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Properties

Label 16.0.286...081.2
Degree $16$
Signature $[0, 8]$
Discriminant $2.864\times 10^{31}$
Root discriminant \(92.48\)
Ramified primes $11,89$
Class number $226$ (GRH)
Class group [226] (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192)
 
gp: K = bnfinit(y^16 - 4*y^15 - 21*y^14 + 2*y^13 + 879*y^12 + 716*y^11 - 5620*y^10 - 41278*y^9 + 22417*y^8 + 112522*y^7 + 1007165*y^6 - 743188*y^5 - 438001*y^4 - 10586538*y^3 + 60329180*y^2 - 91102296*y + 50324192, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192)
 

\( x^{16} - 4 x^{15} - 21 x^{14} + 2 x^{13} + 879 x^{12} + 716 x^{11} - 5620 x^{10} - 41278 x^{9} + \cdots + 50324192 \) 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:   \(28643813255402702732772283462081\) \(\medspace = 11^{4}\cdot 89^{14}\) 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:  \(92.48\)
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:  $11^{1/2}89^{7/8}\approx 168.42734332479344$
Ramified primes:   \(11\), \(89\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{4}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}+\frac{7}{32}a^{5}-\frac{1}{8}a^{4}-\frac{11}{32}a^{3}+\frac{3}{8}a^{2}-\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{64}a^{12}-\frac{1}{64}a^{11}-\frac{1}{64}a^{10}-\frac{3}{64}a^{9}-\frac{1}{16}a^{8}+\frac{1}{16}a^{7}-\frac{5}{64}a^{6}-\frac{11}{64}a^{5}+\frac{9}{64}a^{4}+\frac{7}{64}a^{3}+\frac{1}{4}a^{2}-\frac{7}{16}a+\frac{1}{4}$, $\frac{1}{256}a^{13}-\frac{1}{128}a^{11}+\frac{3}{64}a^{10}+\frac{9}{256}a^{9}-\frac{1}{8}a^{8}-\frac{1}{256}a^{7}+\frac{1}{16}a^{6}+\frac{31}{128}a^{5}-\frac{1}{8}a^{4}-\frac{89}{256}a^{3}-\frac{27}{64}a^{2}+\frac{5}{64}a-\frac{7}{16}$, $\frac{1}{4352}a^{14}-\frac{1}{1088}a^{13}-\frac{9}{2176}a^{12}-\frac{11}{1088}a^{11}+\frac{9}{4352}a^{10}-\frac{41}{1088}a^{9}+\frac{15}{256}a^{8}+\frac{21}{1088}a^{7}-\frac{377}{2176}a^{6}+\frac{93}{544}a^{5}+\frac{951}{4352}a^{4}+\frac{107}{544}a^{3}+\frac{353}{1088}a^{2}-\frac{15}{68}a+\frac{15}{68}$, $\frac{1}{40\!\cdots\!32}a^{15}-\frac{35\!\cdots\!11}{40\!\cdots\!32}a^{14}+\frac{58\!\cdots\!37}{51\!\cdots\!04}a^{13}-\frac{28\!\cdots\!23}{20\!\cdots\!16}a^{12}+\frac{56\!\cdots\!65}{40\!\cdots\!32}a^{11}-\frac{37\!\cdots\!95}{40\!\cdots\!32}a^{10}+\frac{13\!\cdots\!33}{40\!\cdots\!32}a^{9}-\frac{23\!\cdots\!25}{40\!\cdots\!32}a^{8}-\frac{13\!\cdots\!83}{10\!\cdots\!08}a^{7}+\frac{38\!\cdots\!95}{20\!\cdots\!16}a^{6}+\frac{41\!\cdots\!55}{40\!\cdots\!32}a^{5}-\frac{65\!\cdots\!61}{40\!\cdots\!32}a^{4}-\frac{88\!\cdots\!29}{20\!\cdots\!16}a^{3}-\frac{15\!\cdots\!97}{10\!\cdots\!08}a^{2}-\frac{82\!\cdots\!69}{51\!\cdots\!04}a+\frac{16\!\cdots\!71}{45\!\cdots\!72}$ 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

$C_{226}$, which has order $226$ (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:   $\frac{17\!\cdots\!39}{35\!\cdots\!76}a^{15}+\frac{29\!\cdots\!91}{35\!\cdots\!76}a^{14}-\frac{40\!\cdots\!59}{22\!\cdots\!36}a^{13}-\frac{53\!\cdots\!85}{17\!\cdots\!88}a^{12}-\frac{80\!\cdots\!33}{35\!\cdots\!76}a^{11}+\frac{31\!\cdots\!83}{35\!\cdots\!76}a^{10}+\frac{96\!\cdots\!11}{35\!\cdots\!76}a^{9}-\frac{17\!\cdots\!91}{35\!\cdots\!76}a^{8}-\frac{38\!\cdots\!31}{88\!\cdots\!44}a^{7}-\frac{40\!\cdots\!75}{17\!\cdots\!88}a^{6}+\frac{57\!\cdots\!17}{35\!\cdots\!76}a^{5}+\frac{32\!\cdots\!05}{35\!\cdots\!76}a^{4}-\frac{18\!\cdots\!39}{17\!\cdots\!88}a^{3}+\frac{34\!\cdots\!65}{88\!\cdots\!44}a^{2}+\frac{66\!\cdots\!01}{44\!\cdots\!72}a-\frac{58\!\cdots\!63}{39\!\cdots\!96}$, $\frac{34\!\cdots\!31}{35\!\cdots\!76}a^{15}+\frac{68\!\cdots\!39}{35\!\cdots\!76}a^{14}-\frac{33\!\cdots\!43}{88\!\cdots\!44}a^{13}-\frac{12\!\cdots\!89}{17\!\cdots\!88}a^{12}-\frac{24\!\cdots\!25}{35\!\cdots\!76}a^{11}+\frac{71\!\cdots\!27}{35\!\cdots\!76}a^{10}+\frac{22\!\cdots\!39}{35\!\cdots\!76}a^{9}-\frac{35\!\cdots\!91}{35\!\cdots\!76}a^{8}-\frac{21\!\cdots\!33}{22\!\cdots\!36}a^{7}-\frac{97\!\cdots\!55}{17\!\cdots\!88}a^{6}+\frac{12\!\cdots\!77}{35\!\cdots\!76}a^{5}+\frac{73\!\cdots\!85}{35\!\cdots\!76}a^{4}-\frac{43\!\cdots\!29}{17\!\cdots\!88}a^{3}+\frac{14\!\cdots\!21}{88\!\cdots\!44}a^{2}+\frac{95\!\cdots\!67}{44\!\cdots\!72}a-\frac{15\!\cdots\!97}{39\!\cdots\!96}$, $\frac{11\!\cdots\!23}{12\!\cdots\!48}a^{15}+\frac{17\!\cdots\!63}{12\!\cdots\!48}a^{14}+\frac{10\!\cdots\!89}{15\!\cdots\!56}a^{13}-\frac{93\!\cdots\!05}{60\!\cdots\!24}a^{12}+\frac{25\!\cdots\!39}{12\!\cdots\!48}a^{11}+\frac{11\!\cdots\!23}{12\!\cdots\!48}a^{10}+\frac{14\!\cdots\!23}{12\!\cdots\!48}a^{9}-\frac{17\!\cdots\!71}{12\!\cdots\!48}a^{8}-\frac{28\!\cdots\!73}{30\!\cdots\!12}a^{7}-\frac{28\!\cdots\!47}{60\!\cdots\!24}a^{6}+\frac{10\!\cdots\!25}{12\!\cdots\!48}a^{5}+\frac{11\!\cdots\!37}{12\!\cdots\!48}a^{4}+\frac{68\!\cdots\!93}{60\!\cdots\!24}a^{3}-\frac{90\!\cdots\!79}{30\!\cdots\!12}a^{2}-\frac{46\!\cdots\!31}{15\!\cdots\!56}a+\frac{16\!\cdots\!53}{13\!\cdots\!08}$, $\frac{14\!\cdots\!33}{60\!\cdots\!24}a^{15}-\frac{16\!\cdots\!15}{60\!\cdots\!24}a^{14}-\frac{81\!\cdots\!71}{15\!\cdots\!56}a^{13}-\frac{62\!\cdots\!83}{30\!\cdots\!12}a^{12}+\frac{79\!\cdots\!33}{60\!\cdots\!24}a^{11}+\frac{43\!\cdots\!33}{60\!\cdots\!24}a^{10}+\frac{77\!\cdots\!77}{60\!\cdots\!24}a^{9}-\frac{35\!\cdots\!77}{60\!\cdots\!24}a^{8}-\frac{15\!\cdots\!65}{75\!\cdots\!28}a^{7}-\frac{15\!\cdots\!41}{30\!\cdots\!12}a^{6}+\frac{34\!\cdots\!39}{60\!\cdots\!24}a^{5}+\frac{52\!\cdots\!71}{60\!\cdots\!24}a^{4}+\frac{26\!\cdots\!37}{30\!\cdots\!12}a^{3}-\frac{36\!\cdots\!53}{15\!\cdots\!56}a^{2}+\frac{32\!\cdots\!97}{75\!\cdots\!28}a-\frac{89\!\cdots\!99}{66\!\cdots\!04}$, $\frac{49\!\cdots\!03}{20\!\cdots\!16}a^{15}-\frac{15\!\cdots\!25}{20\!\cdots\!16}a^{14}-\frac{14\!\cdots\!25}{25\!\cdots\!52}a^{13}-\frac{42\!\cdots\!85}{10\!\cdots\!08}a^{12}+\frac{43\!\cdots\!91}{20\!\cdots\!16}a^{11}+\frac{72\!\cdots\!91}{20\!\cdots\!16}a^{10}-\frac{23\!\cdots\!53}{20\!\cdots\!16}a^{9}-\frac{22\!\cdots\!75}{20\!\cdots\!16}a^{8}-\frac{18\!\cdots\!61}{51\!\cdots\!04}a^{7}+\frac{29\!\cdots\!13}{10\!\cdots\!08}a^{6}+\frac{56\!\cdots\!13}{20\!\cdots\!16}a^{5}+\frac{93\!\cdots\!57}{20\!\cdots\!16}a^{4}-\frac{16\!\cdots\!55}{10\!\cdots\!08}a^{3}-\frac{14\!\cdots\!39}{51\!\cdots\!04}a^{2}+\frac{31\!\cdots\!77}{25\!\cdots\!52}a-\frac{25\!\cdots\!19}{22\!\cdots\!36}$, $\frac{11\!\cdots\!97}{51\!\cdots\!04}a^{15}-\frac{68\!\cdots\!17}{51\!\cdots\!04}a^{14}-\frac{10\!\cdots\!03}{25\!\cdots\!52}a^{13}+\frac{54\!\cdots\!59}{25\!\cdots\!52}a^{12}+\frac{64\!\cdots\!25}{51\!\cdots\!04}a^{11}-\frac{35\!\cdots\!61}{51\!\cdots\!04}a^{10}-\frac{11\!\cdots\!73}{51\!\cdots\!04}a^{9}-\frac{63\!\cdots\!43}{51\!\cdots\!04}a^{8}+\frac{34\!\cdots\!73}{25\!\cdots\!52}a^{7}+\frac{14\!\cdots\!85}{25\!\cdots\!52}a^{6}-\frac{13\!\cdots\!57}{51\!\cdots\!04}a^{5}+\frac{12\!\cdots\!17}{51\!\cdots\!04}a^{4}-\frac{25\!\cdots\!25}{12\!\cdots\!76}a^{3}+\frac{78\!\cdots\!35}{12\!\cdots\!76}a^{2}-\frac{61\!\cdots\!55}{79\!\cdots\!36}a+\frac{10\!\cdots\!55}{28\!\cdots\!92}$, $\frac{51\!\cdots\!33}{20\!\cdots\!16}a^{15}-\frac{42\!\cdots\!59}{20\!\cdots\!16}a^{14}+\frac{18\!\cdots\!43}{12\!\cdots\!76}a^{13}+\frac{10\!\cdots\!09}{10\!\cdots\!08}a^{12}+\frac{31\!\cdots\!53}{20\!\cdots\!16}a^{11}-\frac{89\!\cdots\!11}{20\!\cdots\!16}a^{10}-\frac{18\!\cdots\!07}{20\!\cdots\!16}a^{9}-\frac{61\!\cdots\!57}{20\!\cdots\!16}a^{8}+\frac{86\!\cdots\!35}{51\!\cdots\!04}a^{7}+\frac{48\!\cdots\!63}{10\!\cdots\!08}a^{6}+\frac{18\!\cdots\!75}{20\!\cdots\!16}a^{5}-\frac{70\!\cdots\!73}{20\!\cdots\!16}a^{4}-\frac{36\!\cdots\!73}{10\!\cdots\!08}a^{3}+\frac{15\!\cdots\!67}{51\!\cdots\!04}a^{2}-\frac{11\!\cdots\!53}{25\!\cdots\!52}a+\frac{50\!\cdots\!59}{22\!\cdots\!36}$ 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:  \( 1882188039.33 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1882188039.33 \cdot 226}{2\cdot\sqrt{28643813255402702732772283462081}}\cr\approx \mathstrut & 96.5305949308 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192)
 
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 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192, 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 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192);
 
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 - 4*x^15 - 21*x^14 + 2*x^13 + 879*x^12 + 716*x^11 - 5620*x^10 - 41278*x^9 + 22417*x^8 + 112522*x^7 + 1007165*x^6 - 743188*x^5 - 438001*x^4 - 10586538*x^3 + 60329180*x^2 - 91102296*x + 50324192);
 
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

$C_2^2:C_8$ (as 16T24):

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)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 4.2.87131.1, 4.2.7754659.1, 8.0.44231334895529.1, 8.4.5351991522359009.1, 8.4.60134736206281.2

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]
 

Sibling fields

Galois closure: deg 32
Degree 16 sibling: 16.8.419374069872350970710519002168327921.1
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 ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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

# 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
\(11\) Copy content Toggle raw display 11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(89\) Copy content Toggle raw display 89.8.7.3$x^{8} + 89$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} + 89$$8$$1$$7$$C_8$$[\ ]_{8}$