Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 3*x^15 + 90*x^14 - 66*x^13 - 198*x^12 - 231*x^11 + 891*x^10 + 55*x^9 - 276*x^8 - 1461*x^7 + 2190*x^6 - 1323*x^5 + 21*x^4 - 2289*x^3 + 3759*x^2 - 1449*x + 1393)
gp: K = bnfinit(y^18 - 3*y^17 - 6*y^16 + 3*y^15 + 90*y^14 - 66*y^13 - 198*y^12 - 231*y^11 + 891*y^10 + 55*y^9 - 276*y^8 - 1461*y^7 + 2190*y^6 - 1323*y^5 + 21*y^4 - 2289*y^3 + 3759*y^2 - 1449*y + 1393, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 6*x^16 + 3*x^15 + 90*x^14 - 66*x^13 - 198*x^12 - 231*x^11 + 891*x^10 + 55*x^9 - 276*x^8 - 1461*x^7 + 2190*x^6 - 1323*x^5 + 21*x^4 - 2289*x^3 + 3759*x^2 - 1449*x + 1393);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 6*x^16 + 3*x^15 + 90*x^14 - 66*x^13 - 198*x^12 - 231*x^11 + 891*x^10 + 55*x^9 - 276*x^8 - 1461*x^7 + 2190*x^6 - 1323*x^5 + 21*x^4 - 2289*x^3 + 3759*x^2 - 1449*x + 1393)
\( x^{18} - 3 x^{17} - 6 x^{16} + 3 x^{15} + 90 x^{14} - 66 x^{13} - 198 x^{12} - 231 x^{11} + 891 x^{10} + \cdots + 1393 \)
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: |
\(-714659620519868840049881088\)
\(\medspace = -\,2^{12}\cdot 3^{31}\cdot 7^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.04\) | 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: | $2^{2/3}3^{97/54}7^{5/6}\approx 57.8077642622062$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13}a^{16}+\frac{5}{13}a^{15}-\frac{2}{13}a^{14}+\frac{2}{13}a^{13}-\frac{4}{13}a^{12}-\frac{1}{13}a^{11}+\frac{3}{13}a^{10}-\frac{2}{13}a^{9}-\frac{3}{13}a^{7}-\frac{1}{13}a^{6}+\frac{4}{13}a^{5}-\frac{4}{13}a^{4}-\frac{4}{13}a^{3}+\frac{3}{13}a^{2}-\frac{2}{13}a-\frac{5}{13}$, $\frac{1}{89\!\cdots\!67}a^{17}-\frac{34\!\cdots\!09}{89\!\cdots\!67}a^{16}-\frac{38\!\cdots\!15}{89\!\cdots\!67}a^{15}+\frac{44\!\cdots\!21}{89\!\cdots\!67}a^{14}+\frac{32\!\cdots\!38}{89\!\cdots\!67}a^{13}-\frac{25\!\cdots\!86}{89\!\cdots\!67}a^{12}-\frac{34\!\cdots\!46}{89\!\cdots\!67}a^{11}-\frac{43\!\cdots\!26}{89\!\cdots\!67}a^{10}-\frac{36\!\cdots\!40}{89\!\cdots\!67}a^{9}-\frac{31\!\cdots\!10}{89\!\cdots\!67}a^{8}-\frac{18\!\cdots\!01}{89\!\cdots\!67}a^{7}+\frac{21\!\cdots\!79}{89\!\cdots\!67}a^{6}+\frac{23\!\cdots\!85}{89\!\cdots\!67}a^{5}+\frac{31\!\cdots\!11}{89\!\cdots\!67}a^{4}+\frac{37\!\cdots\!24}{89\!\cdots\!67}a^{3}+\frac{50\!\cdots\!55}{89\!\cdots\!67}a^{2}-\frac{25\!\cdots\!51}{89\!\cdots\!67}a-\frac{32\!\cdots\!80}{89\!\cdots\!67}$
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
$C_{3}$, which has order $3$ (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: |
\( -\frac{2528067017622}{1429244028260119} a^{17} + \frac{1475478854085}{1429244028260119} a^{16} + \frac{29605592760862}{1429244028260119} a^{15} + \frac{36688980341220}{1429244028260119} a^{14} - \frac{212625976675278}{1429244028260119} a^{13} - \frac{340156683960569}{1429244028260119} a^{12} + \frac{538151807830758}{1429244028260119} a^{11} + \frac{1523265601661526}{1429244028260119} a^{10} - \frac{318090174733398}{1429244028260119} a^{9} - \frac{2688541850613636}{1429244028260119} a^{8} - \frac{862973626533804}{1429244028260119} a^{7} + \frac{1979506434576674}{1429244028260119} a^{6} - \frac{1143349813450698}{1429244028260119} a^{5} - \frac{2404375840676502}{1429244028260119} a^{4} + \frac{2525098701104966}{1429244028260119} a^{3} + \frac{5318464316098794}{1429244028260119} a^{2} + \frac{1686061958396424}{1429244028260119} a + \frac{3173393676903980}{1429244028260119} \)
(order $6$)
| 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: |
$\frac{15\!\cdots\!32}{89\!\cdots\!67}a^{17}-\frac{15\!\cdots\!80}{89\!\cdots\!67}a^{16}-\frac{11\!\cdots\!01}{89\!\cdots\!67}a^{15}-\frac{24\!\cdots\!58}{89\!\cdots\!67}a^{14}+\frac{10\!\cdots\!70}{89\!\cdots\!67}a^{13}+\frac{11\!\cdots\!84}{89\!\cdots\!67}a^{12}-\frac{72\!\cdots\!80}{89\!\cdots\!67}a^{11}-\frac{81\!\cdots\!72}{89\!\cdots\!67}a^{10}+\frac{26\!\cdots\!41}{89\!\cdots\!67}a^{9}+\frac{65\!\cdots\!57}{89\!\cdots\!67}a^{8}+\frac{84\!\cdots\!03}{68\!\cdots\!59}a^{7}-\frac{16\!\cdots\!42}{68\!\cdots\!59}a^{6}+\frac{17\!\cdots\!20}{89\!\cdots\!67}a^{5}+\frac{47\!\cdots\!59}{89\!\cdots\!67}a^{4}+\frac{80\!\cdots\!70}{68\!\cdots\!59}a^{3}-\frac{50\!\cdots\!48}{89\!\cdots\!67}a^{2}+\frac{67\!\cdots\!97}{68\!\cdots\!59}a-\frac{22\!\cdots\!47}{89\!\cdots\!67}$, $\frac{15\!\cdots\!32}{89\!\cdots\!67}a^{17}-\frac{15\!\cdots\!80}{89\!\cdots\!67}a^{16}-\frac{11\!\cdots\!01}{89\!\cdots\!67}a^{15}-\frac{24\!\cdots\!58}{89\!\cdots\!67}a^{14}+\frac{10\!\cdots\!70}{89\!\cdots\!67}a^{13}+\frac{11\!\cdots\!84}{89\!\cdots\!67}a^{12}-\frac{72\!\cdots\!80}{89\!\cdots\!67}a^{11}-\frac{81\!\cdots\!72}{89\!\cdots\!67}a^{10}+\frac{26\!\cdots\!41}{89\!\cdots\!67}a^{9}+\frac{65\!\cdots\!57}{89\!\cdots\!67}a^{8}+\frac{84\!\cdots\!03}{68\!\cdots\!59}a^{7}-\frac{16\!\cdots\!42}{68\!\cdots\!59}a^{6}+\frac{17\!\cdots\!20}{89\!\cdots\!67}a^{5}+\frac{47\!\cdots\!59}{89\!\cdots\!67}a^{4}+\frac{80\!\cdots\!70}{68\!\cdots\!59}a^{3}-\frac{50\!\cdots\!48}{89\!\cdots\!67}a^{2}+\frac{67\!\cdots\!97}{68\!\cdots\!59}a-\frac{13\!\cdots\!80}{89\!\cdots\!67}$, $\frac{24\!\cdots\!85}{89\!\cdots\!67}a^{17}-\frac{68\!\cdots\!93}{89\!\cdots\!67}a^{16}+\frac{15\!\cdots\!72}{89\!\cdots\!67}a^{15}+\frac{40\!\cdots\!79}{89\!\cdots\!67}a^{14}+\frac{70\!\cdots\!27}{89\!\cdots\!67}a^{13}-\frac{50\!\cdots\!37}{89\!\cdots\!67}a^{12}+\frac{26\!\cdots\!76}{89\!\cdots\!67}a^{11}+\frac{98\!\cdots\!96}{89\!\cdots\!67}a^{10}+\frac{91\!\cdots\!82}{89\!\cdots\!67}a^{9}-\frac{33\!\cdots\!44}{89\!\cdots\!67}a^{8}+\frac{47\!\cdots\!80}{89\!\cdots\!67}a^{7}+\frac{91\!\cdots\!57}{89\!\cdots\!67}a^{6}+\frac{16\!\cdots\!66}{89\!\cdots\!67}a^{5}-\frac{61\!\cdots\!11}{89\!\cdots\!67}a^{4}+\frac{51\!\cdots\!11}{89\!\cdots\!67}a^{3}+\frac{10\!\cdots\!05}{89\!\cdots\!67}a^{2}+\frac{13\!\cdots\!31}{89\!\cdots\!67}a+\frac{11\!\cdots\!32}{89\!\cdots\!67}$, $\frac{13\!\cdots\!77}{89\!\cdots\!67}a^{17}-\frac{22\!\cdots\!63}{89\!\cdots\!67}a^{16}-\frac{96\!\cdots\!52}{89\!\cdots\!67}a^{15}-\frac{12\!\cdots\!82}{89\!\cdots\!67}a^{14}+\frac{96\!\cdots\!47}{89\!\cdots\!67}a^{13}+\frac{40\!\cdots\!75}{89\!\cdots\!67}a^{12}-\frac{11\!\cdots\!30}{89\!\cdots\!67}a^{11}-\frac{52\!\cdots\!18}{89\!\cdots\!67}a^{10}+\frac{25\!\cdots\!03}{68\!\cdots\!59}a^{9}+\frac{37\!\cdots\!16}{89\!\cdots\!67}a^{8}+\frac{69\!\cdots\!64}{89\!\cdots\!67}a^{7}-\frac{12\!\cdots\!54}{89\!\cdots\!67}a^{6}+\frac{12\!\cdots\!29}{89\!\cdots\!67}a^{5}-\frac{29\!\cdots\!19}{89\!\cdots\!67}a^{4}+\frac{63\!\cdots\!99}{89\!\cdots\!67}a^{3}-\frac{31\!\cdots\!39}{89\!\cdots\!67}a^{2}+\frac{73\!\cdots\!73}{89\!\cdots\!67}a-\frac{89\!\cdots\!92}{68\!\cdots\!59}$, $\frac{75\!\cdots\!92}{89\!\cdots\!67}a^{17}-\frac{98\!\cdots\!55}{89\!\cdots\!67}a^{16}-\frac{57\!\cdots\!28}{89\!\cdots\!67}a^{15}-\frac{85\!\cdots\!19}{89\!\cdots\!67}a^{14}+\frac{51\!\cdots\!99}{89\!\cdots\!67}a^{13}+\frac{35\!\cdots\!63}{89\!\cdots\!67}a^{12}-\frac{59\!\cdots\!16}{89\!\cdots\!67}a^{11}-\frac{29\!\cdots\!57}{89\!\cdots\!67}a^{10}+\frac{15\!\cdots\!26}{89\!\cdots\!67}a^{9}+\frac{23\!\cdots\!04}{89\!\cdots\!67}a^{8}+\frac{31\!\cdots\!75}{89\!\cdots\!67}a^{7}-\frac{69\!\cdots\!86}{89\!\cdots\!67}a^{6}+\frac{69\!\cdots\!31}{89\!\cdots\!67}a^{5}+\frac{28\!\cdots\!78}{89\!\cdots\!67}a^{4}+\frac{34\!\cdots\!31}{89\!\cdots\!67}a^{3}-\frac{16\!\cdots\!66}{89\!\cdots\!67}a^{2}+\frac{41\!\cdots\!96}{89\!\cdots\!67}a-\frac{68\!\cdots\!51}{89\!\cdots\!67}$, $\frac{19\!\cdots\!34}{89\!\cdots\!67}a^{17}+\frac{49\!\cdots\!91}{89\!\cdots\!67}a^{16}-\frac{35\!\cdots\!47}{89\!\cdots\!67}a^{15}-\frac{59\!\cdots\!99}{89\!\cdots\!67}a^{14}+\frac{10\!\cdots\!97}{68\!\cdots\!59}a^{13}+\frac{60\!\cdots\!01}{89\!\cdots\!67}a^{12}-\frac{66\!\cdots\!43}{89\!\cdots\!67}a^{11}-\frac{13\!\cdots\!72}{89\!\cdots\!67}a^{10}-\frac{29\!\cdots\!22}{89\!\cdots\!67}a^{9}+\frac{36\!\cdots\!23}{89\!\cdots\!67}a^{8}-\frac{19\!\cdots\!60}{89\!\cdots\!67}a^{7}-\frac{11\!\cdots\!44}{89\!\cdots\!67}a^{6}+\frac{91\!\cdots\!75}{89\!\cdots\!67}a^{5}+\frac{62\!\cdots\!79}{89\!\cdots\!67}a^{4}-\frac{69\!\cdots\!15}{89\!\cdots\!67}a^{3}+\frac{12\!\cdots\!26}{89\!\cdots\!67}a^{2}-\frac{15\!\cdots\!68}{89\!\cdots\!67}a-\frac{14\!\cdots\!78}{89\!\cdots\!67}$, $\frac{75\!\cdots\!47}{89\!\cdots\!67}a^{17}-\frac{94\!\cdots\!25}{89\!\cdots\!67}a^{16}-\frac{52\!\cdots\!94}{89\!\cdots\!67}a^{15}-\frac{67\!\cdots\!48}{68\!\cdots\!59}a^{14}+\frac{46\!\cdots\!25}{89\!\cdots\!67}a^{13}+\frac{28\!\cdots\!48}{89\!\cdots\!67}a^{12}-\frac{35\!\cdots\!04}{89\!\cdots\!67}a^{11}-\frac{24\!\cdots\!81}{89\!\cdots\!67}a^{10}+\frac{11\!\cdots\!76}{89\!\cdots\!67}a^{9}+\frac{99\!\cdots\!76}{89\!\cdots\!67}a^{8}+\frac{29\!\cdots\!04}{89\!