sage: x = polygen(QQ); K.<a> = NumberField(x^46 - 21*x^45 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929)
gp: K = bnfinit(y^46 - 21*y^45 + 257*y^44 - 2275*y^43 + 16384*y^42 - 101148*y^41 + 555868*y^40 - 2773061*y^39 + 12764350*y^38 - 54717566*y^37 + 220411975*y^36 - 838622105*y^35 + 3030581020*y^34 - 10433377545*y^33 + 34346625500*y^32 - 108311066940*y^31 + 328058481150*y^30 - 955269123060*y^29 + 2679603110905*y^28 - 7242985668834*y^27 + 18895590411056*y^26 - 47565631805041*y^25 + 115691372227157*y^24 - 271669498342240*y^23 + 616666273593665*y^22 - 1351191092366848*y^21 + 2861502035274188*y^20 - 5844226298666601*y^19 + 11528422276701890*y^18 - 21892211952456407*y^17 + 40102623357336934*y^16 - 70512951104599839*y^15 + 119377822272400223*y^14 - 193124835740771010*y^13 + 300080474989777953*y^12 - 442454098080721470*y^11 + 624685689856470240*y^10 - 827583193065916590*y^9 + 1046376694212400500*y^8 - 1217672042714811795*y^7 + 1349148741219481645*y^6 - 1325965235556309370*y^5 + 1244120066328799465*y^4 - 953228966874939950*y^3 + 714248269821476266*y^2 - 340555683118656924*y + 181351969023278929, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^46 - 21*x^45 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^46 - 21*x^45 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929)
\( x^{46} - 21 x^{45} + 257 x^{44} - 2275 x^{43} + 16384 x^{42} - 101148 x^{41} + 555868 x^{40} + \cdots + 18\!\cdots\!29 \)
sage: K.defining_polynomial()
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree: | | $46$ |
|
Signature: | | $[0, 23]$ |
|
Discriminant: | |
\(-419\!\cdots\!375\)
\(\medspace = -\,3^{23}\cdot 5^{23}\cdot 47^{44}\)
|
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
|
Root discriminant: | | \(153.97\) | 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: | | $3^{1/2}5^{1/2}47^{22/23}\approx 153.97264864641676$
|
Ramified primes: | |
\(3\), \(5\), \(47\)
|
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
|
Discriminant root field: | | \(\Q(\sqrt{-15}) \)
|
$\card{ \Gal(K/\Q) }$: | | $46$ |
|
This field is Galois and abelian over $\Q$. |
Conductor: | | \(705=3\cdot 5\cdot 47\) |
Dirichlet character group:
| |
$\lbrace$$\chi_{705}(256,·)$, $\chi_{705}(1,·)$, $\chi_{705}(136,·)$, $\chi_{705}(524,·)$, $\chi_{705}(269,·)$, $\chi_{705}(14,·)$, $\chi_{705}(271,·)$, $\chi_{705}(16,·)$, $\chi_{705}(659,·)$, $\chi_{705}(404,·)$, $\chi_{705}(149,·)$, $\chi_{705}(89,·)$, $\chi_{705}(284,·)$, $\chi_{705}(541,·)$, $\chi_{705}(286,·)$, $\chi_{705}(674,·)$, $\chi_{705}(676,·)$, $\chi_{705}(166,·)$, $\chi_{705}(554,·)$, $\chi_{705}(299,·)$, $\chi_{705}(314,·)$, $\chi_{705}(571,·)$, $\chi_{705}(316,·)$, $\chi_{705}(61,·)$, $\chi_{705}(194,·)$, $\chi_{705}(451,·)$, $\chi_{705}(196,·)$, $\chi_{705}(74,·)$, $\chi_{705}(331,·)$, $\chi_{705}(209,·)$, $\chi_{705}(526,·)$, $\chi_{705}(601,·)$, $\chi_{705}(346,·)$, $\chi_{705}(479,·)$, $\chi_{705}(224,·)$, $\chi_{705}(59,·)$, $\chi_{705}(614,·)$, $\chi_{705}(361,·)$, $\chi_{705}(106,·)$, $\chi_{705}(494,·)$, $\chi_{705}(239,·)$, $\chi_{705}(241,·)$, $\chi_{705}(629,·)$, $\chi_{705}(119,·)$, $\chi_{705}(121,·)$, $\chi_{705}(661,·)$$\rbrace$
|
This is a CM field. |
Reflex fields: | | unavailable$^{4194304}$ |
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{563}a^{44}+\frac{65}{563}a^{43}+\frac{258}{563}a^{42}+\frac{58}{563}a^{41}-\frac{141}{563}a^{40}+\frac{15}{563}a^{39}+\frac{200}{563}a^{38}+\frac{76}{563}a^{37}+\frac{112}{563}a^{36}+\frac{203}{563}a^{35}-\frac{180}{563}a^{34}-\frac{226}{563}a^{33}+\frac{83}{563}a^{32}+\frac{81}{563}a^{31}+\frac{15}{563}a^{30}-\frac{270}{563}a^{29}+\frac{253}{563}a^{28}-\frac{195}{563}a^{27}-\frac{269}{563}a^{26}-\frac{160}{563}a^{25}-\frac{174}{563}a^{24}+\frac{152}{563}a^{23}-\frac{63}{563}a^{22}+\frac{227}{563}a^{21}-\frac{18}{563}a^{20}+\frac{165}{563}a^{19}-\frac{120}{563}a^{18}+\frac{103}{563}a^{17}+\frac{42}{563}a^{16}-\frac{17}{563}a^{15}+\frac{114}{563}a^{14}+\frac{59}{563}a^{13}+\frac{97}{563}a^{12}+\frac{57}{563}a^{11}+\frac{251}{563}a^{10}-\frac{279}{563}a^{9}+\frac{85}{563}a^{8}-\frac{28}{563}a^{7}+\frac{119}{563}a^{6}+\frac{97}{563}a^{5}-\frac{13}{563}a^{4}+\frac{196}{563}a^{3}+\frac{25}{563}a^{2}-\frac{109}{563}a-\frac{49}{563}$, $\frac{1}{45\!\cdots\!93}a^{45}+\frac{31\!\cdots\!16}{45\!\cdots\!93}a^{44}-\frac{12\!\cdots\!68}{45\!\cdots\!93}a^{43}+\frac{20\!\cdots\!62}{45\!\cdots\!93}a^{42}-\frac{20\!\cdots\!76}{45\!\cdots\!93}a^{41}-\frac{43\!\cdots\!43}{45\!\cdots\!93}a^{40}+\frac{83\!\cdots\!17}{45\!\cdots\!93}a^{39}+\frac{85\!\cdots\!39}{45\!\cdots\!93}a^{38}+\frac{13\!\cdots\!86}{45\!\cdots\!93}a^{37}-\frac{11\!\cdots\!40}{45\!\cdots\!93}a^{36}-\frac{15\!\cdots\!21}{45\!\cdots\!93}a^{35}-\frac{16\!\cdots\!19}{45\!\cdots\!93}a^{34}-\frac{60\!\cdots\!92}{45\!\cdots\!93}a^{33}-\frac{13\!\cdots\!30}{45\!\cdots\!93}a^{32}+\frac{58\!\cdots\!84}{45\!\cdots\!93}a^{31}+\frac{10\!\cdots\!59}{45\!\cdots\!93}a^{30}+\frac{55\!\cdots\!48}{45\!\cdots\!93}a^{29}-\frac{22\!\cdots\!35}{45\!\cdots\!93}a^{28}-\frac{20\!\cdots\!95}{45\!\cdots\!93}a^{27}-\frac{18\!\cdots\!97}{45\!\cdots\!93}a^{26}+\frac{10\!\cdots\!53}{45\!\cdots\!93}a^{25}-\frac{13\!\cdots\!46}{45\!\cdots\!93}a^{24}+\frac{12\!\cdots\!06}{45\!\cdots\!93}a^{23}-\frac{42\!\cdots\!23}{45\!\cdots\!93}a^{22}+\frac{18\!\cdots\!00}{45\!\cdots\!93}a^{21}-\frac{21\!\cdots\!12}{45\!\cdots\!93}a^{20}+\frac{89\!\cdots\!93}{45\!\cdots\!93}a^{19}-\frac{20\!\cdots\!89}{45\!\cdots\!93}a^{18}-\frac{21\!\cdots\!52}{45\!\cdots\!93}a^{17}-\frac{99\!\cdots\!57}{45\!\cdots\!93}a^{16}+\frac{11\!\cdots\!28}{45\!\cdots\!93}a^{15}-\frac{13\!\cdots\!76}{45\!\cdots\!93}a^{14}+\frac{92\!\cdots\!94}{45\!\cdots\!93}a^{13}+\frac{88\!\cdots\!88}{45\!\cdots\!93}a^{12}+\frac{51\!\cdots\!16}{45\!\cdots\!93}a^{11}-\frac{16\!\cdots\!07}{45\!\cdots\!93}a^{10}+\frac{71\!\cdots\!76}{45\!\cdots\!93}a^{9}-\frac{51\!\cdots\!32}{45\!\cdots\!93}a^{8}-\frac{93\!\cdots\!74}{45\!\cdots\!93}a^{7}-\frac{11\!\cdots\!50}{45\!\cdots\!93}a^{6}+\frac{22\!\cdots\!30}{45\!\cdots\!93}a^{5}-\frac{12\!\cdots\!27}{45\!\cdots\!93}a^{4}+\frac{16\!\cdots\!69}{45\!\cdots\!93}a^{3}+\frac{68\!\cdots\!95}{45\!\cdots\!93}a^{2}+\frac{24\!\cdots\!97}{45\!\cdots\!93}a-\frac{28\!\cdots\!99}{45\!\cdots\!93}$
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 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929) 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 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929, 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 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929); 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 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929); 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 |
$23^{2}$ |
R |
R |
$46$ |
$46$ |
$46$ |
$23^{2}$ |
$23^{2}$ |
$23^{2}$ |
$46$ |
$23^{2}$ |
$46$ |
$46$ |
$46$ |
R |
$23^{2}$ |
$46$ |
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]
|