sage: x = polygen(QQ); K.<a> = NumberField(x^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501)
gp: K = bnfinit(y^43 - y^42 - 210*y^41 + 177*y^40 + 19424*y^39 - 12392*y^38 - 1053196*y^37 + 410572*y^36 + 37567316*y^35 - 4029555*y^34 - 936601673*y^33 - 185951871*y^32 + 16894280750*y^31 + 8734312688*y^30 - 224649876009*y^29 - 189102381409*y^28 + 2218265633870*y^27 + 2604819705111*y^26 - 16218499586789*y^25 - 24791417254787*y^24 + 86521587673786*y^23 + 168116844068603*y^22 - 325504806681996*y^21 - 819692515707425*y^20 + 795734675389219*y^19 + 2859558999201013*y^18 - 937302717934781*y^17 - 7011360351157045*y^16 - 902655412698460*y^15 + 11681897146129224*y^14 + 5604720542659268*y^13 - 12486138055465655*y^12 - 9886946345787043*y^11 + 7728522779787490*y^10 + 9075115953805640*y^9 - 2183897418754987*y^8 - 4579013283800046*y^7 - 40649514655288*y^6 + 1249295360564226*y^5 + 160575454035341*y^4 - 166814479770181*y^3 - 29312399288701*y^2 + 7715035819499*y + 1403424452501, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501)
\( x^{43} - x^{42} - 210 x^{41} + 177 x^{40} + 19424 x^{39} - 12392 x^{38} - 1053196 x^{37} + \cdots + 1403424452501 \)
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree: | | $43$ |
|
Signature: | | $[43, 0]$ |
|
Discriminant: | |
\(444\!\cdots\!961\)
\(\medspace = 431^{42}\)
|
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
|
Root discriminant: | | \(374.29\) | 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: | | $431^{42/43}\approx 374.29183811372997$
|
Ramified primes: | |
\(431\)
|
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
|
Discriminant root field: | | \(\Q\)
|
$\card{ \Gal(K/\Q) }$: | | $43$ |
|
This field is Galois and abelian over $\Q$. |
Conductor: | | \(431\) |
Dirichlet character group:
| |
$\lbrace$$\chi_{431}(128,·)$, $\chi_{431}(1,·)$, $\chi_{431}(2,·)$, $\chi_{431}(3,·)$, $\chi_{431}(4,·)$, $\chi_{431}(6,·)$, $\chi_{431}(8,·)$, $\chi_{431}(9,·)$, $\chi_{431}(12,·)$, $\chi_{431}(256,·)$, $\chi_{431}(16,·)$, $\chi_{431}(145,·)$, $\chi_{431}(18,·)$, $\chi_{431}(149,·)$, $\chi_{431}(24,·)$, $\chi_{431}(27,·)$, $\chi_{431}(32,·)$, $\chi_{431}(162,·)$, $\chi_{431}(36,·)$, $\chi_{431}(165,·)$, $\chi_{431}(64,·)$, $\chi_{431}(298,·)$, $\chi_{431}(48,·)$, $\chi_{431}(54,·)$, $\chi_{431}(55,·)$, $\chi_{431}(192,·)$, $\chi_{431}(288,·)$, $\chi_{431}(324,·)$, $\chi_{431}(72,·)$, $\chi_{431}(330,·)$, $\chi_{431}(290,·)$, $\chi_{431}(81,·)$, $\chi_{431}(216,·)$, $\chi_{431}(217,·)$, $\chi_{431}(384,·)$, $\chi_{431}(220,·)$, $\chi_{431}(96,·)$, $\chi_{431}(144,·)$, $\chi_{431}(229,·)$, $\chi_{431}(337,·)$, $\chi_{431}(108,·)$, $\chi_{431}(110,·)$, $\chi_{431}(243,·)$$\rbrace$
|
