sage: x = polygen(QQ); K.<a> = NumberField(x^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031)
gp: K = bnfinit(y^46 - 21*y^45 + 280*y^44 - 2737*y^43 + 22170*y^42 - 154145*y^41 + 956933*y^40 - 5379033*y^39 + 27886243*y^38 - 134248450*y^37 + 606496288*y^36 - 2580676650*y^35 + 10412975743*y^34 - 39920376954*y^33 + 146107747414*y^32 - 510964368992*y^31 + 1713645236702*y^30 - 5512052557476*y^29 + 17054425501759*y^28 - 50730506723382*y^27 + 145445655971166*y^26 - 401467681974859*y^25 + 1069363344649915*y^24 - 2743875089714332*y^23 + 6797884140501079*y^22 - 16220179768462732*y^21 + 37368762579922514*y^20 - 82830583629409339*y^19 + 177181015929801720*y^18 - 363931247102399123*y^17 + 720639735990585495*y^16 - 1365869353643320843*y^15 + 2491905369702459650*y^14 - 4330190064722950604*y^13 + 7228480682418043087*y^12 - 11405749818173784685*y^11 + 17248944398622077393*y^10 - 24356420142161515772*y^9 + 32892937884911140180*y^8 - 40615318559022977115*y^7 + 47943875275560939644*y^6 - 49733323681891310169*y^5 + 49623712340431236341*y^4 - 39861978482127425967*y^3 + 31755951074013931513*y^2 - 15726577420274467740*y + 8942103583744060031, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031)
\( x^{46} - 21 x^{45} + 280 x^{44} - 2737 x^{43} + 22170 x^{42} - 154145 x^{41} + 956933 x^{40} + \cdots + 89\!\cdots\!31 \)
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree: | | $46$ |
|
Signature: | | $[0, 23]$ |
|
Discriminant: | |
\(-963\!\cdots\!579\)
\(\medspace = -\,19^{23}\cdot 47^{44}\)
|
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
|
Root discriminant: | | \(173.29\) | 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: | | $19^{1/2}47^{22/23}\approx 173.29049869947002$
|
Ramified primes: | |
\(19\), \(47\)
|
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
|
Discriminant root field: | | \(\Q(\sqrt{-19}) \)
|
$\card{ \Gal(K/\Q) }$: | | $46$ |
|
This field is Galois and abelian over $\Q$. |
Conductor: | | \(893=19\cdot 47\) |
Dirichlet character group:
| |
$\lbrace$$\chi_{893}(512,·)$, $\chi_{893}(1,·)$, $\chi_{893}(645,·)$, $\chi_{893}(647,·)$, $\chi_{893}(267,·)$, $\chi_{893}(780,·)$, $\chi_{893}(400,·)$, $\chi_{893}(18,·)$, $\chi_{893}(531,·)$, $\chi_{893}(533,·)$, $\chi_{893}(664,·)$, $\chi_{893}(153,·)$, $\chi_{893}(666,·)$, $\chi_{893}(284,·)$, $\chi_{893}(286,·)$, $\chi_{893}(37,·)$, $\chi_{893}(683,·)$, $\chi_{893}(685,·)$, $\chi_{893}(303,·)$, $\chi_{893}(816,·)$, $\chi_{893}(56,·)$, $\chi_{893}(571,·)$, $\chi_{893}(189,·)$, $\chi_{893}(191,·)$, $\chi_{893}(835,·)$, $\chi_{893}(324,·)$, $\chi_{893}(455,·)$, $\chi_{893}(457,·)$, $\chi_{893}(75,·)$, $\chi_{893}(588,·)$, $\chi_{893}(721,·)$, $\chi_{893}(723,·)$, $\chi_{893}(341,·)$, $\chi_{893}(854,·)$, $\chi_{893}(343,·)$, $\chi_{893}(474,·)$, $\chi_{893}(476,·)$, $\chi_{893}(96,·)$, $\chi_{893}(742,·)$, $\chi_{893}(873,·)$, $\chi_{893}(495,·)$, $\chi_{893}(115,·)$, $\chi_{893}(628,·)$, $\chi_{893}(759,·)$, $\chi_{893}(761,·)$, $\chi_{893}(379,·)$$\rbrace$
