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Properties

Label 16.0.503...000.61
Degree $16$
Signature $[0, 8]$
Discriminant $5.032\times 10^{33}$
Root discriminant \(127.75\)
Ramified primes $2,3,5,17$
Class number $4792320$ (GRH)
Class group [2, 4, 4, 24, 6240] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286)
 
gp: K = bnfinit(y^16 + 148*y^14 + 10486*y^12 - 24*y^11 + 460196*y^10 + 3864*y^9 + 13557351*y^8 + 224640*y^7 + 272216812*y^6 + 2718048*y^5 + 3610023514*y^4 - 36903144*y^3 + 28731561716*y^2 - 747884040*y + 104415192286, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286)
 

\( x^{16} + 148 x^{14} + 10486 x^{12} - 24 x^{11} + 460196 x^{10} + 3864 x^{9} + 13557351 x^{8} + \cdots + 104415192286 \) 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:   \(5032238723456334453905817600000000\) \(\medspace = 2^{48}\cdot 3^{8}\cdot 5^{8}\cdot 17^{8}\) 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:  \(127.75\)
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:  $2^{3}3^{1/2}5^{1/2}17^{1/2}\approx 127.74975538137049$
Ramified primes:   \(2\), \(3\), \(5\), \(17\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4080=2^{4}\cdot 3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4080}(1,·)$, $\chi_{4080}(3911,·)$, $\chi_{4080}(2891,·)$, $\chi_{4080}(1871,·)$, $\chi_{4080}(851,·)$, $\chi_{4080}(2041,·)$, $\chi_{4080}(3739,·)$, $\chi_{4080}(2719,·)$, $\chi_{4080}(1699,·)$, $\chi_{4080}(679,·)$, $\chi_{4080}(1021,·)$, $\chi_{4080}(3569,·)$, $\chi_{4080}(3061,·)$, $\chi_{4080}(1529,·)$, $\chi_{4080}(509,·)$, $\chi_{4080}(2549,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{20600506409}a^{14}-\frac{6746824366}{20600506409}a^{13}-\frac{4038700931}{20600506409}a^{12}+\frac{8014398951}{20600506409}a^{11}-\frac{6058036679}{20600506409}a^{10}+\frac{6375823592}{20600506409}a^{9}-\frac{300649151}{2942929487}a^{8}+\frac{2626363545}{20600506409}a^{7}-\frac{3792147596}{20600506409}a^{6}+\frac{930534973}{20600506409}a^{5}+\frac{8264567614}{20600506409}a^{4}+\frac{1185825177}{20600506409}a^{3}+\frac{9903741304}{20600506409}a^{2}+\frac{3765399070}{20600506409}a-\frac{3030760595}{20600506409}$, $\frac{1}{30\!\cdots\!09}a^{15}+\frac{32\!\cdots\!31}{30\!\cdots\!09}a^{14}-\frac{73\!\cdots\!17}{30\!\cdots\!09}a^{13}+\frac{38\!\cdots\!32}{43\!\cdots\!87}a^{12}-\frac{11\!\cdots\!71}{30\!\cdots\!09}a^{11}+\frac{62\!\cdots\!32}{30\!\cdots\!09}a^{10}-\frac{51\!\cdots\!29}{30\!\cdots\!09}a^{9}-\frac{15\!\cdots\!16}{30\!\cdots\!09}a^{8}+\frac{66\!\cdots\!97}{30\!\cdots\!09}a^{7}-\frac{11\!\cdots\!49}{30\!\cdots\!09}a^{6}-\frac{11\!\cdots\!88}{30\!\cdots\!09}a^{5}+\frac{19\!\cdots\!34}{43\!\cdots\!87}a^{4}+\frac{16\!\cdots\!59}{43\!\cdots\!87}a^{3}-\frac{18\!\cdots\!73}{64\!\cdots\!47}a^{2}+\frac{13\!\cdots\!73}{30\!\cdots\!09}a-\frac{12\!\cdots\!09}{30\!\cdots\!09}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{24}\times C_{6240}$, which has order $4792320$ (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:   $\frac{34\!\cdots\!48}{40\!\cdots\!29}a^{15}-\frac{14\!\cdots\!06}{58\!\cdots\!47}a^{14}+\frac{42\!\cdots\!88}{40\!\cdots\!29}a^{13}-\frac{14\!\cdots\!53}{40\!\cdots\!29}a^{12}+\frac{25\!\cdots\!32}{40\!