↑

Properties

Label 15.5.257...000.1
Degree $15$
Signature $(5, 5)$
Discriminant $-2.574\times 10^{21}$
Root discriminant \(26.75\)
Ramified primes $2,5,7$
Class number $1$
Class group trivial
Galois group $F_5 \times S_3$ (as 15T11)

Related objects

Downloads

Learn more

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^15 - 5*x^13 - 5*x^12 - 20*x^11 - 10*x^10 + 30*x^9 + 70*x^8 + 95*x^7 + 110*x^6 + 85*x^5 + 185*x^4 + 45*x^3 + 50*x^2 - 25*x - 5)
 
Copy content gp:K = bnfinit(y^15 - 5*y^13 - 5*y^12 - 20*y^11 - 10*y^10 + 30*y^9 + 70*y^8 + 95*y^7 + 110*y^6 + 85*y^5 + 185*y^4 + 45*y^3 + 50*y^2 - 25*y - 5, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^13 - 5*x^12 - 20*x^11 - 10*x^10 + 30*x^9 + 70*x^8 + 95*x^7 + 110*x^6 + 85*x^5 + 185*x^4 + 45*x^3 + 50*x^2 - 25*x - 5);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 5*x^13 - 5*x^12 - 20*x^11 - 10*x^10 + 30*x^9 + 70*x^8 + 95*x^7 + 110*x^6 + 85*x^5 + 185*x^4 + 45*x^3 + 50*x^2 - 25*x - 5)
 

\( x^{15} - 5 x^{13} - 5 x^{12} - 20 x^{11} - 10 x^{10} + 30 x^{9} + 70 x^{8} + 95 x^{7} + 110 x^{6} + \cdots - 5 \) 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:  $15$
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:  $(5, 5)$
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:   \(-2573571875000000000000\) \(\medspace = -\,2^{12}\cdot 5^{17}\cdot 7^{7}\) 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:  \(26.75\)
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:  $2^{4/5}5^{71/60}7^{1/2}\approx 30.937664227835434$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q(\sqrt{-35}) \)
$\Aut(K/\Q)$:   $C_1$
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 not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{61}a^{12}+\frac{28}{61}a^{11}+\frac{2}{61}a^{10}+\frac{4}{61}a^{9}-\frac{11}{61}a^{8}-\frac{24}{61}a^{7}+\frac{8}{61}a^{6}-\frac{13}{61}a^{5}+\frac{27}{61}a^{4}-\frac{8}{61}a^{3}+\frac{18}{61}a^{2}+\frac{6}{61}a+\frac{8}{61}$, $\frac{1}{61}a^{13}+\frac{11}{61}a^{11}+\frac{9}{61}a^{10}-\frac{1}{61}a^{9}-\frac{21}{61}a^{8}+\frac{9}{61}a^{7}+\frac{7}{61}a^{6}+\frac{25}{61}a^{5}+\frac{29}{61}a^{4}-\frac{2}{61}a^{3}-\frac{10}{61}a^{2}+\frac{23}{61}a+\frac{20}{61}$, $\frac{1}{2300417408251}a^{14}-\frac{8153824941}{2300417408251}a^{13}+\frac{8216104440}{2300417408251}a^{12}-\frac{355227269235}{2300417408251}a^{11}+\frac{714639374387}{2300417408251}a^{10}+\frac{906017418230}{2300417408251}a^{9}+\frac{729559807489}{2300417408251}a^{8}+\frac{235185039888}{2300417408251}a^{7}-\frac{191385374245}{2300417408251}a^{6}+\frac{75252918842}{2300417408251}a^{5}-\frac{848989390883}{2300417408251}a^{4}-\frac{1042558015600}{2300417408251}a^{3}+\frac{655747374462}{2300417408251}a^{2}+\frac{821301266470}{2300417408251}a-\frac{426109395622}{2300417408251}$ 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 $1$
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 $1$
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:  $9$
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:   $\frac{25257214698}{2300417408251}a^{14}+\frac{15671373854}{2300417408251}a^{13}-\frac{132296700598}{2300417408251}a^{12}-\frac{217886872460}{2300417408251}a^{11}-\frac{522979655675}{2300417408251}a^{10}-\frac{501053620141}{2300417408251}a^{9}+\frac{688311446427}{2300417408251}a^{8}+\frac{2509268467651}{2300417408251}a^{7}+\frac{2744093420994}{2300417408251}a^{6}+\frac{3750625875172}{2300417408251}a^{5}+\frac{2672207490081}{2300417408251}a^{4}+\frac{6297021509710}{2300417408251}a^{3}+\frac{4115776702877}{2300417408251}a^{2}+\frac{1367220488614}{2300417408251}a-\frac{921261117459}{2300417408251}$, $\frac{34319669966}{2300417408251}a^{14}-\frac{26741133967}{2300417408251}a^{13}-\frac{139915499017}{2300417408251}a^{12}-\frac{36667827054}{2300417408251}a^{11}-\frac{749695888784}{2300417408251}a^{10}+\frac{62727034981}{2300417408251}a^{9}+\frac{882945105251}{2300417408251}a^{8}+\frac{1253942828775}{2300417408251}a^{7}+\frac{2759237524016}{2300417408251}a^{6}+\frac{3176966399049}{2300417408251}a^{5}+\frac{960433329162}{2300417408251}a^{4}+\frac{5807253280544}{2300417408251}a^{3}-\frac{433436027499}{2300417408251}a^{2}+\frac{6300200017922}{2300417408251}a+\frac{791770979729}{2300417408251}$, $\frac{44356613}{7396840541}a^{14}-\frac{168602701}{7396840541}a^{13}-\frac{176561524}{7396840541}a^{12}+\frac{610455842}{7396840541}a^{11}-\frac{284087940}{7396840541}a^{10}+\frac{2795627224}{7396840541}a^{9}+\frac{2205352878}{7396840541}a^{8}-\frac{2134634750}{7396840541}a^{7}-\frac{6144238165}{7396840541}a^{6}-\frac{9752221789}{7396840541}a^{5}-\frac{10917264285}{7396840541}a^{4}-\frac{3266260180}{7396840541}a^{3}-\frac{24914701595}{7396840541}a^{2}+\frac{6494535395}{7396840541}a-\frac{443863556}{7396840541}$, $\frac{94943173293}{2300417408251}a^{14}-\frac{124034292381}{2300417408251}a^{13}-\frac{446003096817}{2300417408251}a^{12}+\frac{211376670523}{2300417408251}a^{11}-\frac{1463456086506}{2300417408251}a^{10}+\frac{1051981407779}{2300417408251}a^{9}+\frac{3429485328445}{2300417408251}a^{8}+\frac{1522733376101}{2300417408251}a^{7}+\frac{1305095959759}{2300417408251}a^{6}+\frac{2988780459938}{2300417408251}a^{5}-\frac{97775108631}{2300417408251}a^{4}+\frac{13258037477566}{2300417408251}a^{3}-\frac{12910785578483}{2300417408251}a^{2}+\frac{5634451964605}{2300417408251}a+\frac{2164413490919}{2300417408251}$, $\frac{165579762046}{2300417408251}a^{14}-\frac{48224656529}{2300417408251}a^{13}-\frac{803598314165}{2300417408251}a^{12}-\frac{566334681153}{2300417408251}a^{11}-\frac{3223735489765}{2300417408251}a^{10}-\frac{895628135714}{2300417408251}a^{9}+\frac{82077326593}{37711760791}a^{8}+\frac{9513347737225}{2300417408251}a^{7}+\frac{13436721745945}{2300417408251}a^{6}+\frac{16252122015774}{2300417408251}a^{5}+\frac{11900880982400}{2300417408251}a^{4}+\frac{30038881721349}{2300417408251}a^{3}+\frac{2328489156621}{2300417408251}a^{2}+\frac{7460884061550}{2300417408251}a-\frac{5356111345069}{2300417408251}$, $\frac{69676459011}{2300417408251}a^{14}+\frac{35860070187}{2300417408251}a^{13}-\frac{345654028685}{2300417408251}a^{12}-\frac{602272940313}{2300417408251}a^{11}-\frac{1563214559159}{2300417408251}a^{10}-\frac{988821206035}{2300417408251}a^{9}+\frac{1827364814908}{2300417408251}a^{8}+\frac{6998222128428}{2300417408251}a^{7}+\frac{9852496453142}{2300417408251}a^{6}+\frac{7715440029676}{2300417408251}a^{5}+\frac{5912497230005}{2300417408251}a^{4}+\frac{13747160619388}{2300417408251}a^{3}+\frac{3108748605691}{2300417408251}a^{2}+\frac{2307519558352}{2300417408251}a-\frac{2751124949239}{2300417408251}$, $\frac{31488459756}{2300417408251}a^{14}-\frac{11016956744}{2300417408251}a^{13}-\frac{155839655327}{2300417408251}a^{12}-\frac{101352420164}{2300417408251}a^{11}-\frac{574781858322}{2300417408251}a^{10}-\frac{112847074309}{2300417408251}a^{9}+\frac{998017259095}{2300417408251}a^{8}+\frac{1721946324544}{2300417408251}a^{7}+\frac{2162601337383}{2300417408251}a^{6}+\frac{2749267877748}{2300417408251}a^{5}+\frac{2298480323276}{2300417408251}a^{4}+\frac{5635656543005}{2300417408251}a^{3}+\frac{421479473848}{2300417408251}a^{2}+\frac{1131493635786}{2300417408251}a-\frac{1701567419099}{2300417408251}$, $\frac{28712175850}{2300417408251}a^{14}+\frac{53318956093}{2300417408251}a^{13}-\frac{158366574792}{2300417408251}a^{12}-\frac{428887802681}{2300417408251}a^{11}-\frac{728354883227}{2300417408251}a^{10}-\frac{1166001727066}{2300417408251}a^{9}+\frac{535338077219}{2300417408251}a^{8}+\frac{3889541103810}{2300417408251}a^{7}+\frac{5374851338175}{2300417408251}a^{6}+\frac{5718276941566}{2300417408251}a^{5}+\frac{6164130032731}{2300417408251}a^{4}+\frac{7741943631830}{2300417408251}a^{3}+\frac{8658619045313}{2300417408251}a^{2}+\frac{1643362475205}{2300417408251}a-\frac{624003437039}{2300417408251}$, $\frac{43319978665}{2300417408251}a^{14}+\frac{112777966834}{2300417408251}a^{13}-\frac{240079010704}{2300417408251}a^{12}-\frac{934049383164}{2300417408251}a^{11}-\frac{1113860919363}{2300417408251}a^{10}-\frac{1772197450842}{2300417408251}a^{9}+\frac{67152625303}{2300417408251}a^{8}+\frac{8834881900830}{2300417408251}a^{7}+\frac{10769088946203}{2300417408251}a^{6}+\frac{6276448415391}{2300417408251}a^{5}+\frac{12906693988394}{2300417408251}a^{4}+\frac{16489915896477}{2300417408251}a^{3}+\frac{113271441281}{37711760791}a^{2}+\frac{4836821616661}{2300417408251}a-\frac{2673996958934}{2300417408251}$ Copy content Toggle raw display
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:  \( 184544.05094 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 5 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{5}\cdot 184544.05094 \cdot 1}{2\cdot\sqrt{2573571875000000000000}}\cr\approx \mathstrut & 0.56996898381 \end{aligned}\]

