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Properties

Label 16.0.341...969.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.415\times 10^{31}$
Root discriminant \(93.51\)
Ramified primes $23,73$
Class number $89$ (GRH)
Class group [89] (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 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861)
 
gp: K = bnfinit(y^16 - 3*y^15 - 44*y^14 + 209*y^13 + 1029*y^12 - 6688*y^11 - 4322*y^10 + 83350*y^9 - 98968*y^8 - 306130*y^7 + 776806*y^6 - 425376*y^5 + 39313*y^4 - 397603*y^3 + 294148*y^2 - 10639*y + 137861, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861)
 

\( x^{16} - 3 x^{15} - 44 x^{14} + 209 x^{13} + 1029 x^{12} - 6688 x^{11} - 4322 x^{10} + 83350 x^{9} + \cdots + 137861 \) 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:   \(34153198773896725121035596949969\) \(\medspace = 23^{4}\cdot 73^{14}\) 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:  \(93.51\)
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:  $23^{1/2}73^{7/8}\approx 204.77240010459101$
Ramified primes:   \(23\), \(73\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}+\frac{3}{16}a^{3}-\frac{3}{8}a^{2}-\frac{3}{8}a-\frac{3}{16}$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{8}-\frac{1}{16}a^{7}-\frac{1}{8}a^{6}+\frac{3}{16}a^{4}-\frac{3}{8}a^{2}-\frac{3}{16}a-\frac{3}{8}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{3}{16}a^{5}-\frac{3}{16}a^{2}-\frac{3}{8}a-\frac{3}{8}$, $\frac{1}{32}a^{12}+\frac{1}{16}a^{6}-\frac{3}{32}$, $\frac{1}{192}a^{13}-\frac{1}{64}a^{12}-\frac{1}{96}a^{11}-\frac{1}{32}a^{10}+\frac{1}{32}a^{8}+\frac{1}{48}a^{7}+\frac{5}{96}a^{6}+\frac{13}{96}a^{5}-\frac{3}{32}a^{4}+\frac{1}{6}a^{3}+\frac{25}{96}a^{2}+\frac{67}{192}a+\frac{25}{192}$, $\frac{1}{576}a^{14}-\frac{1}{576}a^{13}-\frac{1}{288}a^{12}-\frac{5}{288}a^{11}+\frac{1}{48}a^{10}+\frac{1}{96}a^{9}+\frac{1}{36}a^{8}-\frac{3}{32}a^{7}-\frac{19}{288}a^{6}-\frac{31}{288}a^{5}-\frac{31}{144}a^{4}-\frac{13}{96}a^{3}-\frac{121}{576}a^{2}-\frac{19}{192}a-\frac{1}{9}$, $\frac{1}{68\!\cdots\!12}a^{15}+\frac{40\!\cdots\!35}{17\!\cdots\!28}a^{14}+\frac{70\!\cdots\!75}{34\!\cdots\!56}a^{13}+\frac{76\!\cdots\!29}{68\!\cdots\!12}a^{12}+\frac{10\!\cdots\!93}{86\!\cdots\!64}a^{11}-\frac{39\!\cdots\!83}{57\!\cdots\!76}a^{10}-\frac{62\!\cdots\!11}{34\!\cdots\!56}a^{9}+\frac{10\!\cdots\!01}{86\!\cdots\!64}a^{8}-\frac{13\!\cdots\!13}{21\!\cdots\!16}a^{7}-\frac{10\!\cdots\!81}{12\!\cdots\!28}a^{6}+\frac{68\!\cdots\!99}{86\!\cdots\!64}a^{5}-\frac{38\!\cdots\!47}{17\!\cdots\!28}a^{4}+\frac{12\!\cdots\!05}{68\!\cdots\!12}a^{3}+\frac{50\!\cdots\!87}{17\!\cdots\!28}a^{2}-\frac{74\!\cdots\!87}{34\!\cdots\!56}a+\frac{41\!\cdots\!45}{68\!\cdots\!12}$ 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

$C_{89}$, which has order $89$ (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:   $\frac{11\!\cdots\!17}{68\!\cdots\!12}a^{15}+\frac{88\!\cdots\!41}{17\!\cdots\!28}a^{14}-\frac{26\!\cdots\!03}{34\!\cdots\!56}a^{13}+\frac{62\!\cdots\!45}{68\!\cdots\!12}a^{12}+\frac{19\!\cdots\!27}{86\!\cdots\!64}a^{11}-\frac{22\!\cdots\!79}{57\!\cdots\!76}a^{10}-\frac{99\!\cdots\!43}{34\!\cdots\!56}a^{9}+\frac{54\!\cdots\!61}{86\!\cdots\!64}a^{8}+\frac{12\!\cdots\!19}{86\!\cdots\!64}a^{7}-\frac{40\!\cdots\!89}{11\!\cdots\!52}a^{6}-\frac{16\!\cdots\!27}{86\!\cdots\!64}a^{5}+\frac{44\!\cdots\!05}{17\!\cdots\!28}a^{4}+\frac{24\!\cdots\!49}{68\!\cdots\!12}a^{3}+\frac{45\!\cdots\!21}{17\!\cdots\!28}a^{2}+\frac{46\!\cdots\!15}{34\!\cdots\!56}a+\frac{15\!\cdots\!13}{68\!\cdots\!12}$, $\frac{36\!\cdots\!43}{86\!\cdots\!64}a^{15}-\frac{26\!\cdots\!39}{34\!\cdots\!56}a^{14}-\frac{17\!\cdots\!63}{86\!\cdots\!64}a^{13}+\frac{11\!\cdots\!53}{34\!\cdots\!56}a^{12}+\frac{30\!\cdots\!35}{53\!\cdots\!29}a^{11}-\frac{74\!\cdots\!93}{57\!\cdots\!76}a^{10}-\frac{11\!\cdots\!79}{17\!\cdots\!28}a^{9}+\frac{32\!\cdots\!13}{17\!\cdots\!28}a^{8}+\frac{43\!\cdots\!81}{17\!\cdots\!28}a^{7}-\frac{26\!\cdots\!31}{28\!\cdots\!88}a^{6}+\frac{77\!\cdots\!61}{10\!\cdots\!58}a^{5}+\frac{15\!\cdots\!47}{17\!\cdots\!28}a^{4}+\frac{17\!\cdots\!37}{17\!\cdots\!28}a^{3}+\frac{17\!\cdots\!33}{34\!\cdots\!56}a^{2}+\frac{75\!\cdots\!21}{17\!\cdots\!28}a+\frac{81\!\cdots\!11}{34\!\cdots\!56}$, $\frac{12\!\cdots\!15}{43\!\cdots\!32}a^{15}-\frac{34\!\cdots\!01}{34\!\cdots\!56}a^{14}-\frac{21\!\cdots\!13}{17\!\cdots\!28}a^{13}+\frac{22\!\cdots\!57}{34\!\cdots\!56}a^{12}+\frac{23\!\cdots\!19}{86\!\cdots\!64}a^{11}-\frac{11\!\cdots\!09}{57\!\cdots\!76}a^{10}-\frac{90\!\cdots\!73}{17\!\cdots\!28}a^{9}+\frac{40\!\cdots\!37}{17\!\cdots\!28}a^{8}-\frac{61\!\cdots\!17}{17\!\cdots\!28}a^{7}-\frac{25\!\cdots\!59}{35\!\cdots\!86}a^{6}+\frac{20\!\cdots\!57}{86\!\cdots\!64}a^{5}-\frac{33\!\cdots\!01}{17\!\cdots\!28}a^{4}+\frac{53\!\cdots\!09}{17\!\cdots\!28}a^{3}-\frac{13\!\cdots\!01}{34\!\cdots\!56}a^{2}+\frac{82\!\cdots\!39}{86\!\cdots\!64}a-\frac{20\!\cdots\!29}{34\!\cdots\!56}$, $\frac{26\!\cdots\!01}{68\!\cdots\!12}a^{15}-\frac{13\!\cdots\!73}{10\!\cdots\!58}a^{14}-\frac{54\!\cdots\!45}{34\!\cdots\!56}a^{13}+\frac{57\!\cdots\!17}{68\!\cdots\!12}a^{12}+\frac{18\!\cdots\!85}{53\!\cdots\!29}a^{11}-\frac{14\!\cdots\!79}{57\!\cdots\!76}a^{10}-\frac{53\!\cdots\!07}{34\!\cdots\!56}a^{9}+\frac{25\!\cdots\!63}{86\!\cdots\!64}a^{8}-\frac{23\!\cdots\!69}{43\!\cdots\!32}a^{7}-\frac{84\!\cdots\!89}{11\!\cdots\!52}a^{6}+\frac{37\!\cdots\!05}{10\!\cdots\!58}a^{5}-\frac{69\!\cdots\!23}{17\!\cdots\!28}a^{4}+\frac{13\!\cdots\!57}{68\!\cdots\!12}a^{3}-\frac{75\!\cdots\!55}{86\!\cdots\!64}a^{2}+\frac{17\!\cdots\!65}{34\!\cdots\!56}a+\frac{11\!\cdots\!09}{68\!\cdots\!12}$, $\frac{53\!\cdots\!25}{68\!\cdots\!12}a^{15}-\frac{99\!\cdots\!31}{34\!\cdots\!56}a^{14}-\frac{11\!\cdots\!93}{34\!\cdots\!56}a^{13}+\frac{12\!\cdots\!75}{68\!\cdots\!12}a^{12}+\frac{64\!\cdots\!13}{86\!\cdots\!64}a^{11}-\frac{16\!\cdots\!47}{28\!\cdots\!88}a^{10}-\frac{61\!\cdots\!49}{34\!\cdots\!56}a^{9}+\frac{12\!\cdots\!25}{17\!\cdots\!28}a^{8}-\frac{17\!\cdots\!37}{17\!\cdots\!28}a^{7}-\frac{30\!\cdots\!77}{11\!\cdots\!52}a^{6}+\frac{64\!\cdots\!07}{86\!\cdots\!64}a^{5}-\frac{35\!\cdots\!71}{86\!\cdots\!64}a^{4}-\frac{13\!\cdots\!91}{68\!\cdots\!12}a^{3}-\frac{33\!\cdots\!87}{34\!\cdots\!56}a^{2}+\frac{14\!\cdots\!83}{34\!\cdots\!56}a-\frac{12\!\cdots\!49}{68\!\cdots\!12}$, $\frac{49\!\cdots\!63}{68\!\cdots\!12}a^{15}+\frac{30\!\cdots\!23}{17\!\cdots\!28}a^{14}-\frac{12\!\cdots\!83}{34\!\cdots\!56}a^{13}-\frac{18\!\cdots\!01}{68\!\cdots\!12}a^{12}+\frac{50\!\cdots\!47}{43\!\cdots\!32}a^{11}+\frac{29\!\cdots\!49}{57\!\cdots\!76}a^{10}-\frac{64\!\cdots\!01}{34\!\cdots\!56}a^{9}+\frac{27\!\cdots\!85}{21\!\cdots\!16}a^{8}+\frac{12\!\cdots\!55}{86\!\cdots\!64}a^{7}-\frac{22\!\cdots\!43}{12\!\cdots\!28}a^{6}-\frac{18\!\cdots\!39}{43\!\cdots\!32}a^{5}+\frac{12\!\cdots\!85}{17\!\cdots\!28}a^{4}-\frac{54\!\cdots\!65}{68\!\cdots\!12}a^{3}-\frac{14\!\cdots\!95}{17\!\cdots\!28}a^{2}-\frac{28\!\cdots\!33}{34\!\cdots\!56}a-\frac{92\!\cdots\!97}{68\!\cdots\!12}$, $\frac{37\!\cdots\!67}{22\!\cdots\!04}a^{15}-\frac{16\!\cdots\!39}{11\!\cdots\!52}a^{14}-\frac{86\!\cdots\!65}{11\!\cdots\!52}a^{13}+\frac{41\!\cdots\!45}{22\!\cdots\!04}a^{12}+\frac{30\!\cdots\!79}{14\!\cdots\!44}a^{11}-\frac{15\!\cdots\!81}{23\!\cdots\!24}a^{10}-\frac{25\!\cdots\!63}{11\!\cdots\!52}a^{9}+\frac{53\!\cdots\!19}{57\!\cdots\!76}a^{8}+\frac{27\!\cdots\!73}{57\!\cdots\!76}a^{7}-\frac{16\!\cdots\!03}{38\!\cdots\!84}a^{6}+\frac{40\!\cdots\!41}{14\!\cdots\!44}a^{5}+\frac{81\!\cdots\!99}{71\!\cdots\!72}a^{4}+\frac{61\!\cdots\!67}{22\!\cdots\!04}a^{3}-\frac{13\!\cdots\!11}{11\!\cdots\!52}a^{2}+\frac{55\!\cdots\!39}{11\!\cdots\!52}a-\frac{17\!\cdots\!63}{22\!\cdots\!04}$ 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:  \( 93249594.3322 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 93249594.3322 \cdot 89}{2\cdot\sqrt{34153198773896725121035596949969}}\cr\approx \mathstrut & 1.72476613383 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861)
 
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 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861, 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 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861);
 
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 - 3*x^15 - 44*x^14 + 209*x^13 + 1029*x^12 - 6688*x^11 - 4322*x^10 + 83350*x^9 - 98968*x^8 - 306130*x^7 + 776806*x^6 - 425376*x^5 + 39313*x^4 - 397603*x^3 + 294148*x^2 - 10639*x + 137861);
 
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

\(\Q(\sqrt{73}) \), 4.2.122567.1, 4.4.389017.1, 4.2.8947391.1, 8.4.5844073816602313.1, 8.0.11047398519097.1, 8.4.80055805706881.1

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: 16.8.9557465298086033454595722486076274929.1
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 ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ R ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(23\) Copy content Toggle raw display 23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 98 x^{6} + 38 x^{5} + 3331 x^{4} - 1634 x^{3} + 44919 x^{2} - 57494 x + 224528$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(73\) Copy content Toggle raw display 73.16.14.1$x^{16} + 560 x^{15} + 137240 x^{14} + 19227600 x^{13} + 1684816700 x^{12} + 94599694000 x^{11} + 3327837457000 x^{10} + 67300032450000 x^{9} + 609674268043896 x^{8} + 336500162290880 x^{7} + 83195946420160 x^{6} + 11826359641600 x^{5} + 1175201013000 x^{4} + 6895769020000 x^{3} + 239006660174000 x^{2} + 4775207729180000 x + 41740387870087204$$8$$2$$14$$C_8\times C_2$$[\ ]_{8}^{2}$