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Properties

Label 16.0.297...241.29
Degree $16$
Signature $[0, 8]$
Discriminant $2.970\times 10^{33}$
Root discriminant \(123.61\)
Ramified primes $23,41$
Class number $384$ (GRH)
Class group [2, 4, 48] (GRH)
Galois group $\OD_{16}:C_2$ (as 16T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200)
 
gp: K = bnfinit(y^16 - 6*y^15 + 68*y^14 - 444*y^13 + 1922*y^12 - 9518*y^11 + 46905*y^10 + 40593*y^9 + 619392*y^8 + 556314*y^7 + 7635373*y^6 + 13465847*y^5 + 31633951*y^4 + 112760966*y^3 + 149722596*y^2 + 125046840*y + 534849200, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200)
 

\( x^{16} - 6 x^{15} + 68 x^{14} - 444 x^{13} + 1922 x^{12} - 9518 x^{11} + 46905 x^{10} + 40593 x^{9} + \cdots + 534849200 \) 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:   \(2970275152580737243393920061992241\) \(\medspace = 23^{8}\cdot 41^{14}\) 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:  \(123.61\)
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:  $23^{1/2}41^{7/8}\approx 123.60891451503595$
Ramified primes:   \(23\), \(41\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{460}a^{12}+\frac{5}{23}a^{11}+\frac{111}{460}a^{10}+\frac{27}{230}a^{9}+\frac{179}{460}a^{8}-\frac{1}{2}a^{7}+\frac{1}{46}a^{6}-\frac{189}{460}a^{5}-\frac{53}{115}a^{4}+\frac{11}{92}a^{3}+\frac{99}{460}a^{2}+\frac{71}{230}a-\frac{1}{23}$, $\frac{1}{2300}a^{13}+\frac{1}{2300}a^{11}-\frac{233}{1150}a^{10}-\frac{37}{100}a^{9}+\frac{27}{230}a^{8}-\frac{11}{115}a^{7}-\frac{729}{2300}a^{6}-\frac{273}{575}a^{5}-\frac{27}{460}a^{4}+\frac{1039}{2300}a^{3}+\frac{148}{575}a^{2}-\frac{111}{230}a-\frac{3}{23}$, $\frac{1}{1761137485000}a^{14}+\frac{48224519}{1761137485000}a^{13}+\frac{1085457781}{1761137485000}a^{12}-\frac{130490440197}{1761137485000}a^{11}+\frac{3352638683}{14089099880}a^{10}-\frac{758829522829}{1761137485000}a^{9}+\frac{74722575873}{176113748500}a^{8}-\frac{262610039159}{1761137485000}a^{7}+\frac{530791737657}{1761137485000}a^{6}+\frac{123988199397}{1761137485000}a^{5}-\frac{19089687392}{220142185625}a^{4}-\frac{518349878467}{1761137485000}a^{3}-\frac{88200901273}{440284371250}a^{2}+\frac{15161635491}{88056874250}a+\frac{228299208}{8805687425}$, $\frac{1}{20\!\cdots\!00}a^{15}+\frac{54\!\cdots\!76}{25\!\cdots\!25}a^{14}-\frac{57\!\cdots\!07}{51\!\cdots\!50}a^{13}+\frac{52\!\cdots\!31}{10\!\cdots\!00}a^{12}+\frac{12\!\cdots\!21}{10\!\cdots\!00}a^{11}-\frac{11\!\cdots\!51}{51\!\cdots\!50}a^{10}-\frac{11\!\cdots\!37}{89\!\cdots\!00}a^{9}-\frac{96\!\cdots\!39}{20\!\cdots\!00}a^{8}-\frac{66\!\cdots\!81}{27\!\cdots\!00}a^{7}-\frac{67\!\cdots\!03}{20\!\cdots\!00}a^{6}+\frac{10\!\cdots\!47}{20\!\cdots\!00}a^{5}-\frac{83\!\cdots\!83}{55\!\cdots\!00}a^{4}-\frac{92\!\cdots\!01}{40\!\cdots\!00}a^{3}-\frac{11\!\cdots\!59}{10\!\cdots\!00}a^{2}-\frac{30\!\cdots\!23}{51\!\cdots\!25}a+\frac{33\!\cdots\!52}{33\!\cdots\!75}$ 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

