Properties

Label 16.0.926...625.7
Degree $16$
Signature $[0, 8]$
Discriminant $9.260\times 10^{30}$
Root discriminant \(86.18\)
Ramified primes $5,41$
Class number $600$ (GRH)
Class group [2, 300] (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 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056)
 
gp: K = bnfinit(y^16 - y^15 + 46*y^14 - 105*y^13 + 1270*y^12 - 4449*y^11 + 16060*y^10 - 63735*y^9 + 211046*y^8 - 1759715*y^7 + 8594450*y^6 - 26273899*y^5 + 43579355*y^4 - 18892900*y^3 - 620444*y^2 - 73814496*y + 155285056, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056)
 

\( x^{16} - x^{15} + 46 x^{14} - 105 x^{13} + 1270 x^{12} - 4449 x^{11} + 16060 x^{10} - 63735 x^{9} + 211046 x^{8} - 1759715 x^{7} + 8594450 x^{6} - 26273899 x^{5} + \cdots + 155285056 \) 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:   \(9260065233133681348183837890625\) \(\medspace = 5^{12}\cdot 41^{14}\) 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:  \(86.18\)
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:  $5^{3/4}41^{7/8}\approx 86.18136686908747$
Ramified primes:   \(5\), \(41\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{8}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}+\frac{5}{16}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{9}-\frac{1}{4}a^{7}+\frac{3}{16}a^{6}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{64}a^{13}-\frac{1}{32}a^{12}+\frac{1}{64}a^{11}-\frac{1}{64}a^{9}-\frac{1}{16}a^{8}+\frac{15}{64}a^{7}-\frac{3}{32}a^{6}+\frac{11}{64}a^{5}-\frac{11}{64}a^{3}+\frac{5}{16}a^{2}-\frac{5}{16}a-\frac{1}{4}$, $\frac{1}{128}a^{14}-\frac{3}{128}a^{12}-\frac{1}{64}a^{11}-\frac{5}{128}a^{10}+\frac{3}{64}a^{9}-\frac{9}{128}a^{8}+\frac{1}{16}a^{7}-\frac{1}{128}a^{6}-\frac{11}{64}a^{5}+\frac{9}{128}a^{4}+\frac{1}{64}a^{3}+\frac{9}{32}a^{2}+\frac{3}{16}a+\frac{1}{4}$, $\frac{1}{10\!\cdots\!64}a^{15}-\frac{54\!\cdots\!59}{25\!\cdots\!16}a^{14}-\frac{62\!\cdots\!03}{10\!\cdots\!64}a^{13}+\frac{13\!\cdots\!41}{51\!\cdots\!32}a^{12}-\frac{24\!\cdots\!41}{10\!\cdots\!64}a^{11}-\frac{10\!\cdots\!67}{51\!\cdots\!32}a^{10}+\frac{78\!\cdots\!03}{10\!\cdots\!64}a^{9}+\frac{15\!\cdots\!85}{25\!\cdots\!16}a^{8}-\frac{29\!\cdots\!33}{10\!\cdots\!64}a^{7}-\frac{42\!\cdots\!69}{51\!\cdots\!32}a^{6}+\frac{78\!\cdots\!81}{10\!\cdots\!64}a^{5}-\frac{46\!\cdots\!77}{51\!\cdots\!32}a^{4}-\frac{94\!\cdots\!35}{12\!\cdots\!08}a^{3}-\frac{17\!\cdots\!71}{12\!\cdots\!08}a^{2}-\frac{25\!\cdots\!73}{64\!\cdots\!04}a-\frac{13\!\cdots\!01}{16\!\cdots\!26}$ 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

$C_{2}\times C_{300}$, which has order $600$ (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:   $\frac{42\!\cdots\!86}{17\!\cdots\!33}a^{15}-\frac{13\!\cdots\!19}{56\!\cdots\!56}a^{14}+\frac{51\!\cdots\!71}{56\!\cdots\!56}a^{13}-\frac{15\!\cdots\!57}{56\!\cdots\!56}a^{12}+\frac{13\!\cdots\!01}{56\!\cdots\!56}a^{11}-\frac{57\!\cdots\!01}{56\!\cdots\!56}a^{10}+\frac{12\!\cdots\!51}{56\!\cdots\!56}a^{9}-\frac{59\!\cdots\!01}{56\!\cdots\!56}a^{8}+\frac{26\!\cdots\!33}{56\!\cdots\!56}a^{7}-\frac{19\!\cdots\!87}{56\!\cdots\!56}a^{6}+\frac{10\!\cdots\!63}{56\!\cdots\!56}a^{5}-\frac{23\!\cdots\!99}{56\!\cdots\!56}a^{4}+\frac{31\!\cdots\!81}{56\!\cdots\!56}a^{3}+\frac{85\!\cdots\!53}{35\!\cdots\!66}a^{2}+\frac{12\!\cdots\!47}{14\!\cdots\!64}a-\frac{35\!\cdots\!51}{35\!\cdots\!66}$, $\frac{16\!\cdots\!21}{22\!\cdots\!24}a^{15}+\frac{78\!\cdots\!09}{11\!\cdots\!12}a^{14}+\frac{76\!\cdots\!87}{22\!\cdots\!24}a^{13}-\frac{39\!\cdots\!39}{56\!\cdots\!56}a^{12}+\frac{20\!\cdots\!65}{22\!\cdots\!24}a^{11}-\frac{76\!\cdots\!49}{56\!\cdots\!56}a^{10}+\frac{19\!\cdots\!25}{22\!\cdots\!24}a^{9}-\frac{29\!\cdots\!37}{11\!\cdots\!12}a^{8}+\frac{22\!\cdots\!37}{22\!\cdots\!24}a^{7}-\frac{59\!\cdots\!41}{56\!\cdots\!56}a^{6}+\frac{89\!\cdots\!83}{22\!\cdots\!24}a^{5}-\frac{60\!\cdots\!19}{56\!\cdots\!56}a^{4}+\frac{66\!\cdots\!23}{11\!\cdots\!12}a^{3}+\frac{69\!\cdots\!77}{28\!\cdots\!28}a^{2}+\frac{17\!\cdots\!47}{28\!\cdots\!28}a-\frac{32\!\cdots\!33}{70\!\cdots\!32}$, $\frac{29\!\cdots\!71}{11\!\cdots\!12}a^{15}+\frac{48\!\cdots\!05}{11\!\cdots\!12}a^{14}+\frac{73\!\cdots\!97}{56\!\cdots\!56}a^{13}+\frac{81\!\cdots\!05}{11\!\cdots\!12}a^{12}+\frac{19\!\cdots\!01}{56\!\cdots\!56}a^{11}-\frac{29\!\cdots\!95}{11\!\cdots\!12}a^{10}+\frac{89\!\cdots\!43}{28\!\cdots\!28}a^{9}-\frac{92\!\cdots\!89}{11\!\cdots\!12}a^{8}+\frac{16\!\cdots\!85}{56\!\cdots\!56}a^{7}-\frac{42\!\cdots\!33}{11\!\cdots\!12}a^{6}+\frac{68\!\cdots\!31}{56\!\cdots\!56}a^{5}-\frac{37\!\cdots\!57}{11\!\cdots\!12}a^{4}+\frac{10\!\cdots\!77}{11\!\cdots\!12}a^{3}+\frac{15\!\cdots\!53}{14\!\cdots\!64}a^{2}+\frac{12\!\cdots\!83}{28\!\cdots\!28}a-\frac{15\!\cdots\!69}{70\!\cdots\!32}$, $\frac{12\!\cdots\!87}{22\!\cdots\!24}a^{15}+\frac{12\!\cdots\!41}{11\!\cdots\!12}a^{14}+\frac{70\!\cdots\!03}{22\!\cdots\!24}a^{13}+\frac{15\!\cdots\!49}{56\!\cdots\!56}a^{12}+\frac{18\!\cdots\!45}{22\!\cdots\!24}a^{11}-\frac{11\!\cdots\!43}{28\!\cdots\!28}a^{10}+\frac{19\!\cdots\!01}{22\!\cdots\!24}a^{9}-\frac{21\!\cdots\!17}{11\!\cdots\!12}a^{8}+\frac{12\!\cdots\!37}{22\!\cdots\!24}a^{7}-\frac{48\!\cdots\!57}{56\!\cdots\!56}a^{6}+\frac{55\!\cdots\!59}{22\!\cdots\!24}a^{5}-\frac{24\!\cdots\!37}{28\!\cdots\!28}a^{4}+\frac{26\!\cdots\!47}{28\!\cdots\!28}a^{3}-\frac{42\!\cdots\!15}{35\!\cdots\!66}a^{2}-\frac{15\!\cdots\!23}{14\!\cdots\!64}a+\frac{25\!\cdots\!23}{35\!\cdots\!66}$, $\frac{44\!\cdots\!53}{11\!\cdots\!12}a^{15}+\frac{43\!\cdots\!05}{14\!\cdots\!64}a^{14}+\frac{76\!\cdots\!27}{14\!\cdots\!64}a^{13}+\frac{45\!\cdots\!25}{28\!\cdots\!28}a^{12}-\frac{21\!\cdots\!37}{56\!\cdots\!56}a^{11}+\frac{25\!\cdots\!03}{56\!\cdots\!56}a^{10}-\frac{16\!\cdots\!52}{17\!\cdots\!33}a^{9}+\frac{70\!\cdots\!17}{28\!\cdots\!28}a^{8}+\frac{67\!\cdots\!21}{28\!\cdots\!28}a^{7}+\frac{70\!\cdots\!59}{28\!\cdots\!28}a^{6}-\frac{19\!\cdots\!19}{56\!\cdots\!56}a^{5}+\frac{91\!\cdots\!25}{56\!\cdots\!56}a^{4}-\frac{35\!\cdots\!69}{11\!\cdots\!12}a^{3}+\frac{14\!\cdots\!69}{28\!\cdots\!28}a^{2}+\frac{17\!\cdots\!29}{28\!\cdots\!28}a-\frac{55\!\cdots\!39}{70\!\cdots\!32}$, $\frac{78\!\cdots\!49}{28\!\cdots\!28}a^{15}-\frac{15\!\cdots\!95}{11\!\cdots\!12}a^{14}+\frac{14\!\cdots\!17}{11\!\cdots\!12}a^{13}-\frac{24\!\cdots\!41}{11\!\cdots\!12}a^{12}+\frac{39\!\cdots\!79}{11\!\cdots\!12}a^{11}-\frac{11\!\cdots\!21}{11\!\cdots\!12}a^{10}+\frac{47\!\cdots\!65}{11\!\cdots\!12}a^{9}-\frac{17\!\cdots\!65}{11\!\cdots\!12}a^{8}+\frac{59\!\cdots\!39}{11\!\cdots\!12}a^{7}-\frac{51\!\cdots\!55}{11\!\cdots\!12}a^{6}+\frac{23\!\cdots\!13}{11\!\cdots\!12}a^{5}-\frac{74\!\cdots\!95}{11\!\cdots\!12}a^{4}+\frac{10\!\cdots\!23}{11\!\cdots\!12}a^{3}-\frac{22\!\cdots\!83}{14\!\cdots\!64}a^{2}-\frac{41\!\cdots\!43}{28\!\cdots\!28}a+\frac{74\!\cdots\!29}{70\!\cdots\!32}$, $\frac{15\!\cdots\!79}{11\!\cdots\!12}a^{15}-\frac{28\!\cdots\!65}{11\!\cdots\!12}a^{14}+\frac{42\!\cdots\!55}{28\!\cdots\!28}a^{13}-\frac{14\!\cdots\!85}{11\!\cdots\!12}a^{12}+\frac{32\!\cdots\!61}{35\!\cdots\!66}a^{11}-\frac{39\!\cdots\!97}{11\!\cdots\!12}a^{10}+\frac{24\!\cdots\!95}{56\!\cdots\!56}a^{9}-\frac{33\!\cdots\!91}{11\!\cdots\!12}a^{8}+\frac{76\!\cdots\!17}{70\!\cdots\!32}a^{7}-\frac{48\!\cdots\!31}{11\!\cdots\!12}a^{6}+\frac{15\!\cdots\!59}{35\!\cdots\!66}a^{5}-\frac{14\!\cdots\!91}{11\!\cdots\!12}a^{4}+\frac{29\!\cdots\!75}{11\!\cdots\!12}a^{3}-\frac{74\!\cdots\!47}{14\!\cdots\!64}a^{2}-\frac{13\!\cdots\!99}{28\!\cdots\!28}a-\frac{94\!\cdots\!23}{70\!\cdots\!32}$ 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:  \( 10205497.328 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10205497.328 \cdot 600}{2\cdot\sqrt{9260065233133681348183837890625}}\cr\approx \mathstrut & 2.4439220618 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056)
 
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 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056, 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 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056);
 
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 - x^15 + 46*x^14 - 105*x^13 + 1270*x^12 - 4449*x^11 + 16060*x^10 - 63735*x^9 + 211046*x^8 - 1759715*x^7 + 8594450*x^6 - 26273899*x^5 + 43579355*x^4 - 18892900*x^3 - 620444*x^2 - 73814496*x + 155285056);
 
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

\(\Q(\sqrt{41}) \), 4.0.8405.1, 4.4.1723025.1, 4.0.344605.1, 8.8.3043035529390625.1, 8.0.3043035529390625.1, 8.0.2968815150625.3

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: 16.0.9260065233133681348183837890625.6
Minimal sibling: 16.0.9260065233133681348183837890625.6

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ R ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(5\) Copy content Toggle raw display 5.4.3.4$x^{4} + 15$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 15$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 10 x^{4} - 25$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(41\) Copy content Toggle raw display 41.16.14.1$x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776882 x^{8} + 5977813379504 x^{7} + 3161920977824 x^{6} + 978514601120 x^{5} + 196936323920 x^{4} + 202153692608 x^{3} + 3372805705856 x^{2} + 36445904670848 x + 172395305267889$$8$$2$$14$$C_8\times C_2$$[\ ]_{8}^{2}$