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Properties

Label 16.0.832...361.1
Degree $16$
Signature $[0, 8]$
Discriminant $8.322\times 10^{30}$
Root discriminant \(85.61\)
Ramified primes $11,23$
Class number $3$ (GRH)
Class group [3] (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 + 12*x^14 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063)
 
gp: K = bnfinit(y^16 - 2*y^15 + 12*y^14 - 25*y^13 + 173*y^12 - 1719*y^11 + 7483*y^10 - 10055*y^9 - 26783*y^8 + 105894*y^7 + 125405*y^6 - 1722505*y^5 + 5788287*y^4 - 11178929*y^3 + 13401259*y^2 - 9271368*y + 2808063, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 12*x^14 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 12*x^14 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063)
 

\( x^{16} - 2 x^{15} + 12 x^{14} - 25 x^{13} + 173 x^{12} - 1719 x^{11} + 7483 x^{10} - 10055 x^{9} + \cdots + 2808063 \) 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:   \(8322074981098944220712357040361\) \(\medspace = 11^{14}\cdot 23^{12}\) 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:  \(85.61\)
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:  $11^{7/8}23^{3/4}\approx 85.60802473392998$
Ramified primes:   \(11\), \(23\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{51}a^{11}-\frac{5}{51}a^{10}-\frac{5}{51}a^{9}-\frac{2}{17}a^{8}+\frac{8}{17}a^{7}+\frac{4}{17}a^{6}+\frac{6}{17}a^{5}-\frac{7}{17}a^{4}+\frac{11}{51}a^{3}+\frac{2}{51}a^{2}+\frac{14}{51}a-\frac{7}{17}$, $\frac{1}{8211}a^{12}-\frac{2}{2737}a^{11}+\frac{6}{161}a^{10}+\frac{968}{8211}a^{9}-\frac{173}{1173}a^{8}+\frac{2861}{8211}a^{7}+\frac{2692}{8211}a^{6}-\frac{178}{1173}a^{5}+\frac{583}{2737}a^{4}-\frac{232}{1173}a^{3}-\frac{3422}{8211}a^{2}-\frac{1332}{2737}a-\frac{10}{2737}$, $\frac{1}{8211}a^{13}-\frac{52}{8211}a^{11}-\frac{1060}{8211}a^{10}+\frac{733}{8211}a^{9}+\frac{88}{2737}a^{8}+\frac{394}{2737}a^{7}-\frac{881}{2737}a^{6}+\frac{94}{357}a^{5}+\frac{649}{2737}a^{4}-\frac{3023}{8211}a^{3}-\frac{77}{1173}a^{2}-\frac{1144}{8211}a-\frac{543}{2737}$, $\frac{1}{2652153}a^{14}-\frac{41}{884051}a^{13}+\frac{29}{884051}a^{12}+\frac{6599}{884051}a^{11}+\frac{7111}{52003}a^{10}+\frac{258304}{2652153}a^{9}-\frac{22859}{378879}a^{8}-\frac{483544}{2652153}a^{7}+\frac{303204}{884051}a^{6}+\frac{412907}{2652153}a^{5}-\frac{139946}{2652153}a^{4}+\frac{601184}{2652153}a^{3}+\frac{266017}{2652153}a^{2}+\frac{575077}{2652153}a+\frac{210138}{884051}$, $\frac{1}{60\!\cdots\!89}a^{15}+\frac{15\!\cdots\!87}{60\!\cdots\!89}a^{14}+\frac{23\!\cdots\!94}{13\!\cdots\!97}a^{13}+\frac{98\!\cdots\!57}{60\!\cdots\!89}a^{12}+\frac{43\!\cdots\!12}{20\!\cdots\!63}a^{11}+\frac{29\!\cdots\!24}{28\!\cdots\!09}a^{10}-\frac{27\!\cdots\!94}{86\!\cdots\!27}a^{9}+\frac{97\!\cdots\!45}{60\!\cdots\!89}a^{8}+\frac{47\!\cdots\!07}{20\!\cdots\!63}a^{7}-\frac{80\!\cdots\!77}{35\!\cdots\!59}a^{6}+\frac{28\!\cdots\!45}{60\!\cdots\!89}a^{5}-\frac{71\!\cdots\!36}{31\!\cdots\!31}a^{4}-\frac{47\!\cdots\!10}{60\!\cdots\!89}a^{3}-\frac{10\!\cdots\!42}{60\!\cdots\!89}a^{2}-\frac{46\!\cdots\!70}{20\!\cdots\!63}a+\frac{97\!\cdots\!50}{67\!\cdots\!21}$ 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:  $3$

Class group and class number

$C_{3}$, which has order $3$ (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:   $\frac{16\!\cdots\!69}{60\!\cdots\!89}a^{15}-\frac{12\!\cdots\!12}{60\!\cdots\!89}a^{14}+\frac{18\!\cdots\!30}{60\!\cdots\!89}a^{13}-\frac{18\!\cdots\!20}{60\!\cdots\!89}a^{12}+\frac{88\!\cdots\!15}{20\!\cdots\!63}a^{11}-\frac{40\!\cdots\!22}{96\!\cdots\!03}a^{10}+\frac{13\!\cdots\!20}{86\!\cdots\!27}a^{9}-\frac{50\!\cdots\!23}{60\!\cdots\!89}a^{8}-\frac{56\!\cdots\!27}{67\!\cdots\!21}a^{7}+\frac{37\!\cdots\!08}{20\!\cdots\!63}a^{6}+\frac{35\!\cdots\!46}{60\!\cdots\!89}a^{5}-\frac{24\!\cdots\!04}{60\!\cdots\!89}a^{4}+\frac{66\!\cdots\!65}{60\!\cdots\!89}a^{3}-\frac{10\!\cdots\!26}{60\!\cdots\!89}a^{2}+\frac{31\!\cdots\!24}{20\!\cdots\!63}a-\frac{17\!\cdots\!01}{29\!\cdots\!27}$, $\frac{16\!\cdots\!11}{67\!\cdots\!31}a^{15}-\frac{59\!\cdots\!16}{29\!\cdots\!97}a^{14}+\frac{17\!\cdots\!25}{67\!\cdots\!31}a^{13}-\frac{21\!\cdots\!72}{67\!\cdots\!31}a^{12}+\frac{74\!\cdots\!60}{20\!\cdots\!93}a^{11}-\frac{10\!\cdots\!75}{29\!\cdots\!99}a^{10}+\frac{39\!\cdots\!29}{29\!\cdots\!99}a^{9}-\frac{60\!\cdots\!06}{88\!\cdots\!91}a^{8}-\frac{15\!\cdots\!06}{20\!\cdots\!93}a^{7}+\frac{33\!\cdots\!77}{20\!\cdots\!93}a^{6}+\frac{10\!\cdots\!65}{20\!\cdots\!93}a^{5}-\frac{73\!\cdots\!20}{20\!\cdots\!93}a^{4}+\frac{19\!\cdots\!11}{20\!\cdots\!93}a^{3}-\frac{98\!\cdots\!09}{67\!\cdots\!31}a^{2}+\frac{82\!\cdots\!30}{67\!\cdots\!31}a-\frac{29\!\cdots\!70}{67\!\cdots\!31}$, $\frac{47\!\cdots\!07}{60\!\cdots\!89}a^{15}+\frac{11\!\cdots\!60}{60\!\cdots\!89}a^{14}+\frac{64\!\cdots\!70}{60\!\cdots\!89}a^{13}+\frac{85\!\cdots\!47}{60\!\cdots\!89}a^{12}+\frac{29\!\cdots\!18}{20\!\cdots\!63}a^{11}-\frac{10\!\cdots\!48}{96\!\cdots\!03}a^{10}+\frac{33\!\cdots\!27}{86\!\cdots\!27}a^{9}-\frac{11\!\cdots\!13}{60\!\cdots\!89}a^{8}-\frac{12\!\cdots\!30}{67\!\cdots\!21}a^{7}+\frac{91\!\cdots\!69}{20\!\cdots\!63}a^{6}+\frac{89\!\cdots\!11}{60\!\cdots\!89}a^{5}-\frac{60\!\cdots\!55}{60\!\cdots\!89}a^{4}+\frac{17\!\cdots\!80}{60\!\cdots\!89}a^{3}-\frac{27\!\cdots\!01}{60\!\cdots\!89}a^{2}+\frac{46\!\cdots\!47}{10\!\cdots\!77}a-\frac{13\!\cdots\!08}{67\!\cdots\!21}$, $\frac{72\!\cdots\!74}{86\!\cdots\!27}a^{15}-\frac{95\!\cdots\!91}{45\!\cdots\!33}a^{14}+\frac{26\!\cdots\!89}{37\!\cdots\!49}a^{13}-\frac{25\!\cdots\!50}{86\!\cdots\!27}a^{12}+\frac{31\!\cdots\!39}{28\!\cdots\!09}a^{11}-\frac{63\!\cdots\!05}{41\!\cdots\!87}a^{10}+\frac{78\!\cdots\!72}{12\!\cdots\!61}a^{9}-\frac{58\!\cdots\!57}{86\!\cdots\!27}a^{8}-\frac{30\!\cdots\!04}{96\!\cdots\!03}a^{7}+\frac{11\!\cdots\!82}{12\!\cdots\!83}a^{6}+\frac{14\!\cdots\!68}{86\!\cdots\!27}a^{5}-\frac{13\!\cdots\!47}{86\!\cdots\!27}a^{4}+\frac{40\!\cdots\!64}{86\!\cdots\!27}a^{3}-\frac{66\!\cdots\!58}{86\!\cdots\!27}a^{2}+\frac{20\!\cdots\!45}{28\!\cdots\!09}a-\frac{26\!\cdots\!30}{96\!\cdots\!03}$, $\frac{13\!\cdots\!61}{60\!\cdots\!89}a^{15}-\frac{41\!\cdots\!57}{60\!\cdots\!89}a^{14}+\frac{73\!\cdots\!92}{26\!\cdots\!43}a^{13}-\frac{13\!\cdots\!09}{60\!\cdots\!89}a^{12}+\frac{79\!\cdots\!17}{20\!\cdots\!63}a^{11}-\frac{89\!\cdots\!49}{28\!\cdots\!09}a^{10}+\frac{93\!\cdots\!99}{86\!\cdots\!27}a^{9}-\frac{18\!\cdots\!77}{60\!\cdots\!89}a^{8}-\frac{12\!\cdots\!11}{20\!\cdots\!63}a^{7}+\frac{33\!\cdots\!28}{29\!\cdots\!27}a^{6}+\frac{28\!\cdots\!58}{60\!\cdots\!89}a^{5}-\frac{17\!\cdots\!26}{60\!\cdots\!89}a^{4}+\frac{45\!\cdots\!07}{60\!\cdots\!89}a^{3}-\frac{70\!\cdots\!33}{60\!\cdots\!89}a^{2}+\frac{70\!\cdots\!01}{67\!\cdots\!21}a-\frac{28\!\cdots\!94}{67\!\cdots\!21}$, $\frac{38\!\cdots\!66}{28\!\cdots\!09}a^{15}-\frac{55\!\cdots\!50}{28\!\cdots\!09}a^{14}+\frac{38\!\cdots\!07}{28\!\cdots\!09}a^{13}-\frac{33\!\cdots\!23}{12\!\cdots\!83}a^{12}+\frac{19\!\cdots\!78}{96\!\cdots\!03}a^{11}-\frac{47\!\cdots\!99}{21\!\cdots\!73}a^{10}+\frac{35\!\cdots\!16}{41\!\cdots\!87}a^{9}-\frac{19\!\cdots\!75}{28\!\cdots\!09}a^{8}-\frac{42\!\cdots\!64}{96\!\cdots\!03}a^{7}+\frac{32\!\cdots\!84}{28\!\cdots\!09}a^{6}+\frac{77\!\cdots\!31}{28\!\cdots\!09}a^{5}-\frac{62\!\cdots\!95}{28\!\cdots\!09}a^{4}+\frac{17\!\cdots\!65}{28\!\cdots\!09}a^{3}-\frac{95\!\cdots\!92}{96\!\cdots\!03}a^{2}+\frac{87\!\cdots\!23}{96\!\cdots\!03}a-\frac{35\!\cdots\!62}{96\!\cdots\!03}$, $\frac{22\!\cdots\!02}{96\!\cdots\!03}a^{15}-\frac{46\!\cdots\!21}{28\!\cdots\!09}a^{14}+\frac{73\!\cdots\!29}{28\!\cdots\!09}a^{13}-\frac{23\!\cdots\!19}{96\!\cdots\!03}a^{12}+\frac{10\!\cdots\!36}{28\!\cdots\!09}a^{11}-\frac{47\!\cdots\!48}{13\!\cdots\!29}a^{10}+\frac{17\!\cdots\!34}{13\!\cdots\!29}a^{9}-\frac{63\!\cdots\!43}{96\!\cdots\!03}a^{8}-\frac{67\!\cdots\!86}{96\!\cdots\!03}a^{7}+\frac{43\!\cdots\!45}{28\!\cdots\!09}a^{6}+\frac{13\!\cdots\!74}{28\!\cdots\!09}a^{5}-\frac{31\!\cdots\!71}{96\!\cdots\!03}a^{4}+\frac{25\!\cdots\!14}{28\!\cdots\!09}a^{3}-\frac{13\!\cdots\!20}{96\!\cdots\!03}a^{2}+\frac{12\!\cdots\!87}{96\!\cdots\!03}a-\frac{47\!\cdots\!42}{96\!\cdots\!03}$ 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:  \( 297238964.584 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 297238964.584 \cdot 3}{2\cdot\sqrt{8322074981098944220712357040361}}\cr\approx \mathstrut & 0.375422358312 \end{aligned}\] (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 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063)
 
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 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063, 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 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063);
 
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 - 25*x^13 + 173*x^12 - 1719*x^11 + 7483*x^10 - 10055*x^9 - 26783*x^8 + 105894*x^7 + 125405*x^6 - 1722505*x^5 + 5788287*x^4 - 11178929*x^3 + 13401259*x^2 - 9271368*x + 2808063);
 
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

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{253}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{-23})\), 4.2.704099.1 x2, 4.0.30613.1 x2, 8.0.495755401801.1, 8.2.2884800683080019.1 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: 8.2.2884800683080019.1
Minimal sibling: 8.2.2884800683080019.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{8}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ R ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(11\) Copy content Toggle raw display 11.16.14.1$x^{16} - 198 x^{8} - 10043$$8$$2$$14$$QD_{16}$$[\ ]_{8}^{2}$
\(23\) Copy content Toggle raw display 23.8.6.1$x^{8} - 138 x^{4} - 217948$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
23.8.6.1$x^{8} - 138 x^{4} - 217948$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$