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Properties

Label 16.0.283...441.3
Degree $16$
Signature $[0, 8]$
Discriminant $2.832\times 10^{27}$
Root discriminant \(51.97\)
Ramified primes $7,53$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $QD_{16}$ (as 16T12)

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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 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623)
 
gp: K = bnfinit(y^16 - 2*y^15 - 25*y^14 + 36*y^13 + 216*y^12 - 324*y^11 + 679*y^10 + 6422*y^9 + 2927*y^8 + 1374*y^7 + 14311*y^6 - 22802*y^5 - 16736*y^4 - 47978*y^3 - 157563*y^2 + 76048*y + 231623, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623)
 

\( x^{16} - 2 x^{15} - 25 x^{14} + 36 x^{13} + 216 x^{12} - 324 x^{11} + 679 x^{10} + 6422 x^{9} + \cdots + 231623 \) 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:   \(2832009822518079193000591441\) \(\medspace = 7^{8}\cdot 53^{12}\) 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:  \(51.97\)
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:  $7^{1/2}53^{3/4}\approx 51.97038357490805$
Ramified primes:   \(7\), \(53\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14}a^{8}+\frac{1}{14}a^{7}+\frac{2}{7}a^{6}+\frac{3}{14}a^{5}-\frac{5}{14}a^{4}-\frac{5}{14}a^{3}-\frac{3}{7}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{14}a^{9}+\frac{3}{14}a^{7}-\frac{1}{14}a^{6}+\frac{3}{7}a^{5}-\frac{1}{14}a^{3}-\frac{1}{14}a^{2}-\frac{1}{2}$, $\frac{1}{14}a^{10}-\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{5}{14}a^{5}+\frac{2}{7}a^{2}-\frac{1}{2}$, $\frac{1}{14}a^{11}-\frac{1}{7}a^{7}-\frac{1}{2}a^{6}-\frac{1}{7}a^{5}-\frac{3}{7}a^{4}-\frac{1}{7}a^{3}+\frac{2}{7}a^{2}-\frac{1}{2}a$, $\frac{1}{98}a^{12}-\frac{1}{49}a^{10}+\frac{1}{98}a^{9}-\frac{11}{49}a^{7}+\frac{45}{98}a^{6}+\frac{5}{49}a^{5}-\frac{13}{49}a^{4}-\frac{1}{2}a^{3}-\frac{3}{7}a^{2}-\frac{1}{2}$, $\frac{1}{98}a^{13}-\frac{1}{49}a^{11}+\frac{1}{98}a^{10}-\frac{1}{98}a^{8}-\frac{16}{49}a^{7}-\frac{2}{49}a^{6}+\frac{37}{98}a^{5}+\frac{3}{7}a^{4}-\frac{1}{2}a^{3}-\frac{2}{7}a^{2}-\frac{1}{2}$, $\frac{1}{1173746}a^{14}-\frac{445}{586873}a^{13}+\frac{807}{167678}a^{12}-\frac{14933}{586873}a^{11}-\frac{18451}{586873}a^{10}-\frac{16523}{586873}a^{9}+\frac{36173}{1173746}a^{8}+\frac{519321}{1173746}a^{7}-\frac{16641}{1173746}a^{6}-\frac{179231}{1173746}a^{5}+\frac{168541}{586873}a^{4}-\frac{26753}{83839}a^{3}+\frac{7664}{83839}a^{2}+\frac{437}{23954}a-\frac{143}{413}$, $\frac{1}{13\!\cdots\!78}a^{15}+\frac{19\!\cdots\!47}{13\!\cdots\!78}a^{14}-\frac{12\!\cdots\!19}{66\!\cdots\!39}a^{13}-\frac{28\!\cdots\!03}{66\!\cdots\!39}a^{12}-\frac{18\!\cdots\!21}{66\!\cdots\!39}a^{11}+\frac{35\!\cdots\!11}{18\!\cdots\!54}a^{10}+\frac{23\!\cdots\!90}{94\!\cdots\!77}a^{9}+\frac{55\!\cdots\!74}{39\!\cdots\!67}a^{8}-\frac{14\!\cdots\!15}{13\!\cdots\!78}a^{7}-\frac{65\!\cdots\!61}{13\!\cdots\!78}a^{6}-\frac{56\!\cdots\!95}{13\!\cdots\!78}a^{5}+\frac{13\!\cdots\!43}{13\!\cdots\!78}a^{4}-\frac{60\!\cdots\!77}{18\!\cdots\!54}a^{3}+\frac{85\!\cdots\!29}{18\!\cdots\!54}a^{2}+\frac{20\!\cdots\!02}{13\!\cdots\!11}a-\frac{32\!\cdots\!33}{93\!\cdots\!18}$ 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:  $2$, $7$

Class group and class number

$C_{2}$, which has order $2$ (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:   $\frac{13\!\cdots\!87}{78\!\cdots\!34}a^{15}-\frac{48\!\cdots\!83}{78\!\cdots\!34}a^{14}-\frac{25\!\cdots\!25}{78\!\cdots\!34}a^{13}+\frac{91\!\cdots\!33}{78\!\cdots\!34}a^{12}+\frac{71\!\cdots\!69}{39\!\cdots\!67}a^{11}-\frac{10\!\cdots\!41}{11\!\cdots\!62}a^{10}+\frac{29\!\cdots\!55}{11\!\cdots\!62}a^{9}+\frac{28\!\cdots\!20}{39\!\cdots\!67}a^{8}-\frac{27\!\cdots\!78}{39\!\cdots\!67}a^{7}+\frac{55\!\cdots\!52}{39\!\cdots\!67}a^{6}+\frac{52\!\cdots\!66}{39\!\cdots\!67}a^{5}-\frac{22\!\cdots\!46}{39\!\cdots\!67}a^{4}+\frac{37\!\cdots\!81}{55\!\cdots\!81}a^{3}-\frac{18\!\cdots\!91}{11\!\cdots\!62}a^{2}-\frac{26\!\cdots\!21}{15\!\cdots\!66}a+\frac{19\!\cdots\!81}{54\!\cdots\!54}$, $\frac{58\!\cdots\!69}{22\!\cdots\!42}a^{15}-\frac{31\!\cdots\!55}{22\!\cdots\!42}a^{14}-\frac{15\!\cdots\!73}{22\!\cdots\!42}a^{13}-\frac{11\!\cdots\!41}{22\!\cdots\!42}a^{12}+\frac{66\!\cdots\!33}{11\!\cdots\!21}a^{11}-\frac{28\!\cdots\!79}{32\!\cdots\!06}a^{10}+\frac{49\!\cdots\!91}{32\!\cdots\!06}a^{9}+\frac{13\!\cdots\!87}{66\!\cdots\!13}a^{8}+\frac{37\!\cdots\!66}{11\!\cdots\!21}a^{7}+\frac{52\!\cdots\!56}{11\!\cdots\!21}a^{6}+\frac{12\!\cdots\!84}{11\!\cdots\!21}a^{5}+\frac{69\!\cdots\!33}{11\!\cdots\!21}a^{4}+\frac{53\!\cdots\!10}{16\!\cdots\!03}a^{3}-\frac{16\!\cdots\!93}{32\!\cdots\!06}a^{2}-\frac{25\!\cdots\!43}{45\!\cdots\!58}a-\frac{85\!\cdots\!71}{15\!\cdots\!02}$, $\frac{38\!\cdots\!23}{22\!\cdots\!42}a^{15}-\frac{10\!\cdots\!49}{22\!\cdots\!42}a^{14}-\frac{74\!\cdots\!01}{22\!\cdots\!42}a^{13}+\frac{13\!\cdots\!69}{22\!\cdots\!42}a^{12}+\frac{24\!\cdots\!46}{11\!\cdots\!21}a^{11}-\frac{73\!\cdots\!39}{32\!\cdots\!06}a^{10}+\frac{80\!\cdots\!87}{45\!\cdots\!58}a^{9}+\frac{40\!\cdots\!26}{66\!\cdots\!13}a^{8}+\frac{13\!\cdots\!40}{11\!\cdots\!21}a^{7}+\frac{20\!\cdots\!19}{11\!\cdots\!21}a^{6}-\frac{30\!\cdots\!03}{11\!\cdots\!21}a^{5}+\frac{49\!\cdots\!33}{11\!\cdots\!21}a^{4}+\frac{59\!\cdots\!53}{16\!\cdots\!03}a^{3}-\frac{84\!\cdots\!59}{32\!\cdots\!06}a^{2}-\frac{35\!\cdots\!43}{45\!\cdots\!58}a+\frac{31\!\cdots\!71}{15\!\cdots\!02}$, $\frac{66\!\cdots\!21}{39\!\cdots\!67}a^{15}-\frac{28\!\cdots\!02}{39\!\cdots\!67}a^{14}-\frac{11\!\cdots\!27}{39\!\cdots\!67}a^{13}+\frac{54\!\cdots\!78}{39\!\cdots\!67}a^{12}+\frac{44\!\cdots\!36}{39\!\cdots\!67}a^{11}-\frac{57\!\cdots\!50}{55\!\cdots\!81}a^{10}+\frac{24\!\cdots\!75}{79\!\cdots\!83}a^{9}+\frac{20\!\cdots\!66}{39\!\cdots\!67}a^{8}-\frac{47\!\cdots\!58}{39\!\cdots\!67}a^{7}+\frac{13\!\cdots\!12}{66\!\cdots\!13}a^{6}+\frac{21\!\cdots\!44}{39\!\cdots\!67}a^{5}-\frac{27\!\cdots\!82}{39\!\cdots\!67}a^{4}+\frac{51\!\cdots\!52}{55\!\cdots\!81}a^{3}-\frac{86\!\cdots\!31}{55\!\cdots\!81}a^{2}+\frac{13\!\cdots\!71}{79\!\cdots\!83}a+\frac{36\!\cdots\!55}{27\!\cdots\!27}$, $\frac{52\!\cdots\!45}{66\!\cdots\!39}a^{15}-\frac{57\!\cdots\!31}{13\!\cdots\!78}a^{14}-\frac{17\!\cdots\!41}{66\!\cdots\!39}a^{13}+\frac{80\!\cdots\!63}{66\!\cdots\!39}a^{12}+\frac{59\!\cdots\!23}{13\!\cdots\!78}a^{11}-\frac{12\!\cdots\!83}{94\!\cdots\!77}a^{10}-\frac{35\!\cdots\!69}{94\!\cdots\!77}a^{9}+\frac{14\!\cdots\!25}{39\!\cdots\!67}a^{8}+\frac{78\!\cdots\!81}{13\!\cdots\!78}a^{7}-\frac{97\!\cdots\!43}{66\!\cdots\!39}a^{6}-\frac{90\!\cdots\!83}{66\!\cdots\!39}a^{5}-\frac{11\!\cdots\!69}{13\!\cdots\!78}a^{4}+\frac{93\!\cdots\!91}{18\!\cdots\!54}a^{3}+\frac{23\!\cdots\!19}{18\!\cdots\!54}a^{2}-\frac{11\!\cdots\!09}{27\!\cdots\!22}a-\frac{13\!\cdots\!81}{93\!\cdots\!18}$, $\frac{10\!\cdots\!34}{94\!\cdots\!77}a^{15}-\frac{54\!\cdots\!60}{94\!\cdots\!77}a^{14}+\frac{37\!\cdots\!16}{94\!\cdots\!77}a^{13}+\frac{67\!\cdots\!61}{94\!\cdots\!77}a^{12}-\frac{43\!\cdots\!18}{94\!\cdots\!77}a^{11}-\frac{61\!\cdots\!61}{13\!\cdots\!11}a^{10}+\frac{95\!\cdots\!87}{13\!\cdots\!11}a^{9}+\frac{31\!\cdots\!17}{55\!\cdots\!81}a^{8}-\frac{39\!\cdots\!36}{94\!\cdots\!77}a^{7}+\frac{27\!\cdots\!12}{94\!\cdots\!77}a^{6}-\frac{18\!\cdots\!04}{94\!\cdots\!77}a^{5}-\frac{31\!\cdots\!86}{94\!\cdots\!77}a^{4}-\frac{99\!\cdots\!41}{13\!\cdots\!11}a^{3}-\frac{37\!\cdots\!57}{13\!\cdots\!11}a^{2}+\frac{24\!\cdots\!32}{19\!\cdots\!73}a+\frac{18\!\cdots\!75}{66\!\cdots\!37}$, $\frac{11\!\cdots\!11}{13\!\cdots\!78}a^{15}-\frac{40\!\cdots\!97}{13\!\cdots\!78}a^{14}-\frac{23\!\cdots\!15}{13\!\cdots\!78}a^{13}+\frac{77\!\cdots\!61}{13\!\cdots\!78}a^{12}+\frac{69\!\cdots\!07}{66\!\cdots\!39}a^{11}-\frac{42\!\cdots\!09}{94\!\cdots\!77}a^{10}+\frac{11\!\cdots\!02}{94\!\cdots\!77}a^{9}+\frac{29\!\cdots\!07}{78\!\cdots\!34}a^{8}-\frac{21\!\cdots\!86}{66\!\cdots\!39}a^{7}+\frac{82\!\cdots\!81}{13\!\cdots\!78}a^{6}+\frac{23\!\cdots\!95}{66\!\cdots\!39}a^{5}-\frac{37\!\cdots\!81}{13\!\cdots\!78}a^{4}+\frac{29\!\cdots\!31}{94\!\cdots\!77}a^{3}-\frac{82\!\cdots\!44}{94\!\cdots\!77}a^{2}-\frac{29\!\cdots\!88}{13\!\cdots\!11}a+\frac{58\!\cdots\!17}{46\!\cdots\!59}$ 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:  \( 21203688.514 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 21203688.514 \cdot 2}{2\cdot\sqrt{2832009822518079193000591441}}\cr\approx \mathstrut & 0.96783870250 \end{aligned}\] (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 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623)
 
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 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623, 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 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623);
 
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 - 25*x^14 + 36*x^13 + 216*x^12 - 324*x^11 + 679*x^10 + 6422*x^9 + 2927*x^8 + 1374*x^7 + 14311*x^6 - 22802*x^5 - 16736*x^4 - 47978*x^3 - 157563*x^2 + 76048*x + 231623);
 
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

$\SD_{16}$ (as 16T12):

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 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-371}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{53})\), 4.2.19663.1 x2, 4.0.2597.2 x2, 8.0.18945044881.1, 8.2.7602375867247.2 x4

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

Degree 8 sibling: 8.2.7602375867247.2
Minimal sibling: 8.2.7602375867247.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.1.0.1}{1} }^{16}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ R ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(7\) Copy content Toggle raw display 7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
\(53\) Copy content Toggle raw display 53.4.3.1$x^{4} + 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} + 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} + 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.1$x^{4} + 212$$4$$1$$3$$C_4$$[\ ]_{4}$