Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 - 18*x^14 + 51*x^12 + 72*x^11 + 72*x^10 + 36*x^9 + 72*x^8 + 90*x^7 + 75*x^6 + 72*x^5 + 54*x^4 - 18*x^2 + 3)
gp: K = bnfinit(y^18 - 12*y^15 - 18*y^14 + 51*y^12 + 72*y^11 + 72*y^10 + 36*y^9 + 72*y^8 + 90*y^7 + 75*y^6 + 72*y^5 + 54*y^4 - 18*y^2 + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12*x^15 - 18*x^14 + 51*x^12 + 72*x^11 + 72*x^10 + 36*x^9 + 72*x^8 + 90*x^7 + 75*x^6 + 72*x^5 + 54*x^4 - 18*x^2 + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 12*x^15 - 18*x^14 + 51*x^12 + 72*x^11 + 72*x^10 + 36*x^9 + 72*x^8 + 90*x^7 + 75*x^6 + 72*x^5 + 54*x^4 - 18*x^2 + 3)
\( x^{18} - 12 x^{15} - 18 x^{14} + 51 x^{12} + 72 x^{11} + 72 x^{10} + 36 x^{9} + 72 x^{8} + 90 x^{7} + \cdots + 3 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-118039224225889612726272\)
\(\medspace = -\,2^{18}\cdot 3^{37}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.13\) | 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: | $2^{7/6}3^{37/18}\approx 21.475876974362272$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{63\!\cdots\!85}a^{17}-\frac{31\!\cdots\!94}{63\!\cdots\!85}a^{16}-\frac{24\!\cdots\!64}{63\!\cdots\!85}a^{15}-\frac{24\!\cdots\!91}{63\!\cdots\!85}a^{14}-\frac{88\!\cdots\!84}{63\!\cdots\!85}a^{13}-\frac{18\!\cdots\!44}{63\!\cdots\!85}a^{12}-\frac{27\!\cdots\!23}{63\!\cdots\!85}a^{11}-\frac{11\!\cdots\!41}{63\!\cdots\!85}a^{10}+\frac{25\!\cdots\!06}{63\!\cdots\!85}a^{9}+\frac{177509360287722}{57\!\cdots\!35}a^{8}-\frac{19\!\cdots\!96}{63\!\cdots\!85}a^{7}+\frac{12\!\cdots\!09}{63\!\cdots\!85}a^{6}+\frac{13\!\cdots\!24}{63\!\cdots\!85}a^{5}+\frac{22\!\cdots\!31}{63\!\cdots\!85}a^{4}+\frac{48\!\cdots\!99}{12\!\cdots\!97}a^{3}+\frac{63\!\cdots\!99}{12\!\cdots\!97}a^{2}-\frac{27\!\cdots\!38}{63\!\cdots\!85}a-\frac{61\!\cdots\!73}{63\!\cdots\!85}$
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
$C_{2}$, which has order $2$
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: | $8$ | 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: |
$\frac{12\!\cdots\!75}{12\!\cdots\!97}a^{17}-\frac{77\!\cdots\!49}{12\!\cdots\!97}a^{16}+\frac{51\!\cdots\!92}{12\!\cdots\!97}a^{15}-\frac{15\!\cdots\!41}{12\!\cdots\!97}a^{14}-\frac{13\!\cdots\!10}{12\!\cdots\!97}a^{13}+\frac{77\!\cdots\!10}{12\!\cdots\!97}a^{12}+\frac{60\!\cdots\!01}{12\!\cdots\!97}a^{11}+\frac{57\!\cdots\!33}{12\!\cdots\!97}a^{10}+\frac{61\!\cdots\!56}{12\!\cdots\!97}a^{9}+\frac{12\!\cdots\!11}{11\!\cdots\!27}a^{8}+\frac{89\!\cdots\!60}{12\!\cdots\!97}a^{7}+\frac{66\!\cdots\!74}{12\!\cdots\!97}a^{6}+\frac{61\!\cdots\!86}{12\!\cdots\!97}a^{5}+\frac{62\!\cdots\!93}{12\!\cdots\!97}a^{4}+\frac{37\!\cdots\!56}{12\!\cdots\!97}a^{3}-\frac{17\!\cdots\!12}{12\!\cdots\!97}a^{2}-\frac{92\!\cdots\!20}{12\!\cdots\!97}a+\frac{59\!\cdots\!74}{12\!\cdots\!97}$, $\frac{40\!\cdots\!38}{12\!\cdots\!97}a^{17}-\frac{36\!\cdots\!68}{12\!\cdots\!97}a^{16}+\frac{21\!\cdots\!90}{12\!\cdots\!97}a^{15}-\frac{49\!\cdots\!58}{12\!\cdots\!97}a^{14}-\frac{26\!\cdots\!00}{12\!\cdots\!97}a^{13}+\frac{40\!\cdots\!32}{12\!\cdots\!97}a^{12}+\frac{18\!\cdots\!17}{12\!\cdots\!97}a^{11}+\frac{11\!\cdots\!86}{12\!\cdots\!97}a^{10}+\frac{11\!\cdots\!18}{12\!\cdots\!97}a^{9}-\frac{37\!\cdots\!65}{11\!\cdots\!27}a^{8}+\frac{25\!\cdots\!89}{12\!\cdots\!97}a^{7}+\frac{12\!\cdots\!48}{12\!\cdots\!97}a^{6}+\frac{12\!\cdots\!71}{12\!\cdots\!97}a^{5}+\frac{10\!\cdots\!02}{12\!\cdots\!97}a^{4}+\frac{27\!\cdots\!40}{12\!\cdots\!97}a^{3}-\frac{93\!\cdots\!43}{12\!\cdots\!97}a^{2}-\frac{19\!\cdots\!33}{12\!\cdots\!97}a+\frac{35\!\cdots\!29}{12\!\cdots\!97}$, $\frac{48\!\cdots\!86}{63\!\cdots\!85}a^{17}-\frac{31\!\cdots\!09}{63\!\cdots\!85}a^{16}+\frac{16\!\cdots\!06}{63\!\cdots\!85}a^{15}-\frac{59\!\cdots\!16}{63\!\cdots\!85}a^{14}-\frac{49\!\cdots\!09}{63\!\cdots\!85}a^{13}+\frac{35\!\cdots\!06}{63\!\cdots\!85}a^{12}+\frac{23\!\cdots\!62}{63\!\cdots\!85}a^{11}+\frac{20\!\cdots\!19}{63\!\cdots\!85}a^{10}+\frac{20\!\cdots\!41}{63\!\cdots\!85}a^{9}+\frac{17\!\cdots\!57}{57\!\cdots\!35}a^{8}+\frac{30\!\cdots\!44}{63\!\cdots\!85}a^{7}+\frac{21\!\cdots\!19}{63\!\cdots\!85}a^{6}+\frac{19\!\cdots\!19}{63\!\cdots\!85}a^{5}+\frac{19\!\cdots\!66}{63\!\cdots\!85}a^{4}+\frac{21\!\cdots\!98}{12\!\cdots\!97}a^{3}-\frac{21\!\cdots\!66}{12\!\cdots\!97}a^{2}-\frac{43\!\cdots\!73}{63\!\cdots\!85}a+\frac{23\!\cdots\!67}{63\!\cdots\!85}$, $\frac{18\!\cdots\!62}{63\!\cdots\!85}a^{17}-\frac{13\!\cdots\!88}{63\!\cdots\!85}a^{16}+\frac{90\!\cdots\!72}{63\!\cdots\!85}a^{15}-\frac{23\!\cdots\!22}{63\!\cdots\!85}a^{14}-\frac{16\!\cdots\!73}{63\!\cdots\!85}a^{13}+\frac{13\!\cdots\!22}{63\!\cdots\!85}a^{12}+\frac{86\!\cdots\!19}{63\!\cdots\!85}a^{11}+\frac{70\!\cdots\!88}{63\!\cdots\!85}a^{10}+\frac{78\!\cdots\!27}{63\!\cdots\!85}a^{9}+\frac{37\!\cdots\!99}{57\!\cdots\!35}a^{8}+\frac{12\!\cdots\!73}{63\!\cdots\!85}a^{7}+\frac{73\!\cdots\!38}{63\!\cdots\!85}a^{6}+\frac{78\!\cdots\!43}{63\!\cdots\!85}a^{5}+\frac{76\!\cdots\!87}{63\!\cdots\!85}a^{4}+\frac{85\!\cdots\!58}{12\!\cdots\!97}a^{3}-\frac{76\!\cdots\!49}{12\!\cdots\!97}a^{2}-\frac{92\!\cdots\!16}{63\!\cdots\!85}a+\frac{14\!\cdots\!59}{63\!\cdots\!85}$, $\frac{33\!\cdots\!97}{63\!\cdots\!85}a^{17}-\frac{66\!\cdots\!13}{63\!\cdots\!85}a^{16}+\frac{62\!\cdots\!22}{63\!\cdots\!85}a^{15}-\frac{45\!\cdots\!77}{63\!\cdots\!85}a^{14}+\frac{20\!\cdots\!57}{63\!\cdots\!85}a^{13}+\frac{43\!\cdots\!72}{63\!\cdots\!85}a^{12}+\frac{12\!\cdots\!54}{63\!\cdots\!85}a^{11}-\frac{29\!\cdots\!87}{63\!\cdots\!85}a^{10}+\frac{58\!\cdots\!87}{63\!\cdots\!85}a^{9}-\frac{14\!\cdots\!96}{57\!\cdots\!35}a^{8}+\frac{19\!\cdots\!43}{63\!\cdots\!85}a^{7}-\frac{18\!\cdots\!12}{63\!\cdots\!85}a^{6}+\frac{62\!\cdots\!43}{63\!\cdots\!85}a^{5}+\frac{11\!\cdots\!87}{63\!\cdots\!85}a^{4}-\frac{32\!\cdots\!80}{12\!\cdots\!97}a^{3}-\frac{29\!\cdots\!29}{12\!\cdots\!97}a^{2}+\frac{49\!\cdots\!84}{63\!\cdots\!85}a+\frac{56\!\cdots\!79}{63\!\cdots\!85}$, $\frac{19\!\cdots\!01}{63\!\cdots\!85}a^{17}-\frac{51\!\cdots\!44}{63\!\cdots\!85}a^{16}-\frac{13\!\cdots\!14}{63\!\cdots\!85}a^{15}-\frac{23\!\cdots\!01}{63\!\cdots\!85}a^{14}-\frac{29\!\cdots\!84}{63\!\cdots\!85}a^{13}+\frac{10\!\cdots\!86}{63\!\cdots\!85}a^{12}+\frac{10\!\cdots\!67}{63\!\cdots\!85}a^{11}+\frac{11\!\cdots\!34}{63\!\cdots\!85}a^{10}+\frac{10\!\cdots\!31}{63\!\cdots\!85}a^{9}+\frac{24\!\cdots\!87}{57\!\cdots\!35}a^{8}+\frac{11\!\cdots\!64}{63\!\cdots\!85}a^{7}+\frac{13\!\cdots\!64}{63\!\cdots\!85}a^{6}+\frac{99\!\cdots\!69}{63\!\cdots\!85}a^{5}+\frac{89\!\cdots\!91}{63\!\cdots\!85}a^{4}+\frac{14\!\cdots\!21}{12\!\cdots\!97}a^{3}-\frac{88\!\cdots\!30}{12\!\cdots\!97}a^{2}-\frac{20\!\cdots\!53}{63\!\cdots\!85}a+\frac{11\!\cdots\!52}{63\!\cdots\!85}$, $\frac{27\!\cdots\!17}{63\!\cdots\!85}a^{17}-\frac{13\!\cdots\!08}{63\!\cdots\!85}a^{16}+\frac{96\!\cdots\!42}{63\!\cdots\!85}a^{15}-\frac{33\!\cdots\!62}{63\!\cdots\!85}a^{14}-\frac{32\!\cdots\!83}{63\!\cdots\!85}a^{13}+\frac{12\!\cdots\!07}{63\!\cdots\!85}a^{12}+\frac{12\!\cdots\!29}{63\!\cdots\!85}a^{11}+\frac{13\!\cdots\!08}{63\!\cdots\!85}a^{10}+\frac{14\!\cdots\!92}{63\!\cdots\!85}a^{9}+\frac{48\!\cdots\!44}{57\!\cdots\!35}a^{8}+\frac{19\!\cdots\!08}{63\!\cdots\!85}a^{7}+\frac{16\!\cdots\!28}{63\!\cdots\!85}a^{6}+\frac{14\!\cdots\!38}{63\!\cdots\!85}a^{5}+\frac{15\!\cdots\!52}{63\!\cdots\!85}a^{4}+\frac{19\!\cdots\!92}{12\!\cdots\!97}a^{3}-\frac{45\!\cdots\!57}{12\!\cdots\!97}a^{2}-\frac{11\!\cdots\!41}{63\!\cdots\!85}a+\frac{12\!\cdots\!49}{63\!\cdots\!85}$, $\frac{48\!\cdots\!16}{63\!\cdots\!85}a^{17}-\frac{19\!\cdots\!94}{63\!\cdots\!85}a^{16}+\frac{22\!\cdots\!71}{63\!\cdots\!85}a^{15}-\frac{59\!\cdots\!66}{63\!\cdots\!85}a^{14}-\frac{62\!\cdots\!69}{63\!\cdots\!85}a^{13}+\frac{69\!\cdots\!76}{63\!\cdots\!85}a^{12}+\frac{22\!\cdots\!32}{63\!\cdots\!85}a^{11}+\frac{26\!\cdots\!59}{63\!\cdots\!85}a^{10}+\frac{31\!\cdots\!66}{63\!\cdots\!85}a^{9}+\frac{11\!\cdots\!32}{57\!\cdots\!35}a^{8}+\frac{38\!\cdots\!69}{63\!\cdots\!85}a^{7}+\frac{32\!\cdots\!74}{63\!\cdots\!85}a^{6}+\frac{34\!\cdots\!64}{63\!\cdots\!85}a^{5}+\frac{30\!\cdots\!66}{63\!\cdots\!85}a^{4}+\frac{46\!\cdots\!89}{12\!\cdots\!97}a^{3}+\frac{17\!\cdots\!28}{12\!\cdots\!97}a^{2}-\frac{28\!\cdots\!48}{63\!\cdots\!85}a+\frac{12\!\cdots\!02}{63\!\cdots\!85}$
| 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: | \( 28479.2203221 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 28479.2203221 \cdot 2}{2\cdot\sqrt{118039224225889612726272}}\cr\approx \mathstrut & 1.26512587240 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 - 18*x^14 + 51*x^12 + 72*x^11 + 72*x^10 + 36*x^9 + 72*x^8 + 90*x^7 + 75*x^6 + 72*x^5 + 54*x^4 - 18*x^2 + 3)
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^18 - 12*x^15 - 18*x^14 + 51*x^12 + 72*x^11 + 72*x^10 + 36*x^9 + 72*x^8 + 90*x^7 + 75*x^6 + 72*x^5 + 54*x^4 - 18*x^2 + 3, 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^18 - 12*x^15 - 18*x^14 + 51*x^12 + 72*x^11 + 72*x^10 + 36*x^9 + 72*x^8 + 90*x^7 + 75*x^6 + 72*x^5 + 54*x^4 - 18*x^2 + 3);
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^18 - 12*x^15 - 18*x^14 + 51*x^12 + 72*x^11 + 72*x^10 + 36*x^9 + 72*x^8 + 90*x^7 + 75*x^6 + 72*x^5 + 54*x^4 - 18*x^2 + 3);
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
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 solvable group of order 72 |
| The 9 conjugacy class representatives for $C_3:S_4$ |
| Character table for $C_3:S_4$ |
Intermediate fields
| 3.1.972.2, 3.1.243.1, 3.1.108.1, 3.1.972.1, 6.0.11337408.4, 9.1.24794911296.1 |
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 12 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.2.171382426877952.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.8 | $x^{6} + 2 x + 2$ | $6$ | $1$ | $6$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
| 2.12.12.28 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 130 x^{7} + 159 x^{6} + 132 x^{5} + 10 x^{4} - 100 x^{3} - 53 x^{2} + 22 x + 19$ | $6$ | $2$ | $12$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ | |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $37$ |