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Properties

Label 16.0.345...625.6
Degree $16$
Signature $[0, 8]$
Discriminant $3.455\times 10^{24}$
Root discriminant \(34.17\)
Ramified primes $5,29$
Class number $36$ (GRH)
Class group [3, 12] (GRH)
Galois group $C_2^3:C_4$ (as 16T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856)
 
gp: K = bnfinit(y^16 - 2*y^15 + 6*y^14 + 8*y^13 + 9*y^12 + 62*y^11 + 657*y^10 - 416*y^9 - 2247*y^8 + 2722*y^7 + 5093*y^6 - 5996*y^5 - 1956*y^4 + 5684*y^3 - 2001*y^2 - 1624*y + 1856, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856)
 

\( x^{16} - 2 x^{15} + 6 x^{14} + 8 x^{13} + 9 x^{12} + 62 x^{11} + 657 x^{10} - 416 x^{9} - 2247 x^{8} + \cdots + 1856 \) 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:   \(3455222492240908603515625\) \(\medspace = 5^{10}\cdot 29^{12}\) 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:  \(34.17\)
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:  $5^{3/4}29^{3/4}\approx 41.78553833475025$
Ramified primes:   \(5\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{8}a$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{40}a^{10}+\frac{1}{40}a^{9}-\frac{1}{40}a^{8}-\frac{1}{20}a^{7}+\frac{1}{10}a^{6}+\frac{1}{5}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}+\frac{17}{40}a^{2}-\frac{1}{4}a+\frac{2}{5}$, $\frac{1}{40}a^{11}-\frac{1}{20}a^{9}-\frac{1}{40}a^{8}-\frac{1}{10}a^{7}+\frac{1}{10}a^{6}+\frac{7}{40}a^{5}-\frac{9}{20}a^{3}-\frac{7}{40}a^{2}-\frac{1}{10}a-\frac{2}{5}$, $\frac{1}{480}a^{12}-\frac{1}{80}a^{11}-\frac{1}{480}a^{10}+\frac{1}{40}a^{9}+\frac{7}{160}a^{8}+\frac{13}{240}a^{7}+\frac{17}{480}a^{6}+\frac{3}{40}a^{5}+\frac{67}{480}a^{4}+\frac{113}{240}a^{3}+\frac{41}{96}a^{2}+\frac{17}{120}a+\frac{1}{15}$, $\frac{1}{480}a^{13}-\frac{1}{480}a^{11}-\frac{1}{80}a^{10}+\frac{3}{160}a^{9}+\frac{1}{60}a^{8}+\frac{53}{480}a^{7}-\frac{1}{80}a^{6}-\frac{41}{480}a^{5}-\frac{1}{15}a^{4}-\frac{227}{480}a^{3}+\frac{1}{240}a^{2}+\frac{11}{30}a-\frac{1}{5}$, $\frac{1}{960}a^{14}-\frac{1}{960}a^{13}-\frac{11}{960}a^{11}-\frac{1}{96}a^{10}-\frac{13}{960}a^{9}+\frac{1}{32}a^{8}-\frac{7}{64}a^{7}+\frac{11}{160}a^{6}-\frac{29}{320}a^{5}+\frac{43}{240}a^{4}+\frac{79}{192}a^{3}+\frac{451}{960}a^{2}+\frac{19}{40}a-\frac{4}{15}$, $\frac{1}{12\!\cdots\!00}a^{15}-\frac{51873479265253}{24\!\cdots\!80}a^{14}+\frac{91492530590897}{15\!\cdots\!00}a^{13}+\frac{93138134016617}{24\!\cdots\!80}a^{12}-\frac{36\!\cdots\!13}{60\!\cdots\!00}a^{11}+\frac{809359843969813}{80\!\cdots\!60}a^{10}-\frac{20\!\cdots\!29}{60\!\cdots\!00}a^{9}-\frac{20\!\cdots\!39}{40\!\cdots\!00}a^{8}-\frac{16\!\cdots\!41}{20\!\cdots\!00}a^{7}-\frac{578054057746319}{11\!\cdots\!80}a^{6}-\frac{43\!\cdots\!33}{30\!\cdots\!00}a^{5}+\frac{78\!\cdots\!15}{48\!\cdots\!96}a^{4}+\frac{46\!\cdots\!79}{12\!\cdots\!00}a^{3}+\frac{32\!\cdots\!77}{75\!\cdots\!00}a^{2}+\frac{13\!\cdots\!09}{37\!\cdots\!50}a+\frac{61\!\cdots\!58}{18\!\cdots\!75}$ 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

$C_{3}\times C_{12}$, which has order $36$ (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:   $\frac{48579264421}{92857902243680}a^{15}-\frac{144105060237}{185715804487360}a^{14}+\frac{487494591151}{185715804487360}a^{13}+\frac{521926026893}{92857902243680}a^{12}+\frac{1326539580609}{185715804487360}a^{11}+\frac{1603097747779}{46428951121840}a^{10}+\frac{13337636866051}{37143160897472}a^{9}-\frac{1931564341377}{46428951121840}a^{8}-\frac{237617127760821}{185715804487360}a^{7}+\frac{4713130889549}{6632707303120}a^{6}+\frac{121990379594341}{37143160897472}a^{5}-\frac{124690789503443}{92857902243680}a^{4}-\frac{442634588348191}{185715804487360}a^{3}+\frac{47710182118315}{37143160897472}a^{2}-\frac{2204920505327}{5803618890230}a+\frac{762415453097}{2901809445115}$, $\frac{24430384785733}{12\!\cdots\!00}a^{15}-\frac{7840978286581}{24\!\cdots\!80}a^{14}+\frac{9660368321911}{15\!\cdots\!00}a^{13}+\frac{19336728626783}{48\!\cdots\!96}a^{12}-\frac{71388092760973}{20\!\cdots\!00}a^{11}+\frac{547900781893597}{24\!\cdots\!80}a^{10}+\frac{75\!\cdots\!63}{60\!\cdots\!00}a^{9}-\frac{41\!\cdots\!71}{12\!\cdots\!00}a^{8}-\frac{41\!\cdots\!09}{60\!\cdots\!00}a^{7}+\frac{40\!\cdots\!81}{34\!\cdots\!40}a^{6}+\frac{29\!\cdots\!71}{30\!\cdots\!00}a^{5}-\frac{11\!\cdots\!33}{80\!\cdots\!60}a^{4}+\frac{14\!\cdots\!87}{12\!\cdots\!00}a^{3}-\frac{49\!\cdots\!67}{60\!\cdots\!00}a^{2}+\frac{10\!\cdots\!22}{62\!\cdots\!25}a+\frac{24\!\cdots\!23}{62\!\cdots\!25}$, $\frac{55656448652099}{60\!\cdots\!00}a^{15}+\frac{9184762790123}{40\!\cdots\!80}a^{14}+\frac{14570295833593}{20\!\cdots\!00}a^{13}+\frac{26646155724809}{10\!\cdots\!20}a^{12}+\frac{35\!\cdots\!61}{60\!\cdots\!00}a^{11}+\frac{747632609500729}{60\!\cdots\!20}a^{10}+\frac{54\!\cdots\!53}{60\!\cdots\!00}a^{9}+\frac{75\!\cdots\!01}{30\!\cdots\!00}a^{8}-\frac{30\!\cdots\!63}{20\!\cdots\!00}a^{7}-\frac{66\!\cdots\!49}{85\!\cdots\!60}a^{6}+\frac{40\!\cdots\!97}{60\!\cdots\!00}a^{5}+\frac{17\!\cdots\!49}{10\!\cdots\!20}a^{4}-\frac{56\!\cdots\!29}{10\!\cdots\!00}a^{3}-\frac{92\!\cdots\!17}{60\!\cdots\!00}a^{2}+\frac{22\!\cdots\!13}{15\!\cdots\!00}a+\frac{17\!\cdots\!33}{62\!\cdots\!25}$, $\frac{9464967761403}{28\!\cdots\!00}a^{15}-\frac{8669161259477}{34\!\cdots\!40}a^{14}+\frac{284634027263783}{17\!\cdots\!00}a^{13}+\frac{18694609842163}{57\!\cdots\!40}a^{12}+\frac{18\!\cdots\!57}{17\!\cdots\!00}a^{11}+\frac{32902522191025}{17\!\cdots\!32}a^{10}+\frac{42\!\cdots\!31}{17\!\cdots\!00}a^{9}+\frac{55\!\cdots\!71}{42\!\cdots\!00}a^{8}-\frac{11\!\cdots\!73}{17\!\cdots\!00}a^{7}-\frac{482326178101267}{571739369528944}a^{6}+\frac{31\!\cdots\!29}{17\!\cdots\!00}a^{5}+\frac{49\!\cdots\!03}{57\!\cdots\!40}a^{4}-\frac{47\!\cdots\!03}{17\!\cdots\!00}a^{3}-\frac{46\!\cdots\!29}{17\!\cdots\!00}a^{2}+\frac{13\!\cdots\!59}{10\!\cdots\!00}a-\frac{966483407333771}{26\!\cdots\!25}$, $\frac{1440844627823}{24\!\cdots\!00}a^{15}-\frac{53149058693}{60395003823480}a^{14}+\frac{6747916522333}{24\!\cdots\!00}a^{13}+\frac{3043521579581}{483160030587840}a^{12}+\frac{5893642893479}{805266717646400}a^{11}+\frac{17241148301749}{483160030587840}a^{10}+\frac{322148145648937}{805266717646400}a^{9}-\frac{164727934072951}{24\!\cdots\!00}a^{8}-\frac{13\!\cdots\!91}{805266717646400}a^{7}+\frac{235333130181983}{483160030587840}a^{6}+\frac{31\!\cdots\!93}{805266717646400}a^{5}-\frac{457082266370329}{483160030587840}a^{4}-\frac{38\!\cdots\!19}{12\!\cdots\!00}a^{3}+\frac{20\!\cdots\!31}{24\!\cdots\!00}a^{2}+\frac{141206093137369}{150987509558700}a-\frac{13121139015731}{37746877389675}$, $\frac{690984749014537}{20\!\cdots\!00}a^{15}-\frac{334536186469449}{80\!\cdots\!60}a^{14}+\frac{18\!\cdots\!77}{12\!\cdots\!00}a^{13}+\frac{511953549027353}{12\!\cdots\!24}a^{12}+\frac{64\!\cdots\!43}{12\!\cdots\!00}a^{11}+\frac{27\!\cdots\!83}{12\!\cdots\!40}a^{10}+\frac{96\!\cdots\!63}{40\!\cdots\!00}a^{9}+\frac{19\!\cdots\!73}{60\!\cdots\!00}a^{8}-\frac{36\!\cdots\!69}{40\!\cdots\!00}a^{7}+\frac{37\!\cdots\!39}{17\!\cdots\!20}a^{6}+\frac{30\!\cdots\!51}{12\!\cdots\!00}a^{5}-\frac{13\!\cdots\!31}{500271948337826}a^{4}-\frac{10\!\cdots\!59}{40\!\cdots\!00}a^{3}+\frac{30\!\cdots\!59}{12\!\cdots\!00}a^{2}+\frac{55\!\cdots\!39}{50\!\cdots\!00}a+\frac{10\!\cdots\!56}{18\!\cdots\!75}$, $\frac{4175607454017}{57\!\cdots\!40}a^{15}-\frac{76529405215661}{34\!\cdots\!40}a^{14}+\frac{164796351574033}{34\!\cdots\!40}a^{13}+\frac{28142371971497}{17\!\cdots\!20}a^{12}-\frac{124669197537601}{34\!\cdots\!40}a^{11}+\frac{88352022284801}{42\!\cdots\!80}a^{10}+\frac{13\!\cdots\!69}{34\!\cdots\!40}a^{9}-\frac{77\!\cdots\!49}{85\!\cdots\!60}a^{8}-\frac{24\!\cdots\!21}{11\!\cdots\!80}a^{7}+\frac{64\!\cdots\!11}{21\!\cdots\!40}a^{6}+\frac{15\!\cdots\!31}{34\!\cdots\!40}a^{5}-\frac{10\!\cdots\!67}{17\!\cdots\!20}a^{4}-\frac{15\!\cdots\!65}{68\!\cdots\!28}a^{3}+\frac{49\!\cdots\!93}{11\!\cdots\!80}a^{2}+\frac{422740037137589}{14\!\cdots\!60}a-\frac{592658085580316}{536005658933385}$ 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:  \( 119871.996246 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 119871.996246 \cdot 36}{2\cdot\sqrt{3455222492240908603515625}}\cr\approx \mathstrut & 2.81962388395 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856)
 
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 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856, 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 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856);
 
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 - 2*x^15 + 6*x^14 + 8*x^13 + 9*x^12 + 62*x^11 + 657*x^10 - 416*x^9 - 2247*x^8 + 2722*x^7 + 5093*x^6 - 5996*x^5 - 1956*x^4 + 5684*x^3 - 2001*x^2 - 1624*x + 1856);
 
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

$C_2^3:C_4$ (as 16T33):

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 $C_2^3:C_4$
Character table for $C_2^3:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.121945.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.609725.1 x2, 8.0.371764575625.2 x2, 8.0.371764575625.5, 8.4.64097340625.1 x2

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: 8.4.64097340625.1, 8.0.371764575625.2, 8.4.9294114390625.1, 8.0.11051265625.3
Degree 16 siblings: 16.8.86380562306022715087890625.2, 16.0.102711726879931884765625.6, 16.0.86380562306022715087890625.8
Minimal sibling: 8.0.11051265625.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(5\) Copy content Toggle raw display 5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(29\) Copy content Toggle raw display 29.4.3.1$x^{4} + 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} + 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} + 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} + 116$$4$$1$$3$$C_4$$[\ ]_{4}$