Properties

Label 16.0.436...401.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.369\times 10^{34}$
Root discriminant \(146.23\)
Ramified primes $29,59$
Class number $42$ (GRH)
Class group [42] (GRH)
Galois group $\OD_{16}:C_2$ (as 16T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961)
 
gp: K = bnfinit(y^16 - 4*y^15 - 36*y^14 + 274*y^13 - 16*y^12 - 6092*y^11 + 67960*y^10 - 468046*y^9 + 2156282*y^8 - 6421886*y^7 + 10535912*y^6 - 5431518*y^5 - 7791049*y^4 - 1515286*y^3 + 45995976*y^2 - 27672790*y + 15280961, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961)
 

\( x^{16} - 4 x^{15} - 36 x^{14} + 274 x^{13} - 16 x^{12} - 6092 x^{11} + 67960 x^{10} - 468046 x^{9} + 2156282 x^{8} - 6421886 x^{7} + 10535912 x^{6} - 5431518 x^{5} + \cdots + 15280961 \) 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:   \(43690605516556003276152578767301401\) \(\medspace = 29^{14}\cdot 59^{8}\) 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:  \(146.23\)
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:  $29^{7/8}59^{1/2}\approx 146.22588347393037$
Ramified primes:   \(29\), \(59\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{116}a^{8}-\frac{1}{58}a^{7}-\frac{5}{29}a^{6}+\frac{5}{58}a^{5}-\frac{7}{58}a^{4}+\frac{13}{58}a^{3}+\frac{33}{116}a^{2}+\frac{2}{29}a+\frac{7}{116}$, $\frac{1}{116}a^{9}-\frac{6}{29}a^{7}+\frac{7}{29}a^{6}+\frac{3}{58}a^{5}-\frac{1}{58}a^{4}+\frac{27}{116}a^{3}+\frac{4}{29}a^{2}+\frac{23}{116}a+\frac{7}{58}$, $\frac{1}{116}a^{10}-\frac{5}{29}a^{7}-\frac{5}{58}a^{6}+\frac{3}{58}a^{5}-\frac{19}{116}a^{4}-\frac{14}{29}a^{3}+\frac{3}{116}a^{2}+\frac{8}{29}a-\frac{3}{58}$, $\frac{1}{116}a^{11}+\frac{2}{29}a^{7}+\frac{3}{29}a^{6}+\frac{7}{116}a^{5}+\frac{3}{29}a^{4}-\frac{57}{116}a^{3}-\frac{1}{29}a^{2}+\frac{19}{58}a-\frac{17}{58}$, $\frac{1}{1624}a^{12}-\frac{1}{406}a^{11}-\frac{3}{812}a^{10}-\frac{1}{232}a^{9}-\frac{1}{1624}a^{8}-\frac{59}{406}a^{7}+\frac{351}{1624}a^{6}-\frac{179}{812}a^{5}+\frac{33}{1624}a^{4}+\frac{659}{1624}a^{3}+\frac{613}{1624}a^{2}-\frac{495}{1624}a+\frac{185}{1624}$, $\frac{1}{1624}a^{13}+\frac{3}{812}a^{11}-\frac{3}{1624}a^{10}-\frac{1}{1624}a^{9}-\frac{1}{812}a^{8}+\frac{359}{1624}a^{7}+\frac{187}{812}a^{6}-\frac{111}{1624}a^{5}-\frac{51}{232}a^{4}-\frac{279}{1624}a^{3}+\frac{487}{1624}a^{2}-\frac{535}{1624}a+\frac{27}{812}$, $\frac{1}{1830248}a^{14}-\frac{177}{915124}a^{13}-\frac{5}{261464}a^{12}+\frac{2041}{1830248}a^{11}-\frac{6841}{1830248}a^{10}+\frac{1927}{1830248}a^{9}-\frac{3}{39788}a^{8}-\frac{20299}{457562}a^{7}+\frac{193139}{915124}a^{6}+\frac{377995}{1830248}a^{5}+\frac{51873}{915124}a^{4}+\frac{119263}{915124}a^{3}+\frac{79372}{228781}a^{2}+\frac{322187}{1830248}a+\frac{440213}{1830248}$, $\frac{1}{27\!\cdots\!96}a^{15}-\frac{12\!\cdots\!81}{13\!\cdots\!48}a^{14}-\frac{45\!\cdots\!33}{30\!\cdots\!38}a^{13}+\frac{64\!\cdots\!13}{27\!\cdots\!96}a^{12}+\frac{79\!\cdots\!85}{27\!\cdots\!96}a^{11}-\frac{12\!\cdots\!08}{49\!\cdots\!41}a^{10}-\frac{62\!\cdots\!29}{27\!\cdots\!96}a^{9}+\frac{59\!\cdots\!13}{69\!\cdots\!74}a^{8}-\frac{28\!\cdots\!97}{27\!\cdots\!96}a^{7}+\frac{31\!\cdots\!41}{27\!\cdots\!96}a^{6}+\frac{40\!\cdots\!85}{27\!\cdots\!96}a^{5}-\frac{10\!\cdots\!61}{39\!\cdots\!28}a^{4}-\frac{19\!\cdots\!11}{39\!\cdots\!28}a^{3}-\frac{11\!\cdots\!03}{34\!\cdots\!87}a^{2}+\frac{19\!\cdots\!95}{69\!\cdots\!74}a+\frac{19\!\cdots\!13}{69\!\cdots\!74}$ 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:  $7$

Class group and class number

$C_{42}$, which has order $42$ (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:   $\frac{17\!\cdots\!75}{80\!\cdots\!16}a^{15}-\frac{47\!\cdots\!83}{40\!\cdots\!08}a^{14}-\frac{52\!\cdots\!51}{80\!\cdots\!16}a^{13}+\frac{56\!\cdots\!35}{80\!\cdots\!16}a^{12}-\frac{75\!\cdots\!17}{80\!\cdots\!16}a^{11}-\frac{10\!\cdots\!49}{80\!\cdots\!16}a^{10}+\frac{17\!\cdots\!30}{10\!\cdots\!77}a^{9}-\frac{25\!\cdots\!95}{20\!\cdots\!54}a^{8}+\frac{64\!\cdots\!23}{10\!\cdots\!77}a^{7}-\frac{17\!\cdots\!79}{80\!\cdots\!16}a^{6}+\frac{19\!\cdots\!07}{40\!\cdots\!08}a^{5}-\frac{11\!\cdots\!37}{20\!\cdots\!54}a^{4}+\frac{17\!\cdots\!51}{10\!\cdots\!77}a^{3}+\frac{37\!\cdots\!87}{80\!\cdots\!16}a^{2}-\frac{26\!\cdots\!75}{80\!\cdots\!16}a-\frac{23\!\cdots\!74}{10\!\cdots\!77}$, $\frac{13\!\cdots\!17}{39\!\cdots\!28}a^{15}-\frac{13\!\cdots\!23}{57\!\cdots\!04}a^{14}-\frac{68\!\cdots\!51}{39\!\cdots\!28}a^{13}+\frac{16\!\cdots\!75}{39\!\cdots\!28}a^{12}+\frac{23\!\cdots\!85}{71\!\cdots\!63}a^{11}-\frac{14\!\cdots\!53}{99\!\cdots\!82}a^{10}+\frac{29\!\cdots\!33}{19\!\cdots\!64}a^{9}-\frac{37\!\cdots\!13}{39\!\cdots\!28}a^{8}+\frac{12\!\cdots\!82}{49\!\cdots\!41}a^{7}-\frac{11\!\cdots\!85}{49\!\cdots\!41}a^{6}-\frac{14\!\cdots\!41}{57\!\cdots\!04}a^{5}+\frac{11\!\cdots\!93}{39\!\cdots\!28}a^{4}+\frac{46\!\cdots\!85}{39\!\cdots\!28}a^{3}-\frac{11\!\cdots\!83}{19\!\cdots\!64}a^{2}+\frac{20\!\cdots\!37}{39\!\cdots\!28}a+\frac{33\!\cdots\!99}{99\!\cdots\!82}$, $\frac{15\!\cdots\!87}{13\!\cdots\!48}a^{15}+\frac{89\!\cdots\!27}{12\!\cdots\!52}a^{14}-\frac{39\!\cdots\!83}{69\!\cdots\!74}a^{13}-\frac{31\!\cdots\!01}{13\!\cdots\!48}a^{12}+\frac{52\!\cdots\!51}{27\!\cdots\!96}a^{11}+\frac{24\!\cdots\!03}{27\!\cdots\!96}a^{10}+\frac{23\!\cdots\!51}{13\!\cdots\!48}a^{9}+\frac{28\!\cdots\!89}{39\!\cdots\!28}a^{8}-\frac{87\!\cdots\!83}{69\!\cdots\!74}a^{7}+\frac{15\!\cdots\!49}{27\!\cdots\!96}a^{6}-\frac{35\!\cdots\!91}{27\!\cdots\!96}a^{5}-\frac{46\!\cdots\!31}{27\!\cdots\!96}a^{4}+\frac{75\!\cdots\!19}{27\!\cdots\!96}a^{3}+\frac{63\!\cdots\!57}{17\!\cdots\!36}a^{2}-\frac{74\!\cdots\!84}{34\!\cdots\!87}a+\frac{12\!\cdots\!63}{69\!\cdots\!74}$, $\frac{28\!\cdots\!45}{27\!\cdots\!96}a^{15}+\frac{30\!\cdots\!19}{27\!\cdots\!96}a^{14}-\frac{92\!\cdots\!27}{13\!