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Properties

Label 16.0.141...561.3
Degree $16$
Signature $[0, 8]$
Discriminant $1.411\times 10^{27}$
Root discriminant \(49.76\)
Ramified primes $3,61$
Class number $16$ (GRH)
Class group [2, 8] (GRH)
Galois group $C_2^3:C_4$ (as 16T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524)
 
gp: K = bnfinit(y^16 - y^15 - y^14 + 12*y^13 - 79*y^12 + 484*y^11 + 1411*y^10 - 1635*y^9 + 4582*y^8 + 35484*y^7 + 84259*y^6 + 65558*y^5 - 12802*y^4 + 8304*y^3 + 328913*y^2 + 621070*y + 515524, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524)
 

\( x^{16} - x^{15} - x^{14} + 12 x^{13} - 79 x^{12} + 484 x^{11} + 1411 x^{10} - 1635 x^{9} + 4582 x^{8} + \cdots + 515524 \) 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:   \(1410629873249683485564270561\) \(\medspace = 3^{12}\cdot 61^{12}\) 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:  \(49.76\)
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:  $3^{3/4}61^{3/4}\approx 49.755181783405924$
Ramified primes:   \(3\), \(61\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{15}a^{8}-\frac{2}{15}a^{7}-\frac{1}{15}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{2}{15}a^{2}-\frac{4}{15}a-\frac{2}{15}$, $\frac{1}{15}a^{9}-\frac{1}{3}a^{7}-\frac{7}{15}a^{6}-\frac{1}{3}a^{4}-\frac{1}{5}a^{3}+\frac{1}{3}a-\frac{4}{15}$, $\frac{1}{15}a^{10}-\frac{2}{15}a^{7}-\frac{1}{3}a^{6}+\frac{2}{15}a^{4}-\frac{1}{3}a^{3}+\frac{2}{5}a+\frac{1}{3}$, $\frac{1}{15}a^{11}+\frac{2}{5}a^{7}-\frac{2}{15}a^{6}+\frac{7}{15}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{5}a-\frac{4}{15}$, $\frac{1}{1830}a^{12}+\frac{13}{1830}a^{11}-\frac{9}{610}a^{10}-\frac{1}{366}a^{9}+\frac{14}{915}a^{8}-\frac{283}{610}a^{7}+\frac{54}{305}a^{6}-\frac{49}{1830}a^{5}+\frac{167}{610}a^{4}+\frac{10}{183}a^{3}+\frac{51}{305}a^{2}+\frac{139}{610}a-\frac{173}{915}$, $\frac{1}{1830}a^{13}+\frac{8}{305}a^{11}-\frac{2}{183}a^{10}-\frac{29}{1830}a^{9}+\frac{7}{1830}a^{8}+\frac{249}{610}a^{7}+\frac{25}{122}a^{6}+\frac{203}{915}a^{5}+\frac{35}{366}a^{4}+\frac{296}{915}a^{3}-\frac{511}{1830}a^{2}+\frac{91}{366}a-\frac{26}{183}$, $\frac{1}{40324050}a^{14}+\frac{4382}{20162025}a^{13}-\frac{59}{1550925}a^{12}+\frac{13948}{806481}a^{11}-\frac{651217}{40324050}a^{10}+\frac{194059}{40324050}a^{9}+\frac{1162037}{40324050}a^{8}+\frac{8615003}{40324050}a^{7}-\frac{7595752}{20162025}a^{6}+\frac{3737867}{8064810}a^{5}-\frac{4519027}{20162025}a^{4}+\frac{10377203}{40324050}a^{3}+\frac{47459}{620370}a^{2}-\frac{1962761}{4032405}a+\frac{3980389}{20162025}$, $\frac{1}{26\!\cdots\!50}a^{15}-\frac{24\!\cdots\!39}{26\!\cdots\!50}a^{14}-\frac{76\!\cdots\!57}{89\!\cdots\!50}a^{13}+\frac{41\!\cdots\!87}{26\!\cdots\!50}a^{12}+\frac{10\!\cdots\!54}{13\!\cdots\!25}a^{11}-\frac{11\!\cdots\!67}{59\!\cdots\!70}a^{10}+\frac{82\!\cdots\!53}{89\!\cdots\!55}a^{9}-\frac{69\!\cdots\!23}{44\!\cdots\!50}a^{8}+\frac{86\!\cdots\!39}{89\!\cdots\!50}a^{7}-\frac{74\!\cdots\!19}{13\!\cdots\!25}a^{6}+\frac{54\!\cdots\!83}{13\!\cdots\!25}a^{5}+\frac{10\!\cdots\!81}{17\!\cdots\!10}a^{4}-\frac{17\!\cdots\!73}{14\!\cdots\!25}a^{3}-\frac{12\!\cdots\!38}{26\!\cdots\!65}a^{2}-\frac{16\!\cdots\!99}{24\!\cdots\!25}a+\frac{56\!\cdots\!67}{37\!\cdots\!75}$ 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

$C_{2}\times C_{8}$, which has order $16$ (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:   \( \frac{29105677301655691}{130115855816614863376350} a^{15} + \frac{257933919271321933}{65057927908307431688175} a^{14} - \frac{284088407104872946}{21685975969435810562725} a^{13} + \frac{2849462318596217147}{130115855816614863376350} a^{12} + \frac{204888063937102673}{21685975969435810562725} a^{11} - \frac{3985563973265440427}{13011585581661486337635} a^{10} + \frac{13894166299911984212}{4337195193887162112545} a^{9} - \frac{7231006898509041553}{130115855816614863376350} a^{8} - \frac{406192812673361319851}{43371951938871621125450} a^{7} + \frac{6855112743307157780297}{130115855816614863376350} a^{6} + \frac{11118540052797955381901}{130115855816614863376350} a^{5} + \frac{2390379380427262643473}{13011585581661486337635} a^{4} - \frac{2762666157668337066803}{43371951938871621125450} a^{3} + \frac{70999688606752367062}{867439038777432422509} a^{2} + \frac{78160357599159019556953}{130115855816614863376350} a + \frac{91892297875611360759}{60406618299264096275} \)  (order $6$) 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:   $\frac{11\!\cdots\!39}{13\!\cdots\!25}a^{15}+\frac{18\!\cdots\!93}{10\!\cdots\!66}a^{14}-\frac{77\!\cdots\!32}{53\!\cdots\!33}a^{13}-\frac{11\!\cdots\!77}{26\!\cdots\!50}a^{12}+\frac{75\!\cdots\!19}{26\!\cdots\!50}a^{11}-\frac{90\!\cdots\!72}{13\!\cdots\!25}a^{10}+\frac{55\!\cdots\!34}{13\!\cdots\!25}a^{9}+\frac{49\!\cdots\!99}{53\!\cdots\!30}a^{8}-\frac{79\!\cdots\!29}{13\!\cdots\!25}a^{7}+\frac{15\!\cdots\!79}{13\!\cdots\!25}a^{6}+\frac{24\!\cdots\!04}{13\!\cdots\!25}a^{5}-\frac{14\!\cdots\!43}{26\!\cdots\!50}a^{4}-\frac{54\!\cdots\!01}{26\!\cdots\!50}a^{3}-\frac{43\!\cdots\!99}{53\!\cdots\!30}a^{2}+\frac{90\!\cdots\!33}{20\!\cdots\!50}a+\frac{19\!\cdots\!22}{41\!\cdots\!75}$, $\frac{21\!\cdots\!86}{70\!\cdots\!75}a^{15}-\frac{85\!\cdots\!33}{10\!\cdots\!50}a^{14}+\frac{49\!\cdots\!52}{70\!\cdots\!75}a^{13}+\frac{43\!\cdots\!03}{14\!\cdots\!50}a^{12}-\frac{39\!\cdots\!79}{14\!\cdots\!50}a^{11}+\frac{13\!\cdots\!31}{70\!\cdots\!75}a^{10}+\frac{10\!\cdots\!93}{70\!\cdots\!75}a^{9}-\frac{12\!\cdots\!63}{14\!\cdots\!50}a^{8}+\frac{17\!\cdots\!28}{70\!\cdots\!75}a^{7}+\frac{55\!\cdots\!69}{70\!\cdots\!75}a^{6}+\frac{94\!\cdots\!41}{70\!\cdots\!75}a^{5}-\frac{73\!\cdots\!91}{14\!\cdots\!50}a^{4}+\frac{11\!\cdots\!19}{14\!\cdots\!50}a^{3}+\frac{19\!\cdots\!53}{28\!\cdots\!70}a^{2}+\frac{13\!\cdots\!91}{14\!\cdots\!50}a+\frac{33\!\cdots\!06}{65\!\cdots\!75}$, $\frac{10\!\cdots\!17}{64\!\cdots\!55}a^{15}-\frac{84\!\cdots\!66}{64\!\cdots\!55}a^{14}+\frac{53\!\cdots\!17}{12\!\cdots\!10}a^{13}-\frac{73\!\cdots\!59}{12\!\cdots\!10}a^{12}-\frac{28\!\cdots\!39}{12\!\cdots\!10}a^{11}+\frac{29\!\cdots\!03}{12\!\cdots\!10}a^{10}-\frac{44\!\cdots\!22}{64\!\cdots\!55}a^{9}+\frac{24\!\cdots\!41}{12\!\cdots\!10}a^{8}+\frac{46\!\cdots\!36}{10\!\cdots\!55}a^{7}-\frac{16\!\cdots\!99}{12\!\cdots\!10}a^{6}+\frac{15\!\cdots\!57}{12\!\cdots\!10}a^{5}-\frac{20\!\cdots\!67}{10\!\cdots\!55}a^{4}-\frac{43\!\cdots\!27}{64\!\cdots\!55}a^{3}+\frac{12\!\cdots\!43}{25\!\cdots\!22}a^{2}+\frac{66\!\cdots\!33}{64\!\cdots\!55}a-\frac{14\!\cdots\!41}{64\!\cdots\!55}$, $\frac{39\!\cdots\!53}{29\!\cdots\!50}a^{15}-\frac{78\!\cdots\!77}{14\!\cdots\!25}a^{14}+\frac{26\!\cdots\!19}{29\!\cdots\!50}a^{13}+\frac{65\!\cdots\!27}{89\!\cdots\!50}a^{12}-\frac{64\!\cdots\!19}{44\!\cdots\!75}a^{11}+\frac{15\!\cdots\!17}{14\!\cdots\!25}a^{10}-\frac{18\!\cdots\!73}{29\!\cdots\!50}a^{9}-\frac{18\!\cdots\!62}{44\!\cdots\!75}a^{8}+\frac{11\!\cdots\!68}{68\!\cdots\!35}a^{7}+\frac{58\!\cdots\!66}{44\!\cdots\!75}a^{6}+\frac{83\!\cdots\!03}{68\!\cdots\!50}a^{5}-\frac{73\!\cdots\!71}{89\!\cdots\!50}a^{4}+\frac{76\!\cdots\!81}{89\!\cdots\!50}a^{3}+\frac{37\!\cdots\!01}{59\!\cdots\!70}a^{2}+\frac{66\!\cdots\!16}{44\!\cdots\!75}a-\frac{57\!\cdots\!31}{41\!\cdots\!75}$, $\frac{31\!\cdots\!97}{26\!\cdots\!50}a^{15}+\frac{18\!\cdots\!69}{44\!\cdots\!75}a^{14}-\frac{74\!\cdots\!92}{13\!\cdots\!25}a^{13}+\frac{55\!\cdots\!19}{29\!\cdots\!50}a^{12}-\frac{73\!\cdots\!41}{11\!\cdots\!25}a^{11}+\frac{51\!\cdots\!33}{13\!\cdots\!25}a^{10}+\frac{35\!\cdots\!04}{13\!\cdots\!25}a^{9}-\frac{29\!\cdots\!41}{26\!\cdots\!50}a^{8}-\frac{33\!\cdots\!29}{53\!\cdots\!30}a^{7}+\frac{56\!\cdots\!47}{89\!\cdots\!50}a^{6}+\frac{43\!\cdots\!73}{29\!\cdots\!50}a^{5}+\frac{11\!\cdots\!41}{13\!\cdots\!25}a^{4}+\frac{10\!\cdots\!13}{44\!\cdots\!50}a^{3}+\frac{35\!\cdots\!39}{89\!\cdots\!55}a^{2}+\frac{23\!\cdots\!11}{26\!\cdots\!50}a+\frac{21\!\cdots\!56}{37\!\cdots\!75}$, $\frac{33\!\cdots\!71}{89\!\cdots\!50}a^{15}-\frac{51\!\cdots\!37}{26\!\cdots\!50}a^{14}-\frac{28\!\cdots\!13}{26\!\cdots\!50}a^{13}+\frac{24\!\cdots\!41}{26\!\cdots\!50}a^{12}-\frac{59\!\cdots\!08}{13\!\cdots\!25}a^{11}+\frac{10\!\cdots\!17}{53\!\cdots\!30}a^{10}+\frac{15\!\cdots\!42}{26\!\cdots\!65}a^{9}-\frac{18\!\cdots\!79}{26\!\cdots\!50}a^{8}+\frac{80\!\cdots\!21}{26\!\cdots\!50}a^{7}+\frac{24\!\cdots\!41}{16\!\cdots\!25}a^{6}+\frac{39\!\cdots\!59}{13\!\cdots\!25}a^{5}+\frac{27\!\cdots\!41}{53\!\cdots\!30}a^{4}+\frac{54\!\cdots\!84}{13\!\cdots\!25}a^{3}+\frac{20\!\cdots\!48}{53\!\cdots\!33}a^{2}+\frac{43\!\cdots\!67}{13\!\cdots\!25}a-\frac{12\!\cdots\!29}{37\!\cdots\!75}$, $\frac{36\!\cdots\!67}{23\!\cdots\!50}a^{15}-\frac{93\!\cdots\!87}{26\!\cdots\!65}a^{14}-\frac{13\!\cdots\!08}{26\!\cdots\!65}a^{13}+\frac{80\!\cdots\!71}{26\!\cdots\!50}a^{12}-\frac{22\!\cdots\!71}{13\!\cdots\!25}a^{11}+\frac{12\!\cdots\!51}{13\!\cdots\!25}a^{10}+\frac{19\!\cdots\!48}{13\!\cdots\!25}a^{9}-\frac{35\!\cdots\!81}{53\!\cdots\!30}a^{8}+\frac{37\!\cdots\!79}{26\!\cdots\!50}a^{7}+\frac{14\!\cdots\!81}{26\!\cdots\!50}a^{6}+\frac{60\!\cdots\!81}{26\!\cdots\!50}a^{5}-\frac{20\!\cdots\!03}{13\!\cdots\!25}a^{4}+\frac{44\!\cdots\!83}{26\!\cdots\!50}a^{3}+\frac{45\!\cdots\!20}{53\!\cdots\!33}a^{2}+\frac{84\!\cdots\!93}{26\!\cdots\!50}a+\frac{37\!\cdots\!02}{12\!\cdots\!25}$ 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:  \( 8147753.36594 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8147753.36594 \cdot 16}{6\cdot\sqrt{1410629873249683485564270561}}\cr\approx \mathstrut & 1.40520282786 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524)
 
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 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524, 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 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524);
 
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 - x^15 - x^14 + 12*x^13 - 79*x^12 + 484*x^11 + 1411*x^10 - 1635*x^9 + 4582*x^8 + 35484*x^7 + 84259*x^6 + 65558*x^5 - 12802*x^4 + 8304*x^3 + 328913*x^2 + 621070*x + 515524);
 
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

$C_2^3:C_4$ (as 16T33):

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 $C_2^3:C_4$
Character table for $C_2^3:C_4$

Intermediate fields

\(\Q(\sqrt{-183}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-3}) \), 4.0.2042829.1 x2, 4.2.680943.1 x2, \(\Q(\sqrt{-3}, \sqrt{61})\), 8.0.615710703429.1 x2, 8.0.37558352909169.1 x2, 8.0.4173150323241.2

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]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.0.10093618089.2, 8.0.615710703429.1, 8.4.37558352909169.1, 8.0.37558352909169.1
Degree 16 siblings: 16.0.379099670317033992358041.4, 16.4.1410629873249683485564270561.4, 16.0.1410629873249683485564270561.4
Minimal sibling: 8.0.10093618089.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ R ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(61\) Copy content Toggle raw display 61.4.3.2$x^{4} + 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} + 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} + 61$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.3.2$x^{4} + 61$$4$$1$$3$$C_4$$[\ ]_{4}$