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Properties

Label 16.0.166...601.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.661\times 10^{23}$
Root discriminant \(28.27\)
Ramified primes $17,47$
Class number $10$ (GRH)
Class group [10] (GRH)
Galois group $D_{8}$ (as 16T13)

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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^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 4*y^14 + 15*y^13 + 6*y^12 - 118*y^11 + 317*y^10 - 460*y^9 + 1614*y^8 - 2361*y^7 + 2820*y^6 - 2374*y^5 + 4393*y^4 + 3912*y^3 + 10208*y^2 + 8568*y + 2448, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448)
 

\( x^{16} - 4 x^{15} + 4 x^{14} + 15 x^{13} + 6 x^{12} - 118 x^{11} + 317 x^{10} - 460 x^{9} + 1614 x^{8} + \cdots + 2448 \) 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:  $16$
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:  $[0, 8]$
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:   \(166101760110563345913601\) \(\medspace = 17^{8}\cdot 47^{8}\) 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:  \(28.27\)
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:  $17^{1/2}47^{1/2}\approx 28.26658805020514$
Ramified primes:   \(17\), \(47\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_8$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{17}, \sqrt{-47})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{21}a^{12}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{7}-\frac{1}{7}a^{6}-\frac{2}{7}a^{5}-\frac{1}{21}a^{4}-\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{42}a^{13}-\frac{2}{21}a^{11}-\frac{1}{14}a^{10}-\frac{2}{21}a^{9}-\frac{2}{21}a^{8}-\frac{17}{42}a^{7}+\frac{4}{21}a^{6}+\frac{1}{7}a^{5}-\frac{1}{6}a^{4}-\frac{5}{21}a^{3}+\frac{4}{21}a^{2}-\frac{1}{42}a$, $\frac{1}{12852}a^{14}-\frac{22}{3213}a^{13}-\frac{1}{1071}a^{12}+\frac{1147}{12852}a^{11}+\frac{17}{126}a^{10}+\frac{263}{6426}a^{9}+\frac{605}{12852}a^{8}-\frac{1499}{3213}a^{7}-\frac{505}{6426}a^{6}-\frac{6241}{12852}a^{5}-\frac{1096}{3213}a^{4}-\frac{1}{14}a^{3}-\frac{835}{12852}a^{2}+\frac{1}{21}a-\frac{2}{21}$, $\frac{1}{34\cdots 04}a^{15}+\frac{11\cdots 23}{56\cdots 84}a^{14}-\frac{17\cdots 81}{23\cdots 98}a^{13}-\frac{50\cdots 33}{34\cdots 04}a^{12}-\frac{33\cdots 67}{42\cdots 63}a^{11}-\frac{10\cdots 21}{17\cdots 52}a^{10}-\frac{46\cdots 95}{37\cdots 56}a^{9}+\frac{94\cdots 23}{56\cdots 84}a^{8}-\frac{46\cdots 85}{17\cdots 52}a^{7}-\frac{28\cdots 75}{67\cdots 04}a^{6}-\frac{50\cdots 15}{24\cdots 36}a^{5}+\frac{75\cdots 77}{17\cdots 52}a^{4}-\frac{13\cdots 83}{34\cdots 04}a^{3}+\frac{68\cdots 59}{17\cdots 52}a^{2}-\frac{11\cdots 28}{27\cdots 71}a-\frac{99\cdots 00}{27\cdots 71}$ 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:  $2$, $3$

