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Properties

Label 20.16.206...256.1
Degree $20$
Signature $[16, 2]$
Discriminant $2.065\times 10^{29}$
Root discriminant \(29.22\)
Ramified primes $2,61,397$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^9.S_5$ (as 20T676)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^18 + 9*x^16 + 122*x^14 - 437*x^12 + 515*x^10 - 80*x^8 - 271*x^6 + 182*x^4 - 34*x^2 + 1)
 
gp: K = bnfinit(y^20 - 9*y^18 + 9*y^16 + 122*y^14 - 437*y^12 + 515*y^10 - 80*y^8 - 271*y^6 + 182*y^4 - 34*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 9*x^18 + 9*x^16 + 122*x^14 - 437*x^12 + 515*x^10 - 80*x^8 - 271*x^6 + 182*x^4 - 34*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 9*x^18 + 9*x^16 + 122*x^14 - 437*x^12 + 515*x^10 - 80*x^8 - 271*x^6 + 182*x^4 - 34*x^2 + 1)
 

\( x^{20} - 9x^{18} + 9x^{16} + 122x^{14} - 437x^{12} + 515x^{10} - 80x^{8} - 271x^{6} + 182x^{4} - 34x^{2} + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[16, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(206548481879697235603429344256\) \(\medspace = 2^{10}\cdot 61^{6}\cdot 397^{6}\) 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:  \(29.22\)
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:  $2^{15/8}61^{1/2}397^{1/2}\approx 570.8097895194048$
Ramified primes:   \(2\), \(61\), \(397\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{34}a^{16}+\frac{1}{34}a^{14}-\frac{1}{2}a^{13}+\frac{11}{34}a^{12}+\frac{3}{34}a^{10}-\frac{1}{17}a^{8}-\frac{5}{34}a^{6}-\frac{1}{2}a^{5}-\frac{6}{17}a^{4}-\frac{1}{2}a^{3}+\frac{3}{17}a^{2}-\frac{1}{2}a-\frac{1}{17}$, $\frac{1}{34}a^{17}+\frac{1}{34}a^{15}-\frac{1}{2}a^{14}+\frac{11}{34}a^{13}+\frac{3}{34}a^{11}-\frac{1}{17}a^{9}-\frac{5}{34}a^{7}-\frac{1}{2}a^{6}-\frac{6}{17}a^{5}-\frac{1}{2}a^{4}+\frac{3}{17}a^{3}-\frac{1}{2}a^{2}-\frac{1}{17}a$, $\frac{1}{169354}a^{18}-\frac{1209}{84677}a^{16}-\frac{8871}{84677}a^{14}-\frac{21285}{169354}a^{12}-\frac{1}{2}a^{11}+\frac{2679}{9962}a^{10}-\frac{1}{2}a^{9}-\frac{55857}{169354}a^{8}-\frac{1}{2}a^{7}-\frac{76997}{169354}a^{6}-\frac{1}{2}a^{5}-\frac{41805}{169354}a^{4}-\frac{57203}{169354}a^{2}-\frac{1}{2}a-\frac{52163}{169354}$, $\frac{1}{169354}a^{19}-\frac{1209}{84677}a^{17}-\frac{8871}{84677}a^{15}-\frac{21285}{169354}a^{13}-\frac{1}{2}a^{12}+\frac{2679}{9962}a^{11}-\frac{1}{2}a^{10}-\frac{55857}{169354}a^{9}-\frac{1}{2}a^{8}-\frac{76997}{169354}a^{7}-\frac{1}{2}a^{6}-\frac{41805}{169354}a^{5}-\frac{57203}{169354}a^{3}-\frac{1}{2}a^{2}-\frac{52163}{169354}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:   $a$, $\frac{12553}{84677}a^{18}-\frac{108522}{84677}a^{16}+\frac{84804}{84677}a^{14}+\frac{1484535}{84677}a^{12}-\frac{293467}{4981}a^{10}+\frac{5934820}{84677}a^{8}-\frac{1469610}{84677}a^{6}-\frac{3008453}{84677}a^{4}+\frac{2536992}{84677}a^{2}-\frac{363492}{84677}$, $\frac{12553}{84677}a^{18}-\frac{108522}{84677}a^{16}+\frac{84804}{84677}a^{14}+\frac{1484535}{84677}a^{12}-\frac{293467}{4981}a^{10}+\frac{5934820}{84677}a^{8}-\frac{1469610}{84677}a^{6}-\frac{3008453}{84677}a^{4}+\frac{2536992}{84677}a^{2}-\frac{448169}{84677}$, $\frac{41664}{84677}a^{18}-\frac{371421}{84677}a^{16}+\frac{311439}{84677}a^{14}+\frac{5328670}{84677}a^{12}-\frac{1032147}{4981}a^{10}+\frac{16319509}{84677}a^{8}+\frac{4069557}{84677}a^{6}-\frac{11408949}{84677}a^{4}+\frac{2986707}{84677}a^{2}+\frac{25555}{84677}$, $\frac{91469}{169354}a^{19}+\frac{33753}{169354}a^{18}-\frac{747849}{169354}a^{17}-\frac{260081}{169354}a^{16}+\frac{127850}{84677}a^{15}-\frac{14953}{84677}a^{14}+\frac{5504417}{84677}a^{13}+\frac{2033213}{84677}a^{12}-\frac{913047}{4981}a^{11}-\frac{562175}{9962}a^{10}+\frac{27302963}{169354}a^{9}+\frac{2640953}{84677}a^{8}+\frac{3084767}{169354}a^{7}+\frac{2222445}{84677}a^{6}-\frac{8663277}{84677}a^{5}-\frac{4489461}{169354}a^{4}+\frac{4166709}{84677}a^{3}+\frac{252645}{84677}a^{2}-\frac{953899}{169354}a+\frac{35485}{84677}$, $\frac{117433}{84677}a^{19}-\frac{135339}{169354}a^{18}-\frac{1002308}{84677}a^{17}+\frac{546911}{84677}a^{16}+\frac{618867}{84677}a^{15}-\frac{117629}{84677}a^{14}+\frac{14399105}{84677}a^{13}-\frac{16567807}{169354}a^{12}-\frac{5252963}{9962}a^{11}+\frac{2587445}{9962}a^{10}+\frac{86203995}{169354}a^{9}-\frac{31871729}{169354}a^{8}+\frac{5545203}{169354}a^{7}-\frac{12963869}{169354}a^{6}-\frac{52749815}{169354}a^{5}+\frac{22522811}{169354}a^{4}+\frac{12538327}{84677}a^{3}-\frac{6706555}{169354}a^{2}-\frac{2881747}{169354}a+\frac{465627}{169354}$, $\frac{69465}{169354}a^{19}+\frac{16601}{169354}a^{18}-\frac{689447}{169354}a^{17}-\frac{79035}{169354}a^{16}+\frac{1081677}{169354}a^{15}-\frac{178541}{84677}a^{14}+\frac{8803923}{169354}a^{13}+\frac{1723281}{169354}a^{12}-\frac{2227843}{9962}a^{11}+\frac{55279}{9962}a^{10}+\frac{24709669}{84677}a^{9}-\frac{3643188}{84677}a^{8}-\frac{7878747}{169354}a^{7}+\frac{1908725}{84677}a^{6}-\frac{27476749}{169354}a^{5}+\frac{1680050}{84677}a^{4}+\frac{8179451}{84677}a^{3}-\frac{2200955}{169354}a^{2}-\frac{2116555}{169354}a+\frac{98469}{169354}$, $\frac{18531}{84677}a^{19}-\frac{10378}{84677}a^{18}-\frac{188160}{84677}a^{17}+\frac{243521}{169354}a^{16}+\frac{630355}{169354}a^{15}-\frac{585117}{169354}a^{14}+\frac{2393973}{84677}a^{13}-\frac{1426224}{84677}a^{12}-\frac{1244183}{9962}a^{11}+\frac{921701}{9962}a^{10}+\frac{27381757}{169354}a^{9}-\frac{11122426}{84677}a^{8}-\frac{1607915}{84677}a^{7}+\frac{2929691}{169354}a^{6}-\frac{7963125}{84677}a^{5}+\frac{13388365}{169354}a^{4}+\frac{8239037}{169354}a^{3}-\frac{6296543}{169354}a^{2}-\frac{662011}{169354}a+\frac{69897}{169354}$, $\frac{2478}{84677}a^{19}-\frac{21381}{84677}a^{18}-\frac{4303}{169354}a^{17}+\frac{296797}{169354}a^{16}-\frac{248809}{169354}a^{15}+\frac{133191}{84677}a^{14}+\frac{228705}{84677}a^{13}-\frac{4955017}{169354}a^{12}+\frac{149259}{9962}a^{11}+\frac{487931}{9962}a^{10}-\frac{3478356}{84677}a^{9}-\frac{125435}{84677}a^{8}+\frac{3399301}{169354}a^{7}-\frac{2941224}{84677}a^{6}+\frac{3266651}{169354}a^{5}+\frac{1493753}{169354}a^{4}-\frac{3062787}{169354}a^{3}+\frac{598097}{169354}a^{2}+\frac{596773}{169354}a-\frac{18531}{84677}$, $\frac{18531}{84677}a^{19}+\frac{10378}{84677}a^{18}-\frac{188160}{84677}a^{17}-\frac{243521}{169354}a^{16}+\frac{630355}{169354}a^{15}+\frac{585117}{169354}a^{14}+\frac{2393973}{84677}a^{13}+\frac{1426224}{84677}a^{12}-\frac{1244183}{9962}a^{11}-\frac{921701}{9962}a^{10}+\frac{27381757}{169354}a^{9}+\frac{11122426}{84677}a^{8}-\frac{1607915}{84677}a^{7}-\frac{2929691}{169354}a^{6}-\frac{7963125}{84677}a^{5}-\frac{13388365}{169354}a^{4}+\frac{8239037}{169354}a^{3}+\frac{6296543}{169354}a^{2}-\frac{662011}{169354}a-\frac{69897}{169354}$, $\frac{1274}{4981}a^{19}-\frac{11943}{4981}a^{17}+\frac{15706}{4981}a^{15}+\frac{152127}{4981}a^{13}-\frac{618652}{4981}a^{11}+\frac{852894}{4981}a^{9}-\frac{268803}{4981}a^{7}-\frac{399733}{4981}a^{5}+\frac{340855}{4981}a^{3}-\frac{69480}{4981}a$, $\frac{45259}{84677}a^{19}-\frac{437039}{84677}a^{17}+\frac{619476}{84677}a^{15}+\frac{5670906}{84677}a^{13}-\frac{1369881}{4981}a^{11}+\frac{29681651}{84677}a^{9}-\frac{4428112}{84677}a^{7}-\frac{16611356}{84677}a^{5}+\frac{9736743}{84677}a^{3}-\frac{1275783}{84677}a$, $\frac{206309}{169354}a^{19}-\frac{119033}{169354}a^{18}-\frac{1766305}{169354}a^{17}+\frac{961023}{169354}a^{16}+\frac{1134069}{169354}a^{15}-\frac{95010}{84677}a^{14}+\frac{25256053}{169354}a^{13}-\frac{14632309}{169354}a^{12}-\frac{4650993}{9962}a^{11}+\frac{2264963}{9962}a^{10}+\frac{38989127}{84677}a^{9}-\frac{13367412}{84677}a^{8}+\frac{2004363}{169354}a^{7}-\frac{6625004}{84677}a^{6}-\frac{46200571}{169354}a^{5}+\frac{9992463}{84677}a^{4}+\frac{12056278}{84677}a^{3}-\frac{4764061}{169354}a^{2}-\frac{3214543}{169354}a-\frac{126487}{169354}$, $\frac{206309}{169354}a^{19}+\frac{119033}{169354}a^{18}-\frac{1766305}{169354}a^{17}-\frac{961023}{169354}a^{16}+\frac{1134069}{169354}a^{15}+\frac{95010}{84677}a^{14}+\frac{25256053}{169354}a^{13}+\frac{14632309}{169354}a^{12}-\frac{4650993}{9962}a^{11}-\frac{2264963}{9962}a^{10}+\frac{38989127}{84677}a^{9}+\frac{13367412}{84677}a^{8}+\frac{2004363}{169354}a^{7}+\frac{6625004}{84677}a^{6}-\frac{46200571}{169354}a^{5}-\frac{9992463}{84677}a^{4}+\frac{12056278}{84677}a^{3}+\frac{4764061}{169354}a^{2}-\frac{3214543}{169354}a+\frac{126487}{169354}$, $\frac{210055}{169354}a^{19}+\frac{75163}{169354}a^{18}-\frac{866904}{84677}a^{17}-\frac{307525}{84677}a^{16}+\frac{332903}{84677}a^{15}+\frac{105231}{84677}a^{14}+\frac{25616923}{169354}a^{13}+\frac{4536668}{84677}a^{12}-\frac{2144134}{4981}a^{11}-\frac{751806}{4981}a^{10}+\frac{30955537}{84677}a^{9}+\frac{11083029}{84677}a^{8}+\frac{6294006}{84677}a^{7}+\frac{1685453}{84677}a^{6}-\frac{20417462}{84677}a^{5}-\frac{14788669}{169354}a^{4}+\frac{15859221}{169354}a^{3}+\frac{3195462}{84677}a^{2}-\frac{732090}{84677}a-\frac{361785}{169354}$, $\frac{28557}{169354}a^{19}+\frac{8153}{84677}a^{18}-\frac{238311}{169354}a^{17}-\frac{132799}{169354}a^{16}+\frac{103665}{169354}a^{15}+\frac{22619}{84677}a^{14}+\frac{3542157}{169354}a^{13}+\frac{967749}{84677}a^{12}-\frac{17705}{293}a^{11}-\frac{161241}{4981}a^{10}+\frac{8225741}{169354}a^{9}+\frac{5136905}{169354}a^{8}+\frac{1770420}{84677}a^{7}-\frac{286139}{169354}a^{6}-\frac{3274622}{84677}a^{5}-\frac{2537885}{169354}a^{4}+\frac{482049}{84677}a^{3}+\frac{971247}{84677}a^{2}+\frac{166398}{84677}a-\frac{211380}{84677}$, $\frac{3683}{4981}a^{19}-\frac{26509}{169354}a^{18}-\frac{29645}{4981}a^{17}+\frac{106228}{84677}a^{16}+\frac{6641}{4981}a^{15}-\frac{6614}{84677}a^{14}+\frac{443390}{4981}a^{13}-\frac{3360809}{169354}a^{12}-\frac{2388269}{9962}a^{11}+\frac{482593}{9962}a^{10}+\frac{1916757}{9962}a^{9}-\frac{3764255}{169354}a^{8}+\frac{442921}{9962}a^{7}-\frac{5161023}{169354}a^{6}-\frac{1283295}{9962}a^{5}+\frac{3821425}{169354}a^{4}+\frac{259962}{4981}a^{3}+\frac{98231}{169354}a^{2}-\frac{51937}{9962}a-\frac{76101}{169354}$ 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:  \( 58952585.6348 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{2}\cdot 58952585.6348 \cdot 1}{2\cdot\sqrt{206548481879697235603429344256}}\cr\approx \mathstrut & 0.167803697390 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^18 + 9*x^16 + 122*x^14 - 437*x^12 + 515*x^10 - 80*x^8 - 271*x^6 + 182*x^4 - 34*x^2 + 1)
 
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^20 - 9*x^18 + 9*x^16 + 122*x^14 - 437*x^12 + 515*x^10 - 80*x^8 - 271*x^6 + 182*x^4 - 34*x^2 + 1, 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^20 - 9*x^18 + 9*x^16 + 122*x^14 - 437*x^12 + 515*x^10 - 80*x^8 - 271*x^6 + 182*x^4 - 34*x^2 + 1);
 
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^20 - 9*x^18 + 9*x^16 + 122*x^14 - 437*x^12 + 515*x^10 - 80*x^8 - 271*x^6 + 182*x^4 - 34*x^2 + 1);
 
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

$C_2^9.S_5$ (as 20T676):

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 non-solvable group of order 61440
The 74 conjugacy class representatives for $C_2^9.S_5$
Character table for $C_2^9.S_5$

Intermediate fields

10.10.14202376626313.1

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

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.16.5001984585680627954608248429847552.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}$ ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} + 4 x^{9} + 6 x^{8} + 80 x^{7} + 616 x^{6} + 2352 x^{5} + 6000 x^{4} + 11136 x^{3} + 13776 x^{2} + 9472 x + 2784$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
\(61\) Copy content Toggle raw display 61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.1$x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.8.4.1$x^{8} + 9516 x^{7} + 33958096 x^{6} + 53858928086 x^{5} + 32035798457059 x^{4} + 3448323535094 x^{3} + 98379480266246 x^{2} + 1286299880976982 x + 96975348777163$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(397\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $8$$2$$4$$4$