Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000)
gp: K = bnfinit(y^15 - 99*y^13 - 396*y^12 - 3868*y^11 - 27974*y^10 + 97245*y^9 + 2044656*y^8 + 10660124*y^7 + 28900022*y^6 + 46578665*y^5 + 46949760*y^4 + 29930225*y^3 + 11744850*y^2 + 2593500*y + 247000, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000)
\( x^{15} - 99 x^{13} - 396 x^{12} - 3868 x^{11} - 27974 x^{10} + 97245 x^{9} + 2044656 x^{8} + \cdots + 247000 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[9, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-6381262105849605201341604346805613756416000000\)
\(\medspace = -\,2^{24}\cdot 5^{6}\cdot 13^{4}\cdot 17\cdot 19^{4}\cdot 37^{5}\cdot 74483599^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(1131.52\) | 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: | $2^{8/3}5^{1/2}13^{4/5}17^{1/2}19^{4/5}37^{1/2}74483599^{1/2}\approx 252199876.32092622$ | ||
| Ramified primes: |
\(2\), \(5\), \(13\), \(17\), \(19\), \(37\), \(74483599\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-629}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{5}a^{6}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{8}-\frac{1}{25}a^{6}+\frac{1}{25}a^{5}-\frac{1}{25}a^{4}-\frac{8}{25}a^{3}+\frac{4}{25}a^{2}-\frac{2}{5}$, $\frac{1}{25}a^{9}-\frac{1}{25}a^{7}+\frac{1}{25}a^{6}-\frac{1}{25}a^{5}-\frac{8}{25}a^{4}+\frac{4}{25}a^{3}-\frac{2}{5}a$, $\frac{1}{25}a^{10}+\frac{1}{25}a^{7}-\frac{2}{25}a^{6}-\frac{2}{25}a^{5}+\frac{3}{25}a^{4}+\frac{2}{25}a^{3}+\frac{9}{25}a^{2}-\frac{2}{5}$, $\frac{1}{125}a^{11}+\frac{1}{125}a^{9}-\frac{1}{125}a^{8}-\frac{3}{125}a^{7}-\frac{4}{125}a^{6}+\frac{36}{125}a^{4}-\frac{61}{125}a^{3}-\frac{33}{125}a^{2}-\frac{4}{25}a+\frac{9}{25}$, $\frac{1}{625}a^{12}+\frac{6}{625}a^{10}+\frac{4}{625}a^{9}+\frac{12}{625}a^{8}-\frac{54}{625}a^{7}+\frac{6}{125}a^{6}+\frac{11}{625}a^{5}-\frac{151}{625}a^{4}+\frac{302}{625}a^{3}+\frac{62}{125}a^{2}+\frac{49}{125}a+\frac{2}{25}$, $\frac{1}{2500000}a^{13}-\frac{31}{125000}a^{12}-\frac{7159}{2500000}a^{11}+\frac{5423}{312500}a^{10}-\frac{1244}{78125}a^{9}-\frac{23827}{1250000}a^{8}-\frac{23519}{500000}a^{7}-\frac{52901}{625000}a^{6}-\frac{25837}{312500}a^{5}+\frac{326691}{1250000}a^{4}+\frac{79597}{500000}a^{3}+\frac{49067}{125000}a^{2}+\frac{31653}{100000}a-\frac{4661}{50000}$, $\frac{1}{80000000}a^{14}+\frac{1}{8000000}a^{13}+\frac{10241}{80000000}a^{12}+\frac{76607}{40000000}a^{11}+\frac{33757}{5000000}a^{10}-\frac{787347}{40000000}a^{9}+\frac{207277}{16000000}a^{8}-\frac{344227}{40000000}a^{7}-\frac{92413}{2500000}a^{6}-\frac{2788549}{40000000}a^{5}+\frac{2368129}{16000000}a^{4}+\frac{3870789}{8000000}a^{3}+\frac{1314221}{3200000}a^{2}+\frac{378617}{800000}a-\frac{36043}{160000}$
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
Trivial group, which has order $1$ (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 $2$)
| 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: |
$\frac{230068283}{40000000}a^{14}-\frac{66061281}{20000000}a^{13}-\frac{22701994757}{40000000}a^{12}-\frac{7806844711}{4000000}a^{11}-\frac{52810175177}{2500000}a^{10}-\frac{2975193220833}{20000000}a^{9}+\frac{25794019099323}{40000000}a^{8}+\frac{227811838071879}{20000000}a^{7}+\frac{13692254715679}{250000}a^{6}+\frac{26\!\cdots\!49}{20000000}a^{5}+\frac{76\!\cdots\!91}{40000000}a^{4}+\frac{25640918188119}{160000}a^{3}+\frac{636308129973659}{8000000}a^{2}+\frac{8599869039567}{400000}a+\frac{977524721211}{400000}$, $\frac{712516249}{80000000}a^{14}-\frac{202091843}{40000000}a^{13}-\frac{70312448871}{80000000}a^{12}-\frac{24226743173}{8000000}a^{11}-\frac{163645482731}{5000000}a^{10}-\frac{9222973422299}{40000000}a^{9}+\frac{79761444062969}{80000000}a^{8}+\frac{705833051626837}{40000000}a^{7}+\frac{42464929787647}{500000}a^{6}+\frac{83\!\cdots\!47}{40000000}a^{5}+\frac{23\!\cdots\!73}{80000000}a^{4}+\frac{15981326580609}{64000}a^{3}+\frac{19\!\cdots\!77}{16000000}a^{2}+\frac{26952873015601}{800000}a+\frac{3075138921633}{800000}$, $\frac{33\!\cdots\!01}{40000000}a^{14}-\frac{20\!\cdots\!67}{4000000}a^{13}-\frac{32\!\cdots\!59}{40000000}a^{12}-\frac{55\!\cdots\!33}{20000000}a^{11}-\frac{75\!\cdots\!63}{2500000}a^{10}-\frac{42\!\cdots\!27}{20000000}a^{9}+\frac{75\!\cdots\!21}{8000000}a^{8}+\frac{32\!\cdots\!73}{20000000}a^{7}+\frac{97\!\cdots\!97}{1250000}a^{6}+\frac{38\!\cdots\!91}{20000000}a^{5}+\frac{21\!\cdots\!77}{8000000}a^{4}+\frac{89\!\cdots\!09}{4000000}a^{3}+\frac{17\!\cdots\!73}{1600000}a^{2}+\frac{11\!\cdots\!57}{400000}a+\frac{52\!\cdots\!81}{16000}$, $\frac{354978519975263}{40000000}a^{14}-\frac{106670000940973}{20000000}a^{13}-\frac{35\!\cdots\!97}{40000000}a^{12}-\frac{59\!\cdots\!67}{20000000}a^{11}-\frac{81\!\cdots\!33}{2500000}a^{10}-\frac{45\!\cdots\!57}{20000000}a^{9}+\frac{40\!\cdots\!59}{40000000}a^{8}+\frac{35\!\cdots\!59}{20000000}a^{7}+\frac{10\!\cdots\!03}{1250000}a^{6}+\frac{41\!\cdots\!61}{20000000}a^{5}+\frac{11\!\cdots\!03}{40000000}a^{4}+\frac{97\!\cdots\!71}{4000000}a^{3}+\frac{95\!\cdots\!47}{8000000}a^{2}+\frac{12\!\cdots\!39}{400000}a+\frac{14\!\cdots\!23}{400000}$, $\frac{317714539961143}{16000000}a^{14}-\frac{519993138459377}{40000000}a^{13}-\frac{31\!\cdots\!41}{16000000}a^{12}-\frac{26\!\cdots\!27}{40000000}a^{11}-\frac{36\!\cdots\!41}{5000000}a^{10}-\frac{20\!\cdots\!89}{40000000}a^{9}+\frac{18\!\cdots\!91}{80000000}a^{8}+\frac{31\!\cdots\!27}{8000000}a^{7}+\frac{46\!\cdots\!93}{2500000}a^{6}+\frac{18\!\cdots\!57}{40000000}a^{5}+\frac{50\!\cdots\!47}{80000000}a^{4}+\frac{41\!\cdots\!91}{8000000}a^{3}+\frac{40\!