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Properties

Label 16.0.462...081.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.627\times 10^{44}$
Root discriminant \(618.85\)
Ramified primes $29,41$
Class number $13506148$ (GRH)
Class group [13506148] (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552)
 
gp: K = bnfinit(y^16 - 7*y^15 + 96*y^14 + 91*y^13 + 2772*y^12 + 129667*y^11 - 310036*y^10 + 576733*y^9 - 5430577*y^8 - 152310452*y^7 + 3090172208*y^6 + 409621632*y^5 - 96715161760*y^4 + 166203665088*y^3 - 59137532864*y^2 + 3645586489344*y + 106045326871552, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552)
 

\( x^{16} - 7 x^{15} + 96 x^{14} + 91 x^{13} + 2772 x^{12} + 129667 x^{11} - 310036 x^{10} + \cdots + 106045326871552 \) 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:   \(462732306245995722656474121747020143758680081\) \(\medspace = 29^{14}\cdot 41^{15}\) 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:  \(618.85\)
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:  $29^{7/8}41^{15/16}\approx 618.8467490579351$
Ramified primes:   \(29\), \(41\) 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:  \(\Q(\sqrt{41}) \)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{10}-\frac{1}{32}a^{9}+\frac{1}{32}a^{8}-\frac{1}{16}a^{7}+\frac{3}{32}a^{6}-\frac{5}{32}a^{5}-\frac{5}{32}a^{4}+\frac{1}{4}a$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{8}+\frac{1}{32}a^{7}-\frac{1}{16}a^{6}+\frac{3}{16}a^{5}-\frac{5}{32}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{12}-\frac{1}{64}a^{9}+\frac{1}{64}a^{8}-\frac{1}{32}a^{7}+\frac{3}{32}a^{6}+\frac{11}{64}a^{5}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{13}-\frac{1}{128}a^{11}-\frac{3}{256}a^{10}+\frac{3}{256}a^{9}-\frac{1}{128}a^{8}-\frac{3}{32}a^{7}+\frac{9}{256}a^{6}-\frac{1}{128}a^{5}+\frac{11}{64}a^{4}+\frac{3}{32}a^{3}+\frac{7}{16}a^{2}-\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{42496}a^{14}+\frac{1}{2656}a^{13}+\frac{31}{21248}a^{12}+\frac{397}{42496}a^{11}+\frac{259}{42496}a^{10}-\frac{33}{21248}a^{9}-\frac{167}{5312}a^{8}-\frac{3815}{42496}a^{7}-\frac{1953}{21248}a^{6}-\frac{2129}{10624}a^{5}+\frac{631}{5312}a^{4}-\frac{1217}{2656}a^{3}-\frac{57}{1328}a^{2}+\frac{11}{332}a+\frac{15}{83}$, $\frac{1}{18\!\cdots\!64}a^{15}+\frac{75\!\cdots\!61}{11\!\cdots\!08}a^{14}-\frac{16\!\cdots\!07}{23\!\cdots\!08}a^{13}+\frac{97\!\cdots\!53}{18\!\cdots\!64}a^{12}-\frac{24\!\cdots\!75}{18\!\cdots\!64}a^{11}+\frac{20\!\cdots\!79}{92\!\cdots\!32}a^{10}-\frac{36\!\cdots\!41}{66\!\cdots\!88}a^{9}-\frac{17\!\cdots\!83}{18\!\cdots\!64}a^{8}+\frac{23\!\cdots\!83}{23\!\cdots\!08}a^{7}+\frac{75\!\cdots\!61}{92\!\cdots\!32}a^{6}-\frac{71\!\cdots\!85}{46\!\cdots\!16}a^{5}-\frac{35\!\cdots\!27}{23\!\cdots\!08}a^{4}-\frac{22\!\cdots\!97}{11\!\cdots\!04}a^{3}+\frac{35\!\cdots\!09}{57\!\cdots\!52}a^{2}+\frac{23\!\cdots\!69}{28\!\cdots\!76}a-\frac{22\!\cdots\!11}{72\!\cdots\!94}$ 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

$C_{13506148}$, which has order $13506148$ (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:   $\frac{28\!\cdots\!85}{62\!\cdots\!28}a^{15}-\frac{92\!\cdots\!65}{10\!\cdots\!92}a^{14}+\frac{30\!\cdots\!35}{31\!\cdots\!64}a^{13}-\frac{75\!\cdots\!65}{62\!\cdots\!28}a^{12}+\frac{91\!\cdots\!35}{15\!\cdots\!32}a^{11}-\frac{35\!\cdots\!35}{62\!\cdots\!28}a^{10}-\frac{16\!\cdots\!65}{22\!\cdots\!76}a^{9}+\frac{33\!\cdots\!25}{62\!\cdots\!28}a^{8}-\frac{17\!\cdots\!85}{62\!\cdots\!28}a^{7}+\frac{93\!\cdots\!15}{31\!\cdots\!64}a^{6}-\frac{11\!\cdots\!05}{19\!\cdots\!79}a^{5}-\frac{34\!\cdots\!05}{15\!\cdots\!32}a^{4}+\frac{10\!\cdots\!75}{19\!\cdots\!79}a^{3}+\frac{22\!\cdots\!65}{38\!\cdots\!58}a^{2}+\frac{34\!\cdots\!00}{19\!\cdots\!79}a+\frac{58\!\cdots\!83}{19\!\cdots\!79}$, $\frac{46\!\cdots\!35}{25\!\cdots\!64}a^{15}+\frac{44\!\cdots\!99}{21\!\cdots\!48}a^{14}-\frac{67\!\cdots\!15}{15\!\cdots\!04}a^{13}+\frac{17\!\cdots\!87}{25\!\cdots\!64}a^{12}-\frac{16\!\cdots\!57}{25\!\cdots\!64}a^{11}+\frac{65\!\cdots\!37}{12\!\cdots\!32}a^{10}+\frac{45\!\cdots\!75}{91\!\cdots\!88}a^{9}-\frac{74\!\cdots\!13}{25\!\cdots\!64}a^{8}+\frac{29\!\cdots\!61}{79\!\cdots\!02}a^{7}-\frac{36\!\cdots\!75}{12\!\cdots\!32}a^{6}+\frac{36\!\cdots\!81}{31\!\cdots\!08}a^{5}+\frac{26\!\cdots\!27}{79\!\cdots\!02}a^{4}-\frac{26\!\cdots\!08}{39\!\cdots\!01}a^{3}+\frac{16\!\cdots\!84}{39\!\cdots\!01}a^{2}-\frac{74\!\cdots\!56}{39\!\cdots\!01}a+\frac{20\!\cdots\!91}{39\!\cdots\!01}$, $\frac{66\!\cdots\!11}{50\!\cdots\!28}a^{15}-\frac{78\!\cdots\!09}{86\!\cdots\!92}a^{14}+\frac{21\!\cdots\!23}{25\!\cdots\!64}a^{13}-\frac{58\!\cdots\!91}{50\!\cdots\!28}a^{12}+\frac{97\!\cdots\!83}{25\!\cdots\!64}a^{11}-\frac{22\!\cdots\!91}{50\!\cdots\!28}a^{10}-\frac{16\!\cdots\!99}{18\!\cdots\!76}a^{9}+\frac{13\!\cdots\!27}{50\!\cdots\!28}a^{8}-\frac{17\!\cdots\!25}{50\!\cdots\!28}a^{7}+\frac{57\!\cdots\!95}{25\!\cdots\!64}a^{6}+\frac{16\!\cdots\!05}{31\!\cdots\!08}a^{5}-\frac{90\!\cdots\!53}{63\!\cdots\!16}a^{4}+\frac{10\!\cdots\!39}{31\!\cdots\!08}a^{3}+\frac{33\!\cdots\!77}{79\!\cdots\!02}a^{2}+\frac{19\!\cdots\!46}{39\!\cdots\!01}a+\frac{77\!\cdots\!05}{39\!\cdots\!01}$, $\frac{11\!\cdots\!33}{15\!\cdots\!12}a^{15}-\frac{33\!\cdots\!99}{19\!\cdots\!28}a^{14}+\frac{29\!\cdots\!75}{18\!\cdots\!64}a^{13}-\frac{22\!\cdots\!81}{15\!\cdots\!12}a^{12}-\frac{18\!\cdots\!27}{18\!\cdots\!64}a^{11}+\frac{20\!\cdots\!19}{15\!\cdots\!12}a^{10}-\frac{33\!\cdots\!67}{27\!\cdots\!52}a^{9}-\frac{24\!\cdots\!79}{15\!\cdots\!12}a^{8}+\frac{13\!\cdots\!23}{15\!\cdots\!12}a^{7}-\frac{15\!\cdots\!65}{37\!\cdots\!28}a^{6}-\frac{12\!\cdots\!49}{94\!\cdots\!32}a^{5}+\frac{12\!\cdots\!93}{94\!\cdots\!32}a^{4}-\frac{50\!\cdots\!97}{11\!\cdots\!54}a^{3}+\frac{22\!\cdots\!55}{23\!\cdots\!08}a^{2}+\frac{33\!\cdots\!79}{23\!\cdots\!08}a-\frac{65\!\cdots\!34}{59\!\cdots\!27}$, $\frac{62\!\cdots\!49}{94\!\cdots\!32}a^{15}-\frac{35\!\cdots\!53}{49\!\cdots\!32}a^{14}+\frac{23\!\cdots\!05}{18\!\cdots\!64}a^{13}-\frac{32\!\cdots\!47}{18\!\cdots\!64}a^{12}+\frac{53\!\cdots\!03}{37\!\cdots\!28}a^{11}-\frac{40\!\cdots\!93}{37\!\cdots\!28}a^{10}-\frac{41\!\cdots\!05}{34\!\cdots\!44}a^{9}+\frac{66\!\cdots\!13}{94\!\cdots\!32}a^{8}-\frac{29\!\cdots\!89}{37\!\cdots\!28}a^{7}+\frac{30\!\cdots\!85}{47\!\cdots\!16}a^{6}-\frac{38\!\cdots\!93}{11\!\cdots\!54}a^{5}-\frac{11\!\cdots\!21}{23\!\cdots\!08}a^{4}+\frac{96\!\cdots\!77}{59\!\cdots\!27}a^{3}-\frac{63\!\cdots\!36}{59\!\cdots\!27}a^{2}+\frac{29\!\cdots\!39}{59\!\cdots\!27}a-\frac{10\!\cdots\!29}{59\!\cdots\!27}$, $\frac{40\!\cdots\!45}{75\!\cdots\!56}a^{15}-\frac{72\!\cdots\!21}{98\!\cdots\!64}a^{14}+\frac{38\!\cdots\!11}{37\!\cdots\!28}a^{13}-\frac{47\!\cdots\!97}{75\!\cdots\!56}a^{12}+\frac{21\!\cdots\!37}{37\!\cdots\!28}a^{11}+\frac{26\!\cdots\!43}{75\!\cdots\!56}a^{10}-\frac{11\!\cdots\!67}{27\!\cdots\!52}a^{9}+\frac{24\!\cdots\!77}{75\!\cdots\!56}a^{8}-\frac{19\!\cdots\!67}{75\!\cdots\!56}a^{7}+\frac{35\!\cdots\!31}{37\!\cdots\!28}a^{6}+\frac{51\!\cdots\!69}{47\!\cdots\!16}a^{5}-\frac{67\!\cdots\!21}{94\!\cdots\!32}a^{4}-\frac{21\!\cdots\!69}{47\!\cdots\!16}a^{3}+\frac{15\!\cdots\!93}{11\!\cdots\!54}a^{2}-\frac{55\!\cdots\!05}{59\!\cdots\!27}a+\frac{50\!\cdots\!55}{59\!\cdots\!27}$, $\frac{27\!\cdots\!09}{15\!\cdots\!12}a^{15}-\frac{28\!\cdots\!79}{19\!\cdots\!28}a^{14}+\frac{55\!\cdots\!17}{94\!\cdots\!32}a^{13}+\frac{26\!\cdots\!07}{15\!\cdots\!12}a^{12}+\frac{42\!\cdots\!49}{18\!\cdots\!64}a^{11}+\frac{90\!\cdots\!67}{15\!\cdots\!12}a^{10}-\frac{19\!\cdots\!05}{27\!\cdots\!52}a^{9}-\frac{65\!\cdots\!91}{15\!\cdots\!12}a^{8}+\frac{10\!\cdots\!19}{15\!\cdots\!12}a^{7}+\frac{24\!\cdots\!61}{37\!\cdots\!28}a^{6}-\frac{15\!\cdots\!01}{94\!\cdots\!32}a^{5}+\frac{16\!\cdots\!31}{94\!\cdots\!32}a^{4}-\frac{38\!\cdots\!55}{23\!\cdots\!08}a^{3}+\frac{13\!\cdots\!63}{23\!\cdots\!08}a^{2}+\frac{47\!\cdots\!11}{23\!\cdots\!08}a+\frac{66\!\cdots\!02}{59\!\cdots\!27}$ 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:  \( 276024556298 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 276024556298 \cdot 13506148}{2\cdot\sqrt{462732306245995722656474121747020143758680081}}\cr\approx \mathstrut & 210.486024391043 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552)
 
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 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552, 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 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552);
 
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 - 7*x^15 + 96*x^14 + 91*x^13 + 2772*x^12 + 129667*x^11 - 310036*x^10 + 576733*x^9 - 5430577*x^8 - 152310452*x^7 + 3090172208*x^6 + 409621632*x^5 - 96715161760*x^4 + 166203665088*x^3 - 59137532864*x^2 + 3645586489344*x + 106045326871552);
 
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

$\OD_{32}$ (as 16T22):

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 $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.57962561.1, 8.8.115844383968839978801.2

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
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 ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ R ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ $16$ $16$ ${\href{/padicField/59.1.0.1}{1} }^{16}$

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
\(29\) Copy content Toggle raw display 29.16.14.5$x^{16} - 696 x^{8} + 1682$$8$$2$$14$$C_{16} : C_2$$[\ ]_{8}^{4}$
\(41\) Copy content Toggle raw display 41.16.15.4$x^{16} + 82$$16$$1$$15$$C_{16} : C_2$$[\ ]_{16}^{2}$