This is not a CM field. |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{617}a^{40}-\frac{96}{617}a^{39}-\frac{38}{617}a^{38}-\frac{212}{617}a^{37}-\frac{104}{617}a^{36}-\frac{244}{617}a^{35}+\frac{238}{617}a^{34}-\frac{176}{617}a^{33}-\frac{13}{617}a^{32}+\frac{269}{617}a^{31}-\frac{48}{617}a^{30}-\frac{85}{617}a^{29}+\frac{101}{617}a^{28}-\frac{109}{617}a^{27}-\frac{293}{617}a^{26}-\frac{87}{617}a^{25}-\frac{192}{617}a^{24}-\frac{36}{617}a^{23}-\frac{264}{617}a^{22}-\frac{84}{617}a^{21}-\frac{289}{617}a^{20}+\frac{224}{617}a^{19}+\frac{114}{617}a^{18}+\frac{236}{617}a^{17}-\frac{43}{617}a^{16}+\frac{293}{617}a^{15}+\frac{198}{617}a^{14}-\frac{239}{617}a^{13}+\frac{243}{617}a^{12}-\frac{265}{617}a^{11}+\frac{301}{617}a^{10}-\frac{117}{617}a^{9}+\frac{281}{617}a^{8}-\frac{235}{617}a^{7}+\frac{215}{617}a^{6}+\frac{308}{617}a^{5}-\frac{83}{617}a^{4}-\frac{166}{617}a^{3}+\frac{245}{617}a^{2}-\frac{51}{617}a+\frac{248}{617}$, $\frac{1}{320964466793}a^{41}+\frac{139796710}{320964466793}a^{40}+\frac{28941309337}{320964466793}a^{39}+\frac{126390064522}{320964466793}a^{38}+\frac{66208018678}{320964466793}a^{37}+\frac{14066083648}{320964466793}a^{36}+\frac{87870452426}{320964466793}a^{35}-\frac{15882023825}{320964466793}a^{34}-\frac{117381994781}{320964466793}a^{33}+\frac{51354677571}{320964466793}a^{32}+\frac{31659308681}{320964466793}a^{31}+\frac{26506178368}{320964466793}a^{30}+\frac{72390307907}{320964466793}a^{29}-\frac{156929706337}{320964466793}a^{28}+\frac{30802602707}{320964466793}a^{27}+\frac{81085137460}{320964466793}a^{26}+\frac{158840472175}{320964466793}a^{25}-\frac{153999837168}{320964466793}a^{24}+\frac{131668817904}{320964466793}a^{23}-\frac{123034361111}{320964466793}a^{22}-\frac{85402152911}{320964466793}a^{21}-\frac{117152204418}{320964466793}a^{20}+\frac{117058398997}{320964466793}a^{19}+\frac{90109407825}{320964466793}a^{18}+\frac{145433894181}{320964466793}a^{17}-\frac{34944252620}{320964466793}a^{16}+\frac{95471731337}{320964466793}a^{15}-\frac{77725354228}{320964466793}a^{14}-\frac{99566354704}{320964466793}a^{13}+\frac{41397120030}{320964466793}a^{12}+\frac{119638790936}{320964466793}a^{11}+\frac{55272436394}{320964466793}a^{10}-\frac{77662797205}{320964466793}a^{9}+\frac{58406679660}{320964466793}a^{8}+\frac{88370551543}{320964466793}a^{7}-\frac{61647862697}{320964466793}a^{6}-\frac{33155475203}{320964466793}a^{5}-\frac{33723425418}{320964466793}a^{4}+\frac{93792925557}{320964466793}a^{3}-\frac{97241821493}{320964466793}a^{2}+\frac{23328690925}{320964466793}a-\frac{107274973111}{320964466793}$, $\frac{1}{14\!\cdots\!37}a^{42}-\frac{22\!\cdots\!44}{14\!\cdots\!37}a^{41}+\frac{11\!\cdots\!72}{14\!\cdots\!37}a^{40}-\frac{66\!\cdots\!58}{14\!\cdots\!37}a^{39}+\frac{25\!\cdots\!94}{14\!\cdots\!37}a^{38}-\frac{61\!\cdots\!28}{14\!\cdots\!37}a^{37}+\frac{65\!\cdots\!97}{14\!\cdots\!37}a^{36}+\frac{44\!\cdots\!45}{14\!\cdots\!37}a^{35}-\frac{41\!\cdots\!24}{14\!\cdots\!37}a^{34}-\frac{14\!