Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8)
gp: K = bnfinit(y^19 - 2*y^18 + 8*y^17 - 12*y^16 + 43*y^15 - 74*y^14 + 108*y^13 - 18*y^12 - 25*y^11 + 338*y^9 - 70*y^8 - 264*y^7 - 128*y^6 + 120*y^5 + 98*y^4 + 43*y^3 - 22*y^2 - 31*y - 8, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8)
\( x^{19} - 2 x^{18} + 8 x^{17} - 12 x^{16} + 43 x^{15} - 74 x^{14} + 108 x^{13} - 18 x^{12} - 25 x^{11} + \cdots - 8 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $19$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-75613185918270483380568064\)
\(\medspace = -\,2^{18}\cdot 19^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.02\) | 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: | $2\cdot 19^{8/9}\approx 27.39680150407798$ | ||
| Ramified primes: |
\(2\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{39\!\cdots\!99}a^{18}-\frac{12\!\cdots\!02}{39\!\cdots\!99}a^{17}-\frac{97\!\cdots\!49}{39\!\cdots\!99}a^{16}-\frac{92\!\cdots\!12}{39\!\cdots\!99}a^{15}-\frac{85\!\cdots\!04}{39\!\cdots\!99}a^{14}-\frac{37\!\cdots\!74}{39\!\cdots\!99}a^{13}-\frac{19\!\cdots\!04}{39\!\cdots\!99}a^{12}-\frac{58\!\cdots\!15}{39\!\cdots\!99}a^{11}-\frac{86\!\cdots\!63}{39\!\cdots\!99}a^{10}+\frac{17\!\cdots\!46}{39\!\cdots\!99}a^{9}-\frac{56\!\cdots\!80}{39\!\cdots\!99}a^{8}-\frac{13\!\cdots\!59}{39\!\cdots\!99}a^{7}-\frac{10\!\cdots\!46}{39\!\cdots\!99}a^{6}-\frac{21\!\cdots\!42}{39\!\cdots\!99}a^{5}+\frac{17\!\cdots\!59}{39\!\cdots\!99}a^{4}-\frac{61\!\cdots\!97}{39\!\cdots\!99}a^{3}+\frac{17\!\cdots\!32}{39\!\cdots\!99}a^{2}+\frac{24\!\cdots\!96}{39\!\cdots\!99}a-\frac{16\!\cdots\!50}{39\!\cdots\!99}$
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 $1$
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: | $9$ | 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: |
$\frac{75\!\cdots\!22}{39\!\cdots\!99}a^{18}-\frac{54\!\cdots\!48}{39\!\cdots\!99}a^{17}+\frac{30\!\cdots\!58}{39\!\cdots\!99}a^{16}+\frac{15\!\cdots\!14}{39\!\cdots\!99}a^{15}+\frac{11\!\cdots\!88}{39\!\cdots\!99}a^{14}+\frac{36\!\cdots\!60}{39\!\cdots\!99}a^{13}-\frac{41\!\cdots\!31}{39\!\cdots\!99}a^{12}+\frac{19\!\cdots\!01}{39\!\cdots\!99}a^{11}-\frac{18\!\cdots\!86}{39\!\cdots\!99}a^{10}+\frac{40\!\cdots\!86}{39\!\cdots\!99}a^{9}+\frac{33\!\cdots\!85}{39\!\cdots\!99}a^{8}+\frac{21\!\cdots\!97}{39\!\cdots\!99}a^{7}-\frac{60\!\cdots\!29}{39\!\cdots\!99}a^{6}-\frac{19\!\cdots\!65}{39\!\cdots\!99}a^{5}+\frac{34\!\cdots\!70}{39\!\cdots\!99}a^{4}+\frac{25\!\cdots\!28}{39\!\cdots\!99}a^{3}-\frac{12\!\cdots\!62}{39\!\cdots\!99}a^{2}+\frac{17\!\cdots\!07}{39\!\cdots\!99}a+\frac{27\!\cdots\!27}{39\!\cdots\!99}$, $\frac{12\!\cdots\!06}{39\!\cdots\!99}a^{18}-\frac{35\!