\cdots\!48}a^{13}-\frac{12\!\cdots\!29}{27\!\cdots\!96}a^{12}+\frac{18\!\cdots\!91}{69\!\cdots\!74}a^{11}+\frac{13\!\cdots\!01}{27\!\cdots\!96}a^{10}-\frac{30\!\cdots\!09}{27\!\cdots\!96}a^{9}+\frac{21\!\cdots\!71}{81\!\cdots\!72}a^{8}-\frac{80\!\cdots\!61}{27\!\cdots\!96}a^{7}+\frac{15\!\cdots\!77}{13\!\cdots\!48}a^{6}-\frac{14\!\cdots\!05}{69\!\cdots\!74}a^{5}+\frac{78\!\cdots\!24}{34\!\cdots\!87}a^{4}-\frac{41\!\cdots\!93}{13\!\cdots\!48}a^{3}+\frac{29\!\cdots\!15}{39\!\cdots\!28}a^{2}-\frac{26\!\cdots\!27}{69\!\cdots\!74}a+\frac{28\!\cdots\!63}{13\!\cdots\!48}$, $\frac{87\!\cdots\!01}{69\!\cdots\!74}a^{15}+\frac{18\!\cdots\!31}{12\!\cdots\!52}a^{14}-\frac{15\!\cdots\!81}{27\!\cdots\!96}a^{13}+\frac{10\!\cdots\!71}{34\!\cdots\!87}a^{12}+\frac{30\!\cdots\!79}{27\!\cdots\!96}a^{11}-\frac{35\!\cdots\!01}{13\!\cdots\!48}a^{10}+\frac{20\!\cdots\!35}{39\!\cdots\!28}a^{9}-\frac{76\!\cdots\!97}{27\!\cdots\!96}a^{8}+\frac{21\!\cdots\!45}{39\!\cdots\!28}a^{7}-\frac{25\!\cdots\!35}{27\!\cdots\!96}a^{6}+\frac{32\!\cdots\!89}{13\!\cdots\!48}a^{5}-\frac{16\!\cdots\!25}{34\!\cdots\!87}a^{4}+\frac{16\!\cdots\!45}{34\!\cdots\!87}a^{3}+\frac{29\!\cdots\!07}{30\!\cdots\!38}a^{2}+\frac{71\!\cdots\!63}{27\!\cdots\!96}a-\frac{57\!\cdots\!63}{69\!\cdots\!74}$, $\frac{25\!\cdots\!93}{34\!\cdots\!87}a^{15}-\frac{40\!\cdots\!07}{69\!\cdots\!74}a^{14}-\frac{78\!\cdots\!63}{27\!\cdots\!96}a^{13}+\frac{37\!\cdots\!28}{34\!\cdots\!87}a^{12}+\frac{23\!\cdots\!17}{69\!\cdots\!74}a^{11}-\frac{92\!\cdots\!07}{27\!\cdots\!96}a^{10}+\frac{10\!\cdots\!55}{27\!\cdots\!96}a^{9}-\frac{74\!\cdots\!19}{34\!\cdots\!87}a^{8}+\frac{10\!\cdots\!69}{12\!\cdots\!52}a^{7}-\frac{36\!\cdots\!49}{19\!\cdots\!64}a^{6}+\frac{42\!\cdots\!15}{27\!\cdots\!96}a^{5}+\frac{31\!\cdots\!67}{27\!\cdots\!96}a^{4}-\frac{58\!\cdots\!41}{27\!\cdots\!96}a^{3}-\frac{23\!\cdots\!39}{27\!\cdots\!96}a^{2}+\frac{14\!\cdots\!81}{27\!\cdots\!96}a-\frac{22\!\cdots\!51}{69\!\cdots\!74}$, $\frac{74\!\cdots\!39}{13\!\cdots\!48}a^{15}-\frac{48\!\cdots\!08}{49\!\cdots\!41}a^{14}-\frac{29\!\cdots\!95}{13\!\cdots\!48}a^{13}+\frac{14\!\cdots\!75}{13\!\cdots\!48}a^{12}+\frac{38\!\cdots\!21}{19\!\cdots\!64}a^{11}-\frac{39\!\cdots\!23}{13\!\cdots\!48}a^{10}+\frac{21\!\cdots\!27}{69\!\cdots\!74}a^{9}-\frac{12\!\cdots\!87}{69\!\cdots\!74}a^{8}+\frac{52\!\cdots\!37}{69\!\cdots\!74}a^{7}-\frac{25\!\cdots\!03}{13\!\cdots\!48}a^{6}+\frac{80\!\cdots\!91}{49\!\cdots\!41}a^{5}+\frac{24\!\cdots\!61}{34\!\cdots\!87}a^{4}-\frac{22\!\cdots\!85}{69\!\cdots\!74}a^{3}-\frac{12\!\cdots\!51}{13\!\cdots\!48}a^{2}+\frac{70\!\cdots\!25}{13\!\cdots\!48}a-\frac{12\!\cdots\!37}{34\!\cdots\!87}$ 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:  \( 2123693498.09 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2123693498.09 \cdot 42}{2\cdot\sqrt{43690605516556003276152578767301401}}\cr\approx \mathstrut & 0.518269972159 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961)
 
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 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961, 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 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961);
 
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 - 4*x^15 - 36*x^14 + 274*x^13 - 16*x^12 - 6092*x^11 + 67960*x^10 - 468046*x^9 + 2156282*x^8 - 6421886*x^7 + 10535912*x^6 - 5431518*x^5 - 7791049*x^4 - 1515286*x^3 + 45995976*x^2 - 27672790*x + 15280961);
 
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

$\OD_{16}:C_2$ (as 16T36):

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 11 conjugacy class representatives for $\OD_{16}:C_2$
Character table for $\OD_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{-1711}) \), 4.4.84898109.1, 4.0.24389.1, \(\Q(\sqrt{29}, \sqrt{-59})\), 8.4.209022978441500549.1 x2, 8.0.60046819431629.1 x2, 8.0.7207688911775881.1

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 8 siblings: 8.4.209022978441500549.1, 8.0.60046819431629.1
Degree 16 siblings: 16.0.12551165043538064715930071464321.1, 16.4.43690605516556003276152578767301401.1
Minimal sibling: 8.0.60046819431629.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{8}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ R ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ R

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 29.8.7.2$x^{8} + 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} + 29$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
\(59\) Copy content Toggle raw display 59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 59$$2$$1$$1$$C_2$$[\ ]_{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$