}+\frac{35507451}{12950635}$, $\frac{164744489}{2103597544320}a^{18}+\frac{958137197}{157769815824}a^{16}+\frac{75677938111}{420719508864}a^{14}+\frac{2761547515753}{1051798772160}a^{12}+\frac{1197070836667}{58433265120}a^{10}+\frac{1020732155387}{11686653024}a^{8}+\frac{5798942623171}{29216632560}a^{6}+\frac{1616540839147}{7304158140}a^{4}+\frac{60078973261}{608679845}a^{2}+\frac{28718392953}{2434719380}$, $\frac{58825673}{1051798772160}a^{18}+\frac{13709637251}{3155396316480}a^{16}+\frac{135675213493}{1051798772160}a^{14}+\frac{1985416849849}{1051798772160}a^{12}+\frac{32305019885}{2191247442}a^{10}+\frac{45607999075}{730415814}a^{8}+\frac{669514232379}{4869438760}a^{6}+\frac{2015386582969}{14608316280}a^{4}+\frac{27908284273}{608679845}a^{2}+\frac{2196797879}{243471938}$, $\frac{958033}{7459565760}a^{18}+\frac{222805157}{22378697280}a^{16}+\frac{6595689353}{22378697280}a^{14}+\frac{32053334591}{7459565760}a^{12}+\frac{41553912341}{1243260960}a^{10}+\frac{26313513787}{186489144}a^{8}+\frac{12034493792}{38851905}a^{6}+\frac{94816243661}{310815240}a^{4}+\frac{4447432229}{51802540}a^{2}+\frac{161054027}{25901270}$, $\frac{319921}{29216632560}a^{18}+\frac{2549399089}{3155396316480}a^{16}+\frac{11626104331}{525899386080}a^{14}+\frac{99393104099}{350599590720}a^{12}+\frac{78056009723}{43824948840}a^{10}+\frac{5068149193}{973887752}a^{8}+\frac{3173917066}{608679845}a^{6}-\frac{46701650071}{14608316280}a^{4}-\frac{7384988017}{1217359690}a^{2}-\frac{844172407}{1217359690}$, $\frac{496403683}{1262158526592}a^{18}+\frac{96597040007}{3155396316480}a^{16}+\frac{213132109363}{233733060480}a^{14}+\frac{7057756673417}{525899386080}a^{12}+\frac{2063084526543}{19477755040}a^{10}+\frac{8025922963963}{17529979536}a^{8}+\frac{2049675094015}{1947775504}a^{6}+\frac{8513776360613}{7304158140}a^{4}+\frac{297246889753}{608679845}a^{2}+\frac{67568682953}{1217359690}$, $\frac{900086149}{6310792632960}a^{18}+\frac{17557803713}{1577698158240}a^{16}+\frac{699629298581}{2103597544320}a^{14}+\frac{1723268350031}{350599590720}a^{12}+\frac{2276415780551}{58433265120}a^{10}+\frac{5917364816651}{35059959072}a^{8}+\frac{11312082370717}{29216632560}a^{6}+\frac{1560883489439}{3652079070}a^{4}+\frac{221681635377}{1217359690}a^{2}+\frac{51086027799}{2434719380}$ 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:  \( 255930231.626 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 255930231.626 \cdot 1968}{2\cdot\sqrt{148184566209122179801351864839045120000}}\cr\approx \mathstrut & 1.9838730838 \end{aligned}\] (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 + 78*x^18 + 2331*x^16 + 34488*x^14 + 274536*x^12 + 1203552*x^10 + 2835648*x^8 + 3302208*x^6 + 1586304*x^4 + 279936*x^2 + 15552) 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 + 78*x^18 + 2331*x^16 + 34488*x^14 + 274536*x^12 + 1203552*x^10 + 2835648*x^8 + 3302208*x^6 + 1586304*x^4 + 279936*x^2 + 15552, 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 + 78*x^18 + 2331*x^16 + 34488*x^14 + 274536*x^12 + 1203552*x^10 + 2835648*x^8 + 3302208*x^6 + 1586304*x^4 + 279936*x^2 + 15552); 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 + 78*x^18 + 2331*x^16 + 34488*x^14 + 274536*x^12 + 1203552*x^10 + 2835648*x^8 + 3302208*x^6 + 1586304*x^4 + 279936*x^2 + 15552); 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

$D_5^2.D_4$ (as 20T157):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 800
The 26 conjugacy class representatives for $D_5^2.D_4$
Character table for $D_5^2.D_4$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3 + \sqrt{6}})\), 10.10.24403289466470400.1

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
Arithmetically equivalent 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 R $20$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $20$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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 2.1.4.10a1.8$x^{4} + 4 x^{3} + 12 x^{2} + 8 x + 10$$4$$1$$10$$D_{4}$$$[2, 3, \frac{7}{2}]$$
2.1.16.54o1.297$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$$16$$1$$54$$C_4:C_4$$$[2, 3, \frac{7}{2}, 4]$$
\(3\) Copy content Toggle raw display 3.1.4.3a1.1$x^{4} + 3$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
3.4.4.12a1.3$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
5.2.2.2a1.1$x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
\(173\) Copy content Toggle raw display $\Q_{173}$$x + 171$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{173}$$x + 171$$1$$1$$0$Trivial$$[\ ]$$
173.2.1.0a1.1$x^{2} + 169 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
173.2.2.2a1.1$x^{4} + 338 x^{3} + 28565 x^{2} + 849 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
173.4.1.0a1.1$x^{4} + x^{2} + 102 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
173.4.1.0a1.1$x^{4} + x^{2} + 102 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
173.2.2.2a1.1$x^{4} + 338 x^{3} + 28565 x^{2} + 849 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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