\cdots\!08}{39\!\cdots\!99}a^{17}+\frac{12\!\cdots\!56}{39\!\cdots\!99}a^{16}-\frac{23\!\cdots\!39}{39\!\cdots\!99}a^{15}+\frac{66\!\cdots\!46}{39\!\cdots\!99}a^{14}-\frac{14\!\cdots\!09}{39\!\cdots\!99}a^{13}+\frac{22\!\cdots\!93}{39\!\cdots\!99}a^{12}-\frac{16\!\cdots\!10}{39\!\cdots\!99}a^{11}+\frac{32\!\cdots\!58}{39\!\cdots\!99}a^{10}-\frac{71\!\cdots\!55}{39\!\cdots\!99}a^{9}+\frac{53\!\cdots\!78}{39\!\cdots\!99}a^{8}-\frac{59\!\cdots\!90}{39\!\cdots\!99}a^{7}-\frac{18\!\cdots\!10}{39\!\cdots\!99}a^{6}-\frac{67\!\cdots\!87}{39\!\cdots\!99}a^{5}+\frac{36\!\cdots\!19}{39\!\cdots\!99}a^{4}-\frac{67\!\cdots\!82}{39\!\cdots\!99}a^{3}+\frac{13\!\cdots\!29}{39\!\cdots\!99}a^{2}-\frac{72\!\cdots\!46}{39\!\cdots\!99}a-\frac{41\!\cdots\!27}{39\!\cdots\!99}$, $\frac{46\!\cdots\!10}{39\!\cdots\!99}a^{18}-\frac{21\!\cdots\!91}{39\!\cdots\!99}a^{17}+\frac{64\!\cdots\!42}{39\!\cdots\!99}a^{16}-\frac{16\!\cdots\!10}{39\!\cdots\!99}a^{15}+\frac{37\!\cdots\!46}{39\!\cdots\!99}a^{14}-\frac{94\!\cdots\!46}{39\!\cdots\!99}a^{13}+\frac{15\!\cdots\!11}{39\!\cdots\!99}a^{12}-\frac{17\!\cdots\!34}{39\!\cdots\!99}a^{11}+\frac{64\!\cdots\!65}{39\!\cdots\!99}a^{10}-\frac{13\!\cdots\!17}{39\!\cdots\!99}a^{9}+\frac{15\!\cdots\!58}{39\!\cdots\!99}a^{8}-\frac{43\!\cdots\!43}{39\!\cdots\!99}a^{7}+\frac{36\!\cdots\!27}{39\!\cdots\!99}a^{6}+\frac{10\!\cdots\!35}{39\!\cdots\!99}a^{5}+\frac{20\!\cdots\!65}{39\!\cdots\!99}a^{4}-\frac{73\!\cdots\!25}{39\!\cdots\!99}a^{3}-\frac{31\!\cdots\!92}{39\!\cdots\!99}a^{2}-\frac{69\!\cdots\!56}{39\!\cdots\!99}a-\frac{36\!\cdots\!87}{39\!\cdots\!99}$, $\frac{77\!\cdots\!54}{39\!\cdots\!99}a^{18}-\frac{11\!\cdots\!06}{39\!\cdots\!99}a^{17}+\frac{18\!\cdots\!91}{39\!\cdots\!99}a^{16}-\frac{85\!\cdots\!73}{39\!\cdots\!99}a^{15}+\frac{11\!\cdots\!66}{39\!\cdots\!99}a^{14}-\frac{48\!\cdots\!88}{39\!\cdots\!99}a^{13}+\frac{60\!\cdots\!86}{39\!\cdots\!99}a^{12}-\frac{98\!\cdots\!17}{39\!\cdots\!99}a^{11}-\frac{32\!\cdots\!16}{39\!\cdots\!99}a^{10}-\frac{65\!\cdots\!32}{39\!\cdots\!99}a^{9}+\frac{49\!\cdots\!90}{39\!\cdots\!99}a^{8}-\frac{36\!\cdots\!81}{39\!\cdots\!99}a^{7}-\frac{22\!\cdots\!21}{39\!\cdots\!99}a^{6}-\frac{26\!\cdots\!83}{39\!\cdots\!99}a^{5}+\frac{24\!\cdots\!17}{39\!\cdots\!99}a^{4}+\frac{51\!\cdots\!46}{39\!\cdots\!99}a^{3}-\frac{12\!\cdots\!84}{39\!\cdots\!99}a^{2}-\frac{64\!\cdots\!05}{39\!\cdots\!99}a-\frac{23\!\cdots\!75}{39\!\cdots\!99}$, $\frac{16\!\cdots\!85}{39\!\cdots\!99}a^{18}-\frac{36\!\cdots\!30}{39\!\cdots\!99}a^{17}+\frac{14\!\cdots\!68}{39\!\cdots\!99}a^{16}-\frac{22\!\cdots\!64}{39\!\cdots\!99}a^{15}+\frac{78\!\cdots\!70}{39\!\cdots\!99}a^{14}-\frac{13\!\cdots\!78}{39\!\cdots\!99}a^{13}+\frac{21\!\cdots\!57}{39\!\cdots\!99}a^{12}-\frac{61\!\cdots\!45}{39\!\cdots\!99}a^{11}-\frac{44\!\cdots\!05}{39\!\cdots\!99}a^{10}+\frac{11\!\cdots\!13}{39\!\cdots\!99}a^{9}+\frac{48\!\cdots\!65}{39\!\cdots\!99}a^{8}-\frac{15\!\cdots\!87}{39\!\cdots\!99}a^{7}-\frac{22\!\cdots\!55}{39\!\cdots\!99}a^{6}+\frac{12\!\cdots\!64}{39\!\cdots\!99}a^{5}+\frac{91\!\cdots\!92}{39\!\cdots\!99}a^{4}+\frac{54\!\cdots\!75}{39\!\cdots\!99}a^{3}+\frac{32\!\cdots\!05}{39\!\cdots\!99}a^{2}-\frac{75\!\cdots\!88}{39\!\cdots\!99}a-\frac{16\!\cdots\!65}{39\!\cdots\!99}$, $\frac{98\!\cdots\!32}{39\!\cdots\!99}a^{18}-\frac{11\!\cdots\!00}{39\!\cdots\!99}a^{17}+\frac{52\!\cdots\!26}{39\!\cdots\!99}a^{16}-\frac{24\!\cdots\!31}{39\!\cdots\!99}a^{15}+\frac{23\!\cdots\!25}{39\!\cdots\!99}a^{14}-\frac{19\!\cdots\!78}{39\!\cdots\!99}a^{13}-\frac{71\!\cdots\!38}{39\!\cdots\!99}a^{12}+\frac{17\!\cdots\!90}{39\!\cdots\!99}a^{11}-\frac{19\!\cdots\!55}{39\!\cdots\!99}a^{10}+\frac{77\!\cdots\!32}{39\!\cdots\!99}a^{9}+\frac{32\!\cdots\!05}{39\!\cdots\!99}a^{8}+\frac{22\!\cdots\!75}{39\!\cdots\!99}a^{7}-\frac{66\!\cdots\!24}{39\!\cdots\!99}a^{6}-\frac{74\!\cdots\!22}{39\!\cdots\!99}a^{5}+\frac{21\!\cdots\!64}{39\!\cdots\!99}a^{4}+\frac{14\!\cdots\!62}{39\!\cdots\!99}a^{3}-\frac{10\!\cdots\!76}{39\!\cdots\!99}a^{2}+\frac{37\!\cdots\!95}{39\!\cdots\!99}a-\frac{80\!\cdots\!13}{39\!\cdots\!99}$, $\frac{32\!\cdots\!18}{39\!\cdots\!99}a^{18}-\frac{80\!\cdots\!92}{39\!\cdots\!99}a^{17}+\frac{30\!\cdots\!01}{39\!\cdots\!99}a^{16}-\frac{54\!\cdots\!70}{39\!\cdots\!99}a^{15}+\frac{17\!\cdots\!95}{39\!\cdots\!99}a^{14}-\frac{32\!\cdots\!56}{39\!\cdots\!99}a^{13}+\frac{53\!\cdots\!74}{39\!\cdots\!99}a^{12}-\frac{34\!\cdots\!37}{39\!\cdots\!99}a^{11}+\frac{15\!\cdots\!73}{39\!\cdots\!99}a^{10}-\frac{87\!\cdots\!61}{39\!\cdots\!99}a^{9}+\frac{11\!\cdots\!08}{39\!\cdots\!99}a^{8}-\frac{71\!\cdots\!23}{39\!\cdots\!99}a^{7}-\frac{36\!\cdots\!17}{39\!\cdots\!99}a^{6}-\frac{27\!\cdots\!69}{39\!\cdots\!99}a^{5}+\frac{43\!\cdots\!95}{39\!\cdots\!99}a^{4}+\frac{10\!\cdots\!03}{39\!\cdots\!99}a^{3}+\frac{79\!\cdots\!65}{39\!\cdots\!99}a^{2}-\frac{84\!\cdots\!42}{39\!\cdots\!99}a-\frac{37\!\cdots\!37}{39\!\cdots\!99}$, $\frac{18\!\cdots\!04}{39\!\cdots\!99}a^{18}-\frac{98\!\cdots\!68}{39\!\cdots\!99}a^{17}+\frac{25\!\cdots\!82}{39\!\cdots\!99}a^{16}-\frac{64\!\cdots\!17}{39\!\cdots\!99}a^{15}+\frac{13\!\cdots\!24}{39\!\cdots\!99}a^{14}-\frac{36\!\cdots\!15}{39\!\cdots\!99}a^{13}+\frac{57\!\cdots\!58}{39\!\cdots\!99}a^{12}-\frac{47\!\cdots\!94}{39\!\cdots\!99}a^{11}-\frac{23\!\cdots\!70}{39\!\cdots\!99}a^{10}+\frac{38\!\cdots\!91}{39\!\cdots\!99}a^{9}+\frac{92\!\cdots\!27}{39\!\cdots\!99}a^{8}-\frac{22\!