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Properties

Label 16.0.605...000.8
Degree $16$
Signature $[0, 8]$
Discriminant $6.052\times 10^{23}$
Root discriminant \(30.65\)
Ramified primes $2,3,5,7$
Class number $32$ (GRH)
Class group [4, 8] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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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 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101)
 
gp: K = bnfinit(y^16 - 2*y^15 - 25*y^14 + 50*y^13 + 254*y^12 - 442*y^11 - 1452*y^10 + 2200*y^9 + 4686*y^8 - 7282*y^7 - 5457*y^6 + 3214*y^5 + 30374*y^4 + 11566*y^3 + 3134*y^2 - 9304*y + 2101, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101)
 

\( x^{16} - 2 x^{15} - 25 x^{14} + 50 x^{13} + 254 x^{12} - 442 x^{11} - 1452 x^{10} + 2200 x^{9} + \cdots + 2101 \) 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:   \(605165749776000000000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\) 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:  \(30.65\)
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:  $2\cdot 3^{1/2}5^{3/4}7^{1/2}\approx 30.645530678223075$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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 and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(323,·)$, $\chi_{420}(71,·)$, $\chi_{420}(13,·)$, $\chi_{420}(337,·)$, $\chi_{420}(83,·)$, $\chi_{420}(407,·)$, $\chi_{420}(349,·)$, $\chi_{420}(97,·)$, $\chi_{420}(419,·)$, $\chi_{420}(167,·)$, $\chi_{420}(169,·)$, $\chi_{420}(239,·)$, $\chi_{420}(181,·)$, $\chi_{420}(251,·)$, $\chi_{420}(253,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{44}a^{11}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{21}{44}a-\frac{1}{2}$, $\frac{1}{176}a^{12}-\frac{1}{88}a^{11}-\frac{1}{16}a^{10}+\frac{1}{8}a^{9}-\frac{7}{16}a^{8}+\frac{1}{4}a^{7}-\frac{5}{16}a^{6}-\frac{1}{8}a^{5}-\frac{5}{16}a^{4}-\frac{3}{8}a^{3}-\frac{3}{44}a^{2}+\frac{3}{22}a-\frac{1}{16}$, $\frac{1}{176}a^{13}+\frac{1}{176}a^{11}-\frac{3}{16}a^{9}+\frac{3}{8}a^{8}+\frac{3}{16}a^{7}+\frac{1}{4}a^{6}+\frac{7}{16}a^{5}+\frac{2}{11}a^{3}+\frac{21}{176}a-\frac{1}{8}$, $\frac{1}{296384}a^{14}-\frac{127}{148192}a^{13}-\frac{33}{13472}a^{12}+\frac{43}{9262}a^{11}-\frac{331}{6736}a^{10}-\frac{5719}{13472}a^{9}-\frac{77}{1684}a^{8}-\frac{1297}{13472}a^{7}+\frac{4193}{13472}a^{6}+\frac{483}{6736}a^{5}+\frac{75393}{296384}a^{4}+\frac{36471}{148192}a^{3}-\frac{2373}{26944}a^{2}+\frac{18675}{74096}a+\frac{4195}{26944}$, $\frac{1}{78\!\cdots\!44}a^{15}-\frac{591698762848327}{39\!\cdots\!72}a^{14}+\frac{95\!\cdots\!65}{35\!\cdots\!52}a^{13}-\frac{11\!\cdots\!28}{12\!\cdots\!21}a^{12}+\frac{19\!\cdots\!93}{49\!\cdots\!84}a^{11}+\frac{35\!\cdots\!37}{35\!\cdots\!52}a^{10}+\frac{19\!\cdots\!29}{17\!\cdots\!76}a^{9}-\frac{16\!\cdots\!61}{35\!\cdots\!52}a^{8}+\frac{17\!\cdots\!79}{35\!\cdots\!52}a^{7}-\frac{67\!\cdots\!69}{17\!\cdots\!76}a^{6}-\frac{23\!\cdots\!23}{78\!\cdots\!44}a^{5}-\frac{29\!\cdots\!81}{39\!\cdots\!72}a^{4}-\frac{22\!\cdots\!17}{71\!\cdots\!04}a^{3}+\frac{73\!\cdots\!75}{19\!\cdots\!36}a^{2}-\frac{12\!\cdots\!43}{78\!\cdots\!44}a+\frac{31\!\cdots\!05}{89\!\cdots\!88}$ 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:  $11$

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:   \( -\frac{135658817260507}{620371807485788864} a^{15} + \frac{30976112772923}{310185903742894432} a^{14} + \frac{1887848240981345}{310185903742894432} a^{13} - \frac{88577734674347}{38773237967861804} a^{12} - \frac{11109339251179907}{155092951871447216} a^{11} + \frac{175624209166849}{28198718522081312} a^{10} + \frac{1646174414149053}{3524839815260164} a^{9} + \frac{1274354616739823}{28198718522081312} a^{8} - \frac{51323884261696179}{28198718522081312} a^{7} - \frac{1916306320323637}{14099359261040656} a^{6} + \frac{2530368078893884605}{620371807485788864} a^{5} + \frac{411212021244243957}{310185903742894432} a^{4} - \frac{5883956780913876059}{620371807485788864} a^{3} - \frac{847998471012210605}{77546475935723608} a^{2} - \frac{2947220539064520275}{620371807485788864} a + \frac{35617412783857187}{14099359261040656} \)  (order $10$) 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:   $\frac{33969637999007}{77\!\cdots\!08}a^{15}-\frac{546300843053089}{62\!\cdots\!64}a^{14}-\frac{34\!\cdots\!31}{31\!\cdots\!32}a^{13}+\frac{68\!\cdots\!21}{31\!\cdots\!32}a^{12}+\frac{17\!\cdots\!79}{15\!\cdots\!16}a^{11}-\frac{14\!\cdots\!99}{70\!\cdots\!28}a^{10}-\frac{17\!\cdots\!35}{28\!\cdots\!12}a^{9}+\frac{14\!\cdots\!39}{14\!\cdots\!56}a^{8}+\frac{56\!\cdots\!47}{28\!\cdots\!12}a^{7}-\frac{96\!\cdots\!67}{28\!\cdots\!12}a^{6}-\frac{80\!\cdots\!93}{38\!\cdots\!04}a^{5}+\frac{12\!\cdots\!27}{62\!\cdots\!64}a^{4}+\frac{36\!\cdots\!69}{31\!\cdots\!32}a^{3}+\frac{30\!\cdots\!07}{62\!\cdots\!64}a^{2}+\frac{18\!\cdots\!40}{96\!\cdots\!51}a-\frac{71\!\cdots\!27}{56\!\cdots\!24}$, $\frac{67\!\cdots\!59}{78\!\cdots\!44}a^{15}-\frac{14\!\cdots\!31}{71\!\cdots\!04}a^{14}-\frac{41\!\cdots\!65}{19\!\cdots\!36}a^{13}+\frac{20\!\cdots\!91}{39\!\cdots\!72}a^{12}+\frac{50\!\cdots\!73}{24\!\cdots\!42}a^{11}-\frac{16\!\cdots\!29}{35\!\cdots\!52}a^{10}-\frac{40\!\cdots\!09}{35\!\cdots\!52}a^{9}+\frac{88\!\cdots\!27}{35\!\cdots\!52}a^{8}+\frac{29\!\cdots\!67}{89\!\cdots\!88}a^{7}-\frac{30\!\cdots\!95}{35\!\cdots\!52}a^{6}-\frac{12\!\cdots\!37}{78\!\cdots\!44}a^{5}+\frac{46\!\cdots\!81}{71\!\cdots\!04}a^{4}+\frac{15\!\cdots\!25}{78\!\cdots\!44}a^{3}+\frac{21\!\cdots\!41}{78\!\cdots\!44}a^{2}-\frac{96\!\cdots\!45}{78\!\cdots\!44}a-\frac{59\!\cdots\!57}{71\!\cdots\!04}$, $\frac{69\!\cdots\!59}{78\!\cdots\!44}a^{15}-\frac{48\!\cdots\!35}{78\!\cdots\!44}a^{14}-\frac{53\!\cdots\!23}{49\!\cdots\!84}a^{13}+\frac{59\!\cdots\!53}{39\!\cdots\!72}a^{12}-\frac{61\!\cdots\!83}{98\!\cdots\!68}a^{11}-\frac{50\!\cdots\!01}{35\!\cdots\!52}a^{10}+\frac{51\!\cdots\!89}{35\!\cdots\!52}a^{9}+\frac{25\!\cdots\!35}{35\!\cdots\!52}a^{8}-\frac{18\!\cdots\!43}{17\!\cdots\!76}a^{7}-\frac{71\!\cdots\!57}{35\!\cdots\!52}a^{6}+\frac{35\!\cdots\!59}{78\!\cdots\!44}a^{5}-\frac{24\!\cdots\!83}{78\!\cdots\!44}a^{4}-\frac{11\!\cdots\!87}{78\!\cdots\!44}a^{3}-\frac{50\!\cdots\!77}{78\!\cdots\!44}a^{2}+\frac{94\!\cdots\!35}{78\!\cdots\!44}a+\frac{20\!\cdots\!57}{71\!\cdots\!04}$, $\frac{21\!\cdots\!01}{39\!\cdots\!72}a^{15}-\frac{11\!\cdots\!39}{78\!\cdots\!44}a^{14}-\frac{50\!\cdots\!13}{39\!\cdots\!72}a^{13}+\frac{14\!\cdots\!19}{39\!\cdots\!72}a^{12}+\frac{28\!\cdots\!99}{24\!\cdots\!42}a^{11}-\frac{63\!\cdots\!45}{17\!\cdots\!76}a^{10}-\frac{20\!\cdots\!55}{35\!\cdots\!52}a^{9}+\frac{17\!\cdots\!61}{89\!\cdots\!88}a^{8}+\frac{47\!\cdots\!45}{35\!\cdots\!52}a^{7}-\frac{22\!\cdots\!69}{35\!\cdots\!52}a^{6}+\frac{74\!\cdots\!07}{39\!\cdots\!72}a^{5}+\frac{39\!\cdots\!61}{78\!\cdots\!44}a^{4}+\frac{10\!\cdots\!22}{12\!\cdots\!21}a^{3}-\frac{14\!\cdots\!71}{78\!\cdots\!44}a^{2}-\frac{12\!\cdots\!49}{39\!\cdots\!72}a+\frac{11\!\cdots\!27}{71\!\cdots\!04}$, $\frac{14\!\cdots\!93}{39\!\cdots\!72}a^{15}-\frac{54\!\cdots\!23}{71\!\cdots\!04}a^{14}-\frac{35\!\cdots\!51}{39\!\cdots\!72}a^{13}+\frac{73\!\cdots\!59}{39\!\cdots\!72}a^{12}+\frac{79\!\cdots\!05}{89\!\cdots\!88}a^{11}-\frac{73\!\cdots\!53}{44\!\cdots\!44}a^{10}-\frac{17\!\cdots\!09}{35\!\cdots\!52}a^{9}+\frac{14\!\cdots\!79}{17\!\cdots\!76}a^{8}+\frac{53\!\cdots\!15}{35\!\cdots\!52}a^{7}-\frac{91\!\cdots\!13}{35\!\cdots\!52}a^{6}-\frac{44\!\cdots\!45}{39\!\cdots\!72}a^{5}+\frac{21\!\cdots\!17}{71\!\cdots\!04}a^{4}+\frac{18\!\cdots\!41}{19\!\cdots\!36}a^{3}+\frac{43\!\cdots\!03}{78\!\cdots\!44}a^{2}+\frac{12\!\cdots\!93}{35\!\cdots\!52}a-\frac{12\!\cdots\!23}{71\!\cdots\!04}$, $\frac{56\!\cdots\!71}{39\!\cdots\!72}a^{15}-\frac{19\!\cdots\!39}{78\!\cdots\!44}a^{14}-\frac{13\!\cdots\!33}{39\!\cdots\!72}a^{13}+\frac{25\!\cdots\!33}{39\!\cdots\!72}a^{12}+\frac{17\!\cdots\!47}{49\!\cdots\!84}a^{11}-\frac{56\!\cdots\!13}{89\!\cdots\!88}a^{10}-\frac{68\!\cdots\!19}{35\!\cdots\!52}a^{9}+\frac{63\!\cdots\!67}{17\!\cdots\!76}a^{8}+\frac{20\!\cdots\!89}{35\!\cdots\!52}a^{7}-\frac{47\!\cdots\!51}{35\!\cdots\!52}a^{6}-\frac{14\!\cdots\!59}{39\!\cdots\!72}a^{5}+\frac{10\!\cdots\!41}{78\!\cdots\!44}a^{4}+\frac{43\!\cdots\!19}{19\!\cdots\!36}a^{3}+\frac{55\!\cdots\!89}{78\!\cdots\!44}a^{2}-\frac{13\!\cdots\!63}{39\!\cdots\!72}a+\frac{10\!\cdots\!27}{71\!\cdots\!04}$, $\frac{71\!\cdots\!27}{19\!\cdots\!36}a^{15}-\frac{35\!\cdots\!79}{39\!\cdots\!72}a^{14}-\frac{16\!\cdots\!51}{19\!\cdots\!36}a^{13}+\frac{10\!\cdots\!05}{49\!\cdots\!84}a^{12}+\frac{38\!\cdots\!51}{49\!\cdots\!84}a^{11}-\frac{36\!\cdots\!43}{17\!\cdots\!76}a^{10}-\frac{71\!\cdots\!75}{17\!\cdots\!76}a^{9}+\frac{18\!\cdots\!89}{17\!\cdots\!76}a^{8}+\frac{18\!\cdots\!63}{17\!\cdots\!76}a^{7}-\frac{29\!\cdots\!83}{89\!\cdots\!88}a^{6}+\frac{74\!\cdots\!61}{19\!\cdots\!36}a^{5}+\frac{48\!\cdots\!75}{39\!\cdots\!72}a^{4}+\frac{20\!\cdots\!47}{24\!\cdots\!42}a^{3}+\frac{20\!\cdots\!61}{39\!\cdots\!72}a^{2}+\frac{45\!\cdots\!23}{19\!\cdots\!36}a-\frac{53\!\cdots\!55}{35\!\cdots\!52}$ 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:  \( 76884.6053994 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 76884.6053994 \cdot 32}{10\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.768229791190 \end{aligned}\] (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 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101)
 
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 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101, 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 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101);
 
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 - 25*x^14 + 50*x^13 + 254*x^12 - 442*x^11 - 1452*x^10 + 2200*x^9 + 4686*x^8 - 7282*x^7 - 5457*x^6 + 3214*x^5 + 30374*x^4 + 11566*x^3 + 3134*x^2 - 9304*x + 2101);
 
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

$C_2^2\times C_4$ (as 16T2):

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)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{15}, \sqrt{-21})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-7}, \sqrt{15})\), \(\Q(\zeta_{5})\), 4.4.6125.1, 4.0.18000.1, 4.4.882000.1, 8.0.31116960000.6, 8.0.37515625.1, 8.0.777924000000.4, 8.0.324000000.3, 8.8.777924000000.1, 8.0.777924000000.10, 8.0.777924000000.2

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R R ${\href{/padicField/11.1.0.1}{1} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(2\) Copy content Toggle raw display 2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 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$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$