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Properties

Label 16.0.484...304.235
Degree $16$
Signature $[0, 8]$
Discriminant $4.841\times 10^{23}$
Root discriminant \(30.22\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:D_4$ (as 16T969)

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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 + 8*x^14 + 32*x^12 - 16*x^10 - 96*x^9 - 256*x^8 - 768*x^7 + 128*x^6 + 1152*x^5 - 224*x^4 + 960*x^3 + 3840*x^2 + 2880*x + 648)
 
Copy content gp:K = bnfinit(y^16 + 8*y^14 + 32*y^12 - 16*y^10 - 96*y^9 - 256*y^8 - 768*y^7 + 128*y^6 + 1152*y^5 - 224*y^4 + 960*y^3 + 3840*y^2 + 2880*y + 648, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 8*x^14 + 32*x^12 - 16*x^10 - 96*x^9 - 256*x^8 - 768*x^7 + 128*x^6 + 1152*x^5 - 224*x^4 + 960*x^3 + 3840*x^2 + 2880*x + 648);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 8*x^14 + 32*x^12 - 16*x^10 - 96*x^9 - 256*x^8 - 768*x^7 + 128*x^6 + 1152*x^5 - 224*x^4 + 960*x^3 + 3840*x^2 + 2880*x + 648)
 

\( x^{16} + 8 x^{14} + 32 x^{12} - 16 x^{10} - 96 x^{9} - 256 x^{8} - 768 x^{7} + 128 x^{6} + 1152 x^{5} + \cdots + 648 \) 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:   \(484116351470433472610304\) \(\medspace = 2^{66}\cdot 3^{8}\) 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:  \(30.22\)
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:  $2^{555/128}3^{3/4}\approx 46.035004279893066$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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:  \(\Q(\sqrt{-2}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{4}a^{12}$, $\frac{1}{12}a^{13}+\frac{1}{12}a^{12}+\frac{1}{6}a^{9}+\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{12}a^{14}-\frac{1}{12}a^{12}+\frac{1}{6}a^{10}+\frac{1}{6}a^{8}+\frac{1}{6}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{16\cdots 84}a^{15}-\frac{33\cdots 85}{54\cdots 28}a^{14}-\frac{92\cdots 72}{41\cdots 21}a^{13}-\frac{32\cdots 25}{54\cdots 28}a^{12}-\frac{33\cdots 13}{41\cdots 21}a^{11}-\frac{20\cdots 74}{13\cdots 07}a^{10}-\frac{39\cdots 53}{82\cdots 42}a^{9}-\frac{15\cdots 53}{13\cdots 07}a^{8}-\frac{82\cdots 01}{82\cdots 42}a^{7}-\frac{26\cdots 35}{27\cdots 14}a^{6}-\frac{11\cdots 05}{41\cdots 21}a^{5}-\frac{57\cdots 28}{13\cdots 07}a^{4}+\frac{59\cdots 15}{41\cdots 21}a^{3}+\frac{15\cdots 69}{45\cdots 69}a^{2}+\frac{42\cdots 02}{13\cdots 07}a-\frac{12\cdots 99}{45\cdots 69}$ 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:  $3$

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:   $\frac{41\cdots 55}{16\cdots 84}a^{15}-\frac{36\cdots 99}{27\cdots 14}a^{14}-\frac{28\cdots 09}{16\cdots 84}a^{13}-\frac{13\cdots 33}{91\cdots 38}a^{12}-\frac{22\cdots 90}{41\cdots 21}a^{11}-\frac{22\cdots 65}{27\cdots 14}a^{10}+\frac{13\cdots 39}{82\cdots 42}a^{9}+\frac{80\cdots 89}{91\cdots 38}a^{8}+\frac{34\cdots 05}{41\cdots 21}a^{7}+\frac{25\cdots 67}{13\cdots 07}a^{6}+\frac{13\cdots 00}{41\cdots 21}a^{5}-\frac{66\cdots 77}{13\cdots 07}a^{4}+\frac{84\cdots 86}{41\cdots 21}a^{3}-\frac{24\cdots 57}{13\cdots 07}a^{2}-\frac{18\cdots 04}{13\cdots 07}a-\frac{29\cdots 35}{45\cdots 69}$, $\frac{59\cdots 63}{16\cdots 84}a^{15}+\frac{58\cdots 59}{27\cdots 14}a^{14}-\frac{49\cdots 17}{16\cdots 84}a^{13}+\frac{24\cdots 84}{13\cdots 07}a^{12}-\frac{52\cdots 82}{41\cdots 21}a^{11}+\frac{10\cdots 23}{13\cdots 07}a^{10}+\frac{11\cdots 79}{82\cdots 42}a^{9}+\frac{46\cdots 77}{13\cdots 07}a^{8}+\frac{59\cdots 65}{82\cdots 42}a^{7}+\frac{10\cdots 03}{45\cdots 69}a^{6}-\frac{76\cdots 92}{41\cdots 21}a^{5}-\frac{42\cdots 06}{13\cdots 07}a^{4}+\frac{10\cdots 47}{41\cdots 21}a^{3}-\frac{69\cdots 44}{13\cdots 07}a^{2}-\frac{14\cdots 46}{13\cdots 07}a-\frac{18\cdots 05}{45\cdots 69}$, $\frac{16\cdots 09}{16\cdots 84}a^{15}-\frac{55\cdots 15}{54\cdots 28}a^{14}+\frac{14\cdots 47}{16\cdots 84}a^{13}-\frac{81\cdots 85}{91\cdots 38}a^{12}+\frac{32\cdots 79}{82\cdots 42}a^{11}-\frac{53\cdots 79}{13\cdots 07}a^{10}+\frac{14\cdots 25}{82\cdots 42}a^{9}-\frac{48\cdots 95}{45\cdots 69}a^{8}-\frac{12\cdots 03}{82\cdots 42}a^{7}-\frac{80\cdots 90}{13\cdots 07}a^{6}+\frac{29\cdots 19}{41\cdots 21}a^{5}+\frac{70\cdots 98}{13\cdots 07}a^{4}-\frac{34\cdots 91}{41\cdots 21}a^{3}+\frac{24\cdots 48}{13\cdots 07}a^{2}+\frac{30\cdots 42}{13\cdots 07}a+\frac{12\cdots 01}{45\cdots 69}$, $\frac{32\cdots 55}{45\cdots 69}a^{15}+\frac{23\cdots 03}{54\cdots 28}a^{14}-\frac{10\cdots 55}{18\cdots 76}a^{13}+\frac{99\cdots 09}{27\cdots 14}a^{12}-\frac{11\cdots 85}{45\cdots 69}a^{11}+\frac{20\cdots 74}{13\cdots 07}a^{10}+\frac{16\cdots 99}{91\cdots 38}a^{9}+\frac{91\cdots 41}{13\cdots 07}a^{8}+\frac{12\cdots 09}{91\cdots 38}a^{7}+\frac{62\cdots 66}{13\cdots 07}a^{6}-\frac{16\cdots 71}{45\cdots 69}a^{5}-\frac{80\cdots 66}{13\cdots 07}a^{4}+\frac{23\cdots 80}{45\cdots 69}a^{3}-\frac{13\cdots 30}{13\cdots 07}a^{2}-\frac{95\cdots 78}{45\cdots 69}a-\frac{33\cdots 77}{45\cdots 69}$, $\frac{12\cdots 89}{82\cdots 42}a^{15}+\frac{49\cdots 23}{54\cdots 28}a^{14}-\frac{10\cdots 41}{82\cdots 42}a^{13}+\frac{34\cdots 87}{45\cdots 69}a^{12}-\frac{44\cdots 31}{82\cdots 42}a^{11}+\frac{86\cdots 87}{27\cdots 14}a^{10}+\frac{25\cdots 25}{41\cdots 21}a^{9}+\frac{13\cdots 33}{91\cdots 38}a^{8}+\frac{12\cdots 02}{41\cdots 21}a^{7}+\frac{13\cdots 79}{13\cdots 07}a^{6}-\frac{32\cdots 49}{41\cdots 21}a^{5}-\frac{17\cdots 40}{13\cdots 07}a^{4}+\frac{45\cdots 46}{41\cdots 21}a^{3}-\frac{29\cdots 24}{13\cdots 07}a^{2}-\frac{63\cdots 40}{13\cdots 07}a-\frac{77\cdots 73}{45\cdots 69}$, $\frac{39\cdots 35}{82\cdots 42}a^{15}+\frac{14\cdots 77}{54\cdots 28}a^{14}-\frac{32\cdots 35}{82\cdots 42}a^{13}+\frac{10\cdots 88}{45\cdots 69}a^{12}-\frac{13\cdots 41}{82\cdots 42}a^{11}+\frac{25\cdots 93}{27\cdots 14}a^{10}+\frac{10\cdots 03}{41\cdots 21}a^{9}+\frac{40\cdots 55}{91\cdots 38}a^{8}+\frac{39\cdots 30}{41\cdots 21}a^{7}+\frac{42\cdots 97}{13\cdots 07}a^{6}-\frac{96\cdots 03}{41\cdots 21}a^{5}-\frac{57\cdots 08}{13\cdots 07}a^{4}+\frac{14\cdots 18}{41\cdots 21}a^{3}-\frac{87\cdots 04}{13\cdots 07}a^{2}-\frac{20\cdots 68}{13\cdots 07}a-\frac{25\cdots 97}{45\cdots 69}$, $\frac{25\cdots 48}{45\cdots 69}a^{15}-\frac{15\cdots 91}{54\cdots 28}a^{14}+\frac{42\cdots 09}{91\cdots 38}a^{13}-\frac{64\cdots 53}{27\cdots 14}a^{12}+\frac{17\cdots 45}{91\cdots 38}a^{11}-\frac{26\cdots 41}{27\cdots 14}a^{10}-\frac{17\cdots 23}{45\cdots 69}a^{9}-\frac{70\cdots 32}{13\cdots 07}a^{8}-\frac{53\cdots 08}{45\cdots 69}a^{7}-\frac{50\cdots 67}{13\cdots 07}a^{6}+\frac{11\cdots 37}{45\cdots 69}a^{5}+\frac{70\cdots 59}{13\cdots 07}a^{4}-\frac{17\cdots 22}{45\cdots 69}a^{3}+\frac{10\cdots 96}{13\cdots 07}a^{2}+\frac{81\cdots 58}{45\cdots 69}a+\frac{31\cdots 09}{45\cdots 69}$ 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:  \( 3447328.1967007415 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3447328.1967007415 \cdot 1}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 6.01750842527284 \end{aligned}\] (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 + 8*x^14 + 32*x^12 - 16*x^10 - 96*x^9 - 256*x^8 - 768*x^7 + 128*x^6 + 1152*x^5 - 224*x^4 + 960*x^3 + 3840*x^2 + 2880*x + 648) 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 + 8*x^14 + 32*x^12 - 16*x^10 - 96*x^9 - 256*x^8 - 768*x^7 + 128*x^6 + 1152*x^5 - 224*x^4 + 960*x^3 + 3840*x^2 + 2880*x + 648, 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 + 8*x^14 + 32*x^12 - 16*x^10 - 96*x^9 - 256*x^8 - 768*x^7 + 128*x^6 + 1152*x^5 - 224*x^4 + 960*x^3 + 3840*x^2 + 2880*x + 648); 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 + 8*x^14 + 32*x^12 - 16*x^10 - 96*x^9 - 256*x^8 - 768*x^7 + 128*x^6 + 1152*x^5 - 224*x^4 + 960*x^3 + 3840*x^2 + 2880*x + 648); 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

