Properties

Label 22.2.158...744.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.580\times 10^{42}$
Root discriminant \(82.82\)
Ramified primes $2,7,11$
Class number $8$ (GRH)
Class group [8] (GRH)
Galois group $C_2^{10}.F_{11}$ (as 22T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 55*x^20 + 1199*x^18 + 13299*x^16 + 81114*x^14 + 279642*x^12 + 532422*x^10 + 491194*x^8 + 80201*x^6 - 190157*x^4 - 119053*x^2 - 18225)
 
gp: K = bnfinit(y^22 + 55*y^20 + 1199*y^18 + 13299*y^16 + 81114*y^14 + 279642*y^12 + 532422*y^10 + 491194*y^8 + 80201*y^6 - 190157*y^4 - 119053*y^2 - 18225, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 55*x^20 + 1199*x^18 + 13299*x^16 + 81114*x^14 + 279642*x^12 + 532422*x^10 + 491194*x^8 + 80201*x^6 - 190157*x^4 - 119053*x^2 - 18225);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 55*x^20 + 1199*x^18 + 13299*x^16 + 81114*x^14 + 279642*x^12 + 532422*x^10 + 491194*x^8 + 80201*x^6 - 190157*x^4 - 119053*x^2 - 18225)
 

\( x^{22} + 55 x^{20} + 1199 x^{18} + 13299 x^{16} + 81114 x^{14} + 279642 x^{12} + 532422 x^{10} + 491194 x^{8} + 80201 x^{6} - 190157 x^{4} - 119053 x^{2} + \cdots - 18225 \) Copy content Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1580153645858805893796455628954412194463744\) \(\medspace = 2^{36}\cdot 7^{10}\cdot 11^{22}\) 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:  \(82.82\)
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:  not computed
Ramified primes:   \(2\), \(7\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}-\frac{3}{11}a^{6}+\frac{2}{11}a^{4}-\frac{1}{11}a^{2}+\frac{5}{11}$, $\frac{1}{11}a^{9}-\frac{3}{11}a^{7}+\frac{2}{11}a^{5}-\frac{1}{11}a^{3}+\frac{5}{11}a$, $\frac{1}{11}a^{10}+\frac{4}{11}a^{6}+\frac{5}{11}a^{4}+\frac{2}{11}a^{2}+\frac{4}{11}$, $\frac{1}{22}a^{11}-\frac{1}{22}a^{10}-\frac{1}{22}a^{9}-\frac{1}{22}a^{8}-\frac{2}{11}a^{7}+\frac{5}{11}a^{6}-\frac{4}{11}a^{5}+\frac{2}{11}a^{4}+\frac{3}{22}a^{3}-\frac{1}{22}a^{2}-\frac{1}{22}a-\frac{9}{22}$, $\frac{1}{22}a^{12}-\frac{1}{22}a^{8}-\frac{1}{11}a^{6}+\frac{3}{22}a^{4}-\frac{1}{11}a^{2}-\frac{3}{22}$, $\frac{1}{22}a^{13}-\frac{1}{22}a^{9}-\frac{1}{11}a^{7}+\frac{3}{22}a^{5}-\frac{1}{11}a^{3}-\frac{3}{22}a$, $\frac{1}{22}a^{14}-\frac{1}{22}a^{10}-\frac{3}{22}a^{6}+\frac{1}{11}a^{4}-\frac{5}{22}a^{2}+\frac{5}{11}$, $\frac{1}{22}a^{15}-\frac{1}{22}a^{10}-\frac{1}{22}a^{9}-\frac{1}{22}a^{8}-\frac{7}{22}a^{7}+\frac{5}{11}a^{6}-\frac{3}{11}a^{5}+\frac{2}{11}a^{4}-\frac{1}{11}a^{3}-\frac{1}{22}a^{2}+\frac{9}{22}a-\frac{9}{22}$, $\frac{1}{242}a^{16}+\frac{5}{242}a^{14}+\frac{1}{121}a^{12}-\frac{3}{242}a^{10}-\frac{1}{121}a^{8}+\frac{21}{242}a^{6}+\frac{27}{121}a^{4}-\frac{87}{242}a^{2}+\frac{91}{242}$, $\frac{1}{242}a^{17}+\frac{5}{242}a^{15}+\frac{1}{121}a^{13}-\frac{3}{242}a^{11}-\frac{1}{121}a^{9}+\frac{21}{242}a^{7}+\frac{27}{121}a^{5}-\frac{87}{242}a^{3}+\frac{91}{242}a$, $\frac{1}{242}a^{18}-\frac{1}{242}a^{14}-\frac{1}{121}a^{12}-\frac{9}{242}a^{10}-\frac{1}{121}a^{8}-\frac{73}{242}a^{6}-\frac{41}{121}a^{4}-\frac{34}{121}a^{2}+\frac{53}{121}$, $\frac{1}{242}a^{19}-\frac{1}{242}a^{15}-\frac{1}{121}a^{13}+\frac{1}{121}a^{11}-\frac{1}{22}a^{10}+\frac{9}{242}a^{9}-\frac{1}{22}a^{8}+\frac{59}{242}a^{7}+\frac{5}{11}a^{6}+\frac{58}{121}a^{5}+\frac{2}{11}a^{4}-\frac{57}{242}a^{3}-\frac{1}{22}a^{2}-\frac{37}{242}a-\frac{9}{22}$, $\frac{1}{5108398327516}a^{20}-\frac{4086990131}{2554199163758}a^{18}+\frac{6167743089}{5108398327516}a^{16}+\frac{41300152261}{2554199163758}a^{14}+\frac{8127041165}{2554199163758}a^{12}+\frac{15721243864}{1277099581879}a^{10}-\frac{1}{22}a^{9}-\frac{113176767653}{2554199163758}a^{8}-\frac{4}{11}a^{7}+\frac{743420872933}{2554199163758}a^{6}-\frac{1}{11}a^{5}-\frac{1958117383759}{5108398327516}a^{4}-\frac{5}{11}a^{3}-\frac{985605559167}{2554199163758}a^{2}-\frac{5}{22}a-\frac{1210021761347}{5108398327516}$, $\frac{1}{689633774214660}a^{21}+\frac{51308556782}{34481688710733}a^{19}-\frac{732650196841}{689633774214660}a^{17}+\frac{57896978372}{57469481184555}a^{15}-\frac{81322738798}{57469481184555}a^{13}-\frac{2603527405843}{114938962369110}a^{11}-\frac{1}{22}a^{10}+\frac{809343952942}{19156493728185}a^{9}-\frac{23370731790284}{172408443553665}a^{7}-\frac{2}{11}a^{6}+\frac{4801814144641}{62693979474060}a^{5}+\frac{3}{11}a^{4}+\frac{32092329065699}{344816887107330}a^{3}+\frac{9}{22}a^{2}+\frac{7101628718467}{62693979474060}a-\frac{2}{11}$ 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

$C_{8}$, which has order $8$ (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:  $11$
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:   $\frac{20145560366}{172408443553665}a^{21}+\frac{451322338697}{68963377421466}a^{19}+\frac{4596468339343}{31346989737030}a^{17}+\frac{194999982699701}{114938962369110}a^{15}+\frac{12\!\cdots\!11}{114938962369110}a^{13}+\frac{47\!\cdots\!83}{114938962369110}a^{11}+\frac{33\!\cdots\!01}{38312987456370}a^{9}+\frac{29\!\cdots\!43}{344816887107330}a^{7}+\frac{32\!\cdots\!57}{344816887107330}a^{5}-\frac{69\!\cdots\!97}{172408443553665}a^{3}-\frac{30\!\cdots\!68}{172408443553665}a$, $\frac{15780158843}{344816887107330}a^{21}+\frac{192450086857}{68963377421466}a^{19}+\frac{23390933213377}{344816887107330}a^{17}+\frac{47830536579662}{57469481184555}a^{15}+\frac{617087346076819}{114938962369110}a^{13}+\frac{990796877456821}{57469481184555}a^{11}+\frac{901464748999829}{38312987456370}a^{9}+\frac{811532457787291}{172408443553665}a^{7}-\frac{22\!\cdots\!11}{172408443553665}a^{5}-\frac{18\!\cdots\!71}{344816887107330}a^{3}+\frac{179542921510478}{172408443553665}a$, $\frac{18539009987}{172408443553665}a^{21}+\frac{27337759903}{6269397947406}a^{19}+\frac{71785472848}{1424863169865}a^{17}-\frac{9368686148233}{114938962369110}a^{15}-\frac{587191019272333}{114938962369110}a^{13}-\frac{37\!\cdots\!69}{114938962369110}a^{11}-\frac{30\!\cdots\!53}{38312987456370}a^{9}-\frac{26\!\cdots\!09}{344816887107330}a^{7}+\frac{278759710099919}{344816887107330}a^{5}+\frac{60\!\cdots\!96}{172408443553665}a^{3}+\frac{27\!\cdots\!83}{344816887107330}a$, $\frac{2462346191}{1277099581879}a^{20}+\frac{134296548389}{1277099581879}a^{18}+\frac{5774848785117}{2554199163758}a^{16}+\frac{62522862444153}{2554199163758}a^{14}+\frac{182403678779331}{1277099581879}a^{12}+\frac{11\!\cdots\!37}{2554199163758}a^{10}+\frac{938446841006152}{1277099581879}a^{8}+\frac{105755858786319}{232199923978}a^{6}-\frac{185251424712533}{1277099581879}a^{4}-\frac{58552913438431}{232199923978}a^{2}-\frac{165071819201789}{2554199163758}$, $\frac{4574155358}{1277099581879}a^{20}+\frac{493274019943}{2554199163758}a^{18}+\frac{10437444860397}{2554199163758}a^{16}+\frac{55232551020327}{1277099581879}a^{14}+\frac{312118667859806}{1277099581879}a^{12}+\frac{949073864560498}{1277099581879}a^{10}+\frac{131217737474293}{116099961989}a^{8}+\frac{762492931186490}{1277099581879}a^{6}-\frac{391117818732171}{1277099581879}a^{4}-\frac{909531831674701}{2554199163758}a^{2}-\frac{165847536930431}{2554199163758}$, $\frac{39769560431}{31346989737030}a^{21}+\frac{2346566288434}{34481688710733}a^{19}+\frac{246593794830307}{172408443553665}a^{17}+\frac{862822129613584}{57469481184555}a^{15}+\frac{48\!\cdots\!44}{57469481184555}a^{13}+\frac{14\!\cdots\!57}{57469481184555}a^{11}+\frac{77\!\cdots\!74}{19156493728185}a^{9}+\frac{40\!\cdots\!57}{15673494868515}a^{7}-\frac{22\!\cdots\!69}{344816887107330}a^{5}-\frac{24\!\cdots\!41}{172408443553665}a^{3}-\frac{69\!\cdots\!69}{172408443553665}a$, $\frac{273263248597}{344816887107330}a^{21}+\frac{2758200846317}{68963377421466}a^{19}+\frac{24054128117923}{31346989737030}a^{17}+\frac{802567611490871}{114938962369110}a^{15}+\frac{17\!\cdots\!68}{57469481184555}a^{13}+\frac{77\!\cdots\!13}{114938962369110}a^{11}+\frac{107132008701193}{1741499429835}a^{9}+\frac{23\!\cdots\!63}{344816887107330}a^{7}-\frac{60\!\cdots\!63}{344816887107330}a^{5}-\frac{18\!\cdots\!62}{172408443553665}a^{3}-\frac{20\!\cdots\!31}{344816887107330}a$, $\frac{25925802319}{31346989737030}a^{21}+\frac{3026160688909}{68963377421466}a^{19}+\frac{156752752797353}{172408443553665}a^{17}+\frac{10\!\cdots\!17}{114938962369110}a^{15}+\frac{59\!\cdots\!57}{114938962369110}a^{13}+\frac{17\!\cdots\!71}{114938962369110}a^{11}+\frac{94\!\cdots\!27}{38312987456370}a^{9}+\frac{54\!\cdots\!11}{344816887107330}a^{7}-\frac{78\!\cdots\!78}{172408443553665}a^{5}-\frac{15\!\cdots\!74}{172408443553665}a^{3}-\frac{755658953707357}{31346989737030}a$, $\frac{544954387}{2554199163758}a^{20}+\frac{27618095893}{2554199163758}a^{18}+\frac{531126955633}{2554199163758}a^{16}+\frac{436070988753}{232199923978}a^{14}+\frac{10140926397035}{1277099581879}a^{12}+\frac{32502492862997}{2554199163758}a^{10}-\frac{9711172956048}{1277099581879}a^{8}-\frac{106469055792333}{2554199163758}a^{6}-\frac{563425148967}{21109083998}a^{4}+\frac{14668002582352}{1277099581879}a^{2}+\frac{25520913133951}{2554199163758}$, $\frac{52609464967}{344816887107330}a^{21}+\frac{566141848883}{68963377421466}a^{19}+\frac{29811811313794}{172408443553665}a^{17}+\frac{18926250382531}{10448996579010}a^{15}+\frac{11\!\cdots\!21}{114938962369110}a^{13}+\frac{32\!\cdots\!13}{114938962369110}a^{11}+\frac{13\!\cdots\!31}{38312987456370}a^{9}+\frac{76546988497703}{31346989737030}a^{7}-\frac{51\!\cdots\!14}{172408443553665}a^{5}-\frac{19\!\cdots\!97}{172408443553665}a^{3}+\frac{20\!\cdots\!89}{344816887107330}a$, $\frac{77\!\cdots\!99}{689633774214660}a^{21}-\frac{11\!\cdots\!24}{1277099581879}a^{20}+\frac{21\!\cdots\!15}{34481688710733}a^{19}-\frac{62\!\cdots\!79}{1277099581879}a^{18}+\frac{95\!\cdots\!51}{689633774214660}a^{17}-\frac{27\!\cdots\!27}{2554199163758}a^{16}+\frac{90\!\cdots\!83}{57469481184555}a^{15}-\frac{15\!\cdots\!35}{1277099581879}a^{14}+\frac{95\!\cdots\!01}{949908779910}a^{13}-\frac{20\!\cdots\!77}{2554199163758}a^{12}+\frac{43\!\cdots\!13}{114938962369110}a^{11}-\frac{76\!\cdots\!51}{2554199163758}a^{10}+\frac{31\!\cdots\!21}{38312987456370}a^{9}-\frac{16\!\cdots\!21}{2554199163758}a^{8}+\frac{18\!\cdots\!34}{172408443553665}a^{7}-\frac{10\!\cdots\!28}{1277099581879}a^{6}+\frac{52\!\cdots\!19}{689633774214660}a^{5}-\frac{15\!\cdots\!15}{2554199163758}a^{4}+\frac{82\!\cdots\!21}{31346989737030}a^{3}-\frac{53\!\cdots\!15}{2554199163758}a^{2}+\frac{22\!\cdots\!83}{689633774214660}a-\frac{32\!\cdots\!47}{1277099581879}$ 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:  \( 712861842202 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 712861842202 \cdot 8}{2\cdot\sqrt{1580153645858805893796455628954412194463744}}\cr\approx \mathstrut & 0.870110432986160 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 55*x^20 + 1199*x^18 + 13299*x^16 + 81114*x^14 + 279642*x^12 + 532422*x^10 + 491194*x^8 + 80201*x^6 - 190157*x^4 - 119053*x^2 - 18225)
 
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^22 + 55*x^20 + 1199*x^18 + 13299*x^16 + 81114*x^14 + 279642*x^12 + 532422*x^10 + 491194*x^8 + 80201*x^6 - 190157*x^4 - 119053*x^2 - 18225, 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^22 + 55*x^20 + 1199*x^18 + 13299*x^16 + 81114*x^14 + 279642*x^12 + 532422*x^10 + 491194*x^8 + 80201*x^6 - 190157*x^4 - 119053*x^2 - 18225);
 
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^22 + 55*x^20 + 1199*x^18 + 13299*x^16 + 81114*x^14 + 279642*x^12 + 532422*x^10 + 491194*x^8 + 80201*x^6 - 190157*x^4 - 119053*x^2 - 18225);
 
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

$C_2^{10}.F_{11}$ (as 22T34):

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 112640
The 44 conjugacy class representatives for $C_2^{10}.F_{11}$
Character table for $C_2^{10}.F_{11}$ is not computed

Intermediate fields

11.11.4910318845910094848.1

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 22 sibling: data not computed
Degree 44 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ R R ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.11.0.1}{11} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 Deg $22$$22$$1$$36$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.10.5.1$x^{10} + 2401 x^{2} - 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} + 2401 x^{2} - 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(11\) Copy content Toggle raw display 11.11.11.6$x^{11} + 11 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$
11.11.11.6$x^{11} + 11 x + 11$$11$$1$$11$$F_{11}$$[11/10]_{10}$