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Properties

Label 14.2.282...000.1
Degree $14$
Signature $[2, 6]$
Discriminant $2.823\times 10^{26}$
Root discriminant \(77.51\)
Ramified primes $2,3,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $A_{14}$ (as 14T62)

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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^14 - 3*x^13 - 5*x^12 + 31*x^11 - 50*x^10 + 127*x^9 - 55*x^8 + 1003*x^7 - 5194*x^6 + 13043*x^5 + 6647*x^4 - 129703*x^3 + 382419*x^2 - 490216*x + 319344)
 
Copy content gp:K = bnfinit(y^14 - 3*y^13 - 5*y^12 + 31*y^11 - 50*y^10 + 127*y^9 - 55*y^8 + 1003*y^7 - 5194*y^6 + 13043*y^5 + 6647*y^4 - 129703*y^3 + 382419*y^2 - 490216*y + 319344, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 - 5*x^12 + 31*x^11 - 50*x^10 + 127*x^9 - 55*x^8 + 1003*x^7 - 5194*x^6 + 13043*x^5 + 6647*x^4 - 129703*x^3 + 382419*x^2 - 490216*x + 319344);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 3*x^13 - 5*x^12 + 31*x^11 - 50*x^10 + 127*x^9 - 55*x^8 + 1003*x^7 - 5194*x^6 + 13043*x^5 + 6647*x^4 - 129703*x^3 + 382419*x^2 - 490216*x + 319344)
 