$C_2^6:D_4$ (as 16T969):

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 512
The 44 conjugacy class representatives for $C_2^6:D_4$
Character table for $C_2^6:D_4$

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.6144.1, 8.0.28991029248.5, 8.0.28991029248.10, 8.0.21743271936.5

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: 16.0.30257271966902092038144.4, 16.0.484116351470433472610304.7, 16.0.30257271966902092038144.5, 16.0.30257271966902092038144.6, 16.0.121029087867608368152576.17, 16.0.121029087867608368152576.18, 16.0.30257271966902092038144.11, 16.0.484116351470433472610304.18, 16.0.4357047163233901253492736.147, 16.0.484116351470433472610304.266, 16.0.484116351470433472610304.269, 16.0.4357047163233901253492736.67, 16.0.484116351470433472610304.274, 16.0.272315447702118828343296.127, 16.0.272315447702118828343296.130, 16.0.484116351470433472610304.169, 16.0.30257271966902092038144.54, 16.0.30257271966902092038144.56, 16.0.484116351470433472610304.276, 16.0.484116351470433472610304.277, 16.0.484116351470433472610304.75, 16.0.4357047163233901253492736.264, 16.0.484116351470433472610304.181, 16.0.1089261790808475313373184.138, 16.0.4357047163233901253492736.184, 16.0.1089261790808475313373184.69, 16.0.4357047163233901253492736.78, 16.0.4357047163233901253492736.186, 16.0.4357047163233901253492736.79, 16.0.272315447702118828343296.141, 16.0.272315447702118828343296.160, 16.0.121029087867608368152576.171, 16.0.121029087867608368152576.172, 16.0.484116351470433472610304.217, 16.0.4357047163233901253492736.287, 16.0.4357047163233901253492736.288, 16.0.1089261790808475313373184.147, 16.0.4357047163233901253492736.289, 16.0.4357047163233901253492736.223, 16.0.484116351470433472610304.234, 16.0.4357047163233901253492736.108, 16.0.272315447702118828343296.179, 16.0.272315447702118828343296.184, 16.0.30257271966902092038144.73, 16.0.1089261790808475313373184.95, 16.0.121029087867608368152576.106, 16.0.484116351470433472610304.116, 16.0.30257271966902092038144.78, 16.0.1089261790808475313373184.109, 16.0.121029087867608368152576.123, 16.0.484116351470433472610304.129, 16.0.4357047163233901253492736.141, 16.0.484116351470433472610304.150, 16.0.1089261790808475313373184.18, 16.0.121029087867608368152576.56, 16.0.272315447702118828343296.21, 16.0.1089261790808475313373184.64, 16.0.121029087867608368152576.59, 16.0.4357047163233901253492736.52, 16.0.272315447702118828343296.39, 16.0.1089261790808475313373184.55, 16.0.4357047163233901253492736.304, 16.0.4357047163233901253492736.249
Degree 32 siblings: deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32
Minimal sibling: 16.0.30257271966902092038144.4

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.1.16.66j1.126$x^{16} + 8 x^{14} + 24 x^{12} + 8 x^{10} + 16 x^{9} + 16 x^{3} + 2$$16$$1$$66$16T969$$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.4.6a1.2$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$$4$$2$$6$$D_4$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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