\cdots\!02}{39\!\cdots\!99}a^{7}-\frac{92\!\cdots\!52}{39\!\cdots\!99}a^{6}+\frac{35\!\cdots\!37}{39\!\cdots\!99}a^{5}+\frac{15\!\cdots\!51}{39\!\cdots\!99}a^{4}-\frac{16\!\cdots\!28}{39\!\cdots\!99}a^{3}-\frac{13\!\cdots\!89}{39\!\cdots\!99}a^{2}+\frac{28\!\cdots\!34}{39\!\cdots\!99}a+\frac{20\!\cdots\!29}{39\!\cdots\!99}$, $\frac{38\!\cdots\!95}{39\!\cdots\!99}a^{18}-\frac{86\!\cdots\!16}{39\!\cdots\!99}a^{17}+\frac{33\!\cdots\!22}{39\!\cdots\!99}a^{16}-\frac{53\!\cdots\!58}{39\!\cdots\!99}a^{15}+\frac{17\!\cdots\!35}{39\!\cdots\!99}a^{14}-\frac{32\!\cdots\!88}{39\!\cdots\!99}a^{13}+\frac{48\!\cdots\!59}{39\!\cdots\!99}a^{12}-\frac{16\!\cdots\!49}{39\!\cdots\!99}a^{11}-\frac{93\!\cdots\!20}{39\!\cdots\!99}a^{10}+\frac{62\!\cdots\!99}{39\!\cdots\!99}a^{9}+\frac{13\!\cdots\!19}{39\!\cdots\!99}a^{8}-\frac{63\!\cdots\!45}{39\!\cdots\!99}a^{7}-\frac{84\!\cdots\!20}{39\!\cdots\!99}a^{6}-\frac{15\!\cdots\!66}{39\!\cdots\!99}a^{5}+\frac{49\!\cdots\!29}{39\!\cdots\!99}a^{4}+\frac{14\!\cdots\!56}{39\!\cdots\!99}a^{3}+\frac{10\!\cdots\!34}{39\!\cdots\!99}a^{2}-\frac{10\!\cdots\!99}{39\!\cdots\!99}a-\frac{74\!\cdots\!09}{39\!\cdots\!99}$
| 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: | \( 540383.098056 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 540383.098056 \cdot 1}{2\cdot\sqrt{75613185918270483380568064}}\cr\approx \mathstrut & 0.948466079184 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8)
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^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8, 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^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8);
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^19 - 2*x^18 + 8*x^17 - 12*x^16 + 43*x^15 - 74*x^14 + 108*x^13 - 18*x^12 - 25*x^11 + 338*x^9 - 70*x^8 - 264*x^7 - 128*x^6 + 120*x^5 + 98*x^4 + 43*x^3 - 22*x^2 - 31*x - 8);
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 342 |
| The 19 conjugacy class representatives for $F_{19}$ |
| Character table for $F_{19}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | R | $18{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.9.0.1}{9} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | $18{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $19$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.18.18.115 | $x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4320 x^{14} + 16128 x^{13} + 53696 x^{12} + 165120 x^{11} + 449824 x^{10} + 1006400 x^{9} + 1826368 x^{8} + 2905088 x^{7} + 3317760 x^{6} - 418816 x^{5} - 6684672 x^{4} - 4984832 x^{3} + 2483456 x^{2} + 3566080 x + 1829376$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |