Properties

Label 16.0.107...801.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.077\times 10^{30}$
Root discriminant \(75.34\)
Ramified primes $41,73$
Class number $33$ (GRH)
Class group [33] (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 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424)
 
gp: K = bnfinit(y^16 - 5*y^15 + 20*y^14 - 53*y^13 + 425*y^12 - 1761*y^11 + 8234*y^10 - 26439*y^9 + 85127*y^8 - 227294*y^7 + 576600*y^6 - 1117344*y^5 + 2015984*y^4 - 2860288*y^3 + 4678656*y^2 - 5948416*y + 5607424, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424)
 

\( x^{16} - 5 x^{15} + 20 x^{14} - 53 x^{13} + 425 x^{12} - 1761 x^{11} + 8234 x^{10} - 26439 x^{9} + 85127 x^{8} - 227294 x^{7} + 576600 x^{6} - 1117344 x^{5} + \cdots + 5607424 \) 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:   \(1077123334824966013513439398801\) \(\medspace = 41^{14}\cdot 73^{4}\) 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:  \(75.34\)
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:  $41^{7/8}73^{1/2}\approx 220.21520635937404$
Ramified primes:   \(41\), \(73\) 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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{3}{8}a^{8}-\frac{3}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}+\frac{3}{16}a^{9}-\frac{3}{16}a^{8}+\frac{3}{16}a^{7}-\frac{1}{8}a^{6}-\frac{7}{16}a^{5}+\frac{3}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{5}{64}a^{10}-\frac{11}{64}a^{9}+\frac{3}{64}a^{8}+\frac{3}{32}a^{7}+\frac{1}{64}a^{6}+\frac{11}{64}a^{5}-\frac{1}{32}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{468224}a^{14}-\frac{673}{468224}a^{13}-\frac{167}{14632}a^{12}+\frac{197}{15104}a^{11}+\frac{17077}{468224}a^{10}+\frac{24099}{468224}a^{9}+\frac{15491}{234112}a^{8}-\frac{199583}{468224}a^{7}+\frac{96395}{468224}a^{6}+\frac{116911}{234112}a^{5}+\frac{6463}{14632}a^{4}-\frac{655}{3658}a^{3}-\frac{5793}{29264}a^{2}+\frac{2559}{7316}a+\frac{564}{1829}$, $\frac{1}{14\!\cdots\!28}a^{15}-\frac{77\!\cdots\!77}{14\!\cdots\!28}a^{14}+\frac{59\!\cdots\!97}{37\!\cdots\!32}a^{13}-\frac{53\!\cdots\!17}{14\!\cdots\!28}a^{12}+\frac{86\!\cdots\!45}{14\!\cdots\!28}a^{11}+\frac{15\!\cdots\!43}{14\!\cdots\!28}a^{10}-\frac{13\!\cdots\!99}{74\!\cdots\!64}a^{9}-\frac{15\!\cdots\!91}{14\!\cdots\!28}a^{8}+\frac{63\!\cdots\!15}{14\!\cdots\!28}a^{7}+\frac{76\!\cdots\!85}{74\!\cdots\!64}a^{6}+\frac{16\!\cdots\!37}{18\!\cdots\!16}a^{5}+\frac{91\!\cdots\!89}{23\!\cdots\!52}a^{4}+\frac{12\!\cdots\!51}{92\!\cdots\!08}a^{3}+\frac{99\!\cdots\!93}{58\!\cdots\!88}a^{2}-\frac{21\!\cdots\!79}{58\!\cdots\!88}a+\frac{47\!\cdots\!60}{39\!\cdots\!31}$ 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

$C_{33}$, which has order $33$ (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:   $\frac{78\!\cdots\!87}{35\!\cdots\!52}a^{15}-\frac{70\!\cdots\!01}{35\!\cdots\!52}a^{14}+\frac{12\!\cdots\!17}{17\!\cdots\!76}a^{13}-\frac{57\!\cdots\!51}{35\!\cdots\!52}a^{12}+\frac{30\!\cdots\!85}{35\!\cdots\!52}a^{11}-\frac{22\!\cdots\!61}{35\!\cdots\!52}a^{10}+\frac{23\!\cdots\!43}{89\!\cdots\!88}a^{9}-\frac{31\!\cdots\!01}{35\!\cdots\!52}a^{8}+\frac{77\!\cdots\!31}{35\!\cdots\!52}a^{7}-\frac{56\!\cdots\!03}{89\!\cdots\!88}a^{6}+\frac{12\!\cdots\!69}{89\!\cdots\!88}a^{5}-\frac{30\!\cdots\!71}{11\!\cdots\!36}a^{4}+\frac{76\!\cdots\!57}{22\!\cdots\!72}a^{3}-\frac{22\!\cdots\!43}{36\!\cdots\!56}a^{2}+\frac{25\!\cdots\!21}{28\!\cdots\!84}a-\frac{86\!\cdots\!28}{70\!\cdots\!71}$, $\frac{68\!\cdots\!63}{17\!\cdots\!76}a^{15}-\frac{36\!\cdots\!81}{17\!\cdots\!76}a^{14}+\frac{56\!\cdots\!57}{89\!\cdots\!88}a^{13}-\frac{28\!\cdots\!79}{17\!\cdots\!76}a^{12}+\frac{24\!\cdots\!05}{17\!\cdots\!76}a^{11}-\frac{11\!\cdots\!61}{17\!\cdots\!76}a^{10}+\frac{11\!\cdots\!35}{44\!\cdots\!44}a^{9}-\frac{15\!\cdots\!61}{17\!\cdots\!76}a^{8}+\frac{41\!\cdots\!63}{17\!\cdots\!76}a^{7}-\frac{29\!\cdots\!67}{44\!\cdots\!44}a^{6}+\frac{64\!\cdots\!47}{44\!\cdots\!44}a^{5}-\frac{15\!\cdots\!29}{56\!\cdots\!68}a^{4}+\frac{39\!\cdots\!17}{11\!\cdots\!36}a^{3}-\frac{11\!\cdots\!19}{18\!\cdots\!28}a^{2}+\frac{12\!\cdots\!33}{14\!\cdots\!42}a-\frac{16\!\cdots\!17}{70\!\cdots\!71}$, $\frac{37\!\cdots\!87}{35\!\cdots\!52}a^{15}-\frac{28\!\cdots\!65}{35\!\cdots\!52}a^{14}+\frac{49\!\cdots\!65}{17\!\cdots\!76}a^{13}-\frac{22\!\cdots\!11}{35\!\cdots\!52}a^{12}+\frac{14\!\cdots\!65}{35\!\cdots\!52}a^{11}-\frac{92\!\cdots\!05}{35\!\cdots\!52}a^{10}+\frac{94\!\cdots\!99}{89\!\cdots\!88}a^{9}-\frac{12\!\cdots\!25}{35\!\cdots\!52}a^{8}+\frac{31\!\cdots\!19}{35\!\cdots\!52}a^{7}-\frac{22\!\cdots\!43}{89\!\cdots\!88}a^{6}+\frac{50\!\cdots\!01}{89\!\cdots\!88}a^{5}-\frac{12\!\cdots\!71}{11\!\cdots\!36}a^{4}+\frac{30\!\cdots\!05}{22\!\cdots\!72}a^{3}-\frac{89\!\cdots\!67}{36\!\cdots\!56}a^{2}+\frac{10\!\cdots\!29}{28\!\cdots\!84}a-\frac{56\!\cdots\!43}{70\!\cdots\!71}$, $\frac{28\!\cdots\!65}{74\!\cdots\!64}a^{15}-\frac{55\!\cdots\!63}{74\!\cdots\!64}a^{14}+\frac{95\!\cdots\!75}{37\!\cdots\!32}a^{13}-\frac{50\!\cdots\!37}{74\!\cdots\!64}a^{12}+\frac{11\!\cdots\!91}{74\!\cdots\!64}a^{11}-\frac{18\!\cdots\!19}{74\!\cdots\!64}a^{10}+\frac{15\!\cdots\!11}{18\!\cdots\!16}a^{9}-\frac{25\!\cdots\!95}{74\!\cdots\!64}a^{8}+\frac{73\!\cdots\!21}{74\!\cdots\!64}a^{7}-\frac{52\!\cdots\!75}{18\!\cdots\!16}a^{6}+\frac{12\!\cdots\!37}{18\!\cdots\!16}a^{5}-\frac{71\!\cdots\!93}{46\!\cdots\!04}a^{4}+\frac{79\!\cdots\!91}{46\!\cdots\!04}a^{3}-\frac{60\!\cdots\!95}{23\!\cdots\!52}a^{2}+\frac{67\!\cdots\!98}{14\!\cdots\!47}a-\frac{26\!\cdots\!23}{39\!\cdots\!31}$, $\frac{40\!\cdots\!01}{74\!\cdots\!64}a^{15}+\frac{54\!\cdots\!19}{74\!\cdots\!64}a^{14}-\frac{11\!\cdots\!65}{46\!\cdots\!04}a^{13}+\frac{48\!\cdots\!71}{74\!\cdots\!64}a^{12}-\frac{93\!\cdots\!15}{74\!\cdots\!64}a^{11}+\frac{18\!\cdots\!91}{74\!\cdots\!64}a^{10}-\frac{27\!\cdots\!61}{37\!\cdots\!32}a^{9}+\frac{24\!\cdots\!05}{74\!\cdots\!64}a^{8}-\frac{68\!\cdots\!05}{74\!\cdots\!64}a^{7}+\frac{10\!\cdots\!87}{37\!\cdots\!32}a^{6}-\frac{58\!\cdots\!81}{92\!\cdots\!08}a^{5}+\frac{66\!\cdots\!49}{46\!\cdots\!04}a^{4}-\frac{73\!\cdots\!77}{46\!\cdots\!04}a^{3}+\frac{28\!\cdots\!03}{11\!\cdots\!76}a^{2}-\frac{24\!\cdots\!65}{58\!\cdots\!88}a+\frac{24\!\cdots\!97}{39\!\cdots\!31}$, $\frac{21\!\cdots\!31}{74\!\cdots\!64}a^{15}-\frac{41\!\cdots\!69}{74\!\cdots\!64}a^{14}+\frac{65\!\cdots\!71}{37\!\cdots\!32}a^{13}-\frac{38\!\cdots\!79}{74\!\cdots\!64}a^{12}+\frac{10\!\cdots\!29}{74\!\cdots\!64}a^{11}-\frac{12\!\cdots\!53}{74\!\cdots\!64}a^{10}+\frac{52\!\cdots\!87}{92\!\cdots\!08}a^{9}-\frac{18\!\cdots\!49}{74\!\cdots\!64}a^{8}+\frac{41\!\cdots\!15}{74\!\cdots\!64}a^{7}-\frac{64\!\cdots\!23}{46\!\cdots\!04}a^{6}+\frac{13\!\cdots\!33}{59\!\cdots\!36}a^{5}-\frac{22\!\cdots\!71}{11\!\cdots\!76}a^{4}-\frac{50\!\cdots\!73}{78\!\cdots\!56}a^{3}+\frac{67\!\cdots\!65}{23\!\cdots\!52}a^{2}-\frac{34\!\cdots\!59}{58\!\cdots\!88}a+\frac{20\!\cdots\!95}{39\!\cdots\!31}$, $\frac{29\!\cdots\!23}{74\!\cdots\!64}a^{15}-\frac{35\!\cdots\!97}{74\!\cdots\!64}a^{14}+\frac{87\!\cdots\!37}{37\!\cdots\!32}a^{13}-\frac{25\!\cdots\!97}{12\!\cdots\!96}a^{12}+\frac{94\!\cdots\!17}{74\!\cdots\!64}a^{11}-\frac{12\!\cdots\!57}{74\!\cdots\!64}a^{10}+\frac{26\!\cdots\!57}{18\!\cdots\!16}a^{9}-\frac{14\!\cdots\!53}{74\!\cdots\!64}a^{8}+\frac{67\!\cdots\!15}{74\!\cdots\!64}a^{7}-\frac{27\!\cdots\!85}{18\!\cdots\!16}a^{6}+\frac{26\!\cdots\!79}{59\!\cdots\!36}a^{5}-\frac{13\!\cdots\!49}{11\!\cdots\!76}a^{4}+\frac{60\!\cdots\!01}{46\!\cdots\!04}a^{3}+\frac{35\!\cdots\!75}{23\!\cdots\!52}a^{2}+\frac{15\!\cdots\!63}{58\!\cdots\!88}a+\frac{79\!\cdots\!42}{39\!\cdots\!31}$ 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:  \( 4516212.78969 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4516212.78969 \cdot 33}{2\cdot\sqrt{1077123334824966013513439398801}}\cr\approx \mathstrut & 0.174407264376 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424)
 
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 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424, 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 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424);
 
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 - 5*x^15 + 20*x^14 - 53*x^13 + 425*x^12 - 1761*x^11 + 8234*x^10 - 26439*x^9 + 85127*x^8 - 227294*x^7 + 576600*x^6 - 1117344*x^5 + 2015984*x^4 - 2860288*x^3 + 4678656*x^2 - 5948416*x + 5607424);
 
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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.0.5031233.1, 4.0.122713.1, 8.0.194754273881.1, 8.8.1037845525511849.1, 8.0.25313305500289.1

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: 16.0.30588408049083077668563908786045909041.1
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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ R ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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
\(41\) Copy content Toggle raw display 41.16.14.1$x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776882 x^{8} + 5977813379504 x^{7} + 3161920977824 x^{6} + 978514601120 x^{5} + 196936323920 x^{4} + 202153692608 x^{3} + 3372805705856 x^{2} + 36445904670848 x + 172395305267889$$8$$2$$14$$C_8\times C_2$$[\ ]_{8}^{2}$
\(73\) Copy content Toggle raw display 73.4.0.1$x^{4} + 16 x^{2} + 56 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
73.4.0.1$x^{4} + 16 x^{2} + 56 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
73.8.4.1$x^{8} + 15768 x^{7} + 93236508 x^{6} + 245028523288 x^{5} + 241487187554464 x^{4} + 21812691357056 x^{3} + 3877844534238648 x^{2} + 13666761747168624 x + 1459074653762756$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$