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Properties

Label 16.0.136...881.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.365\times 10^{37}$
Root discriminant \(209.38\)
Ramified primes $17,89$
Class number $51171146$ (GRH)
Class group [51171146] (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608)
 
gp: K = bnfinit(y^16 - 6*y^15 - 6*y^14 + 174*y^13 - 704*y^12 + 266*y^11 - 566*y^10 - 61350*y^9 + 486655*y^8 - 67436*y^7 - 2777412*y^6 - 1040544*y^5 + 7502144*y^4 + 8123904*y^3 - 1187840*y^2 - 12058624*y + 8388608, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608)
 

\( x^{16} - 6 x^{15} - 6 x^{14} + 174 x^{13} - 704 x^{12} + 266 x^{11} - 566 x^{10} - 61350 x^{9} + \cdots + 8388608 \) 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:   \(13647448497711210780660288977472356881\) \(\medspace = 17^{8}\cdot 89^{14}\) 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:  \(209.38\)
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:  $17^{1/2}89^{7/8}\approx 209.38266179210413$
Ramified primes:   \(17\), \(89\) 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:  \(1513=17\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{1513}(1,·)$, $\chi_{1513}(322,·)$, $\chi_{1513}(1412,·)$, $\chi_{1513}(902,·)$, $\chi_{1513}(713,·)$, $\chi_{1513}(611,·)$, $\chi_{1513}(101,·)$, $\chi_{1513}(800,·)$, $\chi_{1513}(1123,·)$, $\chi_{1513}(390,·)$, $\chi_{1513}(1191,·)$, $\chi_{1513}(1512,·)$, $\chi_{1513}(749,·)$, $\chi_{1513}(52,·)$, $\chi_{1513}(1461,·)$, $\chi_{1513}(764,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{32}a^{7}-\frac{1}{16}a^{4}+\frac{7}{32}a^{3}+\frac{1}{16}a^{2}-\frac{1}{4}a$, $\frac{1}{128}a^{8}+\frac{1}{64}a^{6}-\frac{1}{16}a^{5}-\frac{7}{128}a^{4}+\frac{3}{16}a^{3}+\frac{1}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{128}a^{9}-\frac{1}{64}a^{7}+\frac{1}{128}a^{5}-\frac{1}{16}a^{4}+\frac{1}{16}a^{2}$, $\frac{1}{256}a^{10}-\frac{1}{256}a^{9}-\frac{1}{128}a^{7}-\frac{3}{256}a^{6}+\frac{15}{256}a^{5}+\frac{5}{128}a^{4}-\frac{11}{64}a^{3}-\frac{1}{32}a^{2}+\frac{1}{8}a$, $\frac{1}{512}a^{11}+\frac{1}{512}a^{9}+\frac{7}{512}a^{7}-\frac{1}{32}a^{6}+\frac{11}{512}a^{5}-\frac{5}{128}a^{3}+\frac{1}{32}a^{2}$, $\frac{1}{8192}a^{12}-\frac{1}{1024}a^{11}+\frac{13}{8192}a^{10}+\frac{7}{2048}a^{9}-\frac{17}{8192}a^{8}-\frac{3}{256}a^{7}+\frac{151}{8192}a^{6}-\frac{101}{2048}a^{5}-\frac{117}{2048}a^{4}-\frac{17}{256}a^{3}-\frac{3}{128}a^{2}-\frac{7}{16}a$, $\frac{1}{8192}a^{13}-\frac{3}{8192}a^{11}+\frac{1}{2048}a^{10}-\frac{1}{8192}a^{9}+\frac{3}{1024}a^{8}-\frac{25}{8192}a^{7}-\frac{23}{2048}a^{6}-\frac{89}{2048}a^{5}+\frac{5}{128}a^{4}+\frac{15}{64}a^{3}-\frac{3}{32}a^{2}+\frac{1}{4}a$, $\frac{1}{2195456}a^{14}-\frac{129}{2195456}a^{13}-\frac{111}{2195456}a^{12}-\frac{105}{2195456}a^{11}+\frac{2367}{2195456}a^{10}+\frac{2233}{2195456}a^{9}-\frac{2437}{2195456}a^{8}-\frac{13555}{2195456}a^{7}+\frac{5825}{548864}a^{6}-\frac{26247}{548864}a^{5}-\frac{3637}{137216}a^{4}-\frac{2595}{34304}a^{3}-\frac{1359}{8576}a^{2}+\frac{107}{1072}a+\frac{15}{67}$, $\frac{1}{14\!\cdots\!84}a^{15}-\frac{92\!\cdots\!59}{70\!\cdots\!92}a^{14}-\frac{41\!\cdots\!71}{70\!\cdots\!92}a^{13}+\frac{65\!\cdots\!43}{70\!\cdots\!92}a^{12}-\frac{34\!\cdots\!77}{87\!\cdots\!24}a^{11}+\frac{31\!\cdots\!37}{70\!\cdots\!92}a^{10}+\frac{10\!\cdots\!85}{70\!\cdots\!92}a^{9}-\frac{26\!\cdots\!75}{70\!\cdots\!92}a^{8}+\frac{78\!\cdots\!11}{14\!\cdots\!84}a^{7}+\frac{35\!\cdots\!09}{35\!\cdots\!96}a^{6}-\frac{27\!\cdots\!97}{35\!\cdots\!96}a^{5}-\frac{10\!\cdots\!33}{43\!\cdots\!12}a^{4}+\frac{13\!\cdots\!61}{21\!\cdots\!56}a^{3}-\frac{56\!\cdots\!33}{27\!\cdots\!32}a^{2}+\frac{74\!\cdots\!21}{17\!\cdots\!52}a-\frac{23\!\cdots\!91}{53\!\cdots\!86}$ 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

$C_{51171146}$, which has order $51171146$ (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:   $\frac{37\!\cdots\!55}{26\!\cdots\!44}a^{15}-\frac{11\!\cdots\!71}{13\!\cdots\!72}a^{14}-\frac{89\!\cdots\!03}{13\!\cdots\!72}a^{13}+\frac{80\!\cdots\!63}{13\!\cdots\!72}a^{12}-\frac{44\!\cdots\!71}{65\!\cdots\!36}a^{11}-\frac{13\!\cdots\!51}{13\!\cdots\!72}a^{10}+\frac{55\!\cdots\!13}{13\!\cdots\!72}a^{9}-\frac{13\!\cdots\!31}{13\!\cdots\!72}a^{8}+\frac{19\!\cdots\!69}{26\!\cdots\!44}a^{7}+\frac{23\!\cdots\!11}{65\!\cdots\!36}a^{6}-\frac{22\!\cdots\!43}{65\!\cdots\!36}a^{5}+\frac{12\!\cdots\!79}{81\!\cdots\!92}a^{4}+\frac{73\!\cdots\!63}{40\!\cdots\!96}a^{3}+\frac{18\!\cdots\!15}{51\!\cdots\!12}a^{2}-\frac{92\!\cdots\!85}{39\!\cdots\!29}a+\frac{47\!\cdots\!99}{39\!\cdots\!29}$, $\frac{60\!\cdots\!81}{77\!\cdots\!84}a^{15}-\frac{19\!\cdots\!99}{38\!\cdots\!92}a^{14}-\frac{12\!\cdots\!35}{38\!\cdots\!92}a^{13}+\frac{54\!\cdots\!55}{38\!\cdots\!92}a^{12}-\frac{58\!\cdots\!11}{96\!\cdots\!48}a^{11}+\frac{13\!\cdots\!01}{38\!\cdots\!92}a^{10}-\frac{82\!\cdots\!15}{38\!\cdots\!92}a^{9}-\frac{18\!\cdots\!59}{38\!\cdots\!92}a^{8}+\frac{30\!\cdots\!03}{77\!\cdots\!84}a^{7}-\frac{33\!\cdots\!71}{19\!\cdots\!96}a^{6}-\frac{45\!\cdots\!41}{19\!\cdots\!96}a^{5}+\frac{25\!\cdots\!11}{24\!\cdots\!12}a^{4}+\frac{91\!\cdots\!77}{12\!\cdots\!56}a^{3}+\frac{55\!\cdots\!35}{15\!\cdots\!32}a^{2}-\frac{11\!\cdots\!55}{94\!\cdots\!52}a-\frac{14\!\cdots\!53}{59\!\cdots\!97}$, $\frac{13\!\cdots\!23}{17\!\cdots\!48}a^{15}-\frac{48\!\cdots\!43}{87\!\cdots\!24}a^{14}+\frac{26\!\cdots\!33}{87\!\cdots\!24}a^{13}+\frac{10\!\cdots\!71}{87\!\cdots\!24}a^{12}-\frac{31\!\cdots\!81}{43\!\cdots\!12}a^{11}+\frac{11\!\cdots\!37}{87\!\cdots\!24}a^{10}-\frac{26\!\cdots\!67}{87\!\cdots\!24}a^{9}-\frac{41\!\cdots\!19}{87\!\cdots\!24}a^{8}+\frac{75\!\cdots\!81}{17\!\cdots\!48}a^{7}-\frac{27\!\cdots\!01}{43\!\cdots\!12}a^{6}-\frac{24\!\cdots\!11}{43\!\cdots\!12}a^{5}+\frac{48\!\cdots\!67}{54\!\cdots\!64}a^{4}-\frac{48\!\cdots\!49}{27\!\cdots\!32}a^{3}+\frac{70\!\cdots\!63}{34\!\cdots\!04}a^{2}-\frac{57\!\cdots\!39}{53\!\cdots\!86}a+\frac{45\!\cdots\!03}{26\!\cdots\!43}$, $\frac{41\!\cdots\!67}{70\!\cdots\!92}a^{15}-\frac{92\!\cdots\!61}{35\!\cdots\!96}a^{14}-\frac{40\!\cdots\!33}{35\!\cdots\!96}a^{13}+\frac{40\!\cdots\!01}{35\!\cdots\!96}a^{12}-\frac{56\!\cdots\!17}{21\!\cdots\!56}a^{11}-\frac{33\!\cdots\!37}{35\!\cdots\!96}a^{10}+\frac{76\!\cdots\!51}{35\!\cdots\!96}a^{9}-\frac{13\!\cdots\!45}{35\!\cdots\!96}a^{8}+\frac{16\!\cdots\!21}{70\!\cdots\!92}a^{7}+\frac{97\!\cdots\!47}{17\!\cdots\!48}a^{6}-\frac{54\!\cdots\!15}{17\!\cdots\!48}a^{5}-\frac{41\!\cdots\!11}{21\!\cdots\!56}a^{4}+\frac{12\!\cdots\!75}{10\!\cdots\!28}a^{3}+\frac{50\!\cdots\!45}{13\!\cdots\!16}a^{2}-\frac{12\!\cdots\!49}{85\!\cdots\!76}a+\frac{20\!\cdots\!16}{26\!\cdots\!43}$, $\frac{42\!\cdots\!17}{70\!\cdots\!92}a^{15}-\frac{15\!\cdots\!83}{35\!\cdots\!96}a^{14}+\frac{62\!\cdots\!53}{35\!\cdots\!96}a^{13}+\frac{35\!\cdots\!47}{35\!\cdots\!96}a^{12}-\frac{12\!\cdots\!69}{21\!\cdots\!56}a^{11}+\frac{30\!\cdots\!85}{35\!\cdots\!96}a^{10}-\frac{56\!\cdots\!79}{35\!\cdots\!96}a^{9}-\frac{12\!\cdots\!11}{35\!\cdots\!96}a^{8}+\frac{23\!\cdots\!91}{70\!\cdots\!92}a^{7}-\frac{80\!\cdots\!95}{17\!\cdots\!48}a^{6}-\frac{16\!\cdots\!77}{17\!\cdots\!48}a^{5}+\frac{18\!\cdots\!83}{21\!\cdots\!56}a^{4}+\frac{17\!\cdots\!05}{10\!\cdots\!28}a^{3}+\frac{29\!\cdots\!75}{13\!\cdots\!16}a^{2}-\frac{39\!\cdots\!03}{85\!\cdots\!76}a+\frac{61\!\cdots\!68}{26\!\cdots\!43}$, $\frac{20\!\cdots\!81}{17\!\cdots\!48}a^{15}-\frac{81\!\cdots\!95}{87\!\cdots\!24}a^{14}+\frac{16\!\cdots\!53}{87\!\cdots\!24}a^{13}+\frac{59\!\cdots\!51}{87\!\cdots\!24}a^{12}-\frac{99\!\cdots\!11}{21\!\cdots\!56}a^{11}+\frac{12\!\cdots\!65}{87\!\cdots\!24}a^{10}-\frac{30\!\cdots\!27}{87\!\cdots\!24}a^{9}-\frac{46\!\cdots\!33}{13\!\cdots\!72}a^{8}+\frac{76\!\cdots\!35}{17\!\cdots\!48}a^{7}-\frac{92\!\cdots\!79}{43\!\cdots\!12}a^{6}-\frac{11\!\cdots\!09}{43\!\cdots\!12}a^{5}+\frac{41\!\cdots\!15}{54\!\cdots\!64}a^{4}+\frac{35\!\cdots\!51}{40\!\cdots\!96}a^{3}+\frac{13\!\cdots\!43}{34\!\cdots\!04}a^{2}-\frac{29\!\cdots\!81}{21\!\cdots\!44}a+\frac{91\!\cdots\!11}{26\!\cdots\!43}$, $\frac{90\!\cdots\!27}{35\!\cdots\!96}a^{15}-\frac{31\!\cdots\!09}{17\!\cdots\!48}a^{14}-\frac{16\!\cdots\!49}{17\!\cdots\!48}a^{13}+\frac{88\!\cdots\!49}{17\!\cdots\!48}a^{12}-\frac{23\!\cdots\!39}{10\!\cdots\!28}a^{11}+\frac{16\!\cdots\!23}{17\!\cdots\!48}a^{10}+\frac{55\!\cdots\!47}{17\!\cdots\!48}a^{9}-\frac{27\!\cdots\!25}{17\!\cdots\!48}a^{8}+\frac{49\!\cdots\!53}{35\!\cdots\!96}a^{7}-\frac{71\!\cdots\!29}{87\!\cdots\!24}a^{6}-\frac{94\!\cdots\!59}{87\!\cdots\!24}a^{5}+\frac{33\!\cdots\!17}{10\!\cdots\!28}a^{4}+\frac{23\!\cdots\!79}{54\!\cdots\!64}a^{3}+\frac{10\!\cdots\!21}{68\!\cdots\!08}a^{2}-\frac{28\!\cdots\!83}{42\!\cdots\!88}a-\frac{26\!\cdots\!61}{26\!\cdots\!43}$ 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:  \( 651512602.568 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 651512602.568 \cdot 51171146}{2\cdot\sqrt{13647448497711210780660288977472356881}}\cr\approx \mathstrut & 10960.5206189 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608)
 
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 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608, 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 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608);
 
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 - 6*x^15 - 6*x^14 + 174*x^13 - 704*x^12 + 266*x^11 - 566*x^10 - 61350*x^9 + 486655*x^8 - 67436*x^7 - 2777412*x^6 - 1040544*x^5 + 7502144*x^4 + 8123904*x^3 - 1187840*x^2 - 12058624*x + 8388608);
 
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

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{1513}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{89})\), 4.4.704969.1, 4.4.203736041.1, 8.8.41508374402353681.1, 8.0.44231334895529.1, 8.0.3694245321809477609.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{16}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ R ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(17\) Copy content Toggle raw display 17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(89\) Copy content Toggle raw display 89.8.7.3$x^{8} + 89$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} + 89$$8$$1$$7$$C_8$$[\ ]_{8}$