$C_{2}\times C_{4}\times C_{48}$, which has order $384$ (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:   $\frac{18\!\cdots\!17}{72\!\cdots\!50}a^{15}-\frac{35\!\cdots\!31}{14\!\cdots\!00}a^{14}+\frac{37\!\cdots\!91}{14\!\cdots\!00}a^{13}-\frac{55\!\cdots\!97}{28\!\cdots\!00}a^{12}+\frac{15\!\cdots\!19}{14\!\cdots\!00}a^{11}-\frac{77\!\cdots\!11}{14\!\cdots\!00}a^{10}+\frac{36\!\cdots\!53}{14\!\cdots\!00}a^{9}-\frac{39\!\cdots\!33}{72\!\cdots\!50}a^{8}+\frac{31\!\cdots\!81}{14\!\cdots\!00}a^{7}-\frac{22\!\cdots\!91}{14\!\cdots\!00}a^{6}+\frac{36\!\cdots\!57}{14\!\cdots\!00}a^{5}-\frac{22\!\cdots\!53}{72\!\cdots\!50}a^{4}+\frac{10\!\cdots\!81}{14\!\cdots\!00}a^{3}+\frac{38\!\cdots\!07}{72\!\cdots\!50}a^{2}-\frac{39\!\cdots\!12}{72\!\cdots\!25}a+\frac{11\!\cdots\!53}{46\!\cdots\!75}$, $\frac{39\!\cdots\!89}{14\!\cdots\!00}a^{15}-\frac{42\!\cdots\!27}{28\!\cdots\!00}a^{14}+\frac{46\!\cdots\!47}{28\!\cdots\!00}a^{13}-\frac{61\!\cdots\!09}{57\!\cdots\!00}a^{12}+\frac{11\!\cdots\!23}{28\!\cdots\!00}a^{11}-\frac{59\!\cdots\!87}{28\!\cdots\!00}a^{10}+\frac{33\!\cdots\!01}{28\!\cdots\!00}a^{9}+\frac{27\!\cdots\!39}{14\!\cdots\!00}a^{8}+\frac{46\!\cdots\!77}{28\!\cdots\!00}a^{7}-\frac{20\!\cdots\!47}{28\!\cdots\!00}a^{6}+\frac{26\!\cdots\!69}{28\!\cdots\!00}a^{5}+\frac{52\!\cdots\!99}{14\!\cdots\!00}a^{4}-\frac{88\!\cdots\!23}{28\!\cdots\!00}a^{3}-\frac{90\!\cdots\!81}{14\!\cdots\!00}a^{2}+\frac{20\!\cdots\!98}{72\!\cdots\!25}a-\frac{25\!\cdots\!12}{46\!\cdots\!75}$, $\frac{11\!\cdots\!87}{14\!\cdots\!00}a^{15}-\frac{13\!\cdots\!41}{28\!\cdots\!00}a^{14}+\frac{13\!\cdots\!01}{28\!\cdots\!00}a^{13}-\frac{18\!\cdots\!27}{57\!\cdots\!00}a^{12}+\frac{34\!\cdots\!09}{28\!\cdots\!00}a^{11}-\frac{16\!\cdots\!21}{28\!\cdots\!00}a^{10}+\frac{88\!\cdots\!83}{28\!\cdots\!00}a^{9}+\frac{88\!\cdots\!37}{14\!\cdots\!00}a^{8}+\frac{10\!\cdots\!91}{28\!\cdots\!00}a^{7}+\frac{14\!\cdots\!99}{28\!\cdots\!00}a^{6}+\frac{10\!\cdots\!27}{28\!\cdots\!00}a^{5}+\frac{11\!\cdots\!17}{14\!\cdots\!00}a^{4}+\frac{20\!\cdots\!91}{28\!\cdots\!00}a^{3}+\frac{23\!\cdots\!77}{14\!\cdots\!00}a^{2}+\frac{26\!\cdots\!34}{72\!\cdots\!25}a-\frac{23\!\cdots\!71}{46\!\cdots\!75}$, $\frac{39\!\cdots\!61}{51\!\cdots\!50}a^{15}-\frac{48\!\cdots\!73}{10\!\cdots\!00}a^{14}+\frac{49\!\cdots\!53}{10\!\cdots\!00}a^{13}-\frac{16\!\cdots\!79}{51\!\cdots\!25}a^{12}+\frac{12\!\cdots\!77}{10\!\cdots\!00}a^{11}-\frac{29\!\cdots\!69}{51\!\cdots\!50}a^{10}+\frac{31\!\cdots\!99}{10\!\cdots\!00}a^{9}+\frac{61\!\cdots\!47}{10\!\cdots\!00}a^{8}+\frac{93\!\cdots\!29}{27\!\cdots\!00}a^{7}-\frac{18\!\cdots\!53}{10\!\cdots\!00}a^{6}+\frac{19\!\cdots\!53}{51\!\cdots\!50}a^{5}+\frac{62\!\cdots\!24}{69\!\cdots\!25}a^{4}+\frac{78\!\cdots\!87}{25\!\cdots\!25}a^{3}+\frac{21\!\cdots\!87}{10\!\cdots\!00}a^{2}+\frac{27\!\cdots\!54}{51\!\cdots\!25}a-\frac{16\!\cdots\!76}{33\!\cdots\!75}$, $\frac{13\!\cdots\!71}{20\!\cdots\!00}a^{15}-\frac{52\!\cdots\!63}{20\!\cdots\!00}a^{14}+\frac{73\!\cdots\!23}{20\!\cdots\!00}a^{13}-\frac{43\!\cdots\!09}{20\!\cdots\!00}a^{12}+\frac{12\!\cdots\!89}{20\!\cdots\!00}a^{11}-\frac{83\!\cdots\!09}{20\!\cdots\!00}a^{10}+\frac{20\!\cdots\!89}{10\!\cdots\!00}a^{9}+\frac{18\!\cdots\!01}{20\!\cdots\!00}a^{8}+\frac{57\!\cdots\!63}{11\!\cdots\!00}a^{7}+\frac{16\!\cdots\!03}{20\!\cdots\!00}a^{6}+\frac{59\!