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Properties

Label 16.0.988...129.1
Degree $16$
Signature $[0, 8]$
Discriminant $9.886\times 10^{27}$
Root discriminant \(56.19\)
Ramified primes $3,73$
Class number $89$ (GRH)
Class group [89] (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 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352)
 
gp: K = bnfinit(y^16 - 3*y^15 + 12*y^14 - 11*y^13 - 70*y^12 + 243*y^11 - 1544*y^10 + 2471*y^9 - 4787*y^8 - 4508*y^7 + 70900*y^6 - 163440*y^5 + 642304*y^4 - 1005056*y^3 + 2335744*y^2 - 1929216*y + 2916352, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352)
 

\( x^{16} - 3 x^{15} + 12 x^{14} - 11 x^{13} - 70 x^{12} + 243 x^{11} - 1544 x^{10} + 2471 x^{9} + \cdots + 2916352 \) 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:   \(9885646137219473682569328129\) \(\medspace = 3^{4}\cdot 73^{14}\) 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:  \(56.19\)
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:  $3^{1/2}73^{7/8}\approx 73.95510022503493$
Ramified primes:   \(3\), \(73\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{9}-\frac{1}{32}a^{8}-\frac{1}{32}a^{7}-\frac{1}{8}a^{6}+\frac{7}{32}a^{5}+\frac{7}{32}a^{4}-\frac{7}{32}a^{3}-\frac{1}{16}a^{2}$, $\frac{1}{64}a^{10}+\frac{1}{32}a^{8}-\frac{1}{64}a^{7}-\frac{1}{64}a^{6}+\frac{3}{32}a^{5}+\frac{1}{16}a^{4}+\frac{3}{64}a^{3}-\frac{3}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{64}a^{11}+\frac{1}{64}a^{8}+\frac{1}{64}a^{7}-\frac{1}{32}a^{6}+\frac{3}{32}a^{5}-\frac{11}{64}a^{4}+\frac{3}{8}a^{3}+\frac{3}{16}a^{2}-\frac{1}{2}a$, $\frac{1}{512}a^{12}-\frac{1}{512}a^{11}+\frac{1}{256}a^{10}+\frac{1}{512}a^{9}-\frac{1}{128}a^{8}-\frac{13}{512}a^{7}+\frac{7}{256}a^{6}-\frac{117}{512}a^{5}+\frac{99}{512}a^{4}+\frac{49}{256}a^{3}-\frac{7}{32}a^{2}-\frac{3}{16}a-\frac{1}{2}$, $\frac{1}{2048}a^{13}+\frac{1}{2048}a^{12}+\frac{5}{2048}a^{10}+\frac{7}{1024}a^{9}+\frac{91}{2048}a^{8}+\frac{9}{512}a^{7}-\frac{153}{2048}a^{6}+\frac{425}{2048}a^{5}+\frac{11}{256}a^{4}+\frac{25}{512}a^{3}-\frac{13}{64}a^{2}-\frac{13}{32}a+\frac{1}{4}$, $\frac{1}{16384}a^{14}-\frac{3}{16384}a^{13}-\frac{1}{4096}a^{12}-\frac{27}{16384}a^{11}+\frac{29}{8192}a^{10}-\frac{221}{16384}a^{9}-\frac{61}{2048}a^{8}-\frac{393}{16384}a^{7}-\frac{499}{16384}a^{6}+\frac{157}{4096}a^{5}-\frac{39}{4096}a^{4}-\frac{487}{1024}a^{3}-\frac{91}{256}a^{2}+\frac{11}{64}a+\frac{3}{8}$, $\frac{1}{54\!\cdots\!04}a^{15}+\frac{61\!\cdots\!01}{54\!\cdots\!04}a^{14}+\frac{21\!\cdots\!43}{33\!\cdots\!44}a^{13}-\frac{35\!\cdots\!63}{54\!\cdots\!04}a^{12}-\frac{12\!\cdots\!21}{27\!\cdots\!52}a^{11}+\frac{34\!\cdots\!83}{54\!\cdots\!04}a^{10}-\frac{69\!\cdots\!23}{13\!\cdots\!76}a^{9}+\frac{26\!\cdots\!07}{54\!\cdots\!04}a^{8}-\frac{31\!\cdots\!07}{54\!\cdots\!04}a^{7}-\frac{50\!\cdots\!95}{67\!\cdots\!88}a^{6}+\frac{14\!\cdots\!37}{13\!\cdots\!76}a^{5}-\frac{20\!\cdots\!25}{16\!\cdots\!72}a^{4}+\frac{15\!\cdots\!13}{42\!\cdots\!68}a^{3}+\frac{18\!\cdots\!01}{52\!\cdots\!96}a^{2}+\frac{18\!\cdots\!37}{52\!\cdots\!96}a+\frac{89\!\cdots\!91}{65\!\cdots\!12}$ 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