|
This is a CM field. |
Reflex fields: | | unavailable$^{4194304}$ |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{292339}a^{44}+\frac{119321}{292339}a^{43}+\frac{11852}{292339}a^{42}+\frac{68871}{292339}a^{41}-\frac{143641}{292339}a^{40}+\frac{68966}{292339}a^{39}-\frac{87443}{292339}a^{38}+\frac{101125}{292339}a^{37}-\frac{4257}{292339}a^{36}-\frac{139659}{292339}a^{35}+\frac{5458}{292339}a^{34}+\frac{17767}{292339}a^{33}+\frac{89815}{292339}a^{32}+\frac{6949}{292339}a^{31}-\frac{82433}{292339}a^{30}+\frac{86838}{292339}a^{29}-\frac{89222}{292339}a^{28}+\frac{141818}{292339}a^{27}-\frac{83610}{292339}a^{26}-\frac{94682}{292339}a^{25}-\frac{58332}{292339}a^{24}-\frac{57862}{292339}a^{23}+\frac{101356}{292339}a^{22}-\frac{97149}{292339}a^{21}+\frac{85478}{292339}a^{20}+\frac{96989}{292339}a^{19}-\frac{94515}{292339}a^{18}+\frac{82398}{292339}a^{17}+\frac{114440}{292339}a^{16}+\frac{136379}{292339}a^{15}-\frac{32246}{292339}a^{14}-\frac{103592}{292339}a^{13}-\frac{81067}{292339}a^{12}+\frac{6318}{292339}a^{11}-\frac{22347}{292339}a^{10}+\frac{143073}{292339}a^{9}-\frac{85888}{292339}a^{8}+\frac{51027}{292339}a^{7}-\frac{129068}{292339}a^{6}+\frac{122798}{292339}a^{5}+\frac{88836}{292339}a^{4}-\frac{60384}{292339}a^{3}-\frac{98040}{292339}a^{2}+\frac{26881}{292339}a+\frac{80}{1033}$, $\frac{1}{44\!\cdots\!69}a^{45}+\frac{52\!\cdots\!20}{44\!\cdots\!69}a^{44}+\frac{16\!\cdots\!44}{44\!\cdots\!69}a^{43}-\frac{16\!\cdots\!06}{44\!\cdots\!69}a^{42}-\frac{20\!\cdots\!27}{44\!\cdots\!69}a^{41}-\frac{18\!\cdots\!09}{44\!\cdots\!69}a^{40}-\frac{75\!\cdots\!48}{44\!\cdots\!69}a^{39}-\frac{19\!\cdots\!56}{44\!\cdots\!69}a^{38}-\frac{98\!\cdots\!77}{44\!\cdots\!69}a^{37}-\frac{58\!\cdots\!57}{44\!\cdots\!69}a^{36}-\frac{92\!\cdots\!77}{44\!\cdots\!69}a^{35}+\frac{11\!\cdots\!73}{44\!\cdots\!69}a^{34}+\frac{19\!\cdots\!51}{44\!\cdots\!69}a^{33}+\frac{13\!\cdots\!69}{44\!\cdots\!69}a^{32}+\frac{16\!\cdots\!26}{44\!\cdots\!69}a^{31}+\frac{25\!\cdots\!36}{44\!\cdots\!69}a^{30}+\frac{18\!\cdots\!34}{44\!\cdots\!69}a^{29}+\frac{11\!\cdots\!59}{44\!\cdots\!69}a^{28}-\frac{43\!\cdots\!12}{44\!\cdots\!69}a^{27}+\frac{18\!\cdots\!72}{44\!\cdots\!69}a^{26}-\frac{13\!\cdots\!17}{44\!\cdots\!69}a^{25}-\frac{22\!\cdots\!50}{44\!\cdots\!69}a^{24}+\frac{80\!\cdots\!47}{44\!\cdots\!69}a^{23}-\frac{20\!\cdots\!39}{44\!\cdots\!69}a^{22}-\frac{83\!\cdots\!10}{44\!\cdots\!69}a^{21}+\frac{26\!\cdots\!37}{44\!\cdots\!69}a^{20}-\frac{57\!\cdots\!41}{44\!\cdots\!69}a^{19}+\frac{20\!\cdots\!99}{44\!\cdots\!69}a^{18}-\frac{45\!\cdots\!01}{44\!\cdots\!69}a^{17}-\frac{14\!\cdots\!40}{44\!\cdots\!69}a^{16}-\frac{41\!\cdots\!22}{44\!\cdots\!69}a^{15}+\frac{20\!\cdots\!38}{44\!\cdots\!69}a^{14}-\frac{74\!\cdots\!91}{44\!\cdots\!69}a^{13}-\frac{14\!\cdots\!16}{44\!\cdots\!69}a^{12}-\frac{98\!\cdots\!30}{44\!\cdots\!69}a^{11}-\frac{69\!\cdots\!64}{44\!\cdots\!69}a^{10}-\frac{60\!\cdots\!58}{44\!\cdots\!69}a^{9}-\frac{56\!\cdots\!11}{44\!\cdots\!69}a^{8}-\frac{19\!\cdots\!40}{44\!\cdots\!69}a^{7}+\frac{10\!\cdots\!63}{44\!\cdots\!69}a^{6}-\frac{18\!\cdots\!64}{44\!\cdots\!69}a^{5}-\frac{18\!\cdots\!35}{44\!\cdots\!69}a^{4}+\frac{14\!\cdots\!36}{44\!\cdots\!69}a^{3}-\frac{12\!\cdots\!93}{44\!\cdots\!69}a^{2}+\frac{10\!\cdots\!29}{44\!\cdots\!69}a+\frac{48\!\cdots\!34}{15\!\cdots\!43}$
not computed
sage: K.class_group().invariants()
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank: | | $22$
|
|
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^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031) 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^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031, 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^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031); 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^46 - 21*x^45 + 280*x^44 - 2737*x^43 + 22170*x^42 - 154145*x^41 + 956933*x^40 - 5379033*x^39 + 27886243*x^38 - 134248450*x^37 + 606496288*x^36 - 2580676650*x^35 + 10412975743*x^34 - 39920376954*x^33 + 146107747414*x^32 - 510964368992*x^31 + 1713645236702*x^30 - 5512052557476*x^29 + 17054425501759*x^28 - 50730506723382*x^27 + 145445655971166*x^26 - 401467681974859*x^25 + 1069363344649915*x^24 - 2743875089714332*x^23 + 6797884140501079*x^22 - 16220179768462732*x^21 + 37368762579922514*x^20 - 82830583629409339*x^19 + 177181015929801720*x^18 - 363931247102399123*x^17 + 720639735990585495*x^16 - 1365869353643320843*x^15 + 2491905369702459650*x^14 - 4330190064722950604*x^13 + 7228480682418043087*x^12 - 11405749818173784685*x^11 + 17248944398622077393*x^10 - 24356420142161515772*x^9 + 32892937884911140180*x^8 - 40615318559022977115*x^7 + 47943875275560939644*x^6 - 49733323681891310169*x^5 + 49623712340431236341*x^4 - 39861978482127425967*x^3 + 31755951074013931513*x^2 - 15726577420274467740*x + 8942103583744060031); 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))))
$C_{46}$ (as 46T1):
sage: K.galois_group(type='pari')
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
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]
$p$ |
$2$ |
$3$ |
$5$ |
$7$ |
$11$ |
$13$ |
$17$ |
$19$ |
$23$ |
$29$ |
$31$ |
$37$ |
$41$ |
$43$ |
$47$ |
$53$ |
$59$ |
Cycle type |
$46$ |
$46$ |
$23^{2}$ |
$23^{2}$ |
$23^{2}$ |
$46$ |
$23^{2}$ |
R |
$23^{2}$ |
$46$ |
$46$ |
$46$ |
$46$ |
$23^{2}$ |
R |
$46$ |
$46$ |
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]
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