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Properties

Label 16.0.410...689.5
Degree $16$
Signature $[0, 8]$
Discriminant $4.106\times 10^{44}$
Root discriminant \(614.24\)
Ramified primes $29,53$
Class number $38887696$ (GRH)
Class group [2, 3118, 6236] (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 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072)
 
gp: K = bnfinit(y^16 - 5*y^15 + 146*y^14 - 1265*y^13 + 2855*y^12 + 46132*y^11 - 283221*y^10 + 11221001*y^9 + 76707714*y^8 - 1902825239*y^7 + 20717625413*y^6 - 188022386120*y^5 + 1304389721708*y^4 - 8627965710464*y^3 + 49990424556304*y^2 - 169299154243200*y + 231877632771072, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072)
 

\( x^{16} - 5 x^{15} + 146 x^{14} - 1265 x^{13} + 2855 x^{12} + 46132 x^{11} - 283221 x^{10} + \cdots + 231877632771072 \) 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:   \(410614370925299326221754150224933859206386689\) \(\medspace = 29^{14}\cdot 53^{14}\) 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:  \(614.24\)
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:  $29^{7/8}53^{7/8}\approx 614.2421770815839$
Ramified primes:   \(29\), \(53\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{12}a^{8}+\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{1}{12}a^{2}-\frac{1}{3}a$, $\frac{1}{12}a^{9}+\frac{1}{4}a^{3}-\frac{1}{3}a$, $\frac{1}{24}a^{10}-\frac{1}{24}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{5}{12}a^{2}+\frac{1}{6}a$, $\frac{1}{72}a^{11}-\frac{1}{72}a^{10}+\frac{1}{36}a^{9}-\frac{1}{36}a^{8}+\frac{1}{36}a^{7}+\frac{5}{36}a^{6}+\frac{11}{72}a^{5}+\frac{7}{72}a^{4}-\frac{5}{18}a^{3}+\frac{7}{36}a^{2}+\frac{1}{3}a$, $\frac{1}{144}a^{12}-\frac{1}{72}a^{10}-\frac{1}{48}a^{9}-\frac{1}{24}a^{8}-\frac{1}{4}a^{7}-\frac{7}{48}a^{6}+\frac{1}{6}a^{5}+\frac{13}{72}a^{4}+\frac{7}{16}a^{3}-\frac{5}{72}a^{2}-\frac{1}{12}a$, $\frac{1}{864}a^{13}-\frac{1}{864}a^{12}+\frac{1}{288}a^{10}-\frac{5}{864}a^{9}-\frac{17}{432}a^{8}-\frac{125}{864}a^{7}+\frac{101}{864}a^{6}-\frac{1}{72}a^{5}-\frac{49}{288}a^{4}-\frac{383}{864}a^{3}+\frac{199}{432}a^{2}-\frac{23}{72}a-\frac{1}{3}$, $\frac{1}{1728}a^{14}-\frac{1}{1728}a^{13}-\frac{1}{288}a^{12}+\frac{1}{576}a^{11}-\frac{29}{1728}a^{10}-\frac{13}{432}a^{9}-\frac{17}{1728}a^{8}+\frac{29}{1728}a^{7}+\frac{31}{288}a^{6}+\frac{23}{576}a^{5}+\frac{73}{1728}a^{4}+\frac{17}{108}a^{3}+\frac{1}{8}a^{2}-\frac{11}{24}a$, $\frac{1}{41\!\cdots\!76}a^{15}+\frac{64\!\cdots\!11}{41\!\cdots\!76}a^{14}+\frac{11\!\cdots\!31}{20\!\cdots\!88}a^{13}+\frac{55\!\cdots\!19}{41\!\cdots\!76}a^{12}-\frac{26\!\cdots\!49}{41\!\cdots\!76}a^{11}+\frac{40\!\cdots\!43}{51\!\cdots\!72}a^{10}-\frac{30\!\cdots\!87}{13\!\cdots\!92}a^{9}-\frac{36\!\cdots\!09}{15\!\cdots\!88}a^{8}-\frac{80\!\cdots\!13}{20\!\cdots\!88}a^{7}-\frac{92\!\cdots\!67}{41\!\cdots\!76}a^{6}-\frac{26\!\cdots\!39}{41\!\cdots\!76}a^{5}-\frac{19\!\cdots\!03}{10\!\cdots\!44}a^{4}+\frac{13\!\cdots\!03}{10\!\cdots\!44}a^{3}-\frac{17\!\cdots\!65}{12\!\cdots\!68}a^{2}+\frac{54\!\cdots\!25}{86\!\cdots\!12}a+\frac{22\!\cdots\!81}{35\!\cdots\!63}$ 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:  $2$, $3$