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^15 - 5*x^13 - 5*x^12 - 20*x^11 - 10*x^10 + 30*x^9 + 70*x^8 + 95*x^7 + 110*x^6 + 85*x^5 + 185*x^4 + 45*x^3 + 50*x^2 - 25*x - 5) 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^15 - 5*x^13 - 5*x^12 - 20*x^11 - 10*x^10 + 30*x^9 + 70*x^8 + 95*x^7 + 110*x^6 + 85*x^5 + 185*x^4 + 45*x^3 + 50*x^2 - 25*x - 5, 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^15 - 5*x^13 - 5*x^12 - 20*x^11 - 10*x^10 + 30*x^9 + 70*x^8 + 95*x^7 + 110*x^6 + 85*x^5 + 185*x^4 + 45*x^3 + 50*x^2 - 25*x - 5); 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^15 - 5*x^13 - 5*x^12 - 20*x^11 - 10*x^10 + 30*x^9 + 70*x^8 + 95*x^7 + 110*x^6 + 85*x^5 + 185*x^4 + 45*x^3 + 50*x^2 - 25*x - 5); 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

$S_3\times F_5$ (as 15T11):

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 solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.175.1, 5.5.2450000.1

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 30 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 ${\href{/padicField/3.4.0.1}{4} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ R R $15$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$

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 2.3.5.12a1.1$x^{15} + 5 x^{13} + 5 x^{12} + 10 x^{11} + 20 x^{10} + 20 x^{9} + 30 x^{8} + 35 x^{7} + 30 x^{6} + 31 x^{5} + 25 x^{4} + 15 x^{3} + 10 x^{2} + 5 x + 3$$5$$3$$12$$F_5\times C_3$$$[\ ]_{5}^{12}$$
\(5\) Copy content Toggle raw display 5.1.15.17a1.4$x^{15} + 20 x^{3} + 5$$15$$1$$17$$F_5 \times S_3$$$[\frac{5}{4}]_{12}^{2}$$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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