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Properties

Label 16.0.720...881.13
Degree $16$
Signature $[0, 8]$
Discriminant $7.203\times 10^{31}$
Root discriminant \(97.97\)
Ramified primes $37,149$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_4^2.(C_2\times C_4)$ (as 16T362)

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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 + 15*x^14 - 146*x^13 + 530*x^12 - 1742*x^11 + 11886*x^10 - 43533*x^9 + 86109*x^8 - 161859*x^7 + 528801*x^6 - 1060517*x^5 + 466404*x^4 + 647937*x^3 + 862606*x^2 - 3001470*x + 1730321)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 15*y^14 - 146*y^13 + 530*y^12 - 1742*y^11 + 11886*y^10 - 43533*y^9 + 86109*y^8 - 161859*y^7 + 528801*y^6 - 1060517*y^5 + 466404*y^4 + 647937*y^3 + 862606*y^2 - 3001470*y + 1730321, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 15*x^14 - 146*x^13 + 530*x^12 - 1742*x^11 + 11886*x^10 - 43533*x^9 + 86109*x^8 - 161859*x^7 + 528801*x^6 - 1060517*x^5 + 466404*x^4 + 647937*x^3 + 862606*x^2 - 3001470*x + 1730321);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 15*x^14 - 146*x^13 + 530*x^12 - 1742*x^11 + 11886*x^10 - 43533*x^9 + 86109*x^8 - 161859*x^7 + 528801*x^6 - 1060517*x^5 + 466404*x^4 + 647937*x^3 + 862606*x^2 - 3001470*x + 1730321)
 

\( x^{16} - 4 x^{15} + 15 x^{14} - 146 x^{13} + 530 x^{12} - 1742 x^{11} + 11886 x^{10} - 43533 x^{9} + \cdots + 1730321 \) 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:   \(72034127342222967481030341869881\) \(\medspace = 37^{12}\cdot 149^{6}\) 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:  \(97.97\)
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:  $37^{3/4}149^{1/2}\approx 183.12364628723972$
Ramified primes:   \(37\), \(149\) 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.
Maximal CM subfield:  4.0.50653.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{1}{7}a^{10}-\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{2}{7}a^{4}-\frac{1}{7}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{49}a^{13}+\frac{1}{49}a^{12}+\frac{3}{49}a^{11}-\frac{17}{49}a^{10}-\frac{2}{7}a^{9}-\frac{3}{49}a^{8}-\frac{16}{49}a^{7}-\frac{22}{49}a^{6}-\frac{19}{49}a^{5}-\frac{16}{49}a^{4}-\frac{22}{49}a^{2}+\frac{3}{7}a-\frac{1}{49}$, $\frac{1}{49}a^{14}+\frac{2}{49}a^{12}-\frac{20}{49}a^{11}+\frac{3}{49}a^{10}+\frac{11}{49}a^{9}-\frac{13}{49}a^{8}-\frac{6}{49}a^{7}+\frac{3}{49}a^{6}+\frac{3}{49}a^{5}+\frac{16}{49}a^{4}-\frac{22}{49}a^{3}-\frac{6}{49}a^{2}-\frac{22}{49}a+\frac{1}{49}$, $\frac{1}{17\cdots 13}a^{15}-\frac{16\cdots 48}{17\cdots 13}a^{14}+\frac{15\cdots 50}{17\cdots 13}a^{13}-\frac{28\cdots 53}{17\cdots 13}a^{12}-\frac{38\cdots 97}{17\cdots 13}a^{11}+\frac{98\cdots 64}{17\cdots 13}a^{10}+\frac{12\cdots 14}{17\cdots 13}a^{9}+\frac{35\cdots 09}{17\cdots 13}a^{8}+\frac{10\cdots 04}{17\cdots 13}a^{7}+\frac{57\cdots 09}{17\cdots 13}a^{6}+\frac{29\cdots 57}{17\cdots 13}a^{5}-\frac{76\cdots 13}{17\cdots 13}a^{4}-\frac{46\cdots 27}{17\cdots 13}a^{3}-\frac{43\cdots 26}{17\cdots 13}a^{2}+\frac{73\cdots 71}{17\cdots 13}a+\frac{64\cdots 19}{17\cdots 13}$ 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:  $C_{4}$, which has order $4$ (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:  $C_{4}$, 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:  $7$
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:   $\frac{26\cdots 44}{27\cdots 37}a^{15}-\frac{62\cdots 10}{27\cdots 37}a^{14}+\frac{29\cdots 24}{27\cdots 37}a^{13}-\frac{34\cdots 70}{27\cdots 37}a^{12}+\frac{84\cdots 66}{27\cdots 37}a^{11}-\frac{32\cdots 66}{27\cdots 37}a^{10}+\frac{26\cdots 72}{27\cdots 37}a^{9}-\frac{72\cdots 74}{27\cdots 37}a^{8}+\frac{10\cdots 80}{27\cdots 37}a^{7}-\frac{24\cdots 00}{27\cdots 37}a^{6}+\frac{99\cdots 42}{27\cdots 37}a^{5}-\frac{11\cdots 28}{27\cdots 37}a^{4}-\frac{86\cdots 48}{27\cdots 37}a^{3}+\frac{41\cdots 76}{27\cdots 37}a^{2}+\frac{33\cdots 26}{27\cdots 37}a-\frac{23\cdots 35}{27\cdots 37}$, $\frac{59\cdots 13}{17\cdots 13}a^{15}-\frac{45\cdots 78}{17\cdots 13}a^{14}+\frac{52\cdots 49}{17\cdots 13}a^{13}-\frac{66\cdots 70}{17\cdots 13}a^{12}+\frac{79\cdots 35}{17\cdots 13}a^{11}-\frac{52\cdots 20}{17\cdots 13}a^{10}+\frac{48\cdots 29}{17\cdots 13}a^{9}-\frac{79\cdots 77}{17\cdots 13}a^{8}+\frac{65\cdots 25}{17\cdots 13}a^{7}-\frac{31\cdots 02}{17\cdots 13}a^{6}+\frac{16\cdots 93}{17\cdots 13}a^{5}-\frac{23\cdots 26}{17\cdots 13}a^{4}-\frac{34\cdots 46}{17\cdots 13}a^{3}-\frac{45\cdots 36}{17\cdots 13}a^{2}+\frac{90\cdots 15}{17\cdots 13}a+\frac{16\cdots 24}{17\cdots 13}$, $\frac{27\cdots 04}{17\cdots 13}a^{15}-\frac{13\cdots 38}{17\cdots 13}a^{14}+\frac{41\cdots 25}{17\cdots 13}a^{13}-\frac{42\cdots 25}{17\cdots 13}a^{12}+\frac{17\cdots 04}{17\cdots 13}a^{11}-\frac{49\cdots 88}{17\cdots 13}a^{10}+\frac{35\cdots 68}{17\cdots 13}a^{9}-\frac{14\cdots 39}{17\cdots 13}a^{8}+\frac{25\cdots 99}{17\cdots 13}a^{7}-\frac{46\cdots 67}{17\cdots 13}a^{6}+\frac{16\cdots 83}{17\cdots 13}a^{5}-\frac{34\cdots 48}{17\cdots 13}a^{4}+\frac{27\cdots 32}{17\cdots 13}a^{3}+\frac{38\cdots 67}{17\cdots 13}a^{2}+\frac{54\cdots 49}{17\cdots 13}a-\frac{10\cdots 07}{17\cdots 13}$, $\frac{56\cdots 41}{17\cdots 13}a^{15}-\frac{19\cdots 09}{17\cdots 13}a^{14}+\frac{43\cdots 72}{17\cdots 13}a^{13}-\frac{78\cdots 75}{17\cdots 13}a^{12}+\frac{24\cdots 33}{17\cdots 13}a^{11}-\frac{50\cdots 58}{17\cdots 13}a^{10}+\frac{62\cdots 56}{17\cdots 13}a^{9}-\frac{19\cdots 88}{17\cdots 13}a^{8}+\frac{12\cdots 04}{17\cdots 13}a^{7}-\frac{61\cdots 38}{17\cdots 13}a^{6}+\frac{27\cdots 81}{17\cdots 13}a^{5}-\frac{22\cdots 25}{17\cdots 13}a^{4}-\frac{23\cdots 27}{17\cdots 13}a^{3}-\frac{15\cdots 03}{17\cdots 13}a^{2}+\frac{73\cdots 44}{17\cdots 13}a-\frac{49\cdots 03}{17\cdots 13}$, $\frac{30\cdots 16}{17\cdots 13}a^{15}+\frac{24\cdots 60}{17\cdots 13}a^{14}-\frac{13\cdots 84}{17\cdots 13}a^{13}+\frac{20\cdots 14}{17\cdots 13}a^{12}-\frac{28\cdots 48}{17\cdots 13}a^{11}-\frac{11\cdots 22}{17\cdots 13}a^{10}-\frac{62\cdots 91}{17\cdots 13}a^{9}+\frac{26\cdots 05}{17\cdots 13}a^{8}-\frac{23\cdots 96}{17\cdots 13}a^{7}-\frac{49\cdots 88}{17\cdots 13}a^{6}-\frac{15\cdots 28}{17\cdots 13}a^{5}+\frac{39\cdots 46}{17\cdots 13}a^{4}+\frac{90\cdots 73}{17\cdots 13}a^{3}-\frac{26\cdots 34}{17\cdots 13}a^{2}+\frac{16\cdots 39}{17\cdots 13}a+\frac{18\cdots 75}{17\cdots 13}$, $\frac{80\cdots 62}{17\cdots 13}a^{15}-\frac{18\cdots 84}{17\cdots 13}a^{14}+\frac{70\cdots 22}{17\cdots 13}a^{13}-\frac{10\cdots 16}{17\cdots 13}a^{12}+\frac{17\cdots 20}{17\cdots 13}a^{11}-\frac{10\cdots 02}{17\cdots 13}a^{10}+\frac{67\cdots 27}{17\cdots 13}a^{9}-\frac{19\cdots 63}{17\cdots 13}a^{8}+\frac{18\cdots 24}{17\cdots 13}a^{7}-\frac{28\cdots 74}{17\cdots 13}a^{6}+\frac{13\cdots 45}{17\cdots 13}a^{5}-\frac{42\cdots 69}{17\cdots 13}a^{4}-\frac{89\cdots 47}{17\cdots 13}a^{3}+\frac{19\cdots 26}{17\cdots 13}a^{2}-\frac{14\cdots 98}{17\cdots 13}a+\frac{36\cdots 80}{17\cdots 13}$, $\frac{33\cdots 50}{25\cdots 59}a^{15}-\frac{87\cdots 84}{25\cdots 59}a^{14}+\frac{37\cdots 28}{25\cdots 59}a^{13}-\frac{43\cdots 62}{25\cdots 59}a^{12}+\frac{11\cdots 99}{25\cdots 59}a^{11}-\frac{41\cdots 84}{25\cdots 59}a^{10}+\frac{33\cdots 87}{25\cdots 59}a^{9}-\frac{97\cdots 87}{25\cdots 59}a^{8}+\frac{14\cdots 95}{25\cdots 59}a^{7}-\frac{32\cdots 28}{25\cdots 59}a^{6}+\frac{12\cdots 78}{25\cdots 59}a^{5}-\frac{17\cdots 61}{25\cdots 59}a^{4}-\frac{94\cdots 02}{25\cdots 59}a^{3}+\frac{89\cdots 23}{25\cdots 59}a^{2}+\frac{42\cdots 13}{25\cdots 59}a-\frac{39\cdots 52}{25\cdots 59}$ 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:  \( 597275788.994 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 597275788.994 \cdot 4}{2\cdot\sqrt{72034127342222967481030341869881}}\cr\approx \mathstrut & 0.341880793716 \end{aligned}\] (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 + 15*x^14 - 146*x^13 + 530*x^12 - 1742*x^11 + 11886*x^10 - 43533*x^9 + 86109*x^8 - 161859*x^7 + 528801*x^6 - 1060517*x^5 + 466404*x^4 + 647937*x^3 + 862606*x^2 - 3001470*x + 1730321) 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 + 15*x^14 - 146*x^13 + 530*x^12 - 1742*x^11 + 11886*x^10 - 43533*x^9 + 86109*x^8 - 161859*x^7 + 528801*x^6 - 1060517*x^5 + 466404*x^4 + 647937*x^3 + 862606*x^2 - 3001470*x + 1730321, 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 + 15*x^14 - 146*x^13 + 530*x^12 - 1742*x^11 + 11886*x^10 - 43533*x^9 + 86109*x^8 - 161859*x^7 + 528801*x^6 - 1060517*x^5 + 466404*x^4 + 647937*x^3 + 862606*x^2 - 3001470*x + 1730321); 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^16 - 4*x^15 + 15*x^14 - 146*x^13 + 530*x^12 - 1742*x^11 + 11886*x^10 - 43533*x^9 + 86109*x^8 - 161859*x^7 + 528801*x^6 - 1060517*x^5 + 466404*x^4 + 647937*x^3 + 862606*x^2 - 3001470*x + 1730321); 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

$C_4^2.(C_2\times C_4)$ (as 16T362):

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 solvable group of order 128
The 14 conjugacy class representatives for $C_4^2.(C_2\times C_4)$
Character table for $C_4^2.(C_2\times C_4)$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.0.50653.1, 8.0.56961692006209.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 32 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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ R ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(37\) Copy content Toggle raw display 37.1.4.3a1.1$x^{4} + 37$$4$$1$$3$$C_4$$$[\ ]_{4}$$
37.1.4.3a1.1$x^{4} + 37$$4$$1$$3$$C_4$$$[\ ]_{4}$$
37.1.4.3a1.1$x^{4} + 37$$4$$1$$3$$C_4$$$[\ ]_{4}$$
37.1.4.3a1.1$x^{4} + 37$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(149\) Copy content Toggle raw display 149.2.1.0a1.1$x^{2} + 145 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
149.1.2.1a1.2$x^{2} + 298$$2$$1$$1$$C_2$$$[\ ]_{2}$$
149.1.2.1a1.2$x^{2} + 298$$2$$1$$1$$C_2$$$[\ ]_{2}$$
149.2.1.0a1.1$x^{2} + 145 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
149.2.2.2a1.2$x^{4} + 290 x^{3} + 21029 x^{2} + 580 x + 153$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
149.2.2.2a1.2$x^{4} + 290 x^{3} + 21029 x^{2} + 580 x + 153$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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