LMFDB - Number field 16.0.95038694188787201488441.2

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27)

gp:K = bnfinit(y^16 - y^15 + 6*y^14 + 13*y^13 + 60*y^12 + 33*y^11 + 31*y^10 + 28*y^9 + 330*y^8 + 704*y^7 + 811*y^6 + 579*y^5 + 307*y^4 + 208*y^3 + 162*y^2 + 90*y + 27, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27)

$ x^{16} - x^{15} + 6 x^{14} + 13 x^{13} + 60 x^{12} + 33 x^{11} + 31 x^{10} + 28 x^{9} + 330 x^{8} + \cdots + 27 $ x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$16$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[0, 8]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$95038694188787201488441$ 95038694188787201488441 $\medspace = 13^{14}\cdot 17^{6}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$27.30$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$13^{7/8}17^{1/2}\approx 38.897787010151276$
Ramified primes:	$13$, $17$ 13, 17	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q$
$\Aut(K/\Q)$:	$C_4$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:	8.0.1394947801.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{2}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}-\frac{1}{9}a^{3}-\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{4}{9}a^{6}+\frac{4}{9}a^{5}-\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{567}a^{14}-\frac{11}{567}a^{13}-\frac{1}{63}a^{12}+\frac{92}{567}a^{11}-\frac{50}{567}a^{10}-\frac{5}{567}a^{9}-\frac{32}{567}a^{8}+\frac{31}{81}a^{7}-\frac{4}{21}a^{6}-\frac{5}{189}a^{5}-\frac{92}{567}a^{4}-\frac{22}{81}a^{3}+\frac{3}{7}a^{2}+\frac{23}{63}a-\frac{1}{21}$, $\frac{1}{3498453071379}a^{15}+\frac{1366803722}{3498453071379}a^{14}-\frac{95480202887}{3498453071379}a^{13}+\frac{31965109175}{3498453071379}a^{12}+\frac{25414905175}{166593003399}a^{11}+\frac{348345798095}{3498453071379}a^{10}+\frac{507339268805}{3498453071379}a^{9}+\frac{143339424749}{3498453071379}a^{8}-\frac{1357439084816}{3498453071379}a^{7}-\frac{301517250803}{1166151023793}a^{6}+\frac{283577465134}{3498453071379}a^{5}-\frac{188539524569}{388717007931}a^{4}+\frac{34184415326}{3498453071379}a^{3}+\frac{86584678972}{388717007931}a^{2}-\frac{3122586307}{9039930417}a+\frac{47647982864}{129572335977}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/3*a^8 + 1/3*a^7 + 1/3*a^6 + 1/3*a^5 + 1/3*a^4 + 1/3*a^3 + 1/3*a^2 + 1/3*a, 1/3*a^9 - 1/3*a, 1/3*a^10 - 1/3*a^2, 1/3*a^11 - 1/3*a^3, 1/9*a^12 - 1/9*a^9 - 1/9*a^8 + 2/9*a^7 - 1/9*a^6 - 1/9*a^5 - 2/9*a^4 - 1/9*a^3 - 4/9*a^2 + 1/3*a, 1/9*a^13 - 1/9*a^10 - 1/9*a^9 - 1/9*a^8 - 4/9*a^7 - 4/9*a^6 + 4/9*a^5 - 4/9*a^4 + 2/9*a^3 - 1/3*a, 1/567*a^14 - 11/567*a^13 - 1/63*a^12 + 92/567*a^11 - 50/567*a^10 - 5/567*a^9 - 32/567*a^8 + 31/81*a^7 - 4/21*a^6 - 5/189*a^5 - 92/567*a^4 - 22/81*a^3 + 3/7*a^2 + 23/63*a - 1/21, 1/3498453071379*a^15 + 1366803722/3498453071379*a^14 - 95480202887/3498453071379*a^13 + 31965109175/3498453071379*a^12 + 25414905175/166593003399*a^11 + 348345798095/3498453071379*a^10 + 507339268805/3498453071379*a^9 + 143339424749/3498453071379*a^8 - 1357439084816/3498453071379*a^7 - 301517250803/1166151023793*a^6 + 283577465134/3498453071379*a^5 - 188539524569/388717007931*a^4 + 34184415326/3498453071379*a^3 + 86584678972/388717007931*a^2 - 3122586307/9039930417*a + 47647982864/129572335977

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

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)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$7$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity sage:UK.torsion_generator() gp:K.tu[2] magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar:torsion_units_generator(OK)
Fundamental units:	$\frac{19045343}{108623997}a^{15}-\frac{8922911}{36207999}a^{14}+\frac{118261462}{108623997}a^{13}+\frac{211635193}{108623997}a^{12}+\frac{1010221960}{108623997}a^{11}+\frac{5851319}{4023111}a^{10}+\frac{20536847}{12069333}a^{9}+\frac{171974197}{36207999}a^{8}+\frac{5991742762}{108623997}a^{7}+\frac{3603955781}{36207999}a^{6}+\frac{8993525309}{108623997}a^{5}+\frac{4218945995}{108623997}a^{4}+\frac{1960782725}{108623997}a^{3}+\frac{262148869}{12069333}a^{2}+\frac{187684088}{12069333}a+\frac{13677086}{4023111}$, $\frac{153101849393}{1166151023793}a^{15}-\frac{42271229960}{166593003399}a^{14}+\frac{1176953586629}{1166151023793}a^{13}+\frac{133678355947}{166593003399}a^{12}+\frac{2721046981558}{388717007931}a^{11}-\frac{2604350865962}{1166151023793}a^{10}+\frac{6305511318055}{1166151023793}a^{9}-\frac{774935747900}{1166151023793}a^{8}+\frac{50498891299253}{1166151023793}a^{7}+\frac{20347395141767}{388717007931}a^{6}+\frac{61370664968360}{1166151023793}a^{5}+\frac{9179491341268}{388717007931}a^{4}+\frac{18880098875485}{1166151023793}a^{3}+\frac{1690554462241}{129572335977}a^{2}+\frac{26938027495}{3013310139}a+\frac{145314814745}{43190778659}$, $\frac{249855513455}{3498453071379}a^{15}-\frac{725719789652}{3498453071379}a^{14}+\frac{2358717536855}{3498453071379}a^{13}-\frac{178107853307}{3498453071379}a^{12}+\frac{3785756276837}{1166151023793}a^{11}-\frac{2363108294594}{499779010197}a^{10}+\frac{1868507800954}{499779010197}a^{9}-\frac{5801285210429}{3498453071379}a^{8}+\frac{78400011749354}{3498453071379}a^{7}+\frac{1136050585733}{166593003399}a^{6}-\frac{7837975691440}{3498453071379}a^{5}-\frac{3621527403566}{388717007931}a^{4}+\frac{2128407704650}{3498453071379}a^{3}+\frac{121925887525}{55531001133}a^{2}-\frac{22253185652}{9039930417}a-\frac{241626796079}{129572335977}$, $\frac{116707953841}{3498453071379}a^{15}-\frac{36279762137}{499779010197}a^{14}+\frac{336678686851}{1166151023793}a^{13}+\frac{57920576816}{499779010197}a^{12}+\frac{6406298764189}{3498453071379}a^{11}-\frac{2855807991896}{3498453071379}a^{10}+\frac{8082203684602}{3498453071379}a^{9}-\frac{1621512258848}{3498453071379}a^{8}+\frac{12773075972101}{1166151023793}a^{7}+\frac{13091823612967}{1166151023793}a^{6}+\frac{49554537096784}{3498453071379}a^{5}+\frac{37840985389013}{3498453071379}a^{4}+\frac{8184537768545}{1166151023793}a^{3}+\frac{1768833231698}{388717007931}a^{2}+\frac{1024493453}{1004436713}a+\frac{57699063131}{43190778659}$, $\frac{354167086703}{3498453071379}a^{15}-\frac{527514423967}{3498453071379}a^{14}+\frac{225132148163}{388717007931}a^{13}+\frac{4325122647109}{3498453071379}a^{12}+\frac{16593743525915}{3498453071379}a^{11}+\frac{117598421339}{499779010197}a^{10}-\frac{838641132100}{499779010197}a^{9}+\frac{19747027141349}{3498453071379}a^{8}+\frac{3836704305926}{129572335977}a^{7}+\frac{8844562293806}{166593003399}a^{6}+\frac{95394964757867}{3498453071379}a^{5}+\frac{18233313432223}{3498453071379}a^{4}+\frac{4262381195930}{388717007931}a^{3}+\frac{422418847675}{55531001133}a^{2}+\frac{6386093243}{1004436713}a-\frac{87092604615}{43190778659}$, $\frac{30402889817}{3498453071379}a^{15}+\frac{23244859663}{3498453071379}a^{14}+\frac{90104484899}{3498453071379}a^{13}+\frac{729139550509}{3498453071379}a^{12}+\frac{29269865151}{43190778659}a^{11}+\frac{3424463180521}{3498453071379}a^{10}+\frac{427057830448}{3498453071379}a^{9}-\frac{15437376455}{499779010197}a^{8}+\frac{11534235225425}{3498453071379}a^{7}+\frac{11300120913212}{1166151023793}a^{6}+\frac{46757621304563}{3498453071379}a^{5}+\frac{1141201599872}{166593003399}a^{4}+\frac{10381200427306}{3498453071379}a^{3}+\frac{886359835660}{388717007931}a^{2}+\frac{18989675974}{9039930417}a+\frac{22359406744}{18510333711}$, $\frac{2218574759}{129572335977}a^{15}-\frac{106072769765}{3498453071379}a^{14}+\frac{494642740000}{3498453071379}a^{13}+\frac{29908364329}{388717007931}a^{12}+\frac{3801062637698}{3498453071379}a^{11}-\frac{735124058990}{3498453071379}a^{10}+\frac{4697761647682}{3498453071379}a^{9}-\frac{4240939752701}{3498453071379}a^{8}+\frac{22568003331775}{3498453071379}a^{7}+\frac{2587135074445}{388717007931}a^{6}+\frac{2151500401708}{166593003399}a^{5}+\frac{14375805850390}{3498453071379}a^{4}+\frac{4328650356230}{3498453071379}a^{3}-\frac{1973085949883}{388717007931}a^{2}-\frac{26793098602}{9039930417}a-\frac{219240063016}{129572335977}$ 19045343/108623997a^15 - 8922911/36207999a^14 + 118261462/108623997a^13 + 211635193/108623997a^12 + 1010221960/108623997a^11 + 5851319/4023111a^10 + 20536847/12069333a^9 + 171974197/36207999a^8 + 5991742762/108623997a^7 + 3603955781/36207999a^6 + 8993525309/108623997a^5 + 4218945995/108623997a^4 + 1960782725/108623997a^3 + 262148869/12069333a^2 + 187684088/12069333a + 13677086/4023111, 153101849393/1166151023793a^15 - 42271229960/166593003399a^14 + 1176953586629/1166151023793a^13 + 133678355947/166593003399a^12 + 2721046981558/388717007931a^11 - 2604350865962/1166151023793a^10 + 6305511318055/1166151023793a^9 - 774935747900/1166151023793a^8 + 50498891299253/1166151023793a^7 + 20347395141767/388717007931a^6 + 61370664968360/1166151023793a^5 + 9179491341268/388717007931a^4 + 18880098875485/1166151023793a^3 + 1690554462241/129572335977a^2 + 26938027495/3013310139a + 145314814745/43190778659, 249855513455/3498453071379a^15 - 725719789652/3498453071379a^14 + 2358717536855/3498453071379a^13 - 178107853307/3498453071379a^12 + 3785756276837/1166151023793a^11 - 2363108294594/499779010197a^10 + 1868507800954/499779010197a^9 - 5801285210429/3498453071379a^8 + 78400011749354/3498453071379a^7 + 1136050585733/166593003399a^6 - 7837975691440/3498453071379a^5 - 3621527403566/388717007931a^4 + 2128407704650/3498453071379a^3 + 121925887525/55531001133a^2 - 22253185652/9039930417a - 241626796079/129572335977, 116707953841/3498453071379a^15 - 36279762137/499779010197a^14 + 336678686851/1166151023793a^13 + 57920576816/499779010197a^12 + 6406298764189/3498453071379a^11 - 2855807991896/3498453071379a^10 + 8082203684602/3498453071379a^9 - 1621512258848/3498453071379a^8 + 12773075972101/1166151023793a^7 + 13091823612967/1166151023793a^6 + 49554537096784/3498453071379a^5 + 37840985389013/3498453071379a^4 + 8184537768545/1166151023793a^3 + 1768833231698/388717007931a^2 + 1024493453/1004436713a + 57699063131/43190778659, 354167086703/3498453071379a^15 - 527514423967/3498453071379a^14 + 225132148163/388717007931a^13 + 4325122647109/3498453071379a^12 + 16593743525915/3498453071379a^11 + 117598421339/499779010197a^10 - 838641132100/499779010197a^9 + 19747027141349/3498453071379a^8 + 3836704305926/129572335977a^7 + 8844562293806/166593003399a^6 + 95394964757867/3498453071379a^5 + 18233313432223/3498453071379a^4 + 4262381195930/388717007931a^3 + 422418847675/55531001133a^2 + 6386093243/1004436713a - 87092604615/43190778659, 30402889817/3498453071379a^15 + 23244859663/3498453071379a^14 + 90104484899/3498453071379a^13 + 729139550509/3498453071379a^12 + 29269865151/43190778659a^11 + 3424463180521/3498453071379a^10 + 427057830448/3498453071379a^9 - 15437376455/499779010197a^8 + 11534235225425/3498453071379a^7 + 11300120913212/1166151023793a^6 + 46757621304563/3498453071379a^5 + 1141201599872/166593003399a^4 + 10381200427306/3498453071379a^3 + 886359835660/388717007931a^2 + 18989675974/9039930417a + 22359406744/18510333711, 2218574759/129572335977a^15 - 106072769765/3498453071379a^14 + 494642740000/3498453071379a^13 + 29908364329/388717007931a^12 + 3801062637698/3498453071379a^11 - 735124058990/3498453071379a^10 + 4697761647682/3498453071379a^9 - 4240939752701/3498453071379a^8 + 22568003331775/3498453071379a^7 + 2587135074445/388717007931a^6 + 2151500401708/166593003399a^5 + 14375805850390/3498453071379a^4 + 4328650356230/3498453071379a^3 - 1973085949883/388717007931a^2 - 26793098602/9039930417*a - 219240063016/129572335977 (assuming GRH)	comment:Fundamental units sage:UK.fundamental_units() gp:K.fu magma:[K\|fUK(g): g in Generators(UK)]; oscar:[K(fUK(a)) for a in gens(UK)]
Regulator:	$ 408026.14455 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 408026.14455 \cdot 1}{2\cdot\sqrt{95038694188787201488441}}\cr\approx \mathstrut & 1.6074841988 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27); 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

$C_4\wr C_2$ (as 16T28):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A solvable group of order 32

The 14 conjugacy class representatives for $C_4\wr C_2$

Character table for $C_4\wr C_2$

Intermediate fields

$\Q(\sqrt{13}) $, 4.0.2197.1, 4.4.37349.1, 4.0.2873.1, 8.0.1394947801.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Galois closure:	deg 32
Degree 8 siblings:	8.4.5240818888357.2, 8.4.5240818888357.1
Degree 16 sibling:	16.8.27466182620559501230159449.1
Minimal sibling:	8.4.5240818888357.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.8.0.1}{8} }^{2}$	${\href{/padicField/3.2.0.1}{2} }^{6}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$	${\href{/padicField/5.4.0.1}{4} }^{4}$	${\href{/padicField/7.4.0.1}{4} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{4}$	R	R	${\href{/padicField/19.8.0.1}{8} }^{2}$	${\href{/padicField/23.4.0.1}{4} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$	${\href{/padicField/31.4.0.1}{4} }^{4}$	${\href{/padicField/37.4.0.1}{4} }^{4}$	${\href{/padicField/41.4.0.1}{4} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{8}$	${\href{/padicField/47.8.0.1}{8} }^{2}$	${\href{/padicField/53.4.0.1}{4} }^{4}$	${\href{/padicField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

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)]

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]])

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))];

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$13$ 13	13.2.8.14a1.4	$x^{16} + 96 x^{15} + 4048 x^{14} + 98112 x^{13} + 1500016 x^{12} + 14910336 x^{11} + 95462080 x^{10} + 374159616 x^{9} + 799894624 x^{8} + 748319232 x^{7} + 381848320 x^{6} + 119282688 x^{5} + 24000256 x^{4} + 3139584 x^{3} + 259072 x^{2} + 12418 x + 282$	$8$	$2$	$14$	$C_8: C_2$	$$[\ ]_{8}^{2}$$
$17$ 17	17.4.1.0a1.1	$x^{4} + 7 x^{2} + 10 x + 3$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	17.2.2.2a1.1	$x^{4} + 32 x^{3} + 262 x^{2} + 113 x + 9$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
	17.4.2.4a1.2	$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$	$2$	$4$	$4$	$C_4\times C_2$	$$[\ ]_{2}^{4}$$

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