\cdots\!03}{16000000}a^{2}+\frac{53\!\cdots\!27}{800000}a+\frac{59\!\cdots\!47}{800000}$, $\frac{66\!\cdots\!57}{16000000}a^{14}-\frac{37\!\cdots\!79}{40000000}a^{13}-\frac{62\!\cdots\!71}{16000000}a^{12}-\frac{30\!\cdots\!69}{40000000}a^{11}-\frac{71\!\cdots\!47}{5000000}a^{10}-\frac{33\!\cdots\!63}{40000000}a^{9}+\frac{47\!\cdots\!57}{80000000}a^{8}+\frac{57\!\cdots\!37}{8000000}a^{7}+\frac{70\!\cdots\!71}{2500000}a^{6}+\frac{22\!\cdots\!19}{40000000}a^{5}+\frac{52\!\cdots\!69}{80000000}a^{4}+\frac{37\!\cdots\!77}{8000000}a^{3}+\frac{31\!\cdots\!81}{16000000}a^{2}+\frac{36\!\cdots\!89}{800000}a+\frac{36\!\cdots\!69}{800000}$, $\frac{10\!\cdots\!89}{80000000}a^{14}-\frac{337359926518439}{8000000}a^{13}-\frac{95\!\cdots\!51}{80000000}a^{12}-\frac{62\!\cdots\!17}{40000000}a^{11}-\frac{23\!\cdots\!47}{5000000}a^{10}-\frac{91\!\cdots\!63}{40000000}a^{9}+\frac{32\!\cdots\!77}{16000000}a^{8}+\frac{84\!\cdots\!97}{40000000}a^{7}+\frac{19\!\cdots\!03}{2500000}a^{6}+\frac{58\!\cdots\!79}{40000000}a^{5}+\frac{26\!\cdots\!89}{16000000}a^{4}+\frac{90\!\cdots\!41}{8000000}a^{3}+\frac{14\!\cdots\!61}{3200000}a^{2}+\frac{85\!\cdots\!53}{800000}a+\frac{16\!\cdots\!81}{160000}$, $\frac{12\!\cdots\!93}{80000000}a^{14}-\frac{35\!\cdots\!99}{8000000}a^{13}-\frac{12\!\cdots\!87}{80000000}a^{12}-\frac{23\!\cdots\!09}{40000000}a^{11}-\frac{29\!\cdots\!79}{5000000}a^{10}-\frac{16\!\cdots\!91}{40000000}a^{9}+\frac{24\!\cdots\!97}{16000000}a^{8}+\frac{12\!\cdots\!89}{40000000}a^{7}+\frac{40\!\cdots\!31}{2500000}a^{6}+\frac{16\!\cdots\!03}{40000000}a^{5}+\frac{10\!\cdots\!09}{16000000}a^{4}+\frac{47\!\cdots\!57}{8000000}a^{3}+\frac{10\!\cdots\!41}{3200000}a^{2}+\frac{75\!\cdots\!41}{800000}a+\frac{18\!\cdots\!13}{160000}$, $\frac{15\!\cdots\!19}{80000000}a^{14}-\frac{24\!\cdots\!53}{40000000}a^{13}-\frac{13\!\cdots\!01}{80000000}a^{12}-\frac{17\!\cdots\!67}{8000000}a^{11}-\frac{33\!\cdots\!21}{5000000}a^{10}-\frac{12\!\cdots\!09}{40000000}a^{9}+\frac{23\!\cdots\!99}{80000000}a^{8}+\frac{12\!\cdots\!47}{40000000}a^{7}+\frac{54\!\cdots\!93}{500000}a^{6}+\frac{83\!\cdots\!77}{40000000}a^{5}+\frac{18\!\cdots\!83}{80000000}a^{4}+\frac{25\!\cdots\!67}{1600000}a^{3}+\frac{10\!\cdots\!67}{16000000}a^{2}+\frac{12\!\cdots\!51}{800000}a+\frac{12\!\cdots\!43}{800000}$, $\frac{40\!\cdots\!69}{40000000}a^{14}-\frac{13\!\cdots\!59}{20000000}a^{13}-\frac{39\!\cdots\!11}{40000000}a^{12}-\frac{65\!\cdots\!81}{20000000}a^{11}-\frac{91\!\cdots\!59}{2500000}a^{10}-\frac{51\!\cdots\!11}{20000000}a^{9}+\frac{45\!\cdots\!97}{40000000}a^{8}+\frac{39\!\cdots\!17}{20000000}a^{7}+\frac{11\!\cdots\!29}{1250000}a^{6}+\frac{45\!\cdots\!03}{20000000}a^{5}+\frac{12\!\cdots\!49}{40000000}a^{4}+\frac{10\!\cdots\!53}{4000000}a^{3}+\frac{99\!