Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717)
gp: K = bnfinit(y^18 - y^17 + 20*y^16 - 21*y^15 + 120*y^14 - 142*y^13 + 174*y^12 - 302*y^11 - 405*y^10 - 1196*y^9 + 785*y^8 - 6238*y^7 + 9767*y^6 + 981*y^5 - 3327*y^4 + 13403*y^3 + 21875*y^2 + 620*y + 22717, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717)
\( x^{18} - x^{17} + 20 x^{16} - 21 x^{15} + 120 x^{14} - 142 x^{13} + 174 x^{12} - 302 x^{11} + \cdots + 22717 \)
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: |
\(-8985052139278849963819823767311\)
\(\medspace = -\,3^{9}\cdot 37^{17}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.44\) | 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: | $3^{1/2}37^{17/18}\approx 52.43726518793858$ | ||
| Ramified primes: |
\(3\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-111}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(111=3\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(70,·)$, $\chi_{111}(65,·)$, $\chi_{111}(10,·)$, $\chi_{111}(11,·)$, $\chi_{111}(77,·)$, $\chi_{111}(16,·)$, $\chi_{111}(46,·)$, $\chi_{111}(95,·)$, $\chi_{111}(34,·)$, $\chi_{111}(100,·)$, $\chi_{111}(101,·)$, $\chi_{111}(104,·)$, $\chi_{111}(41,·)$, $\chi_{111}(7,·)$, $\chi_{111}(110,·)$, $\chi_{111}(49,·)$, $\chi_{111}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{74\!\cdots\!09}a^{17}-\frac{24\!\cdots\!11}{74\!\cdots\!09}a^{16}+\frac{77\!\cdots\!35}{74\!\cdots\!09}a^{15}-\frac{15\!\cdots\!70}{74\!\cdots\!09}a^{14}+\frac{23\!\cdots\!55}{74\!\cdots\!09}a^{13}+\frac{45\!\cdots\!90}{74\!\cdots\!09}a^{12}-\frac{36\!\cdots\!51}{74\!\cdots\!09}a^{11}+\frac{20\!\cdots\!33}{74\!\cdots\!09}a^{10}+\frac{19\!\cdots\!31}{74\!\cdots\!09}a^{9}-\frac{12\!\cdots\!63}{74\!\cdots\!09}a^{8}+\frac{12\!\cdots\!92}{74\!\cdots\!09}a^{7}-\frac{48\!\cdots\!46}{74\!\cdots\!09}a^{6}+\frac{26\!\cdots\!76}{74\!\cdots\!09}a^{5}-\frac{65\!\cdots\!14}{74\!\cdots\!09}a^{4}+\frac{30\!\cdots\!20}{74\!\cdots\!09}a^{3}-\frac{56\!\cdots\!14}{74\!\cdots\!09}a^{2}-\frac{36\!\cdots\!79}{74\!\cdots\!09}a-\frac{24\!\cdots\!86}{74\!\cdots\!09}$
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
$C_{152}$, which has order $152$ (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: | $8$ | 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: |
$\frac{64\!\cdots\!16}{74\!\cdots\!09}a^{17}-\frac{17\!\cdots\!54}{74\!\cdots\!09}a^{16}+\frac{16\!\cdots\!77}{74\!\cdots\!09}a^{15}-\frac{34\!\cdots\!51}{74\!\cdots\!09}a^{14}+\frac{18\!\cdots\!62}{74\!\cdots\!09}a^{13}-\frac{20\!\cdots\!53}{74\!\cdots\!09}a^{12}+\frac{11\!\cdots\!22}{74\!\cdots\!09}a^{11}-\frac{25\!\cdots\!03}{74\!\cdots\!09}a^{10}+\frac{35\!\cdots\!84}{74\!\cdots\!09}a^{9}+\frac{65\!\cdots\!92}{74\!\cdots\!09}a^{8}+\frac{28\!\cdots\!06}{74\!\cdots\!09}a^{7}-\frac{77\!\cdots\!24}{74\!\cdots\!09}a^{6}+\frac{14\!\cdots\!89}{74\!\cdots\!09}a^{5}-\frac{10\!\cdots\!57}{74\!\cdots\!09}a^{4}-\frac{17\!\cdots\!04}{74\!\cdots\!09}a^{3}+\frac{27\!\cdots\!76}{74\!\cdots\!09}a^{2}-\frac{14\!\cdots\!41}{74\!\cdots\!09}a+\frac{60\!\cdots\!95}{74\!\cdots\!09}$, $\frac{23\!\cdots\!21}{74\!\cdots\!09}a^{17}+\frac{35\!\cdots\!58}{74\!\cdots\!09}a^{16}+\frac{41\!\cdots\!71}{74\!\cdots\!09}a^{15}+\frac{66\!\cdots\!10}{74\!\cdots\!09}a^{14}+\frac{18\!\cdots\!23}{74\!\cdots\!09}a^{13}+\frac{29\!\cdots\!98}{74\!\cdots\!09}a^{12}-\frac{31\!\cdots\!60}{74\!\cdots\!09}a^{11}-\frac{40\!\cdots\!77}{74\!\cdots\!09}a^{10}-\frac{24\!\cdots\!95}{74\!\cdots\!09}a^{9}-\frac{69\!\cdots\!38}{74\!\cdots\!09}a^{8}-\frac{35\!\cdots\!55}{74\!\cdots\!09}a^{7}-\frac{99\!\cdots\!79}{74\!\cdots\!09}a^{6}+\frac{73\!\cdots\!56}{74\!\cdots\!09}a^{5}+\frac{56\!\cdots\!60}{74\!\cdots\!09}a^{4}+\frac{47\!\cdots\!34}{74\!\cdots\!09}a^{3}+\frac{82\!\cdots\!45}{74\!\cdots\!09}a^{2}+\frac{47\!\cdots\!75}{74\!\cdots\!09}a+\frac{13\!\cdots\!44}{74\!\cdots\!09}$, $\frac{21\!\cdots\!86}{74\!\cdots\!09}a^{17}+\frac{13\!\cdots\!03}{74\!\cdots\!09}a^{16}+\frac{13\!\cdots\!31}{74\!\cdots\!09}a^{15}+\frac{24\!\cdots\!22}{74\!\cdots\!09}a^{14}-\frac{34\!\cdots\!56}{74\!\cdots\!09}a^{13}+\frac{10\!\cdots\!61}{74\!\cdots\!09}a^{12}-\frac{33\!\cdots\!04}{74\!\cdots\!09}a^{11}-\frac{55\!\cdots\!85}{74\!\cdots\!09}a^{10}-\frac{56\!\cdots\!32}{74\!\cdots\!09}a^{9}-\frac{10\!\cdots\!57}{74\!\cdots\!09}a^{8}-\frac{25\!\cdots\!25}{74\!\cdots\!09}a^{7}+\frac{28\!\cdots\!91}{74\!\cdots\!09}a^{6}-\frac{18\!\cdots\!52}{74\!\cdots\!09}a^{5}+\frac{28\!\cdots\!34}{74\!\cdots\!09}a^{4}+\frac{25\!\cdots\!10}{74\!\cdots\!09}a^{3}-\frac{67\!\cdots\!31}{74\!\cdots\!09}a^{2}+\frac{25\!\cdots\!67}{74\!\cdots\!09}a+\frac{28\!\cdots\!49}{74\!\cdots\!09}$, $\frac{61\!\cdots\!40}{74\!\cdots\!09}a^{17}-\frac{56\!\cdots\!89}{74\!\cdots\!09}a^{16}+\frac{12\!\cdots\!36}{74\!\cdots\!09}a^{15}-\frac{30\!\cdots\!05}{74\!\cdots\!09}a^{14}+\frac{70\!\cdots\!21}{74\!\cdots\!09}a^{13}-\frac{45\!\cdots\!74}{74\!\cdots\!09}a^{12}+\frac{68\!\cdots\!31}{74\!\cdots\!09}a^{11}-\frac{22\!\cdots\!23}{74\!\cdots\!09}a^{10}-\frac{43\!\cdots\!06}{74\!\cdots\!09}a^{9}-\frac{10\!\cdots\!62}{74\!\cdots\!09}a^{8}-\frac{44\!\cdots\!05}{74\!\cdots\!09}a^{7}-\frac{28\!\cdots\!37}{74\!\cdots\!09}a^{6}+\frac{47\!\cdots\!91}{74\!\cdots\!09}a^{5}+\frac{52\!\cdots\!17}{74\!\cdots\!09}a^{4}+\frac{65\!\cdots\!88}{74\!\cdots\!09}a^{3}+\frac{10\!\cdots\!15}{74\!\cdots\!09}a^{2}+\frac{24\!\cdots\!21}{74\!\cdots\!09}a+\frac{64\!\cdots\!90}{74\!\cdots\!09}$, $\frac{39\!\cdots\!30}{74\!\cdots\!09}a^{17}-\frac{49\!\cdots\!28}{74\!\cdots\!09}a^{16}+\frac{83\!\cdots\!17}{74\!\cdots\!09}a^{15}-\frac{10\!\cdots\!49}{74\!\cdots\!09}a^{14}+\frac{54\!\cdots\!96}{74\!\cdots\!09}a^{13}-\frac{72\!\cdots\!38}{74\!\cdots\!09}a^{12}+\frac{10\!\cdots\!84}{74\!\cdots\!09}a^{11}-\frac{17\!\cdots\!45}{74\!\cdots\!09}a^{10}-\frac{18\!\cdots\!64}{74\!\cdots\!09}a^{9}-\frac{42\!\cdots\!73}{74\!\cdots\!09}a^{8}+\frac{11\!\cdots\!11}{74\!\cdots\!09}a^{7}-\frac{22\!\cdots\!04}{74\!\cdots\!09}a^{6}+\frac{48\!\cdots\!74}{74\!\cdots\!09}a^{5}-\frac{84\!\cdots\!04}{74\!\cdots\!09}a^{4}+\frac{11\!\cdots\!52}{74\!\cdots\!09}a^{3}+\frac{96\!\cdots\!14}{74\!\cdots\!09}a^{2}-\frac{28\!\cdots\!38}{74\!\cdots\!09}a+\frac{36\!\cdots\!95}{74\!\cdots\!09}$, $\frac{22\!\cdots\!55}{74\!\cdots\!09}a^{17}+\frac{27\!\cdots\!41}{74\!\cdots\!09}a^{16}+\frac{43\!\cdots\!12}{74\!\cdots\!09}a^{15}+\frac{49\!\cdots\!42}{74\!\cdots\!09}a^{14}+\frac{21\!\cdots\!36}{74\!\cdots\!09}a^{13}+\frac{18\!\cdots\!90}{74\!\cdots\!09}a^{12}-\frac{11\!\cdots\!77}{74\!\cdots\!09}a^{11}-\frac{57\!\cdots\!86}{74\!\cdots\!09}a^{10}-\frac{26\!\cdots\!12}{74\!\cdots\!09}a^{9}-\frac{57\!\cdots\!85}{74\!\cdots\!09}a^{8}-\frac{53\!\cdots\!11}{74\!\cdots\!09}a^{7}-\frac{74\!\cdots\!30}{74\!\cdots\!09}a^{6}+\frac{72\!\cdots\!77}{74\!\cdots\!09}a^{5}+\frac{54\!\cdots\!37}{74\!\cdots\!09}a^{4}+\frac{49\!\cdots\!90}{74\!\cdots\!09}a^{3}+\frac{10\!\cdots\!59}{74\!\cdots\!09}a^{2}+\frac{47\!\cdots\!97}{74\!\cdots\!09}a+\frac{53\!\cdots\!34}{74\!\cdots\!09}$, $\frac{29\!\cdots\!48}{74\!\cdots\!09}a^{17}-\frac{19\!\cdots\!33}{74\!\cdots\!09}a^{16}+\frac{61\!\cdots\!54}{74\!\cdots\!09}a^{15}-\frac{45\!\cdots\!25}{74\!\cdots\!09}a^{14}+\frac{37\!\cdots\!13}{74\!\cdots\!09}a^{13}-\frac{34\!\cdots\!84}{74\!\cdots\!09}a^{12}+\frac{57\!\cdots\!99}{74\!\cdots\!09}a^{11}-\frac{94\!\cdots\!79}{74\!\cdots\!09}a^{10}-\frac{13\!\cdots\!16}{74\!\cdots\!09}a^{9}-\frac{41\!\cdots\!37}{74\!\cdots\!09}a^{8}+\frac{76\!\cdots\!53}{74\!\cdots\!09}a^{7}-\frac{21\!\cdots\!85}{74\!\cdots\!09}a^{6}+\frac{27\!\cdots\!11}{74\!\cdots\!09}a^{5}-\frac{12\!\cdots\!96}{74\!\cdots\!09}a^{4}+\frac{18\!\cdots\!59}{74\!\cdots\!09}a^{3}+\frac{48\!\cdots\!79}{74\!\cdots\!09}a^{2}-\frac{15\!\cdots\!21}{74\!\cdots\!09}a+\frac{13\!\cdots\!37}{74\!\cdots\!09}$, $\frac{30\!\cdots\!64}{74\!\cdots\!09}a^{17}-\frac{37\!\cdots\!87}{74\!\cdots\!09}a^{16}+\frac{62\!\cdots\!31}{74\!\cdots\!09}a^{15}-\frac{79\!\cdots\!76}{74\!\cdots\!09}a^{14}+\frac{39\!\cdots\!75}{74\!\cdots\!09}a^{13}-\frac{55\!\cdots\!37}{74\!\cdots\!09}a^{12}+\frac{69\!\cdots\!21}{74\!\cdots\!09}a^{11}-\frac{12\!\cdots\!82}{74\!\cdots\!09}a^{10}-\frac{98\!\cdots\!32}{74\!\cdots\!09}a^{9}-\frac{35\!\cdots\!45}{74\!\cdots\!09}a^{8}+\frac{35\!\cdots\!59}{74\!\cdots\!09}a^{7}-\frac{21\!\cdots\!09}{74\!\cdots\!09}a^{6}+\frac{41\!\cdots\!00}{74\!\cdots\!09}a^{5}-\frac{12\!\cdots\!53}{74\!\cdots\!09}a^{4}+\frac{11\!\cdots\!55}{74\!\cdots\!09}a^{3}+\frac{75\!\cdots\!55}{74\!\cdots\!09}a^{2}-\frac{30\!\cdots\!62}{74\!\cdots\!09}a+\frac{93\!\cdots\!41}{74\!\cdots\!09}$
| 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: | \( 409151.310213 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 409151.310213 \cdot 152}{2\cdot\sqrt{8985052139278849963819823767311}}\cr\approx \mathstrut & 0.158327390139 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717)
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 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717, 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 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717);
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 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717);
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 cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-111}) \), 3.3.1369.1, 6.0.1872286839.1, 9.9.3512479453921.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | R | $18$ | ${\href{/padicField/43.2.0.1}{2} }^{9}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $2$ | $9$ | $9$ | |||
|
\(37\)
| 37.18.17.1 | $x^{18} + 37$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |