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Properties

Label 22.0.189...904.1
Degree $22$
Signature $[0, 11]$
Discriminant $-1.898\times 10^{48}$
Root discriminant \(156.48\)
Ramified primes $2,11$
Class number $14411$ (GRH)
Class group [14411] (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049)
 
gp: K = bnfinit(y^22 + 110*y^20 + 4675*y^18 + 101783*y^16 + 1279201*y^14 + 9871136*y^12 + 48063862*y^10 + 147421505*y^8 + 274663048*y^6 + 281918538*y^4 + 121207185*y^2 + 59049, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049)
 

\( x^{22} + 110 x^{20} + 4675 x^{18} + 101783 x^{16} + 1279201 x^{14} + 9871136 x^{12} + 48063862 x^{10} + \cdots + 59049 \) 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:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1898310766666226673632489495745922930975549947904\) \(\medspace = -\,2^{22}\cdot 11^{40}\) 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:  \(156.48\)
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:  $2\cdot 11^{20/11}\approx 156.48446931720272$
Ramified primes:   \(2\), \(11\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $22$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(484=2^{2}\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{484}(1,·)$, $\chi_{484}(67,·)$, $\chi_{484}(133,·)$, $\chi_{484}(199,·)$, $\chi_{484}(265,·)$, $\chi_{484}(331,·)$, $\chi_{484}(397,·)$, $\chi_{484}(463,·)$, $\chi_{484}(23,·)$, $\chi_{484}(89,·)$, $\chi_{484}(155,·)$, $\chi_{484}(221,·)$, $\chi_{484}(287,·)$, $\chi_{484}(353,·)$, $\chi_{484}(419,·)$, $\chi_{484}(45,·)$, $\chi_{484}(111,·)$, $\chi_{484}(177,·)$, $\chi_{484}(243,·)$, $\chi_{484}(309,·)$, $\chi_{484}(375,·)$, $\chi_{484}(441,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1024}$

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

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{9}a^{6}-\frac{1}{9}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{5}+\frac{1}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{2}$, $\frac{1}{81}a^{9}+\frac{1}{27}a^{7}+\frac{1}{27}a^{5}+\frac{10}{81}a^{3}+\frac{1}{9}a$, $\frac{1}{81}a^{10}+\frac{1}{27}a^{8}+\frac{1}{27}a^{6}+\frac{10}{81}a^{4}+\frac{1}{9}a^{2}$, $\frac{1}{81}a^{11}+\frac{1}{27}a^{7}-\frac{8}{81}a^{5}-\frac{4}{27}a^{3}$, $\frac{1}{243}a^{12}+\frac{1}{243}a^{10}-\frac{1}{81}a^{8}+\frac{4}{243}a^{6}-\frac{11}{243}a^{4}-\frac{2}{27}a^{2}$, $\frac{1}{243}a^{13}+\frac{1}{243}a^{11}+\frac{13}{243}a^{7}-\frac{2}{243}a^{5}+\frac{4}{81}a^{3}+\frac{1}{9}a$, $\frac{1}{729}a^{14}-\frac{1}{729}a^{12}+\frac{4}{729}a^{10}-\frac{17}{729}a^{8}+\frac{8}{729}a^{6}+\frac{13}{729}a^{4}-\frac{2}{81}a^{2}$, $\frac{1}{729}a^{15}-\frac{1}{729}a^{13}+\frac{4}{729}a^{11}+\frac{1}{729}a^{9}-\frac{19}{729}a^{7}-\frac{95}{729}a^{5}+\frac{1}{9}a^{3}+\frac{2}{9}a$, $\frac{1}{6561}a^{16}-\frac{1}{2187}a^{14}-\frac{4}{2187}a^{12}+\frac{38}{6561}a^{10}-\frac{22}{2187}a^{8}+\frac{56}{2187}a^{6}+\frac{721}{6561}a^{4}+\frac{130}{729}a^{2}+\frac{1}{9}$, $\frac{1}{6561}a^{17}-\frac{1}{2187}a^{15}-\frac{4}{2187}a^{13}+\frac{38}{6561}a^{11}+\frac{5}{2187}a^{9}-\frac{106}{2187}a^{7}-\frac{494}{6561}a^{5}-\frac{104}{729}a^{3}-\frac{1}{9}a$, $\frac{1}{19683}a^{18}+\frac{1}{19683}a^{16}+\frac{1}{6561}a^{14}-\frac{37}{19683}a^{12}-\frac{49}{19683}a^{10}+\frac{301}{6561}a^{8}+\frac{880}{19683}a^{6}+\frac{1975}{19683}a^{4}-\frac{182}{2187}a^{2}-\frac{5}{27}$, $\frac{1}{59049}a^{19}-\frac{2}{59049}a^{17}-\frac{5}{19683}a^{15}+\frac{26}{59049}a^{13}-\frac{28}{59049}a^{11}+\frac{34}{19683}a^{9}+\frac{889}{59049}a^{7}+\frac{8452}{59049}a^{5}+\frac{211}{6561}a^{3}-\frac{8}{81}a$, $\frac{1}{26\!\cdots\!03}a^{20}+\frac{138754934398}{26\!\cdots\!03}a^{18}-\frac{442186375951}{88\!\cdots\!01}a^{16}-\frac{11464582516660}{26\!\cdots\!03}a^{14}+\frac{9487050996848}{26\!\cdots\!03}a^{12}-\frac{5481345992918}{977860961863389}a^{10}+\frac{623585795482060}{26\!\cdots\!03}a^{8}-\frac{10\!\cdots\!13}{26\!\cdots\!03}a^{6}+\frac{161857180309537}{88\!\cdots\!01}a^{4}+\frac{389337864680260}{977860961863389}a^{2}-\frac{25077976301}{12072357553869}$, $\frac{1}{79\!\cdots\!09}a^{21}+\frac{585879288245}{79\!\cdots\!09}a^{19}+\frac{5827430533699}{79\!\cdots\!09}a^{17}-\frac{6099090270496}{79\!\cdots\!09}a^{15}-\frac{111683648895689}{79\!\cdots\!09}a^{13}-\frac{35768128993189}{79\!\cdots\!09}a^{11}-\frac{477681488043101}{79\!\cdots\!09}a^{9}+\frac{53693859497165}{79\!\cdots\!09}a^{7}+\frac{43\!\cdots\!15}{79\!\cdots\!09}a^{5}-\frac{262784338269620}{88\!\cdots\!01}a^{3}-\frac{19748705498171}{108651217984821}a$ 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:  $3$

Class group and class number

$C_{14411}$, which has order $14411$ (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:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{4825964}{6419218568031} a^{21} + \frac{505973402}{6419218568031} a^{19} + \frac{19919812957}{6419218568031} a^{17} + \frac{385002011911}{6419218568031} a^{15} + \frac{4046996472904}{6419218568031} a^{13} + \frac{23912359716185}{6419218568031} a^{11} + \frac{76235839889033}{6419218568031} a^{9} + \frac{100272578425481}{6419218568031} a^{7} - \frac{77790813933239}{6419218568031} a^{5} - \frac{41046756604106}{713246507559} a^{3} - \frac{132988481978}{2935170813} a \)  (order $4$) 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:   $\frac{3213235804019}{26\!\cdots\!03}a^{20}+\frac{342493955012510}{26\!\cdots\!03}a^{18}+\frac{46\!\cdots\!23}{88\!\cdots\!01}a^{16}+\frac{27\!\cdots\!33}{26\!\cdots\!03}a^{14}+\frac{31\!\cdots\!51}{26\!\cdots\!03}a^{12}+\frac{69\!\cdots\!43}{88\!\cdots\!01}a^{10}+\frac{83\!\cdots\!73}{26\!\cdots\!03}a^{8}+\frac{19\!\cdots\!68}{26\!\cdots\!03}a^{6}+\frac{25\!\cdots\!67}{29\!\cdots\!67}a^{4}+\frac{47\!\cdots\!20}{108651217984821}a^{2}+\frac{1442502203678}{1341373061541}$, $\frac{5660300017022}{26\!\cdots\!03}a^{20}+\frac{603784439898341}{26\!\cdots\!03}a^{18}+\frac{81\!\cdots\!06}{88\!\cdots\!01}a^{16}+\frac{49\!\cdots\!76}{26\!\cdots\!03}a^{14}+\frac{55\!\cdots\!43}{26\!\cdots\!03}a^{12}+\frac{41\!\cdots\!35}{29\!\cdots\!67}a^{10}+\frac{14\!\cdots\!23}{26\!\cdots\!03}a^{8}+\frac{34\!\cdots\!35}{26\!\cdots\!03}a^{6}+\frac{13\!\cdots\!71}{88\!\cdots\!01}a^{4}+\frac{75\!\cdots\!42}{977860961863389}a^{2}+\frac{28602104097260}{12072357553869}$, $\frac{39531992434}{26\!\cdots\!03}a^{20}+\frac{4552247909278}{26\!\cdots\!03}a^{18}+\frac{7601111176616}{977860961863389}a^{16}+\frac{47\!\cdots\!39}{26\!\cdots\!03}a^{14}+\frac{63\!\cdots\!94}{26\!\cdots\!03}a^{12}+\frac{16\!\cdots\!43}{88\!\cdots\!01}a^{10}+\frac{23\!\cdots\!54}{26\!\cdots\!03}a^{8}+\frac{63\!\cdots\!38}{26\!\cdots\!03}a^{6}+\frac{29\!\cdots\!07}{88\!\cdots\!01}a^{4}+\frac{18\!\cdots\!77}{977860961863389}a^{2}+\frac{4940021374070}{12072357553869}$, $\frac{14887371542023}{26\!\cdots\!03}a^{20}+\frac{15\!\cdots\!32}{26\!\cdots\!03}a^{18}+\frac{21\!\cdots\!38}{88\!\cdots\!01}a^{16}+\frac{13\!\cdots\!38}{26\!\cdots\!03}a^{14}+\frac{14\!\cdots\!70}{26\!\cdots\!03}a^{12}+\frac{12\!\cdots\!27}{325953653954463}a^{10}+\frac{39\!\cdots\!37}{26\!\cdots\!03}a^{8}+\frac{90\!\cdots\!69}{26\!\cdots\!03}a^{6}+\frac{36\!\cdots\!78}{88\!\cdots\!01}a^{4}+\frac{20\!\cdots\!18}{977860961863389}a^{2}+\frac{119710897645543}{12072357553869}$, $\frac{2537028351073}{88\!\cdots\!01}a^{20}+\frac{270817562025460}{88\!\cdots\!01}a^{18}+\frac{406640171996329}{325953653954463}a^{16}+\frac{22\!\cdots\!37}{88\!\cdots\!01}a^{14}+\frac{25\!\cdots\!70}{88\!\cdots\!01}a^{12}+\frac{56\!\cdots\!54}{29\!\cdots\!67}a^{10}+\frac{67\!\cdots\!57}{88\!\cdots\!01}a^{8}+\frac{15\!\cdots\!74}{88\!\cdots\!01}a^{6}+\frac{63\!\cdots\!66}{29\!\cdots\!67}a^{4}+\frac{34\!\cdots\!41}{325953653954463}a^{2}+\frac{41027964312374}{4024119184623}$, $\frac{924539413048}{29\!\cdots\!67}a^{20}+\frac{98678714044666}{29\!\cdots\!67}a^{18}+\frac{39\!\cdots\!17}{29\!\cdots\!67}a^{16}+\frac{81\!\cdots\!64}{29\!\cdots\!67}a^{14}+\frac{91\!\cdots\!17}{29\!\cdots\!67}a^{12}+\frac{61\!\cdots\!20}{29\!\cdots\!67}a^{10}+\frac{24\!\cdots\!84}{29\!\cdots\!67}a^{8}+\frac{56\!\cdots\!15}{29\!\cdots\!67}a^{6}+\frac{69\!\cdots\!51}{29\!\cdots\!67}a^{4}+\frac{38\!\cdots\!27}{325953653954463}a^{2}+\frac{49360658371892}{4024119184623}$, $\frac{1157218833626}{26\!\cdots\!03}a^{20}+\frac{123636634581062}{26\!\cdots\!03}a^{18}+\frac{16\!\cdots\!25}{88\!\cdots\!01}a^{16}+\frac{10\!\cdots\!30}{26\!\cdots\!03}a^{14}+\frac{11\!\cdots\!46}{26\!\cdots\!03}a^{12}+\frac{25\!\cdots\!43}{88\!\cdots\!01}a^{10}+\frac{31\!\cdots\!39}{26\!\cdots\!03}a^{8}+\frac{72\!\cdots\!19}{26\!\cdots\!03}a^{6}+\frac{10\!\cdots\!84}{29\!\cdots\!67}a^{4}+\frac{56\!\cdots\!74}{325953653954463}a^{2}-\frac{6943689641014}{4024119184623}$, $\frac{453176620148}{88\!\cdots\!01}a^{20}+\frac{48625520889263}{88\!\cdots\!01}a^{18}+\frac{662159067326713}{29\!\cdots\!67}a^{16}+\frac{40\!\cdots\!77}{88\!\cdots\!01}a^{14}+\frac{46\!\cdots\!97}{88\!\cdots\!01}a^{12}+\frac{11\!\cdots\!37}{325953653954463}a^{10}+\frac{12\!\cdots\!30}{88\!\cdots\!01}a^{8}+\frac{29\!\cdots\!90}{88\!\cdots\!01}a^{6}+\frac{11\!\cdots\!60}{29\!\cdots\!67}a^{4}+\frac{62\!\cdots\!20}{325953653954463}a^{2}-\frac{89013297270859}{4024119184623}$, $\frac{8430320958326}{26\!\cdots\!03}a^{20}+\frac{901575211788992}{26\!\cdots\!03}a^{18}+\frac{12\!\cdots\!89}{88\!\cdots\!01}a^{16}+\frac{74\!\cdots\!56}{26\!\cdots\!03}a^{14}+\frac{84\!\cdots\!75}{26\!\cdots\!03}a^{12}+\frac{19\!\cdots\!25}{88\!\cdots\!01}a^{10}+\frac{22\!\cdots\!53}{26\!\cdots\!03}a^{8}+\frac{53\!\cdots\!52}{26\!\cdots\!03}a^{6}+\frac{72\!\cdots\!48}{29\!\cdots\!67}a^{4}+\frac{13\!\cdots\!97}{108651217984821}a^{2}+\frac{8086350631442}{1341373061541}$, $\frac{9515555908064}{26\!\cdots\!03}a^{20}+\frac{10\!\cdots\!48}{26\!\cdots\!03}a^{18}+\frac{13\!\cdots\!48}{88\!\cdots\!01}a^{16}+\frac{83\!\cdots\!75}{26\!\cdots\!03}a^{14}+\frac{94\!\cdots\!51}{26\!\cdots\!03}a^{12}+\frac{20\!\cdots\!46}{88\!\cdots\!01}a^{10}+\frac{25\!\cdots\!79}{26\!\cdots\!03}a^{8}+\frac{57\!\cdots\!74}{26\!\cdots\!03}a^{6}+\frac{78\!\cdots\!63}{29\!\cdots\!67}a^{4}+\frac{43\!\cdots\!26}{325953653954463}a^{2}+\frac{25987769018384}{4024119184623}$ 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:  \( 285114946276.13544 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 285114946276.13544 \cdot 14411}{4\cdot\sqrt{1898310766666226673632489495745922930975549947904}}\cr\approx \mathstrut & 0.449209578116448 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^22 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049)
 
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 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049, 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 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049);
 
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 + 110*x^20 + 4675*x^18 + 101783*x^16 + 1279201*x^14 + 9871136*x^12 + 48063862*x^10 + 147421505*x^8 + 274663048*x^6 + 281918538*x^4 + 121207185*x^2 + 59049);
 
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

$C_{22}$ (as 22T1):

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 cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 11.11.672749994932560009201.1

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 R ${\href{/padicField/3.2.0.1}{2} }^{11}$ ${\href{/padicField/5.11.0.1}{11} }^{2}$ $22$ R ${\href{/padicField/13.11.0.1}{11} }^{2}$ ${\href{/padicField/17.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/padicField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/padicField/37.11.0.1}{11} }^{2}$ ${\href{/padicField/41.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/padicField/53.11.0.1}{11} }^{2}$ $22$

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$$2$$11$$22$
\(11\) Copy content Toggle raw display Deg $22$$11$$2$$40$