\( x^{14} - 3 x^{13} - 5 x^{12} + 31 x^{11} - 50 x^{10} + 127 x^{9} - 55 x^{8} + 1003 x^{7} - 5194 x^{6} + \cdots + 319344 \) 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:  $14$
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:  $[2, 6]$
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:   \(282346132215490560000000000\) \(\medspace = 2^{20}\cdot 3^{14}\cdot 5^{10}\cdot 7^{8}\) 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:  \(77.51\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{11/4}3^{121/54}5^{23/24}7^{19/20}\approx 2342.2158414001506$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{63}a^{12}-\frac{2}{21}a^{11}+\frac{2}{63}a^{10}+\frac{4}{9}a^{9}-\frac{10}{21}a^{8}-\frac{4}{9}a^{7}-\frac{19}{63}a^{6}-\frac{2}{7}a^{5}-\frac{17}{63}a^{4}-\frac{1}{63}a^{3}-\frac{10}{21}a^{2}-\frac{11}{63}a-\frac{2}{21}$, $\frac{1}{58\cdots 20}a^{13}+\frac{83\cdots 47}{19\cdots 40}a^{12}-\frac{28\cdots 83}{83\cdots 60}a^{11}+\frac{68\cdots 07}{58\cdots 20}a^{10}+\frac{45\cdots 61}{32\cdots 40}a^{9}-\frac{12\cdots 01}{58\cdots 20}a^{8}-\frac{67\cdots 19}{58\cdots 20}a^{7}+\frac{53\cdots 49}{19\cdots 40}a^{6}-\frac{15\cdots 03}{29\cdots 60}a^{5}-\frac{38\cdots 01}{58\cdots 20}a^{4}+\frac{13\cdots 63}{27\cdots 20}a^{3}+\frac{37\cdots 27}{83\cdots 60}a^{2}+\frac{35\cdots 09}{38\cdots 08}a-\frac{22\cdots 21}{16\cdots 20}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $7$
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:   $\frac{14\cdots 37}{58\cdots 20}a^{13}+\frac{35\cdots 59}{19\cdots 40}a^{12}+\frac{31\cdots 29}{83\cdots 60}a^{11}+\frac{25\cdots 39}{58\cdots 20}a^{10}+\frac{88\cdots 51}{96\cdots 20}a^{9}-\frac{21\cdots 57}{58\cdots 20}a^{8}+\frac{10\cdots 17}{58\cdots 20}a^{7}+\frac{10\cdots 71}{64\cdots 80}a^{6}+\frac{17\cdots 29}{29\cdots 60}a^{5}-\frac{67\cdots 97}{58\cdots 20}a^{4}+\frac{83\cdots 11}{27\cdots 20}a^{3}+\frac{71\cdots 39}{83\cdots 60}a^{2}-\frac{29\cdots 53}{12\cdots 36}a+\frac{41\cdots 43}{16\cdots 20}$, $\frac{27\cdots 35}{16\cdots 32}a^{13}-\frac{52\cdots 81}{16\cdots 32}a^{12}-\frac{21\cdots 67}{16\cdots 32}a^{11}+\frac{59\cdots 45}{16\cdots 32}a^{10}-\frac{15\cdots 01}{83\cdots 16}a^{9}+\frac{24\cdots 85}{16\cdots 32}a^{8}+\frac{49\cdots 45}{55\cdots 44}a^{7}+\frac{25\cdots 49}{16\cdots 32}a^{6}-\frac{58\cdots 93}{83\cdots 16}a^{5}+\frac{19\cdots 61}{16\cdots 32}a^{4}+\frac{48\cdots 09}{16\cdots 32}a^{3}-\frac{29\cdots 25}{16\cdots 32}a^{2}+\frac{59\cdots 01}{16\cdots 32}a-\frac{24\cdots 93}{13\cdots 36}$, $\frac{52\cdots 71}{58\cdots 20}a^{13}+\frac{28\cdots 17}{19\cdots 40}a^{12}-\frac{18\cdots 73}{83\cdots 60}a^{11}+\frac{57\cdots 57}{58\cdots 20}a^{10}+\frac{12\cdots 91}{32\cdots 40}a^{9}-\frac{13\cdots 51}{58\cdots 20}a^{8}-\frac{12\cdots 09}{58\cdots 20}a^{7}+\frac{78\cdots 99}{19\cdots 40}a^{6}+\frac{34\cdots 67}{29\cdots 60}a^{5}-\frac{19\cdots 91}{58\cdots 20}a^{4}-\frac{11\cdots 67}{27\cdots 20}a^{3}+\frac{27\cdots 17}{83\cdots 60}a^{2}-\frac{19\cdots 45}{38\cdots 08}a+\frac{56\cdots 09}{16\cdots 20}$, $\frac{14\cdots 93}{12\cdots 36}a^{13}+\frac{24\cdots 61}{11\cdots 24}a^{12}-\frac{13\cdots 07}{18\cdots 48}a^{11}-\frac{14\cdots 89}{11\cdots 24}a^{10}-\frac{21\cdots 07}{58\cdots 12}a^{9}-\frac{83\cdots 35}{38\cdots 08}a^{8}+\frac{19\cdots 21}{11\cdots 24}a^{7}+\frac{20\cdots 19}{11\cdots 24}a^{6}+\frac{24\cdots 15}{19\cdots 04}a^{5}+\frac{53\cdots 31}{11\cdots 24}a^{4}+\frac{61\cdots 89}{16\cdots 32}a^{3}-\frac{42\cdots 65}{18\cdots 48}a^{2}+\frac{20\cdots 43}{11\cdots 24}a+\frac{79\cdots 49}{96\cdots 52}$, $\frac{51\cdots 37}{29\cdots 56}a^{13}-\frac{17\cdots 87}{29\cdots 56}a^{12}-\frac{15\cdots 71}{41\cdots 08}a^{11}+\frac{59\cdots 03}{32\cdots 84}a^{10}-\frac{10\cdots 09}{14\cdots 78}a^{9}-\frac{32\cdots 57}{29\cdots 56}a^{8}+\frac{14\cdots 61}{29\cdots 56}a^{7}-\frac{32\cdots 25}{29\cdots 56}a^{6}-\frac{10\cdots 29}{14\cdots 78}a^{5}+\frac{84\cdots 41}{96\cdots 52}a^{4}-\frac{85\cdots 95}{41\cdots 08}a^{3}+\frac{32\cdots 43}{41\cdots 08}a^{2}+\frac{16\cdots 67}{29\cdots 56}a-\frac{22\cdots 16}{24\cdots 63}$, $\frac{10\cdots 41}{58\cdots 20}a^{13}-\frac{10\cdots 39}{58\cdots 20}a^{12}+\frac{51\cdots 77}{83\cdots 60}a^{11}-\frac{42\cdots 51}{19\cdots 40}a^{10}-\frac{19\cdots 91}{29\cdots 60}a^{9}+\frac{19\cdots 39}{58\cdots 20}a^{8}-\frac{53\cdots 79}{58\cdots 20}a^{7}+\frac{63\cdots 07}{58\cdots 20}a^{6}+\frac{59\cdots 77}{29\cdots 60}a^{5}-\frac{87\cdots 49}{64\cdots 80}a^{4}+\frac{39\cdots 27}{11\cdots 80}a^{3}-\frac{57\cdots 59}{11\cdots 80}a^{2}+\frac{48\cdots 91}{11\cdots 24}a-\frac{83\cdots 63}{48\cdots 60}$, $\frac{15\cdots 15}{29\cdots 56}a^{13}-\frac{45\cdots 93}{29\cdots 56}a^{12}-\frac{27\cdots 65}{41\cdots 08}a^{11}+\frac{47\cdots 47}{96\cdots 52}a^{10}-\frac{69\cdots 31}{14\cdots 78}a^{9}-\frac{94\cdots 75}{29\cdots 56}a^{8}-\frac{15\cdots 97}{29\cdots 56}a^{7}+\frac{15\cdots 05}{29\cdots 56}a^{6}-\frac{47\cdots 67}{14\cdots 78}a^{5}+\frac{31\cdots 31}{96\cdots 52}a^{4}+\frac{53\cdots 35}{41\cdots 08}a^{3}-\frac{30\cdots 51}{41\cdots 08}a^{2}+\frac{31\cdots 61}{29\cdots 56}a-\frac{21\cdots 74}{24\cdots 63}$ 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:  \( 874877246.508 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 874877246.508 \cdot 1}{2\cdot\sqrt{282346132215490560000000000}}\cr\approx \mathstrut & 6.40715759150 \end{aligned}\] (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^14 - 3*x^13 - 5*x^12 + 31*x^11 - 50*x^10 + 127*x^9 - 55*x^8 + 1003*x^7 - 5194*x^6 + 13043*x^5 + 6647*x^4 - 129703*x^3 + 382419*x^2 - 490216*x + 319344) 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^14 - 3*x^13 - 5*x^12 + 31*x^11 - 50*x^10 + 127*x^9 - 55*x^8 + 1003*x^7 - 5194*x^6 + 13043*x^5 + 6647*x^4 - 129703*x^3 + 382419*x^2 - 490216*x + 319344, 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^14 - 3*x^13 - 5*x^12 + 31*x^11 - 50*x^10 + 127*x^9 - 55*x^8 + 1003*x^7 - 5194*x^6 + 13043*x^5 + 6647*x^4 - 129703*x^3 + 382419*x^2 - 490216*x + 319344); 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 = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 - 5*x^12 + 31*x^11 - 50*x^10 + 127*x^9 - 55*x^8 + 1003*x^7 - 5194*x^6 + 13043*x^5 + 6647*x^4 - 129703*x^3 + 382419*x^2 - 490216*x + 319344); 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

$A_{14}$ (as 14T62):

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); G, transitive_group_identification(G)
 
A non-solvable group of order 43589145600
The 72 conjugacy class representatives for $A_{14}$
Character table for $A_{14}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-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 R R R ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.2.4.20a1.15$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 47 x^{4} + 60 x^{3} + 54 x^{2} + 32 x + 11$$4$$2$$20$$D_4\times C_2$$$[2, 3, \frac{7}{2}]^{2}$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.1.3.3a1.2$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.6.11a2.6$x^{6} + 18 x^{2} + 9 x + 6$$6$$1$$11$$S_3^2$$$[2, \frac{5}{2}]_{2}^{2}$$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.3.2a1.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.1.8.7a1.4$x^{8} + 20$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
\(7\) 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$$$[\ ]_{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.1.4.3a1.2$x^{4} + 21$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
7.1.5.4a1.1$x^{5} + 7$$5$$1$$4$$F_5$$$[\ ]_{5}^{4}$$

Spectrum of ring of integers

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