\cdots\!29}a^{11}-\frac{94\!\cdots\!53}{40\!\cdots\!29}a^{10}+\frac{91\!\cdots\!10}{40\!\cdots\!29}a^{9}-\frac{37\!\cdots\!91}{40\!\cdots\!29}a^{8}+\frac{21\!\cdots\!80}{40\!\cdots\!29}a^{7}-\frac{97\!\cdots\!27}{40\!\cdots\!29}a^{6}+\frac{32\!\cdots\!76}{40\!\cdots\!29}a^{5}-\frac{35\!\cdots\!57}{86\!\cdots\!07}a^{4}+\frac{29\!\cdots\!32}{40\!\cdots\!29}a^{3}-\frac{16\!\cdots\!97}{40\!\cdots\!29}a^{2}+\frac{11\!\cdots\!70}{40\!\cdots\!29}a-\frac{76\!\cdots\!03}{40\!\cdots\!29}$, $\frac{32\!\cdots\!80}{30\!\cdots\!09}a^{15}-\frac{67\!\cdots\!16}{30\!\cdots\!09}a^{14}+\frac{40\!\cdots\!32}{30\!\cdots\!09}a^{13}-\frac{95\!\cdots\!49}{30\!\cdots\!09}a^{12}+\frac{24\!\cdots\!64}{30\!\cdots\!09}a^{11}-\frac{62\!\cdots\!62}{30\!\cdots\!09}a^{10}+\frac{12\!\cdots\!04}{43\!\cdots\!87}a^{9}-\frac{24\!\cdots\!39}{30\!\cdots\!09}a^{8}+\frac{21\!\cdots\!44}{30\!\cdots\!09}a^{7}-\frac{65\!\cdots\!00}{30\!\cdots\!09}a^{6}+\frac{33\!\cdots\!68}{30\!\cdots\!09}a^{5}-\frac{23\!\cdots\!57}{64\!\cdots\!47}a^{4}+\frac{30\!\cdots\!44}{30\!\cdots\!09}a^{3}-\frac{11\!\cdots\!66}{30\!\cdots\!09}a^{2}+\frac{12\!\cdots\!29}{30\!\cdots\!09}a-\frac{74\!\cdots\!21}{43\!\cdots\!87}$, $\frac{20\!\cdots\!98}{30\!\cdots\!09}a^{15}-\frac{47\!\cdots\!10}{30\!\cdots\!09}a^{14}+\frac{24\!\cdots\!59}{30\!\cdots\!09}a^{13}-\frac{66\!\cdots\!25}{30\!\cdots\!09}a^{12}+\frac{14\!\cdots\!69}{30\!\cdots\!09}a^{11}-\frac{44\!\cdots\!03}{30\!\cdots\!09}a^{10}+\frac{78\!\cdots\!72}{43\!\cdots\!87}a^{9}-\frac{17\!\cdots\!73}{30\!\cdots\!09}a^{8}+\frac{13\!\cdots\!17}{30\!\cdots\!09}a^{7}-\frac{46\!\cdots\!62}{30\!\cdots\!09}a^{6}+\frac{20\!\cdots\!30}{30\!\cdots\!09}a^{5}-\frac{16\!\cdots\!33}{64\!\cdots\!47}a^{4}+\frac{18\!\cdots\!79}{30\!\cdots\!09}a^{3}-\frac{80\!\cdots\!05}{30\!\cdots\!09}a^{2}+\frac{78\!\cdots\!07}{30\!\cdots\!09}a-\frac{53\!\cdots\!89}{43\!\cdots\!87}$, $\frac{27\!\cdots\!02}{15\!\cdots\!81}a^{15}-\frac{59\!\cdots\!34}{15\!\cdots\!81}a^{14}+\frac{34\!\cdots\!19}{15\!\cdots\!81}a^{13}-\frac{84\!\cdots\!66}{15\!\cdots\!81}a^{12}+\frac{20\!\cdots\!97}{15\!\cdots\!81}a^{11}-\frac{55\!\cdots\!85}{15\!\cdots\!81}a^{10}+\frac{10\!\cdots\!84}{22\!\cdots\!83}a^{9}-\frac{22\!\cdots\!08}{15\!\cdots\!81}a^{8}+\frac{18\!\cdots\!49}{15\!\cdots\!81}a^{7}-\frac{58\!\cdots\!58}{15\!\cdots\!81}a^{6}+\frac{28\!\cdots\!82}{15\!\cdots\!81}a^{5}-\frac{21\!\cdots\!10}{33\!\cdots\!23}a^{4}+\frac{25\!\cdots\!07}{15\!\cdots\!81}a^{3}-\frac{10\!\cdots\!39}{15\!\cdots\!81}a^{2}+\frac{10\!\cdots\!24}{15\!\cdots\!81}a-\frac{66\!\cdots\!73}{22\!\cdots\!83}$, $\frac{17\!\cdots\!40}{15\!\cdots\!81}a^{15}-\frac{32\!\cdots\!97}{15\!\cdots\!81}a^{14}+\frac{21\!\cdots\!10}{15\!\cdots\!81}a^{13}-\frac{65\!\cdots\!96}{22\!\cdots\!83}a^{12}+\frac{13\!\cdots\!72}{15\!\cdots\!81}a^{11}-\frac{30\!\cdots\!47}{15\!\cdots\!81}a^{10}+\frac{48\!\cdots\!77}{15\!\cdots\!81}a^{9}-\frac{11\!\cdots\!44}{15\!\cdots\!81}a^{8}+\frac{11\!\cdots\!80}{15\!\cdots\!81}a^{7}-\frac{31\!\cdots\!43}{15\!\cdots\!81}a^{6}+\frac{18\!\cdots\!46}{15\!\cdots\!81}a^{5}-\frac{16\!\cdots\!06}{48\!\cdots\!89}a^{4}+\frac{24\!\cdots\!09}{22\!\cdots\!83}a^{3}-\frac{54\!\cdots\!71}{15\!\cdots\!81}a^{2}+\frac{73\!\cdots\!84}{15\!\cdots\!81}a-\frac{25\!\cdots\!87}{15\!\cdots\!81}$, $\frac{54\!\cdots\!42}{30\!\cdots\!09}a^{15}-\frac{30\!\cdots\!72}{30\!\cdots\!09}a^{14}+\frac{63\!\cdots\!29}{30\!\cdots\!09}a^{13}-\frac{43\!\cdots\!48}{30\!\cdots\!09}a^{12}+\frac{35\!\cdots\!93}{30\!\cdots\!09}a^{11}-\frac{28\!\cdots\!12}{30\!\cdots\!09}a^{10}+\frac{17\!\cdots\!12}{43\!\cdots\!87}a^{9}-\frac{11\!\cdots\!53}{30\!\cdots\!09}a^{8}+\frac{27\!\cdots\!17}{30\!\cdots\!09}a^{7}-\frac{29\!\cdots\!45}{30\!\cdots\!09}a^{6}+\frac{36\!\cdots\!60}{30\!\cdots\!09}a^{5}-\frac{10\!\cdots\!32}{64\!\cdots\!47}a^{4}+\frac{27\!\cdots\!55}{30\!\cdots\!09}a^{3}-\frac{51\!\cdots\!87}{30\!\cdots\!09}a^{2}+\frac{87\!\cdots\!95}{30\!\cdots\!09}a-\frac{33\!\cdots\!19}{43\!\cdots\!87}$, $\frac{31\!\cdots\!66}{30\!\cdots\!09}a^{15}-\frac{65\!\cdots\!81}{30\!\cdots\!09}a^{14}+\frac{38\!\cdots\!33}{30\!\cdots\!09}a^{13}-\frac{92\!\cdots\!39}{30\!\cdots\!09}a^{12}+\frac{23\!\cdots\!45}{30\!\cdots\!09}a^{11}-\frac{60\!\cdots\!01}{30\!\cdots\!09}a^{10}+\frac{85\!\cdots\!26}{30\!\cdots\!09}a^{9}-\frac{24\!\cdots\!91}{30\!\cdots\!09}a^{8}+\frac{20\!\cdots\!63}{30\!\cdots\!09}a^{7}-\frac{89\!\cdots\!44}{43\!\cdots\!87}a^{6}+\frac{45\!\cdots\!57}{43\!\cdots\!87}a^{5}-\frac{22\!\cdots\!36}{64\!\cdots\!47}a^{4}+\frac{29\!\cdots\!18}{30\!\cdots\!09}a^{3}-\frac{10\!\cdots\!11}{30\!\cdots\!09}a^{2}+\frac{12\!\cdots\!95}{30\!\cdots\!09}a-\frac{49\!\cdots\!25}{30\!\cdots\!09}$ 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:  \( 11964.310642723332 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11964.310642723332 \cdot 4792320}{2\cdot\sqrt{5032238723456334453905817600000000}}\cr\approx \mathstrut & 0.981661647338185 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286)
 
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 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286, 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 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286);
 
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 + 148*x^14 + 10486*x^12 - 24*x^11 + 460196*x^10 + 3864*x^9 + 13557351*x^8 + 224640*x^7 + 272216812*x^6 + 2718048*x^5 + 3610023514*x^4 - 36903144*x^3 + 28731561716*x^2 - 747884040*x + 104415192286);
 
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

$C_2^2\times C_4$ (as 16T2):

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)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-510}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{3}, \sqrt{-85})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-170})\), \(\Q(\sqrt{2}, \sqrt{-255})\), \(\Q(\sqrt{6}, \sqrt{-170})\), \(\Q(\sqrt{2}, \sqrt{-85})\), \(\Q(\sqrt{6}, \sqrt{-85})\), 4.4.18432.1, \(\Q(\zeta_{16})^+\), 4.0.14796800.5, 4.0.133171200.5, 8.0.277102632960000.225, \(\Q(\zeta_{48})^+\), 8.0.70938274037760000.16, 8.0.17734568509440000.260, 8.0.17734568509440000.473, 8.0.70938274037760000.13, 8.0.875781160960000.56

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.1.0.1}{1} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(2\) Copy content Toggle raw display 2.8.24.9$x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(17\) Copy content Toggle raw display 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$