\cdots\!33}{14\!\cdots\!37}a^{33}-\frac{47\!\cdots\!66}{14\!\cdots\!37}a^{32}-\frac{17\!\cdots\!80}{14\!\cdots\!37}a^{31}-\frac{64\!\cdots\!37}{14\!\cdots\!37}a^{30}+\frac{62\!\cdots\!71}{14\!\cdots\!37}a^{29}+\frac{29\!\cdots\!34}{14\!\cdots\!37}a^{28}-\frac{19\!\cdots\!95}{14\!\cdots\!37}a^{27}+\frac{95\!\cdots\!04}{14\!\cdots\!37}a^{26}+\frac{10\!\cdots\!42}{14\!\cdots\!37}a^{25}+\frac{66\!\cdots\!21}{14\!\cdots\!37}a^{24}+\frac{66\!\cdots\!88}{14\!\cdots\!37}a^{23}-\frac{52\!\cdots\!52}{14\!\cdots\!37}a^{22}+\frac{49\!\cdots\!09}{14\!\cdots\!37}a^{21}+\frac{41\!\cdots\!80}{14\!\cdots\!37}a^{20}+\frac{48\!\cdots\!27}{14\!\cdots\!37}a^{19}-\frac{25\!\cdots\!98}{14\!\cdots\!37}a^{18}+\frac{55\!\cdots\!45}{14\!\cdots\!37}a^{17}+\frac{62\!\cdots\!99}{14\!\cdots\!37}a^{16}+\frac{57\!\cdots\!15}{14\!\cdots\!37}a^{15}-\frac{60\!\cdots\!54}{14\!\cdots\!37}a^{14}-\frac{71\!\cdots\!58}{14\!\cdots\!37}a^{13}-\frac{77\!\cdots\!19}{23\!\cdots\!61}a^{12}-\frac{42\!\cdots\!38}{14\!\cdots\!37}a^{11}+\frac{33\!\cdots\!80}{14\!\cdots\!37}a^{10}+\frac{97\!\cdots\!15}{23\!\cdots\!61}a^{9}+\frac{21\!\cdots\!41}{14\!\cdots\!37}a^{8}-\frac{37\!\cdots\!20}{14\!\cdots\!37}a^{7}-\frac{44\!\cdots\!01}{14\!\cdots\!37}a^{6}+\frac{19\!\cdots\!19}{14\!\cdots\!37}a^{5}+\frac{26\!\cdots\!23}{14\!\cdots\!37}a^{4}+\frac{26\!\cdots\!89}{14\!\cdots\!37}a^{3}-\frac{36\!\cdots\!53}{14\!\cdots\!37}a^{2}+\frac{16\!\cdots\!33}{14\!\cdots\!37}a-\frac{22\!\cdots\!52}{45\!\cdots\!79}$
not computed
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank: | | $42$
|
|
Torsion generator: | |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
|
Fundamental units: | | not computed
| sage: UK.fundamental_units()
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
|
Regulator: | | not computed
|
|
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
not computed
\end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501) 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^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501, 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^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501); 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^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501); 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))))
$C_{43}$ (as 43T1):
sage: K.galois_group(type='pari')
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
The extension is primitive: there are no intermediate fields
between this field and $\Q$.
|
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]
$p$ |
$2$ |
$3$ |
$5$ |
$7$ |
$11$ |
$13$ |
$17$ |
$19$ |
$23$ |
$29$ |
$31$ |
$37$ |
$41$ |
$43$ |
$47$ |
$53$ |
$59$ |
Cycle type |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
$43$ |
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]
|