\cdots\!01}{8000000}a^{2}+\frac{13\!\cdots\!17}{400000}a+\frac{14\!\cdots\!09}{400000}$, $\frac{94\!\cdots\!51}{80000000}a^{14}-\frac{34\!\cdots\!93}{40000000}a^{13}-\frac{94\!\cdots\!89}{80000000}a^{12}-\frac{15\!\cdots\!11}{40000000}a^{11}-\frac{21\!\cdots\!97}{5000000}a^{10}-\frac{11\!\cdots\!13}{40000000}a^{9}+\frac{11\!\cdots\!19}{80000000}a^{8}+\frac{96\!\cdots\!83}{40000000}a^{7}+\frac{28\!\cdots\!49}{2500000}a^{6}+\frac{11\!\cdots\!09}{40000000}a^{5}+\frac{31\!\cdots\!23}{80000000}a^{4}+\frac{26\!\cdots\!83}{8000000}a^{3}+\frac{25\!\cdots\!27}{16000000}a^{2}+\frac{69\!\cdots\!59}{160000}a+\frac{39\!\cdots\!63}{800000}$
| 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: | \( 1801375611110000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{3}\cdot 1801375611110000000 \cdot 1}{2\cdot\sqrt{6381262105849605201341604346805613756416000000}}\cr\approx \mathstrut & 1.43195891379680 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000)
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^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000, 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^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000);
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^15 - 99*x^13 - 396*x^12 - 3868*x^11 - 27974*x^10 + 97245*x^9 + 2044656*x^8 + 10660124*x^7 + 28900022*x^6 + 46578665*x^5 + 46949760*x^4 + 29930225*x^3 + 11744850*x^2 + 2593500*x + 247000);
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
$S_5^3.S_3$ (as 15T102):
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 non-solvable group of order 10368000 |
| The 140 conjugacy class representatives for $S_5^3.S_3$ |
| Character table for $S_5^3.S_3$ |
Intermediate fields
| 3.3.148.1 |
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 18 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Degree 45 sibling: | 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | R | R | R | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | R | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.12.22.38 | $x^{12} + 8 x^{11} + 28 x^{10} + 44 x^{9} + 18 x^{8} - 32 x^{7} + 16 x^{6} + 64 x^{5} + 76 x^{4} - 24 x^{3} - 12 x^{2} + 36$ | $6$ | $2$ | $22$ | 12T50 | $[2, 8/3, 8/3, 3]_{3}^{2}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.5.4.1 | $x^{5} + 13$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.3.0.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.5.4.1 | $x^{5} + 19$ | $5$ | $1$ | $4$ | $D_{5}$ | $[\ ]_{5}^{2}$ | |
| 19.8.0.1 | $x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.6.3.1 | $x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(74483599\)
| $\Q_{74483599}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{74483599}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |