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Properties

Label 16.0.400...161.28
Degree $16$
Signature $[0, 8]$
Discriminant $4.007\times 10^{29}$
Root discriminant \(70.82\)
Ramified primes $13,53$
Class number $16$ (GRH)
Class group [2, 8] (GRH)
Galois group $C_2^3.D_4$ (as 16T153)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 27*x^14 - 86*x^13 + 482*x^12 - 2230*x^11 + 9081*x^10 - 30942*x^9 + 90574*x^8 - 218582*x^7 + 469209*x^6 - 873386*x^5 + 1373544*x^4 - 1780692*x^3 + 2036236*x^2 - 1574750*x + 873203)
 
Copy content gp:K = bnfinit(y^16 - 6*y^15 + 27*y^14 - 86*y^13 + 482*y^12 - 2230*y^11 + 9081*y^10 - 30942*y^9 + 90574*y^8 - 218582*y^7 + 469209*y^6 - 873386*y^5 + 1373544*y^4 - 1780692*y^3 + 2036236*y^2 - 1574750*y + 873203, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 27*x^14 - 86*x^13 + 482*x^12 - 2230*x^11 + 9081*x^10 - 30942*x^9 + 90574*x^8 - 218582*x^7 + 469209*x^6 - 873386*x^5 + 1373544*x^4 - 1780692*x^3 + 2036236*x^2 - 1574750*x + 873203);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 27*x^14 - 86*x^13 + 482*x^12 - 2230*x^11 + 9081*x^10 - 30942*x^9 + 90574*x^8 - 218582*x^7 + 469209*x^6 - 873386*x^5 + 1373544*x^4 - 1780692*x^3 + 2036236*x^2 - 1574750*x + 873203)
 

\( x^{16} - 6 x^{15} + 27 x^{14} - 86 x^{13} + 482 x^{12} - 2230 x^{11} + 9081 x^{10} - 30942 x^{9} + \cdots + 873203 \) 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:   \(400734980167009195224860426161\) \(\medspace = 13^{8}\cdot 53^{12}\) 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:  \(70.82\)
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:  $13^{3/4}53^{3/4}\approx 134.48207963630105$
Ramified primes:   \(13\), \(53\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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:  8.0.3745777030801.2

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{26}a^{12}-\frac{2}{13}a^{11}-\frac{5}{26}a^{9}-\frac{3}{13}a^{8}+\frac{5}{13}a^{7}+\frac{6}{13}a^{6}-\frac{1}{26}a^{5}+\frac{5}{13}a^{4}-\frac{9}{26}a^{3}+\frac{11}{26}a^{2}+\frac{5}{26}a+\frac{1}{13}$, $\frac{1}{26}a^{13}-\frac{3}{26}a^{11}-\frac{5}{26}a^{10}-\frac{1}{26}a^{8}-\frac{5}{26}a^{6}+\frac{3}{13}a^{5}-\frac{4}{13}a^{4}+\frac{1}{26}a^{3}+\frac{5}{13}a^{2}+\frac{9}{26}a-\frac{5}{26}$, $\frac{1}{26}a^{14}-\frac{2}{13}a^{11}-\frac{3}{26}a^{9}-\frac{5}{26}a^{8}+\frac{6}{13}a^{7}+\frac{3}{26}a^{6}+\frac{1}{13}a^{5}+\frac{5}{26}a^{4}-\frac{2}{13}a^{3}+\frac{3}{26}a^{2}+\frac{5}{13}a+\frac{3}{13}$, $\frac{1}{11\cdots 78}a^{15}+\frac{44\cdots 05}{11\cdots 78}a^{14}-\frac{91\cdots 29}{11\cdots 78}a^{13}+\frac{71\cdots 41}{56\cdots 89}a^{12}-\frac{13\cdots 40}{56\cdots 89}a^{11}+\frac{15\cdots 77}{11\cdots 78}a^{10}+\frac{26\cdots 79}{11\cdots 78}a^{9}-\frac{15\cdots 97}{11\cdots 78}a^{8}+\frac{25\cdots 34}{56\cdots 89}a^{7}-\frac{15\cdots 33}{56\cdots 89}a^{6}+\frac{26\cdots 33}{56\cdots 89}a^{5}-\frac{15\cdots 13}{56\cdots 89}a^{4}-\frac{27\cdots 54}{56\cdots 89}a^{3}-\frac{37\cdots 91}{87\cdots 06}a^{2}+\frac{23\cdots 43}{56\cdots 89}a+\frac{19\cdots 42}{56\cdots 89}$ 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:  $C_{2}\times C_{8}$, which has order $16$ (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:  $C_{2}\times C_{8}$, which has order $16$ (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:   $\frac{79\cdots 05}{19\cdots 62}a^{15}-\frac{10\cdots 94}{95\cdots 31}a^{14}+\frac{10\cdots 23}{19\cdots 62}a^{13}-\frac{18\cdots 91}{19\cdots 62}a^{12}+\frac{25\cdots 83}{19\cdots 62}a^{11}-\frac{60\cdots 79}{14\cdots 74}a^{10}+\frac{25\cdots 45}{14\cdots 74}a^{9}-\frac{65\cdots 73}{14\cdots 74}a^{8}+\frac{11\cdots 71}{95\cdots 31}a^{7}-\frac{36\cdots 01}{19\cdots 62}a^{6}+\frac{74\cdots 33}{19\cdots 62}a^{5}-\frac{41\cdots 62}{95\cdots 31}a^{4}+\frac{67\cdots 85}{19\cdots 62}a^{3}-\frac{55\cdots 05}{95\cdots 31}a^{2}-\frac{21\cdots 96}{95\cdots 31}a+\frac{80\cdots 79}{19\cdots 62}$, $\frac{23\cdots 03}{19\cdots 62}a^{15}+\frac{83\cdots 31}{19\cdots 62}a^{14}-\frac{41\cdots 15}{19\cdots 62}a^{13}+\frac{45\cdots 61}{95\cdots 31}a^{12}+\frac{33\cdots 45}{19\cdots 62}a^{11}+\frac{31\cdots 91}{95\cdots 31}a^{10}-\frac{14\cdots 74}{95\cdots 31}a^{9}+\frac{11\cdots 74}{95\cdots 31}a^{8}-\frac{91\cdots 09}{19\cdots 62}a^{7}+\frac{16\cdots 45}{95\cdots 31}a^{6}-\frac{76\cdots 59}{19\cdots 62}a^{5}+\frac{17\cdots 93}{19\cdots 62}a^{4}-\frac{16\cdots 66}{95\cdots 31}a^{3}+\frac{49\cdots 31}{19\cdots 62}a^{2}-\frac{33\cdots 73}{14\cdots 74}a+\frac{83\cdots 21}{19\cdots 62}$, $\frac{46\cdots 43}{19\cdots 62}a^{15}-\frac{11\cdots 74}{95\cdots 31}a^{14}+\frac{45\cdots 45}{19\cdots 62}a^{13}-\frac{41\cdots 29}{19\cdots 62}a^{12}+\frac{11\cdots 93}{19\cdots 62}a^{11}-\frac{66\cdots 61}{19\cdots 62}a^{10}+\frac{15\cdots 11}{19\cdots 62}a^{9}-\frac{26\cdots 31}{19\cdots 62}a^{8}+\frac{18\cdots 67}{95\cdots 31}a^{7}+\frac{70\cdots 09}{14\cdots 74}a^{6}-\frac{23\cdots 01}{14\cdots 74}a^{5}+\frac{41\cdots 94}{95\cdots 31}a^{4}-\frac{12\cdots 29}{14\cdots 74}a^{3}+\frac{14\cdots 09}{95\cdots 31}a^{2}-\frac{15\cdots 82}{95\cdots 31}a+\frac{20\cdots 39}{19\cdots 62}$, $\frac{71\cdots 83}{11\cdots 78}a^{15}-\frac{84\cdots 03}{11\cdots 78}a^{14}+\frac{28\cdots 85}{11\cdots 78}a^{13}-\frac{12\cdots 75}{11\cdots 78}a^{12}+\frac{44\cdots 11}{11\cdots 78}a^{11}-\frac{30\cdots 59}{11\cdots 78}a^{10}+\frac{10\cdots 03}{11\cdots 78}a^{9}-\frac{41\cdots 05}{11\cdots 78}a^{8}+\frac{56\cdots 47}{56\cdots 89}a^{7}-\frac{30\cdots 29}{11\cdots 78}a^{6}+\frac{59\cdots 97}{11\cdots 78}a^{5}-\frac{63\cdots 87}{56\cdots 89}a^{4}+\frac{16\cdots 81}{11\cdots 78}a^{3}-\frac{19\cdots 89}{87\cdots 06}a^{2}+\frac{21\cdots 37}{11\cdots 78}a-\frac{17\cdots 75}{11\cdots 78}$, $\frac{12\cdots 78}{56\cdots 89}a^{15}+\frac{92\cdots 88}{56\cdots 89}a^{14}-\frac{14\cdots 23}{56\cdots 89}a^{13}+\frac{11\cdots 52}{56\cdots 89}a^{12}+\frac{47\cdots 09}{11\cdots 78}a^{11}+\frac{24\cdots 28}{56\cdots 89}a^{10}-\frac{53\cdots 48}{56\cdots 89}a^{9}+\frac{62\cdots 43}{11\cdots 78}a^{8}-\frac{58\cdots 93}{56\cdots 89}a^{7}+\frac{16\cdots 58}{56\cdots 89}a^{6}-\frac{72\cdots 80}{56\cdots 89}a^{5}+\frac{68\cdots 63}{11\cdots 78}a^{4}-\frac{10\cdots 93}{56\cdots 89}a^{3}+\frac{81\cdots 87}{87\cdots 06}a^{2}-\frac{43\cdots 61}{11\cdots 78}a+\frac{36\cdots 77}{11\cdots 78}$, $\frac{23\cdots 01}{11\cdots 78}a^{15}-\frac{59\cdots 85}{56\cdots 89}a^{14}+\frac{37\cdots 79}{11\cdots 78}a^{13}-\frac{58\cdots 30}{56\cdots 89}a^{12}+\frac{40\cdots 56}{56\cdots 89}a^{11}-\frac{20\cdots 17}{56\cdots 89}a^{10}+\frac{64\cdots 42}{56\cdots 89}a^{9}-\frac{21\cdots 87}{56\cdots 89}a^{8}+\frac{53\cdots 08}{56\cdots 89}a^{7}-\frac{11\cdots 06}{56\cdots 89}a^{6}+\frac{22\cdots 05}{56\cdots 89}a^{5}-\frac{38\cdots 39}{56\cdots 89}a^{4}+\frac{98\cdots 09}{11\cdots 78}a^{3}-\frac{10\cdots 49}{87\cdots 06}a^{2}+\frac{11\cdots 53}{11\cdots 78}a-\frac{91\cdots 83}{11\cdots 78}$, $\frac{67\cdots 51}{87\cdots 06}a^{15}-\frac{83\cdots 50}{33\cdots 81}a^{14}+\frac{10\cdots 03}{87\cdots 06}a^{13}-\frac{15\cdots 62}{43\cdots 53}a^{12}+\frac{12\cdots 86}{43\cdots 53}a^{11}-\frac{43\cdots 66}{43\cdots 53}a^{10}+\frac{17\cdots 18}{43\cdots 53}a^{9}-\frac{58\cdots 20}{43\cdots 53}a^{8}+\frac{15\cdots 11}{43\cdots 53}a^{7}-\frac{35\cdots 40}{43\cdots 53}a^{6}+\frac{77\cdots 35}{43\cdots 53}a^{5}-\frac{12\cdots 21}{43\cdots 53}a^{4}+\frac{39\cdots 65}{87\cdots 06}a^{3}-\frac{51\cdots 31}{87\cdots 06}a^{2}+\frac{46\cdots 93}{87\cdots 06}a-\frac{25\cdots 67}{87\cdots 06}$ 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:  \( 7655259.72698 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7655259.72698 \cdot 16}{2\cdot\sqrt{400734980167009195224860426161}}\cr\approx \mathstrut & 0.234995873952 \end{aligned}\] (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 - 6*x^15 + 27*x^14 - 86*x^13 + 482*x^12 - 2230*x^11 + 9081*x^10 - 30942*x^9 + 90574*x^8 - 218582*x^7 + 469209*x^6 - 873386*x^5 + 1373544*x^4 - 1780692*x^3 + 2036236*x^2 - 1574750*x + 873203) 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 - 6*x^15 + 27*x^14 - 86*x^13 + 482*x^12 - 2230*x^11 + 9081*x^10 - 30942*x^9 + 90574*x^8 - 218582*x^7 + 469209*x^6 - 873386*x^5 + 1373544*x^4 - 1780692*x^3 + 2036236*x^2 - 1574750*x + 873203, 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 - 6*x^15 + 27*x^14 - 86*x^13 + 482*x^12 - 2230*x^11 + 9081*x^10 - 30942*x^9 + 90574*x^8 - 218582*x^7 + 469209*x^6 - 873386*x^5 + 1373544*x^4 - 1780692*x^3 + 2036236*x^2 - 1574750*x + 873203); 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 - 6*x^15 + 27*x^14 - 86*x^13 + 482*x^12 - 2230*x^11 + 9081*x^10 - 30942*x^9 + 90574*x^8 - 218582*x^7 + 469209*x^6 - 873386*x^5 + 1373544*x^4 - 1780692*x^3 + 2036236*x^2 - 1574750*x + 873203); 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

$C_2^3.D_4$ (as 16T153):

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 64
The 13 conjugacy class representatives for $C_2^3.D_4$
Character table for $C_2^3.D_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.36517.1, 4.0.148877.1, 4.0.1935401.1, 8.0.3745777030801.2

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.0.67724211648224553993001412021209.23

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ R ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{13}$$x + 11$$1$$1$$0$Trivial$$[\ ]$$
13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(53\) Copy content Toggle raw display 53.2.4.6a1.2$x^{8} + 196 x^{7} + 14414 x^{6} + 471772 x^{5} + 5822449 x^{4} + 943544 x^{3} + 57656 x^{2} + 1568 x + 69$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
53.2.4.6a1.2$x^{8} + 196 x^{7} + 14414 x^{6} + 471772 x^{5} + 5822449 x^{4} + 943544 x^{3} + 57656 x^{2} + 1568 x + 69$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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