$C_{89}$, which has order $89$ (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:   $\frac{14\!\cdots\!31}{74\!\cdots\!36}a^{15}-\frac{94\!\cdots\!69}{74\!\cdots\!36}a^{14}+\frac{39\!\cdots\!23}{92\!\cdots\!92}a^{13}-\frac{42\!\cdots\!41}{74\!\cdots\!36}a^{12}-\frac{97\!\cdots\!91}{37\!\cdots\!68}a^{11}+\frac{11\!\cdots\!77}{74\!\cdots\!36}a^{10}-\frac{91\!\cdots\!55}{18\!\cdots\!84}a^{9}+\frac{69\!\cdots\!53}{74\!\cdots\!36}a^{8}-\frac{10\!\cdots\!45}{74\!\cdots\!36}a^{7}-\frac{12\!\cdots\!79}{23\!\cdots\!48}a^{6}+\frac{47\!\cdots\!99}{18\!\cdots\!84}a^{5}-\frac{85\!\cdots\!55}{11\!\cdots\!24}a^{4}+\frac{18\!\cdots\!67}{14\!\cdots\!28}a^{3}-\frac{30\!\cdots\!01}{14\!\cdots\!28}a^{2}+\frac{12\!\cdots\!09}{72\!\cdots\!64}a-\frac{14\!\cdots\!69}{90\!\cdots\!08}$, $\frac{22\!\cdots\!75}{16\!\cdots\!72}a^{15}+\frac{23\!\cdots\!79}{16\!\cdots\!72}a^{14}+\frac{10\!\cdots\!77}{21\!\cdots\!84}a^{13}+\frac{40\!\cdots\!39}{16\!\cdots\!72}a^{12}-\frac{65\!\cdots\!63}{84\!\cdots\!36}a^{11}-\frac{36\!\cdots\!27}{16\!\cdots\!72}a^{10}-\frac{51\!\cdots\!59}{42\!\cdots\!68}a^{9}-\frac{45\!\cdots\!27}{16\!\cdots\!72}a^{8}-\frac{49\!\cdots\!57}{16\!\cdots\!72}a^{7}-\frac{55\!\cdots\!17}{10\!\cdots\!92}a^{6}+\frac{25\!\cdots\!51}{42\!\cdots\!68}a^{5}+\frac{17\!\cdots\!81}{13\!\cdots\!24}a^{4}+\frac{36\!\cdots\!57}{65\!\cdots\!12}a^{3}+\frac{15\!\cdots\!41}{32\!\cdots\!56}a^{2}+\frac{19\!\cdots\!13}{16\!\cdots\!28}a+\frac{10\!\cdots\!95}{20\!\cdots\!16}$, $\frac{69\!\cdots\!21}{84\!\cdots\!36}a^{15}-\frac{22\!\cdots\!07}{84\!\cdots\!36}a^{14}-\frac{14\!\cdots\!91}{21\!\cdots\!84}a^{13}+\frac{10\!\cdots\!77}{84\!\cdots\!36}a^{12}+\frac{11\!\cdots\!45}{42\!\cdots\!68}a^{11}-\frac{14\!\cdots\!25}{84\!\cdots\!36}a^{10}+\frac{17\!\cdots\!65}{32\!\cdots\!56}a^{9}+\frac{94\!\cdots\!19}{84\!\cdots\!36}a^{8}+\frac{37\!\cdots\!89}{84\!\cdots\!36}a^{7}+\frac{10\!\cdots\!91}{21\!\cdots\!84}a^{6}-\frac{70\!\cdots\!95}{21\!\cdots\!84}a^{5}+\frac{20\!\cdots\!71}{52\!\cdots\!96}a^{4}-\frac{81\!\cdots\!41}{13\!\cdots\!24}a^{3}-\frac{54\!\cdots\!03}{32\!\cdots\!56}a^{2}+\frac{39\!\cdots\!49}{20\!\cdots\!16}a-\frac{32\!\cdots\!41}{51\!\cdots\!04}$, $\frac{15\!\cdots\!11}{27\!\cdots\!52}a^{15}-\frac{11\!\cdots\!57}{27\!\cdots\!52}a^{14}+\frac{40\!\cdots\!45}{16\!\cdots\!72}a^{13}-\frac{28\!\cdots\!01}{27\!\cdots\!52}a^{12}+\frac{52\!\cdots\!85}{13\!\cdots\!76}a^{11}-\frac{34\!\cdots\!43}{27\!\cdots\!52}a^{10}+\frac{23\!\cdots\!35}{67\!\cdots\!88}a^{9}-\frac{23\!\cdots\!11}{27\!\cdots\!52}a^{8}+\frac{50\!\cdots\!95}{27\!\cdots\!52}a^{7}-\frac{11\!\cdots\!61}{33\!\cdots\!44}a^{6}+\frac{37\!\cdots\!19}{67\!\cdots\!88}a^{5}-\frac{43\!\cdots\!59}{84\!\cdots\!36}a^{4}+\frac{10\!\cdots\!03}{21\!\cdots\!84}a^{3}+\frac{18\!\cdots\!87}{26\!\cdots\!48}a^{2}-\frac{23\!\cdots\!05}{26\!\cdots\!48}a+\frac{97\!\cdots\!77}{32\!\cdots\!56}$, $\frac{34\!\cdots\!15}{27\!\cdots\!52}a^{15}-\frac{16\!\cdots\!89}{27\!\cdots\!52}a^{14}+\frac{35\!\cdots\!69}{16\!\cdots\!72}a^{13}-\frac{42\!\cdots\!97}{27\!\cdots\!52}a^{12}-\frac{26\!\cdots\!63}{13\!\cdots\!76}a^{11}+\frac{21\!\cdots\!21}{27\!\cdots\!52}a^{10}-\frac{16\!\cdots\!97}{67\!\cdots\!88}a^{9}+\frac{69\!\cdots\!53}{27\!\cdots\!52}a^{8}+\frac{84\!\cdots\!99}{27\!\cdots\!52}a^{7}-\frac{10\!\cdots\!77}{33\!\cdots\!44}a^{6}+\frac{97\!\cdots\!39}{67\!\cdots\!88}a^{5}-\frac{23\!\cdots\!75}{84\!\cdots\!36}a^{4}+\frac{10\!\cdots\!07}{21\!\cdots\!84}a^{3}-\frac{97\!\cdots\!45}{26\!\cdots\!48}a^{2}+\frac{97\!\cdots\!27}{26\!\cdots\!48}a+\frac{11\!\cdots\!49}{32\!\cdots\!56}$, $\frac{11\!\cdots\!85}{27\!\cdots\!52}a^{15}-\frac{16\!\cdots\!67}{27\!\cdots\!52}a^{14}-\frac{21\!\cdots\!37}{16\!\cdots\!72}a^{13}+\frac{48\!\cdots\!57}{27\!\cdots\!52}a^{12}-\frac{88\!\cdots\!77}{13\!\cdots\!76}a^{11}+\frac{65\!\cdots\!11}{27\!\cdots\!52}a^{10}+\frac{90\!\cdots\!21}{67\!\cdots\!88}a^{9}-\frac{35\!\cdots\!89}{27\!\cdots\!52}a^{8}+\frac{85\!\cdots\!33}{27\!\cdots\!52}a^{7}-\frac{18\!\cdots\!91}{33\!\cdots\!44}a^{6}+\frac{72\!\cdots\!77}{67\!\cdots\!88}a^{5}+\frac{38\!\cdots\!55}{84\!\cdots\!36}a^{4}-\frac{24\!\cdots\!47}{21\!\cdots\!84}a^{3}+\frac{10\!\cdots\!05}{26\!\cdots\!48}a^{2}-\frac{10\!\cdots\!91}{26\!\cdots\!48}a+\frac{19\!\cdots\!59}{32\!\cdots\!56}$, $\frac{22\!\cdots\!89}{10\!\cdots\!92}a^{15}-\frac{19\!\cdots\!01}{21\!\cdots\!84}a^{14}+\frac{84\!\cdots\!97}{21\!\cdots\!84}a^{13}-\frac{25\!\cdots\!09}{10\!\cdots\!92}a^{12}-\frac{58\!\cdots\!39}{21\!\cdots\!84}a^{11}+\frac{17\!\cdots\!91}{13\!\cdots\!24}a^{10}-\frac{10\!\cdots\!21}{21\!\cdots\!84}a^{9}+\frac{40\!\cdots\!27}{10\!\cdots\!92}a^{8}-\frac{82\!\cdots\!23}{21\!\cdots\!84}a^{7}-\frac{12\!\cdots\!63}{21\!\cdots\!84}a^{6}+\frac{13\!\cdots\!75}{52\!\cdots\!96}a^{5}-\frac{24\!\cdots\!55}{52\!\cdots\!96}a^{4}+\frac{17\!\cdots\!05}{13\!\cdots\!24}a^{3}-\frac{18\!\cdots\!99}{32\!\cdots\!56}a^{2}+\frac{13\!\cdots\!27}{82\!\cdots\!64}a+\frac{18\!\cdots\!35}{10\!\cdots\!08}$ 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:  \( 103600962.089 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 103600962.089 \cdot 89}{2\cdot\sqrt{9885646137219473682569328129}}\cr\approx \mathstrut & 112.631590272 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352)
 
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 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352, 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 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352);
 
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 - 3*x^15 + 12*x^14 - 11*x^13 - 70*x^12 + 243*x^11 - 1544*x^10 + 2471*x^9 - 4787*x^8 - 4508*x^7 + 70900*x^6 - 163440*x^5 + 642304*x^4 - 1005056*x^3 + 2335744*x^2 - 1929216*x + 2916352);
 
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

\(\Q(\sqrt{73}) \), 4.2.15987.1, 4.4.389017.1, 4.2.1167051.1, 8.4.99426586671873.1, 8.0.11047398519097.1, 8.4.1362008036601.1

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
Degree 16 sibling: 16.8.800737337114777368288115578449.1
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 ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }^{8}$ R ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(3\) Copy content Toggle raw display 3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{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}$
\(73\) Copy content Toggle raw display 73.16.14.1$x^{16} + 560 x^{15} + 137240 x^{14} + 19227600 x^{13} + 1684816700 x^{12} + 94599694000 x^{11} + 3327837457000 x^{10} + 67300032450000 x^{9} + 609674268043896 x^{8} + 336500162290880 x^{7} + 83195946420160 x^{6} + 11826359641600 x^{5} + 1175201013000 x^{4} + 6895769020000 x^{3} + 239006660174000 x^{2} + 4775207729180000 x + 41740387870087204$$8$$2$$14$$C_8\times C_2$$[\ ]_{8}^{2}$