Class group and class number

$C_{2}\times C_{3118}\times C_{6236}$, which has order $38887696$ (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:   $\frac{12\!\cdots\!11}{17\!\cdots\!68}a^{15}+\frac{23\!\cdots\!47}{19\!\cdots\!52}a^{14}+\frac{22\!\cdots\!59}{21\!\cdots\!96}a^{13}-\frac{11\!\cdots\!69}{57\!\cdots\!56}a^{12}-\frac{37\!\cdots\!53}{17\!\cdots\!68}a^{11}+\frac{87\!\cdots\!37}{28\!\cdots\!28}a^{10}+\frac{72\!\cdots\!99}{64\!\cdots\!84}a^{9}+\frac{44\!\cdots\!19}{57\!\cdots\!56}a^{8}+\frac{45\!\cdots\!19}{43\!\cdots\!92}a^{7}-\frac{39\!\cdots\!31}{57\!\cdots\!56}a^{6}+\frac{15\!\cdots\!05}{17\!\cdots\!68}a^{5}-\frac{21\!\cdots\!63}{32\!\cdots\!92}a^{4}+\frac{21\!\cdots\!57}{54\!\cdots\!24}a^{3}-\frac{24\!\cdots\!11}{80\!\cdots\!48}a^{2}+\frac{50\!\cdots\!51}{40\!\cdots\!24}a-\frac{88\!\cdots\!49}{50\!\cdots\!03}$, $\frac{39\!\cdots\!23}{42\!\cdots\!12}a^{15}+\frac{21\!\cdots\!15}{12\!\cdots\!36}a^{14}+\frac{99\!\cdots\!81}{70\!\cdots\!52}a^{13}-\frac{34\!\cdots\!01}{12\!\cdots\!36}a^{12}-\frac{15\!\cdots\!39}{42\!\cdots\!12}a^{11}+\frac{13\!\cdots\!65}{31\!\cdots\!84}a^{10}+\frac{25\!\cdots\!23}{14\!\cdots\!04}a^{9}+\frac{14\!\cdots\!57}{14\!\cdots\!04}a^{8}+\frac{29\!\cdots\!15}{21\!\cdots\!56}a^{7}-\frac{11\!\cdots\!03}{12\!\cdots\!36}a^{6}+\frac{50\!\cdots\!71}{42\!\cdots\!12}a^{5}-\frac{14\!\cdots\!65}{15\!\cdots\!92}a^{4}+\frac{55\!\cdots\!85}{10\!\cdots\!28}a^{3}-\frac{15\!\cdots\!97}{39\!\cdots\!48}a^{2}+\frac{44\!\cdots\!23}{26\!\cdots\!32}a-\frac{26\!\cdots\!95}{10\!\cdots\!43}$, $\frac{13\!\cdots\!37}{14\!\cdots\!68}a^{15}+\frac{22\!\cdots\!09}{16\!\cdots\!52}a^{14}+\frac{49\!\cdots\!47}{36\!\cdots\!92}a^{13}-\frac{13\!\cdots\!23}{48\!\cdots\!56}a^{12}-\frac{20\!\cdots\!87}{14\!\cdots\!68}a^{11}+\frac{96\!\cdots\!61}{24\!\cdots\!28}a^{10}+\frac{32\!\cdots\!77}{48\!\cdots\!56}a^{9}+\frac{48\!\cdots\!65}{48\!\cdots\!56}a^{8}+\frac{12\!\cdots\!53}{90\!\cdots\!48}a^{7}-\frac{43\!\cdots\!25}{48\!\cdots\!56}a^{6}+\frac{17\!\cdots\!59}{14\!\cdots\!68}a^{5}-\frac{21\!\cdots\!15}{24\!\cdots\!28}a^{4}+\frac{59\!\cdots\!37}{11\!\cdots\!31}a^{3}-\frac{24\!\cdots\!05}{60\!\cdots\!32}a^{2}+\frac{17\!\cdots\!61}{10\!\cdots\!72}a-\frac{10\!\cdots\!00}{41\!\cdots\!53}$, $\frac{10\!\cdots\!77}{50\!\cdots\!36}a^{15}-\frac{59\!\cdots\!11}{72\!\cdots\!84}a^{14}+\frac{25\!\cdots\!11}{72\!\cdots\!84}a^{13}-\frac{10\!\cdots\!21}{12\!\cdots\!64}a^{12}+\frac{15\!\cdots\!49}{24\!\cdots\!28}a^{11}+\frac{40\!\cdots\!03}{72\!\cdots\!84}a^{10}-\frac{10\!\cdots\!91}{18\!\cdots\!96}a^{9}+\frac{83\!\cdots\!67}{72\!\cdots\!84}a^{8}-\frac{32\!\cdots\!27}{72\!\cdots\!84}a^{7}-\frac{13\!\cdots\!01}{12\!\cdots\!64}a^{6}+\frac{44\!\cdots\!55}{24\!\cdots\!28}a^{5}-\frac{59\!\cdots\!43}{72\!\cdots\!84}a^{4}+\frac{54\!\cdots\!35}{90\!\cdots\!48}a^{3}-\frac{63\!\cdots\!37}{10\!\cdots\!72}a^{2}+\frac{93\!\cdots\!47}{33\!\cdots\!24}a-\frac{13\!\cdots\!60}{41\!\cdots\!53}$, $\frac{14\!\cdots\!07}{14\!\cdots\!68}a^{15}+\frac{14\!\cdots\!27}{14\!\cdots\!68}a^{14}+\frac{27\!\cdots\!99}{18\!\cdots\!96}a^{13}-\frac{51\!\cdots\!59}{14\!\cdots\!68}a^{12}+\frac{78\!\cdots\!43}{14\!\cdots\!68}a^{11}+\frac{31\!\cdots\!23}{72\!\cdots\!84}a^{10}-\frac{13\!\cdots\!81}{48\!\cdots\!56}a^{9}+\frac{52\!\cdots\!91}{48\!\cdots\!56}a^{8}+\frac{50\!\cdots\!47}{36\!\cdots\!92}a^{7}-\frac{14\!\cdots\!93}{14\!\cdots\!68}a^{6}+\frac{20\!\cdots\!73}{14\!\cdots\!68}a^{5}-\frac{76\!\cdots\!73}{72\!\cdots\!84}a^{4}+\frac{23\!\cdots\!77}{36\!\cdots\!92}a^{3}-\frac{44\!\cdots\!07}{90\!\cdots\!48}a^{2}+\frac{30\!\cdots\!33}{15\!\cdots\!08}a-\frac{61\!\cdots\!89}{12\!\cdots\!59}$, $\frac{30\!\cdots\!83}{48\!\cdots\!56}a^{15}+\frac{75\!\cdots\!89}{14\!\cdots\!68}a^{14}-\frac{66\!\cdots\!43}{12\!\cdots\!64}a^{13}+\frac{88\!\cdots\!27}{14\!\cdots\!68}a^{12}-\frac{13\!\cdots\!73}{48\!\cdots\!56}a^{11}-\frac{32\!\cdots\!49}{72\!\cdots\!84}a^{10}+\frac{12\!\cdots\!71}{16\!\cdots\!52}a^{9}+\frac{11\!\cdots\!39}{16\!\cdots\!52}a^{8}-\frac{16\!\cdots\!53}{30\!\cdots\!16}a^{7}+\frac{99\!\cdots\!29}{14\!\cdots\!68}a^{6}-\frac{10\!\cdots\!21}{16\!\cdots\!52}a^{5}+\frac{36\!\cdots\!39}{72\!\cdots\!84}a^{4}-\frac{14\!\cdots\!59}{40\!\cdots\!88}a^{3}+\frac{23\!\cdots\!67}{11\!\cdots\!31}a^{2}-\frac{10\!\cdots\!89}{15\!\cdots\!08}a+\frac{11\!\cdots\!77}{12\!\cdots\!59}$, $\frac{71\!\cdots\!31}{35\!\cdots\!36}a^{15}+\frac{55\!\cdots\!43}{31\!\cdots\!24}a^{14}+\frac{26\!\cdots\!79}{26\!\cdots\!52}a^{13}+\frac{10\!\cdots\!81}{31\!\cdots\!24}a^{12}-\frac{40\!\cdots\!59}{10\!\cdots\!08}a^{11}+\frac{61\!\cdots\!01}{15\!\cdots\!12}a^{10}+\frac{50\!\cdots\!53}{35\!\cdots\!36}a^{9}-\frac{22\!\cdots\!41}{11\!\cdots\!12}a^{8}+\frac{93\!\cdots\!58}{16\!\cdots\!97}a^{7}-\frac{12\!\cdots\!89}{31\!\cdots\!24}a^{6}-\frac{28\!\cdots\!73}{10\!\cdots\!08}a^{5}+\frac{11\!\cdots\!49}{15\!\cdots\!12}a^{4}-\frac{10\!\cdots\!01}{14\!\cdots\!64}a^{3}+\frac{17\!\cdots\!17}{39\!\cdots\!28}a^{2}-\frac{89\!\cdots\!25}{66\!\cdots\!88}a+\frac{92\!\cdots\!03}{55\!\cdots\!99}$ 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:  \( 1358079502.6 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1358079502.6 \cdot 38887696}{2\cdot\sqrt{410614370925299326221754150224933859206386689}}\cr\approx \mathstrut & 3.1654050160 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072)
 
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 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072, 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 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072);
 
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 - 5*x^15 + 146*x^14 - 1265*x^13 + 2855*x^12 + 46132*x^11 - 283221*x^10 + 11221001*x^9 + 76707714*x^8 - 1902825239*x^7 + 20717625413*x^6 - 188022386120*x^5 + 1304389721708*x^4 - 8627965710464*x^3 + 49990424556304*x^2 - 169299154243200*x + 231877632771072);
 
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

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{1537}) \), \(\Q(\sqrt{29}) \), 4.4.3630961153.1, 4.4.3630961153.2, \(\Q(\sqrt{29}, \sqrt{53})\), 8.0.20263621860992652421633.1 x2, 8.0.20263621860992652421633.2 x2, 8.8.13183878894595089409.1

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: 8.0.20263621860992652421633.2, 8.0.20263621860992652421633.1
Degree 16 siblings: 16.0.410614370925299326221754150224933859206386689.3, 16.0.410614370925299326221754150224933859206386689.2
Minimal sibling: 8.0.20263621860992652421633.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.2.0.1}{2} }^{8}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ R ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 29.8.7.1$x^{8} + 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.1$x^{8} + 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
\(53\) Copy content Toggle raw display 53.8.7.2$x^{8} + 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} + 53$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$