Class group and class number

Ideal class group:  $C_{10}$, which has order $10$ (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:  $C_{10}$, which has order $10$ (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:   $\frac{53\cdots 17}{17\cdots 52}a^{15}-\frac{34\cdots 49}{17\cdots 52}a^{14}+\frac{27\cdots 25}{77\cdots 66}a^{13}+\frac{12\cdots 83}{17\cdots 52}a^{12}-\frac{33\cdots 09}{18\cdots 28}a^{11}-\frac{53\cdots 93}{85\cdots 26}a^{10}+\frac{38\cdots 35}{17\cdots 52}a^{9}-\frac{31\cdots 09}{17\cdots 52}a^{8}+\frac{16\cdots 28}{42\cdots 63}a^{7}-\frac{26\cdots 99}{17\cdots 52}a^{6}+\frac{22\cdots 41}{17\cdots 52}a^{5}+\frac{22\cdots 78}{14\cdots 21}a^{4}+\frac{32\cdots 95}{17\cdots 52}a^{3}-\frac{12\cdots 99}{56\cdots 84}a^{2}-\frac{22\cdots 91}{55\cdots 42}a+\frac{63\cdots 93}{93\cdots 57}$, $\frac{48\cdots 55}{56\cdots 52}a^{15}-\frac{20\cdots 35}{39\cdots 64}a^{14}+\frac{25\cdots 46}{26\cdots 93}a^{13}+\frac{33\cdots 83}{23\cdots 92}a^{12}-\frac{18\cdots 71}{39\cdots 64}a^{11}-\frac{10\cdots 70}{99\cdots 41}a^{10}+\frac{23\cdots 83}{39\cdots 64}a^{9}-\frac{34\cdots 07}{39\cdots 64}a^{8}+\frac{82\cdots 65}{73\cdots 66}a^{7}-\frac{95\cdots 03}{39\cdots 64}a^{6}+\frac{42\cdots 57}{13\cdots 88}a^{5}+\frac{23\cdots 91}{19\cdots 82}a^{4}-\frac{17\cdots 83}{56\cdots 52}a^{3}-\frac{37\cdots 87}{39\cdots 64}a^{2}+\frac{46\cdots 87}{64\cdots 97}a+\frac{86\cdots 29}{64\cdots 97}$, $\frac{30\cdots 68}{14\cdots 21}a^{15}-\frac{14\cdots 15}{17\cdots 52}a^{14}+\frac{13\cdots 17}{33\cdots 14}a^{13}+\frac{26\cdots 58}{47\cdots 07}a^{12}-\frac{57\cdots 05}{17\cdots 52}a^{11}-\frac{14\cdots 33}{47\cdots 07}a^{10}+\frac{67\cdots 83}{85\cdots 26}a^{9}-\frac{87\cdots 15}{17\cdots 52}a^{8}+\frac{98\cdots 97}{85\cdots 26}a^{7}-\frac{10\cdots 71}{12\cdots 18}a^{6}-\frac{62\cdots 77}{17\cdots 52}a^{5}+\frac{11\cdots 69}{85\cdots 26}a^{4}-\frac{43\cdots 59}{28\cdots 42}a^{3}+\frac{60\cdots 51}{24\cdots 36}a^{2}+\frac{34\cdots 31}{55\cdots 42}a+\frac{31\cdots 53}{27\cdots 71}$, $\frac{70\cdots 85}{56\cdots 84}a^{15}+\frac{16\cdots 08}{15\cdots 69}a^{14}-\frac{29\cdots 35}{77\cdots 66}a^{13}-\frac{19\cdots 69}{56\cdots 84}a^{12}+\frac{13\cdots 63}{28\cdots 42}a^{11}+\frac{29\cdots 05}{14\cdots 21}a^{10}-\frac{25\cdots 01}{11\cdots 84}a^{9}+\frac{80\cdots 36}{47\cdots 07}a^{8}+\frac{10\cdots 36}{14\cdots 21}a^{7}+\frac{29\cdots 27}{63\cdots 76}a^{6}-\frac{33\cdots 53}{20\cdots 03}a^{5}-\frac{68\cdots 96}{14\cdots 21}a^{4}+\frac{47\cdots 89}{56\cdots 84}a^{3}+\frac{58\cdots 20}{14\cdots 21}a^{2}+\frac{57\cdots 21}{55\cdots 42}a+\frac{18\cdots 16}{93\cdots 57}$, $\frac{11\cdots 39}{67\cdots 04}a^{15}-\frac{62\cdots 93}{16\cdots 26}a^{14}-\frac{56\cdots 55}{50\cdots 22}a^{13}+\frac{40\cdots 77}{67\cdots 04}a^{12}+\frac{46\cdots 15}{11\cdots 84}a^{11}-\frac{95\cdots 81}{33\cdots 52}a^{10}+\frac{11\cdots 47}{67\cdots 04}a^{9}+\frac{22\cdots 09}{23\cdots 18}a^{8}-\frac{11\cdots 95}{33\cdots 52}a^{7}+\frac{18\cdots 61}{67\cdots 04}a^{6}-\frac{16\cdots 55}{16\cdots 26}a^{5}+\frac{27\cdots 91}{15\cdots 12}a^{4}-\frac{58\cdots 85}{67\cdots 04}a^{3}+\frac{16\cdots 90}{93\cdots 57}a^{2}+\frac{37\cdots 69}{13\cdots 51}a+\frac{11\cdots 04}{93\cdots 57}$, $\frac{53\cdots 79}{56\cdots 52}a^{15}-\frac{13\cdots 11}{39\cdots 64}a^{14}+\frac{25\cdots 06}{26\cdots 93}a^{13}+\frac{76\cdots 59}{39\cdots 64}a^{12}+\frac{32\cdots 93}{39\cdots 64}a^{11}-\frac{11\cdots 79}{99\cdots 41}a^{10}+\frac{97\cdots 71}{39\cdots 64}a^{9}-\frac{92\cdots 15}{39\cdots 64}a^{8}+\frac{67\cdots 09}{73\cdots 66}a^{7}-\frac{28\cdots 79}{39\cdots 64}a^{6}+\frac{35\cdots 85}{13\cdots 88}a^{5}+\frac{24\cdots 11}{19\cdots 82}a^{4}-\frac{60\cdots 71}{56\cdots 52}a^{3}+\frac{29\cdots 01}{39\cdots 64}a^{2}+\frac{16\cdots 29}{21\cdots 99}a+\frac{69\cdots 90}{64\cdots 97}$, $\frac{29\cdots 05}{13\cdots 02}a^{15}-\frac{20\cdots 09}{24\cdots 36}a^{14}+\frac{22\cdots 53}{23\cdots 98}a^{13}+\frac{49\cdots 01}{16\cdots 26}a^{12}+\frac{25\cdots 19}{17\cdots 52}a^{11}-\frac{33\cdots 72}{14\cdots 21}a^{10}+\frac{30\cdots 10}{42\cdots 63}a^{9}-\frac{18\cdots 91}{17\cdots 52}a^{8}+\frac{31\cdots 61}{85\cdots 26}a^{7}-\frac{22\cdots 55}{42\cdots 63}a^{6}+\frac{12\cdots 67}{17\cdots 52}a^{5}-\frac{56\cdots 27}{85\cdots 26}a^{4}+\frac{58\cdots 70}{47\cdots 07}a^{3}+\frac{61\cdots 57}{17\cdots 52}a^{2}+\frac{41\cdots 23}{18\cdots 14}a+\frac{31\cdots 12}{27\cdots 71}$ 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:  \( 25496.198852 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 25496.198852 \cdot 10}{2\cdot\sqrt{166101760110563345913601}}\cr\approx \mathstrut & 0.75979647019 \end{aligned}\] (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^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448) 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^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448, 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^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448); 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^16 - 4*x^15 + 4*x^14 + 15*x^13 + 6*x^12 - 118*x^11 + 317*x^10 - 460*x^9 + 1614*x^8 - 2361*x^7 + 2820*x^6 - 2374*x^5 + 4393*x^4 + 3912*x^3 + 10208*x^2 + 8568*x + 2448); 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

$D_8$ (as 16T13):

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 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}, \sqrt{-47})\), 4.2.13583.1 x2, 4.0.37553.1 x2, 8.0.407555836801.2, 8.2.8671400783.1 x4, 8.0.23973872753.1 x4

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 8 siblings: 8.2.8671400783.1, 8.0.23973872753.1
Minimal sibling: 8.2.8671400783.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.2.0.1}{2} }^{8}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ R ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(17\) Copy content Toggle raw display 17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
17.1.2.1a1.1$x^{2} + 17$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(47\) Copy content Toggle raw display 47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_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)