\cdots\!53}{20\!\cdots\!00}a^{5}+\frac{21\!\cdots\!47}{44\!\cdots\!80}a^{4}-\frac{49\!\cdots\!89}{10\!\cdots\!00}a^{3}+\frac{15\!\cdots\!87}{10\!\cdots\!00}a^{2}+\frac{34\!\cdots\!69}{51\!\cdots\!25}a-\frac{19\!\cdots\!66}{33\!\cdots\!75}$, $\frac{21\!\cdots\!67}{66\!\cdots\!00}a^{15}-\frac{15\!\cdots\!83}{66\!\cdots\!00}a^{14}+\frac{16\!\cdots\!13}{66\!\cdots\!00}a^{13}-\frac{23\!\cdots\!47}{13\!\cdots\!00}a^{12}+\frac{55\!\cdots\!07}{66\!\cdots\!00}a^{11}-\frac{26\!\cdots\!43}{66\!\cdots\!00}a^{10}+\frac{16\!\cdots\!23}{83\!\cdots\!75}a^{9}-\frac{32\!\cdots\!71}{28\!\cdots\!00}a^{8}+\frac{12\!\cdots\!23}{66\!\cdots\!00}a^{7}-\frac{37\!\cdots\!43}{66\!\cdots\!00}a^{6}+\frac{17\!\cdots\!07}{83\!\cdots\!75}a^{5}+\frac{10\!\cdots\!77}{66\!\cdots\!00}a^{4}+\frac{12\!\cdots\!69}{33\!\cdots\!00}a^{3}+\frac{52\!\cdots\!23}{16\!\cdots\!50}a^{2}-\frac{67\!\cdots\!61}{33\!\cdots\!50}a+\frac{84\!\cdots\!92}{10\!\cdots\!25}$, $\frac{48\!\cdots\!07}{20\!\cdots\!00}a^{15}-\frac{20\!\cdots\!37}{20\!\cdots\!00}a^{14}+\frac{20\!\cdots\!37}{20\!\cdots\!00}a^{13}-\frac{11\!\cdots\!49}{20\!\cdots\!00}a^{12}+\frac{17\!\cdots\!83}{40\!\cdots\!00}a^{11}-\frac{15\!\cdots\!53}{20\!\cdots\!00}a^{10}+\frac{26\!\cdots\!67}{10\!\cdots\!50}a^{9}+\frac{96\!\cdots\!87}{20\!\cdots\!00}a^{8}+\frac{50\!\cdots\!17}{55\!\cdots\!00}a^{7}-\frac{37\!\cdots\!11}{20\!\cdots\!00}a^{6}+\frac{17\!\cdots\!29}{10\!\cdots\!00}a^{5}+\frac{15\!\cdots\!23}{55\!\cdots\!00}a^{4}-\frac{23\!\cdots\!77}{10\!\cdots\!00}a^{3}+\frac{75\!\cdots\!89}{10\!\cdots\!50}a^{2}+\frac{27\!\cdots\!33}{20\!\cdots\!50}a-\frac{22\!\cdots\!52}{66\!\cdots\!95}$ 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:  \( 31804349.3331 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 31804349.3331 \cdot 384}{2\cdot\sqrt{2970275152580737243393920061992241}}\cr\approx \mathstrut & 0.272162540046 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200)
 
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 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200, 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 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200);
 
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 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200);
 
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

$\OD_{16}:C_2$ (as 16T36):

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 11 conjugacy class representatives for $\OD_{16}:C_2$
Character table for $\OD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-943}) \), 4.0.36459209.2, 4.4.68921.1, \(\Q(\sqrt{-23}, \sqrt{41})\), 8.0.54500230757132921.4 x2, 8.4.103025010883049.2 x2, 8.0.1329273920905681.3

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 8 siblings: 8.0.54500230757132921.4, 8.4.103025010883049.2
Degree 16 siblings: 16.4.5614886866882301027209678756129.7, 16.0.2970275152580737243393920061992241.24
Minimal sibling: 8.4.103025010883049.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ R ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ R ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(23\) Copy content Toggle raw display 23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(41\) Copy content Toggle raw display 41.8.7.3$x^{8} + 41$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} + 41$$8$$1$$7$$C_8$$[\ ]_{8}$