\cdots\!67}a^{7}-\frac{50\!\cdots\!83}{89\!\cdots\!67}a^{6}+\frac{43\!\cdots\!82}{68\!\cdots\!59}a^{5}-\frac{11\!\cdots\!45}{89\!\cdots\!67}a^{4}+\frac{43\!\cdots\!13}{89\!\cdots\!67}a^{3}-\frac{11\!\cdots\!97}{89\!\cdots\!67}a^{2}+\frac{29\!\cdots\!01}{89\!\cdots\!67}a-\frac{42\!\cdots\!46}{89\!\cdots\!67}$, $\frac{27\!\cdots\!73}{89\!\cdots\!67}a^{17}-\frac{39\!\cdots\!16}{89\!\cdots\!67}a^{16}-\frac{22\!\cdots\!49}{89\!\cdots\!67}a^{15}-\frac{28\!\cdots\!20}{89\!\cdots\!67}a^{14}+\frac{20\!\cdots\!91}{89\!\cdots\!67}a^{13}+\frac{13\!\cdots\!58}{89\!\cdots\!67}a^{12}-\frac{28\!\cdots\!48}{89\!\cdots\!67}a^{11}-\frac{11\!\cdots\!93}{89\!\cdots\!67}a^{10}+\frac{60\!\cdots\!05}{89\!\cdots\!67}a^{9}+\frac{10\!\cdots\!89}{89\!\cdots\!67}a^{8}+\frac{11\!\cdots\!98}{89\!\cdots\!67}a^{7}-\frac{22\!\cdots\!85}{89\!\cdots\!67}a^{6}+\frac{24\!\cdots\!79}{89\!\cdots\!67}a^{5}+\frac{51\!\cdots\!65}{89\!\cdots\!67}a^{4}+\frac{57\!\cdots\!68}{89\!\cdots\!67}a^{3}-\frac{59\!\cdots\!72}{89\!\cdots\!67}a^{2}+\frac{10\!\cdots\!12}{89\!\cdots\!67}a-\frac{23\!\cdots\!73}{89\!\cdots\!67}$
| 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: | \( 1566158.489890443 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1566158.489890443 \cdot 3}{6\cdot\sqrt{714659620519868840049881088}}\cr\approx \mathstrut & 0.447069359996086 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 6*x^16 + 3*x^15 + 90*x^14 - 66*x^13 - 198*x^12 - 231*x^11 + 891*x^10 + 55*x^9 - 276*x^8 - 1461*x^7 + 2190*x^6 - 1323*x^5 + 21*x^4 - 2289*x^3 + 3759*x^2 - 1449*x + 1393)
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 - 3*x^17 - 6*x^16 + 3*x^15 + 90*x^14 - 66*x^13 - 198*x^12 - 231*x^11 + 891*x^10 + 55*x^9 - 276*x^8 - 1461*x^7 + 2190*x^6 - 1323*x^5 + 21*x^4 - 2289*x^3 + 3759*x^2 - 1449*x + 1393, 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 - 3*x^17 - 6*x^16 + 3*x^15 + 90*x^14 - 66*x^13 - 198*x^12 - 231*x^11 + 891*x^10 + 55*x^9 - 276*x^8 - 1461*x^7 + 2190*x^6 - 1323*x^5 + 21*x^4 - 2289*x^3 + 3759*x^2 - 1449*x + 1393);
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 - 3*x^17 - 6*x^16 + 3*x^15 + 90*x^14 - 66*x^13 - 198*x^12 - 231*x^11 + 891*x^10 + 55*x^9 - 276*x^8 - 1461*x^7 + 2190*x^6 - 1323*x^5 + 21*x^4 - 2289*x^3 + 3759*x^2 - 1449*x + 1393);
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
$C_3\times S_3^2$ (as 18T46):
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 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.964467.2, 6.0.1714608.1 |
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 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.571750917736368384.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $31$ | |||
|
\(7\)
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.6.5.6 | $x^{6} + 28$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.3.2 | $x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |