Properties

Label 16.0.392...849.7
Degree $16$
Signature $[0, 8]$
Discriminant $3.923\times 10^{30}$
Root discriminant \(81.68\)
Ramified primes $13,17$
Class number $9792$ (GRH)
Class group [2, 12, 408] (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 - 2*x^15 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832)
 
gp: K = bnfinit(y^16 - 2*y^15 + 12*y^14 - 22*y^13 + 865*y^12 + 68*y^11 + 5879*y^10 + 75224*y^9 - 163742*y^8 + 335966*y^7 + 4327649*y^6 - 4700058*y^5 - 33955384*y^4 - 22705032*y^3 + 79867792*y^2 + 138718144*y + 66232832, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832)
 

\( x^{16} - 2 x^{15} + 12 x^{14} - 22 x^{13} + 865 x^{12} + 68 x^{11} + 5879 x^{10} + 75224 x^{9} - 163742 x^{8} + 335966 x^{7} + 4327649 x^{6} - 4700058 x^{5} + \cdots + 66232832 \) 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:   \(3922880935919264967742950184849\) \(\medspace = 13^{12}\cdot 17^{14}\) 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:  \(81.68\)
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:  $13^{3/4}17^{7/8}\approx 81.67710206860912$
Ramified primes:   \(13\), \(17\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{7}{32}a^{5}+\frac{5}{32}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{12}-\frac{1}{64}a^{11}-\frac{1}{16}a^{9}-\frac{3}{32}a^{8}-\frac{5}{32}a^{7}-\frac{13}{64}a^{6}+\frac{15}{64}a^{5}+\frac{1}{32}a^{4}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{13}-\frac{1}{64}a^{11}-\frac{1}{16}a^{10}-\frac{1}{32}a^{9}-\frac{7}{64}a^{7}-\frac{7}{32}a^{6}-\frac{15}{64}a^{5}+\frac{5}{32}a^{4}+\frac{3}{8}a^{3}-\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{512}a^{14}+\frac{3}{512}a^{12}-\frac{1}{64}a^{11}+\frac{3}{256}a^{10}+\frac{1}{32}a^{9}-\frac{47}{512}a^{8}-\frac{35}{256}a^{7}-\frac{99}{512}a^{6}+\frac{43}{256}a^{5}+\frac{5}{64}a^{4}+\frac{3}{64}a^{3}-\frac{11}{32}a^{2}+\frac{3}{8}a$, $\frac{1}{25\!\cdots\!52}a^{15}-\frac{84\!\cdots\!09}{25\!\cdots\!52}a^{14}+\frac{14\!\cdots\!99}{25\!\cdots\!52}a^{13}+\frac{17\!\cdots\!57}{25\!\cdots\!52}a^{12}+\frac{19\!\cdots\!67}{12\!\cdots\!76}a^{11}-\frac{24\!\cdots\!59}{12\!\cdots\!76}a^{10}-\frac{14\!\cdots\!23}{25\!\cdots\!52}a^{9}-\frac{28\!\cdots\!59}{25\!\cdots\!52}a^{8}+\frac{23\!\cdots\!27}{25\!\cdots\!52}a^{7}-\frac{44\!\cdots\!03}{25\!\cdots\!52}a^{6}-\frac{76\!\cdots\!79}{12\!\cdots\!76}a^{5}+\frac{26\!\cdots\!49}{16\!\cdots\!72}a^{4}+\frac{10\!\cdots\!11}{32\!\cdots\!44}a^{3}-\frac{31\!\cdots\!61}{16\!\cdots\!72}a^{2}-\frac{49\!\cdots\!31}{40\!\cdots\!68}a-\frac{10\!\cdots\!17}{50\!\cdots\!96}$ 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

$C_{2}\times C_{12}\times C_{408}$, which has order $9792$ (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:   $\frac{12\!\cdots\!77}{30\!\cdots\!32}a^{15}-\frac{92\!\cdots\!47}{30\!\cdots\!32}a^{14}+\frac{41\!\cdots\!39}{30\!\cdots\!32}a^{13}-\frac{14\!\cdots\!73}{30\!\cdots\!32}a^{12}+\frac{72\!\cdots\!51}{15\!\cdots\!16}a^{11}-\frac{30\!\cdots\!89}{15\!\cdots\!16}a^{10}+\frac{18\!\cdots\!49}{30\!\cdots\!32}a^{9}+\frac{47\!\cdots\!91}{30\!\cdots\!32}a^{8}-\frac{63\!\cdots\!09}{30\!\cdots\!32}a^{7}+\frac{24\!\cdots\!79}{30\!\cdots\!32}a^{6}+\frac{55\!\cdots\!95}{15\!\cdots\!16}a^{5}-\frac{35\!\cdots\!55}{37\!\cdots\!04}a^{4}+\frac{49\!\cdots\!77}{37\!\cdots\!04}a^{3}+\frac{41\!\cdots\!31}{18\!\cdots\!52}a^{2}-\frac{11\!\cdots\!67}{47\!\cdots\!63}a-\frac{15\!\cdots\!47}{47\!\cdots\!63}$, $\frac{30\!\cdots\!27}{10\!\cdots\!32}a^{15}-\frac{87\!\cdots\!83}{10\!\cdots\!32}a^{14}+\frac{40\!\cdots\!21}{10\!\cdots\!32}a^{13}-\frac{87\!\cdots\!33}{10\!\cdots\!32}a^{12}+\frac{13\!\cdots\!73}{54\!\cdots\!16}a^{11}-\frac{98\!\cdots\!85}{54\!\cdots\!16}a^{10}+\frac{16\!\cdots\!19}{10\!\cdots\!32}a^{9}+\frac{21\!\cdots\!83}{10\!\cdots\!32}a^{8}-\frac{71\!\cdots\!47}{10\!\cdots\!32}a^{7}+\frac{14\!\cdots\!11}{10\!\cdots\!32}a^{6}+\frac{64\!\cdots\!67}{54\!\cdots\!16}a^{5}-\frac{18\!\cdots\!69}{67\!\cdots\!52}a^{4}-\frac{10\!\cdots\!27}{13\!\cdots\!04}a^{3}+\frac{34\!\cdots\!53}{67\!\cdots\!52}a^{2}+\frac{32\!\cdots\!55}{16\!\cdots\!88}a+\frac{22\!\cdots\!53}{21\!\cdots\!36}$, $\frac{81\!\cdots\!75}{64\!\cdots\!88}a^{15}+\frac{84\!\cdots\!45}{64\!\cdots\!88}a^{14}+\frac{95\!\cdots\!85}{64\!\cdots\!88}a^{13}+\frac{16\!\cdots\!51}{64\!\cdots\!88}a^{12}+\frac{35\!\cdots\!33}{32\!\cdots\!44}a^{11}+\frac{11\!\cdots\!87}{32\!\cdots\!44}a^{10}+\frac{88\!\cdots\!51}{64\!\cdots\!88}a^{9}+\frac{90\!\cdots\!71}{64\!\cdots\!88}a^{8}+\frac{97\!\cdots\!25}{64\!\cdots\!88}a^{7}+\frac{39\!\cdots\!99}{64\!\cdots\!88}a^{6}+\frac{24\!\cdots\!03}{32\!\cdots\!44}a^{5}+\frac{55\!\cdots\!83}{40\!\cdots\!68}a^{4}-\frac{13\!\cdots\!67}{80\!\cdots\!36}a^{3}-\frac{35\!\cdots\!43}{40\!\cdots\!68}a^{2}-\frac{10\!\cdots\!53}{10\!\cdots\!92}a-\frac{59\!\cdots\!15}{12\!\cdots\!24}$, $\frac{14\!\cdots\!57}{12\!\cdots\!76}a^{15}-\frac{33\!\cdots\!05}{12\!\cdots\!76}a^{14}+\frac{23\!\cdots\!83}{12\!\cdots\!76}a^{13}-\frac{72\!\cdots\!07}{12\!\cdots\!76}a^{12}+\frac{64\!\cdots\!71}{64\!\cdots\!88}a^{11}-\frac{55\!\cdots\!51}{64\!\cdots\!88}a^{10}+\frac{14\!\cdots\!29}{12\!\cdots\!76}a^{9}+\frac{84\!\cdots\!93}{12\!\cdots\!76}a^{8}-\frac{25\!\cdots\!93}{12\!\cdots\!76}a^{7}+\frac{44\!\cdots\!85}{12\!\cdots\!76}a^{6}+\frac{16\!\cdots\!97}{64\!\cdots\!88}a^{5}-\frac{61\!\cdots\!59}{80\!\cdots\!36}a^{4}-\frac{31\!\cdots\!93}{16\!\cdots\!72}a^{3}+\frac{97\!\cdots\!71}{80\!\cdots\!36}a^{2}+\frac{12\!\cdots\!81}{20\!\cdots\!84}a+\frac{99\!\cdots\!15}{25\!\cdots\!48}$, $\frac{21\!\cdots\!39}{64\!\cdots\!88}a^{15}-\frac{65\!\cdots\!43}{64\!\cdots\!88}a^{14}+\frac{32\!\cdots\!29}{64\!\cdots\!88}a^{13}-\frac{78\!\cdots\!85}{64\!\cdots\!88}a^{12}+\frac{97\!\cdots\!17}{32\!\cdots\!44}a^{11}-\frac{88\!\cdots\!33}{32\!\cdots\!44}a^{10}+\frac{14\!\cdots\!23}{64\!\cdots\!88}a^{9}+\frac{15\!\cdots\!99}{64\!\cdots\!88}a^{8}-\frac{50\!\cdots\!27}{64\!\cdots\!88}a^{7}+\frac{12\!\cdots\!75}{64\!\cdots\!88}a^{6}+\frac{41\!\cdots\!39}{32\!\cdots\!44}a^{5}-\frac{11\!\cdots\!01}{40\!\cdots\!68}a^{4}-\frac{70\!\cdots\!59}{80\!\cdots\!36}a^{3}+\frac{56\!\cdots\!29}{40\!\cdots\!68}a^{2}+\frac{26\!\cdots\!31}{10\!\cdots\!92}a+\frac{25\!\cdots\!65}{12\!\cdots\!24}$, $\frac{92\!\cdots\!25}{84\!\cdots\!88}a^{15}-\frac{27\!\cdots\!37}{84\!\cdots\!88}a^{14}+\frac{22\!\cdots\!23}{84\!\cdots\!88}a^{13}-\frac{77\!\cdots\!59}{84\!\cdots\!88}a^{12}+\frac{32\!\cdots\!15}{42\!\cdots\!44}a^{11}-\frac{14\!\cdots\!99}{42\!\cdots\!44}a^{10}+\frac{55\!\cdots\!29}{84\!\cdots\!88}a^{9}+\frac{30\!\cdots\!85}{84\!\cdots\!88}a^{8}-\frac{31\!\cdots\!25}{84\!\cdots\!88}a^{7}+\frac{70\!\cdots\!45}{84\!\cdots\!88}a^{6}+\frac{29\!\cdots\!57}{42\!\cdots\!44}a^{5}-\frac{55\!\cdots\!71}{52\!\cdots\!68}a^{4}+\frac{20\!\cdots\!75}{10\!\cdots\!36}a^{3}+\frac{12\!\cdots\!47}{52\!\cdots\!68}a^{2}-\frac{62\!\cdots\!91}{13\!\cdots\!92}a-\frac{54\!\cdots\!85}{16\!\cdots\!24}$, $\frac{19\!\cdots\!25}{84\!\cdots\!88}a^{15}-\frac{17\!\cdots\!01}{84\!\cdots\!88}a^{14}+\frac{73\!\cdots\!43}{84\!\cdots\!88}a^{13}-\frac{28\!\cdots\!87}{84\!\cdots\!88}a^{12}+\frac{11\!\cdots\!07}{42\!\cdots\!44}a^{11}-\frac{61\!\cdots\!23}{42\!\cdots\!44}a^{10}+\frac{32\!\cdots\!13}{84\!\cdots\!88}a^{9}+\frac{42\!\cdots\!65}{84\!\cdots\!88}a^{8}-\frac{11\!\cdots\!73}{84\!\cdots\!88}a^{7}+\frac{43\!\cdots\!57}{84\!\cdots\!88}a^{6}-\frac{10\!\cdots\!11}{42\!\cdots\!44}a^{5}-\frac{31\!\cdots\!59}{52\!\cdots\!68}a^{4}+\frac{95\!\cdots\!59}{10\!\cdots\!36}a^{3}+\frac{10\!\cdots\!71}{52\!\cdots\!68}a^{2}-\frac{62\!\cdots\!71}{13\!\cdots\!92}a-\frac{10\!\cdots\!45}{16\!\cdots\!24}$ 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:  \( 32676067.5694 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 32676067.5694 \cdot 9792}{2\cdot\sqrt{3922880935919264967742950184849}}\cr\approx \mathstrut & 196.203873904 \end{aligned}\] (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 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832)
 
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 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832, 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 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832);
 
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 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832);
 
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

\(\Q(\sqrt{17}) \), 4.0.3757.1, 4.4.830297.1, 4.0.63869.1, 8.0.1980626399884457.1, 8.8.1980626399884457.1, 8.0.689393108209.2

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: 16.0.3922880935919264967742950184849.8
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 ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ R R ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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
\(13\) Copy content Toggle raw display 13.8.6.1$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(17\) Copy content Toggle raw display 17.8.7.1$x^{8} + 68$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} + 68$$8$$1$$7$$C_8$$[\ ]_{8}$