↑

Properties

Label 16.0.249...000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2.497\times 10^{31}$
Root discriminant \(91.69\)
Ramified primes $2,5,47$
Class number $176800$ (GRH)
Class group [10, 17680] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561)
 
gp: K = bnfinit(y^16 - 8*y^15 + 108*y^14 - 616*y^13 + 4916*y^12 - 21852*y^11 + 129218*y^10 - 459508*y^9 + 2174117*y^8 - 6156340*y^7 + 24095662*y^6 - 52453964*y^5 + 172046412*y^4 - 263040692*y^3 + 722028526*y^2 - 598345980*y + 1353436561, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561)
 

\( x^{16} - 8 x^{15} + 108 x^{14} - 616 x^{13} + 4916 x^{12} - 21852 x^{11} + 129218 x^{10} + \cdots + 1353436561 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(24967943722642702336000000000000\) \(\medspace = 2^{32}\cdot 5^{12}\cdot 47^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(91.69\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2}5^{3/4}47^{1/2}\approx 91.6930509657041$
Ramified primes:   \(2\), \(5\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1880=2^{3}\cdot 5\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{1880}(1409,·)$, $\chi_{1880}(1221,·)$, $\chi_{1880}(1,·)$, $\chi_{1880}(847,·)$, $\chi_{1880}(469,·)$, $\chi_{1880}(281,·)$, $\chi_{1880}(283,·)$, $\chi_{1880}(1503,·)$, $\chi_{1880}(187,·)$, $\chi_{1880}(1127,·)$, $\chi_{1880}(1129,·)$, $\chi_{1880}(1223,·)$, $\chi_{1880}(941,·)$, $\chi_{1880}(563,·)$, $\chi_{1880}(1787,·)$, $\chi_{1880}(189,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{12}-\frac{1}{10}a^{11}+\frac{1}{5}a^{10}-\frac{1}{10}a^{8}+\frac{3}{10}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{10}a^{4}+\frac{1}{10}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{10}a^{13}+\frac{1}{10}a^{11}+\frac{1}{5}a^{10}-\frac{1}{10}a^{9}+\frac{1}{5}a^{8}-\frac{1}{10}a^{7}-\frac{1}{2}a^{5}+\frac{1}{5}a^{4}-\frac{3}{10}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{74\!\cdots\!10}a^{14}-\frac{7}{74\!\cdots\!10}a^{13}+\frac{15\!\cdots\!96}{37\!\cdots\!05}a^{12}-\frac{18\!\cdots\!61}{74\!\cdots\!10}a^{11}-\frac{79\!\cdots\!14}{37\!\cdots\!05}a^{10}-\frac{54\!\cdots\!18}{37\!\cdots\!05}a^{9}+\frac{17\!\cdots\!99}{74\!\cdots\!10}a^{8}+\frac{28\!\cdots\!33}{14\!\cdots\!62}a^{7}+\frac{10\!\cdots\!08}{37\!\cdots\!05}a^{6}-\frac{58\!\cdots\!62}{37\!\cdots\!05}a^{5}-\frac{26\!\cdots\!71}{74\!\cdots\!10}a^{4}+\frac{20\!\cdots\!81}{74\!\cdots\!10}a^{3}+\frac{22\!\cdots\!07}{74\!\cdots\!10}a^{2}+\frac{22\!\cdots\!71}{74\!\cdots\!10}a+\frac{47\!\cdots\!49}{74\!\cdots\!10}$, $\frac{1}{66\!\cdots\!50}a^{15}+\frac{890989}{13\!\cdots\!10}a^{14}+\frac{17\!\cdots\!93}{66\!\cdots\!50}a^{13}+\frac{15\!\cdots\!54}{33\!\cdots\!25}a^{12}-\frac{22\!\cdots\!21}{66\!\cdots\!05}a^{11}+\frac{22\!\cdots\!19}{33\!\cdots\!25}a^{10}-\frac{13\!\cdots\!03}{66\!\cdots\!50}a^{9}+\frac{19\!\cdots\!69}{33\!\cdots\!25}a^{8}-\frac{34\!\cdots\!52}{33\!\cdots\!25}a^{7}-\frac{15\!\cdots\!86}{33\!\cdots\!25}a^{6}-\frac{31\!\cdots\!29}{66\!\cdots\!50}a^{5}-\frac{59\!\cdots\!88}{33\!\cdots\!25}a^{4}+\frac{12\!\cdots\!49}{66\!\cdots\!50}a^{3}-\frac{16\!\cdots\!39}{70\!\cdots\!90}a^{2}+\frac{35\!\cdots\!58}{33\!\cdots\!25}a-\frac{51\!\cdots\!26}{33\!\cdots\!25}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{10}\times C_{17680}$, which has order $176800$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{329376}{2775260459905}a^{14}-\frac{2305632}{2775260459905}a^{13}+\frac{68017729}{5550520919810}a^{12}-\frac{174079971}{2775260459905}a^{11}+\frac{3039255529}{5550520919810}a^{10}-\frac{6057356651}{2775260459905}a^{9}+\frac{79141559053}{5550520919810}a^{8}-\frac{24751931269}{555052091981}a^{7}+\frac{1295751894087}{5550520919810}a^{6}-\frac{1534093973789}{2775260459905}a^{5}+\frac{13420801445853}{5550520919810}a^{4}-\frac{10997157193119}{2775260459905}a^{3}+\frac{81787259107729}{5550520919810}a^{2}-\frac{35631786403859}{2775260459905}a+\frac{112112240468194}{2775260459905}$, $\frac{40\!\cdots\!43}{33\!\cdots\!25}a^{15}-\frac{12\!\cdots\!29}{13\!\cdots\!10}a^{14}+\frac{80\!\cdots\!13}{66\!\cdots\!50}a^{13}-\frac{21\!\cdots\!26}{33\!\cdots\!25}a^{12}+\frac{13\!\cdots\!59}{26\!\cdots\!22}a^{11}-\frac{69\!\cdots\!81}{33\!\cdots\!25}a^{10}+\frac{78\!\cdots\!47}{66\!\cdots\!50}a^{9}-\frac{13\!\cdots\!23}{35\!\cdots\!50}a^{8}+\frac{11\!\cdots\!41}{66\!\cdots\!50}a^{7}-\frac{14\!\cdots\!71}{33\!\cdots\!25}a^{6}+\frac{98\!\cdots\!51}{66\!\cdots\!50}a^{5}-\frac{18\!\cdots\!11}{66\!\cdots\!50}a^{4}+\frac{49\!\cdots\!09}{66\!\cdots\!50}a^{3}-\frac{11\!\cdots\!11}{13\!\cdots\!10}a^{2}+\frac{54\!\cdots\!48}{33\!\cdots\!25}a-\frac{45\!\cdots\!57}{66\!\cdots\!50}$, $\frac{66\!\cdots\!94}{33\!\cdots\!25}a^{15}-\frac{99\!\cdots\!41}{66\!\cdots\!05}a^{14}+\frac{57\!\cdots\!02}{33\!\cdots\!25}a^{13}-\frac{59\!\cdots\!41}{66\!\cdots\!50}a^{12}+\frac{83\!\cdots\!22}{13\!\cdots\!61}a^{11}-\frac{84\!\cdots\!23}{33\!\cdots\!25}a^{10}+\frac{89\!\cdots\!51}{66\!\cdots\!50}a^{9}-\frac{74\!\cdots\!42}{17\!\cdots\!75}a^{8}+\frac{60\!\cdots\!39}{33\!\cdots\!25}a^{7}-\frac{14\!\cdots\!43}{33\!\cdots\!25}a^{6}+\frac{10\!\cdots\!83}{66\!\cdots\!50}a^{5}-\frac{95\!\cdots\!94}{33\!\cdots\!25}a^{4}+\frac{26\!\cdots\!36}{33\!\cdots\!25}a^{3}-\frac{62\!\cdots\!74}{66\!\cdots\!05}a^{2}+\frac{12\!\cdots\!93}{66\!\cdots\!50}a-\frac{53\!\cdots\!81}{66\!\cdots\!50}$, $\frac{89\!\cdots\!91}{33\!\cdots\!25}a^{15}-\frac{11\!\cdots\!61}{13\!\cdots\!10}a^{14}+\frac{13\!\cdots\!71}{66\!\cdots\!50}a^{13}-\frac{98\!\cdots\!22}{33\!\cdots\!25}a^{12}+\frac{83\!\cdots\!53}{13\!\cdots\!10}a^{11}+\frac{64\!\cdots\!58}{33\!\cdots\!25}a^{10}+\frac{36\!\cdots\!51}{35\!\cdots\!50}a^{9}+\frac{11\!\cdots\!78}{33\!\cdots\!25}a^{8}+\frac{55\!\cdots\!87}{66\!\cdots\!50}a^{7}+\frac{31\!\cdots\!58}{33\!\cdots\!25}a^{6}+\frac{27\!\cdots\!87}{66\!\cdots\!50}a^{5}+\frac{44\!\cdots\!19}{33\!\cdots\!25}a^{4}-\frac{29\!\cdots\!27}{66\!\cdots\!50}a^{3}+\frac{13\!\cdots\!47}{13\!\cdots\!10}a^{2}-\frac{79\!\cdots\!09}{33\!\cdots\!25}a+\frac{12\!\cdots\!88}{33\!\cdots\!25}$, $\frac{1027592014}{68\!\cdots\!25}a^{15}-\frac{1541388021}{13\!\cdots\!05}a^{14}+\frac{103085825542}{68\!\cdots\!25}a^{13}-\frac{1106338548861}{13\!\cdots\!50}a^{12}+\frac{173563218385}{275742672533381}a^{11}-\frac{36074470821411}{13\!\cdots\!50}a^{10}+\frac{103331768894773}{68\!\cdots\!25}a^{9}-\frac{17804733750357}{362819305964975}a^{8}+\frac{15\!\cdots\!19}{68\!\cdots\!25}a^{7}-\frac{77\!\cdots\!81}{13\!\cdots\!50}a^{6}+\frac{14\!\cdots\!94}{68\!\cdots\!25}a^{5}-\frac{26\!\cdots\!99}{68\!\cdots\!25}a^{4}+\frac{75\!\cdots\!56}{68\!\cdots\!25}a^{3}-\frac{35\!\cdots\!73}{27\!\cdots\!10}a^{2}+\frac{18\!\cdots\!14}{68\!\cdots\!25}a-\frac{14\!\cdots\!41}{13\!\cdots\!50}$, $\frac{56\!\cdots\!17}{33\!\cdots\!25}a^{15}-\frac{26\!\cdots\!61}{13\!\cdots\!10}a^{14}+\frac{12\!\cdots\!47}{66\!\cdots\!50}a^{13}-\frac{95\!\cdots\!53}{66\!\cdots\!50}a^{12}+\frac{11\!\cdots\!33}{13\!\cdots\!10}a^{11}-\frac{15\!\cdots\!54}{33\!\cdots\!25}a^{10}+\frac{68\!\cdots\!59}{33\!\cdots\!25}a^{9}-\frac{62\!\cdots\!63}{66\!\cdots\!50}a^{8}+\frac{20\!\cdots\!09}{66\!\cdots\!50}a^{7}-\frac{38\!\cdots\!19}{33\!\cdots\!25}a^{6}+\frac{97\!\cdots\!92}{33\!\cdots\!25}a^{5}-\frac{61\!\cdots\!49}{66\!\cdots\!50}a^{4}+\frac{10\!\cdots\!81}{66\!\cdots\!50}a^{3}-\frac{56\!\cdots\!87}{13\!\cdots\!10}a^{2}+\frac{26\!\cdots\!79}{66\!\cdots\!50}a-\frac{16\!\cdots\!71}{18\!\cdots\!75}$, $\frac{1027592014}{68\!\cdots\!25}a^{15}-\frac{493215152001231}{49\!\cdots\!39}a^{14}+\frac{91\!\cdots\!82}{64\!\cdots\!25}a^{13}-\frac{83\!\cdots\!52}{12\!\cdots\!75}a^{12}+\frac{13\!\cdots\!26}{24\!\cdots\!95}a^{11}-\frac{25\!\cdots\!77}{12\!\cdots\!75}a^{10}+\frac{15\!\cdots\!42}{12\!\cdots\!75}a^{9}-\frac{85\!\cdots\!19}{24\!\cdots\!50}a^{8}+\frac{21\!\cdots\!86}{12\!\cdots\!75}a^{7}-\frac{40\!\cdots\!87}{12\!\cdots\!75}a^{6}+\frac{18\!\cdots\!31}{12\!\cdots\!75}a^{5}-\frac{34\!\cdots\!27}{24\!\cdots\!50}a^{4}+\frac{86\!\cdots\!59}{12\!\cdots\!75}a^{3}+\frac{44\!\cdots\!82}{24\!\cdots\!95}a^{2}+\frac{17\!\cdots\!61}{12\!\cdots\!75}a+\frac{74\!\cdots\!81}{24\!\cdots\!50}$ Copy content Toggle raw display (assuming GRH)
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:  \( 7114.135357253273 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7114.135357253273 \cdot 176800}{2\cdot\sqrt{24967943722642702336000000000000}}\cr\approx \mathstrut & 0.305718660128616 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 108*x^14 - 616*x^13 + 4916*x^12 - 21852*x^11 + 129218*x^10 - 459508*x^9 + 2174117*x^8 - 6156340*x^7 + 24095662*x^6 - 52453964*x^5 + 172046412*x^4 - 263040692*x^3 + 722028526*x^2 - 598345980*x + 1353436561);
 
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

$C_2^2\times C_4$ (as 16T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-94}) \), \(\Q(\sqrt{-47}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-470}) \), \(\Q(\sqrt{-235}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-47})\), \(\Q(\sqrt{5}, \sqrt{-94})\), \(\Q(\sqrt{10}, \sqrt{-94})\), \(\Q(\sqrt{5}, \sqrt{-47})\), \(\Q(\sqrt{10}, \sqrt{-47})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-235})\), 4.0.4418000.1, 4.4.8000.1, \(\Q(\zeta_{20})^+\), 4.0.17672000.5, 8.0.12491983360000.26, 8.0.4996793344000000.29, 8.0.4996793344000000.12, 8.0.19518724000000.17, 8.0.312299584000000.18, 8.0.4996793344000000.20, \(\Q(\zeta_{40})^+\)

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ R ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 2.8.16.3$x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{5} + 36 x^{4} - 32 x^{3} + 88 x^{2} - 32 x + 124$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{5} + 36 x^{4} - 32 x^{3} + 88 x^{2} - 32 x + 124$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(47\) Copy content Toggle raw display 47.8.4.1$x^{8} + 204 x^{6} + 80 x^{5} + 14080 x^{4} - 6880 x^{3} + 384824 x^{2} - 499680 x + 3453444$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47.8.4.1$x^{8} + 204 x^{6} + 80 x^{5} + 14080 x^{4} - 6880 x^{3} + 384824 x^{2} - 499680 x + 3453444$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$