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Properties

Label 16.0.612...689.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.123\times 10^{39}$
Root discriminant \(306.68\)
Ramified primes $13,41$
Class number $45284$ (GRH)
Class group [45284] (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 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600)
 
gp: K = bnfinit(y^16 - 7*y^15 + 55*y^14 + 419*y^13 - 877*y^12 + 13350*y^11 + 109599*y^10 + 897435*y^9 + 5128604*y^8 + 9987966*y^7 + 61086386*y^6 + 217020565*y^5 + 407235395*y^4 + 860961746*y^3 + 4064864216*y^2 + 670732880*y + 11844163600, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600)
 

\( x^{16} - 7 x^{15} + 55 x^{14} + 419 x^{13} - 877 x^{12} + 13350 x^{11} + 109599 x^{10} + \cdots + 11844163600 \) 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:   \(6123007382888435990757129497254763904689\) \(\medspace = 13^{14}\cdot 41^{15}\) 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:  \(306.68\)
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:  $13^{7/8}41^{15/16}\approx 306.6798900930961$
Ramified primes:   \(13\), \(41\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5}a^{8}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{50}a^{9}-\frac{3}{50}a^{8}+\frac{3}{50}a^{7}-\frac{12}{25}a^{6}-\frac{23}{50}a^{5}+\frac{2}{5}a^{2}+\frac{17}{50}a-\frac{2}{5}$, $\frac{1}{50}a^{10}+\frac{2}{25}a^{8}-\frac{1}{10}a^{7}-\frac{1}{2}a^{6}+\frac{21}{50}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{7}{50}a^{2}+\frac{1}{50}a-\frac{1}{5}$, $\frac{1}{50}a^{11}-\frac{3}{50}a^{8}+\frac{3}{50}a^{7}-\frac{3}{50}a^{6}-\frac{9}{25}a^{5}+\frac{1}{5}a^{4}-\frac{13}{50}a^{3}-\frac{9}{50}a^{2}+\frac{1}{25}a-\frac{2}{5}$, $\frac{1}{250}a^{12}+\frac{1}{250}a^{11}-\frac{1}{250}a^{10}-\frac{1}{250}a^{9}+\frac{21}{250}a^{7}-\frac{62}{125}a^{6}-\frac{17}{50}a^{5}-\frac{53}{250}a^{4}-\frac{46}{125}a^{3}+\frac{28}{125}a^{2}+\frac{17}{50}a+\frac{1}{5}$, $\frac{1}{1250}a^{13}-\frac{1}{1250}a^{12}+\frac{6}{625}a^{11}+\frac{11}{1250}a^{10}+\frac{6}{625}a^{9}-\frac{57}{625}a^{8}+\frac{9}{1250}a^{7}+\frac{214}{625}a^{6}-\frac{73}{1250}a^{5}+\frac{207}{625}a^{4}+\frac{99}{250}a^{3}+\frac{29}{625}a^{2}+\frac{39}{125}a+\frac{6}{25}$, $\frac{1}{912500}a^{14}-\frac{33}{912500}a^{13}+\frac{119}{912500}a^{12}+\frac{5277}{912500}a^{11}+\frac{937}{182500}a^{10}+\frac{4301}{456250}a^{9}-\frac{37943}{912500}a^{8}+\frac{14543}{182500}a^{7}-\frac{171497}{456250}a^{6}-\frac{5353}{18250}a^{5}+\frac{73443}{228125}a^{4}-\frac{426657}{912500}a^{3}-\frac{163991}{912500}a^{2}-\frac{13579}{45625}a+\frac{169}{9125}$, $\frac{1}{18\!\cdots\!00}a^{15}+\frac{33\!\cdots\!69}{37\!\cdots\!00}a^{14}+\frac{77\!\cdots\!39}{37\!\cdots\!00}a^{13}-\frac{33\!\cdots\!41}{18\!\cdots\!00}a^{12}-\frac{22\!\cdots\!09}{18\!\cdots\!00}a^{11}+\frac{79\!\cdots\!17}{41\!\cdots\!00}a^{10}+\frac{16\!\cdots\!63}{18\!\cdots\!00}a^{9}-\frac{85\!\cdots\!89}{18\!\cdots\!00}a^{8}-\frac{35\!\cdots\!31}{47\!\cdots\!50}a^{7}+\frac{18\!\cdots\!33}{41\!\cdots\!00}a^{6}-\frac{57\!\cdots\!39}{94\!\cdots\!00}a^{5}+\frac{23\!\cdots\!09}{18\!\cdots\!00}a^{4}-\frac{38\!\cdots\!37}{18\!\cdots\!00}a^{3}-\frac{40\!\cdots\!89}{94\!\cdots\!00}a^{2}-\frac{89\!\cdots\!77}{20\!\cdots\!25}a+\frac{32\!\cdots\!41}{94\!\cdots\!75}$ 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:  $2$, $5$

Class group and class number

$C_{45284}$, which has order $45284$ (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:   $\frac{18\!\cdots\!87}{28\!\cdots\!75}a^{15}-\frac{18\!\cdots\!71}{11\!\cdots\!75}a^{14}+\frac{47\!\cdots\!22}{57\!\cdots\!75}a^{13}-\frac{40\!\cdots\!77}{28\!\cdots\!75}a^{12}-\frac{20\!\cdots\!38}{28\!\cdots\!75}a^{11}+\frac{72\!\cdots\!34}{28\!\cdots\!75}a^{10}-\frac{15\!\cdots\!99}{28\!\cdots\!75}a^{9}-\frac{29\!\cdots\!98}{28\!\cdots\!75}a^{8}-\frac{32\!\cdots\!63}{28\!\cdots\!75}a^{7}-\frac{24\!\cdots\!74}{28\!\cdots\!75}a^{6}-\frac{69\!\cdots\!11}{28\!\cdots\!75}a^{5}-\frac{19\!\cdots\!22}{28\!\cdots\!75}a^{4}-\frac{73\!\cdots\!64}{28\!\cdots\!75}a^{3}-\frac{18\!\cdots\!96}{28\!\cdots\!75}a^{2}-\frac{40\!\cdots\!76}{57\!\cdots\!75}a-\frac{28\!\cdots\!29}{11\!\cdots\!75}$, $\frac{11\!\cdots\!19}{82\!\cdots\!00}a^{15}-\frac{78\!\cdots\!19}{16\!\cdots\!00}a^{14}+\frac{36\!\cdots\!11}{16\!\cdots\!00}a^{13}-\frac{59\!\cdots\!29}{82\!\cdots\!00}a^{12}-\frac{18\!\cdots\!21}{82\!\cdots\!00}a^{11}-\frac{73\!\cdots\!23}{20\!\cdots\!50}a^{10}-\frac{19\!\cdots\!03}{82\!\cdots\!00}a^{9}-\frac{32\!\cdots\!41}{82\!\cdots\!00}a^{8}-\frac{16\!\cdots\!53}{41\!\cdots\!00}a^{7}-\frac{11\!\cdots\!79}{41\!\cdots\!00}a^{6}-\frac{33\!\cdots\!91}{41\!\cdots\!00}a^{5}-\frac{19\!\cdots\!29}{82\!\cdots\!00}a^{4}-\frac{70\!\cdots\!53}{82\!\cdots\!00}a^{3}-\frac{44\!\cdots\!33}{20\!\cdots\!50}a^{2}-\frac{95\!\cdots\!23}{41\!\cdots\!50}a-\frac{25\!\cdots\!46}{41\!\cdots\!25}$, $\frac{78\!\cdots\!31}{41\!\cdots\!00}a^{15}-\frac{20\!\cdots\!59}{82\!\cdots\!00}a^{14}+\frac{18\!\cdots\!43}{82\!\cdots\!00}a^{13}-\frac{17\!\cdots\!81}{41\!\cdots\!00}a^{12}-\frac{44\!\cdots\!09}{41\!\cdots\!00}a^{11}+\frac{22\!\cdots\!48}{10\!\cdots\!25}a^{10}+\frac{22\!\cdots\!73}{41\!\cdots\!00}a^{9}+\frac{60\!\cdots\!07}{56\!\cdots\!00}a^{8}-\frac{18\!\cdots\!97}{20\!\cdots\!50}a^{7}-\frac{21\!\cdots\!91}{20\!\cdots\!50}a^{6}-\frac{28\!\cdots\!59}{20\!\cdots\!50}a^{5}-\frac{17\!\cdots\!01}{41\!\cdots\!00}a^{4}-\frac{21\!\cdots\!17}{41\!\cdots\!00}a^{3}-\frac{43\!\cdots\!57}{10\!\cdots\!25}a^{2}-\frac{37\!\cdots\!67}{20\!\cdots\!25}a-\frac{65\!\cdots\!43}{41\!\cdots\!25}$, $\frac{41\!\cdots\!33}{16\!\cdots\!00}a^{15}-\frac{37\!\cdots\!41}{16\!\cdots\!00}a^{14}+\frac{30\!\cdots\!33}{16\!\cdots\!00}a^{13}+\frac{10\!\cdots\!13}{16\!\cdots\!00}a^{12}-\frac{88\!\cdots\!03}{22\!\cdots\!00}a^{11}+\frac{18\!\cdots\!09}{41\!\cdots\!50}a^{10}+\frac{28\!\cdots\!07}{16\!\cdots\!00}a^{9}+\frac{29\!\cdots\!21}{16\!\cdots\!00}a^{8}+\frac{78\!\cdots\!89}{82\!\cdots\!00}a^{7}+\frac{42\!\cdots\!79}{82\!\cdots\!00}a^{6}+\frac{13\!\cdots\!03}{82\!\cdots\!00}a^{5}+\frac{77\!\cdots\!57}{32\!\cdots\!00}a^{4}+\frac{57\!\cdots\!61}{16\!\cdots\!00}a^{3}+\frac{13\!\cdots\!53}{41\!\cdots\!50}a^{2}-\frac{35\!\cdots\!77}{82\!\cdots\!50}a+\frac{71\!\cdots\!41}{82\!\cdots\!25}$, $\frac{44\!\cdots\!09}{82\!\cdots\!00}a^{15}-\frac{28\!\cdots\!37}{32\!\cdots\!00}a^{14}+\frac{11\!\cdots\!69}{16\!\cdots\!00}a^{13}+\frac{31\!\cdots\!61}{82\!\cdots\!00}a^{12}+\frac{58\!\cdots\!09}{82\!\cdots\!00}a^{11}+\frac{32\!\cdots\!86}{10\!\cdots\!25}a^{10}+\frac{77\!\cdots\!07}{82\!\cdots\!00}a^{9}+\frac{64\!\cdots\!89}{82\!\cdots\!00}a^{8}+\frac{20\!\cdots\!67}{41\!\cdots\!00}a^{7}+\frac{70\!\cdots\!91}{41\!\cdots\!00}a^{6}+\frac{17\!\cdots\!99}{41\!\cdots\!00}a^{5}+\frac{18\!\cdots\!21}{82\!\cdots\!00}a^{4}+\frac{55\!\cdots\!77}{82\!\cdots\!00}a^{3}+\frac{15\!\cdots\!57}{20\!\cdots\!50}a^{2}+\frac{88\!\cdots\!67}{41\!\cdots\!50}a+\frac{37\!\cdots\!59}{41\!\cdots\!25}$, $\frac{77\!\cdots\!21}{82\!\cdots\!00}a^{15}-\frac{25\!\cdots\!01}{16\!\cdots\!00}a^{14}+\frac{13\!\cdots\!29}{16\!\cdots\!00}a^{13}+\frac{10\!\cdots\!89}{82\!\cdots\!00}a^{12}-\frac{49\!\cdots\!39}{82\!\cdots\!00}a^{11}+\frac{46\!\cdots\!34}{10\!\cdots\!25}a^{10}+\frac{11\!\cdots\!23}{82\!\cdots\!00}a^{9}-\frac{33\!\cdots\!19}{82\!\cdots\!00}a^{8}-\frac{27\!\cdots\!77}{41\!\cdots\!00}a^{7}-\frac{26\!\cdots\!61}{41\!\cdots\!00}a^{6}-\frac{76\!\cdots\!69}{41\!\cdots\!00}a^{5}-\frac{38\!\cdots\!11}{82\!\cdots\!00}a^{4}-\frac{20\!\cdots\!99}{11\!\cdots\!00}a^{3}-\frac{10\!\cdots\!47}{20\!\cdots\!50}a^{2}-\frac{20\!\cdots\!57}{41\!\cdots\!50}a-\frac{64\!\cdots\!89}{41\!\cdots\!25}$, $\frac{72\!\cdots\!61}{82\!\cdots\!00}a^{15}-\frac{29\!\cdots\!09}{16\!\cdots\!00}a^{14}+\frac{24\!\cdots\!73}{16\!\cdots\!00}a^{13}+\frac{49\!\cdots\!89}{82\!\cdots\!00}a^{12}+\frac{86\!\cdots\!21}{82\!\cdots\!00}a^{11}+\frac{55\!\cdots\!94}{10\!\cdots\!25}a^{10}+\frac{12\!\cdots\!63}{82\!\cdots\!00}a^{9}+\frac{10\!\cdots\!41}{82\!\cdots\!00}a^{8}+\frac{32\!\cdots\!93}{41\!\cdots\!00}a^{7}+\frac{10\!\cdots\!79}{41\!\cdots\!00}a^{6}+\frac{27\!\cdots\!21}{41\!\cdots\!00}a^{5}+\frac{29\!\cdots\!69}{82\!\cdots\!00}a^{4}+\frac{86\!\cdots\!73}{82\!\cdots\!00}a^{3}+\frac{24\!\cdots\!33}{20\!\cdots\!50}a^{2}+\frac{13\!\cdots\!23}{41\!\cdots\!50}a+\frac{58\!\cdots\!71}{41\!\cdots\!25}$ 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:  \( 1702357630.14 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1702357630.14 \cdot 45284}{2\cdot\sqrt{6123007382888435990757129497254763904689}}\cr\approx \mathstrut & 1.19652600803 \end{aligned}\] (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 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600)
 
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 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600, 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 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600);
 
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 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600);
 
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

\(\Q(\sqrt{41}) \), 4.4.11647649.1, 8.8.940041681957275729.1

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 ${\href{/padicField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{8}$ $16$ $16$ R $16$ $16$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ R ${\href{/padicField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 13.16.14.6$x^{16} - 1014 x^{8} - 24505$$8$$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}$