Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125)
gp: K = bnfinit(y^18 - 9*y^17 + 42*y^16 - 119*y^15 + 219*y^14 - 219*y^13 - 63*y^12 + 720*y^11 - 1461*y^10 + 1613*y^9 - 378*y^8 - 2184*y^7 + 5316*y^6 - 7317*y^5 + 7290*y^4 - 5308*y^3 + 2691*y^2 - 834*y + 125, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125)
\( x^{18} - 9 x^{17} + 42 x^{16} - 119 x^{15} + 219 x^{14} - 219 x^{13} - 63 x^{12} + 720 x^{11} + \cdots + 125 \)
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: |
\(-54783205459591112303652864\)
\(\medspace = -\,2^{12}\cdot 3^{18}\cdot 11^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.91\) | 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: | $2^{2/3}3^{7/6}11^{5/6}\approx 42.18473888411525$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{1}{5}a^{9}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{14}-\frac{2}{5}a^{11}+\frac{2}{5}a^{10}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{25}a^{15}+\frac{2}{25}a^{14}-\frac{2}{25}a^{13}-\frac{2}{25}a^{12}-\frac{4}{25}a^{11}+\frac{2}{25}a^{10}+\frac{6}{25}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{25}a^{6}+\frac{1}{25}a^{5}-\frac{2}{25}a^{4}+\frac{7}{25}a^{3}-\frac{12}{25}a^{2}+\frac{4}{25}a$, $\frac{1}{25}a^{16}-\frac{1}{25}a^{14}+\frac{2}{25}a^{13}+\frac{12}{25}a^{10}+\frac{8}{25}a^{9}-\frac{2}{5}a^{8}-\frac{1}{25}a^{7}+\frac{8}{25}a^{6}+\frac{11}{25}a^{5}+\frac{1}{25}a^{4}+\frac{9}{25}a^{3}-\frac{2}{25}a^{2}+\frac{12}{25}a$, $\frac{1}{78\!\cdots\!25}a^{17}-\frac{23\!\cdots\!91}{78\!\cdots\!25}a^{16}+\frac{10\!\cdots\!44}{78\!\cdots\!25}a^{15}+\frac{21\!\cdots\!88}{78\!\cdots\!25}a^{14}+\frac{16\!\cdots\!68}{78\!\cdots\!25}a^{13}-\frac{222415676958069}{15\!\cdots\!25}a^{12}+\frac{56\!\cdots\!72}{78\!\cdots\!25}a^{11}+\frac{11\!\cdots\!56}{78\!\cdots\!25}a^{10}-\frac{26\!\cdots\!93}{78\!\cdots\!25}a^{9}-\frac{15\!\cdots\!01}{78\!\cdots\!25}a^{8}+\frac{24\!\cdots\!54}{78\!\cdots\!25}a^{7}-\frac{82\!\cdots\!77}{78\!\cdots\!25}a^{6}+\frac{53\!\cdots\!37}{15\!\cdots\!25}a^{5}+\frac{27\!\cdots\!43}{78\!\cdots\!25}a^{4}+\frac{16\!\cdots\!24}{78\!\cdots\!25}a^{3}+\frac{21\!\cdots\!14}{78\!\cdots\!25}a^{2}-\frac{26\!\cdots\!92}{78\!\cdots\!25}a+\frac{25\!\cdots\!62}{62\!\cdots\!21}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
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: |
\( -1 \)
(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: |
$\frac{41\!\cdots\!64}{78\!\cdots\!25}a^{17}-\frac{36\!\cdots\!14}{78\!\cdots\!25}a^{16}+\frac{16\!\cdots\!26}{78\!\cdots\!25}a^{15}-\frac{44\!\cdots\!58}{78\!\cdots\!25}a^{14}+\frac{75\!\cdots\!02}{78\!\cdots\!25}a^{13}-\frac{25\!\cdots\!63}{31\!\cdots\!05}a^{12}-\frac{53\!\cdots\!32}{78\!\cdots\!25}a^{11}+\frac{28\!\cdots\!24}{78\!\cdots\!25}a^{10}-\frac{50\!\cdots\!12}{78\!\cdots\!25}a^{9}+\frac{47\!\cdots\!36}{78\!\cdots\!25}a^{8}+\frac{44\!\cdots\!46}{78\!\cdots\!25}a^{7}-\frac{93\!\cdots\!08}{78\!\cdots\!25}a^{6}+\frac{37\!\cdots\!32}{15\!\cdots\!25}a^{5}-\frac{23\!\cdots\!08}{78\!\cdots\!25}a^{4}+\frac{21\!\cdots\!96}{78\!\cdots\!25}a^{3}-\frac{13\!\cdots\!44}{78\!\cdots\!25}a^{2}+\frac{53\!\cdots\!22}{78\!\cdots\!25}a-\frac{79\!\cdots\!30}{62\!\cdots\!21}$, $\frac{26\!\cdots\!58}{78\!\cdots\!25}a^{17}-\frac{22\!\cdots\!83}{78\!\cdots\!25}a^{16}+\frac{96\!\cdots\!17}{78\!\cdots\!25}a^{15}-\frac{24\!\cdots\!11}{78\!\cdots\!25}a^{14}+\frac{40\!\cdots\!54}{78\!\cdots\!25}a^{13}-\frac{57\!\cdots\!53}{15\!\cdots\!25}a^{12}-\frac{38\!\cdots\!34}{78\!\cdots\!25}a^{11}+\frac{16\!\cdots\!18}{78\!\cdots\!25}a^{10}-\frac{26\!\cdots\!19}{78\!\cdots\!25}a^{9}+\frac{23\!\cdots\!17}{78\!\cdots\!25}a^{8}+\frac{69\!\cdots\!37}{78\!\cdots\!25}a^{7}-\frac{53\!\cdots\!71}{78\!\cdots\!25}a^{6}+\frac{20\!\cdots\!33}{15\!\cdots\!25}a^{5}-\frac{12\!\cdots\!41}{78\!\cdots\!25}a^{4}+\frac{10\!\cdots\!27}{78\!\cdots\!25}a^{3}-\frac{64\!\cdots\!58}{78\!\cdots\!25}a^{2}+\frac{24\!\cdots\!39}{78\!\cdots\!25}a-\frac{36\!\cdots\!80}{62\!\cdots\!21}$, $\frac{518467873860093}{78\!\cdots\!25}a^{17}+\frac{32\!\cdots\!07}{78\!\cdots\!25}a^{16}-\frac{28\!\cdots\!18}{78\!\cdots\!25}a^{15}+\frac{12\!\cdots\!44}{78\!\cdots\!25}a^{14}-\frac{31\!\cdots\!66}{78\!\cdots\!25}a^{13}+\frac{99\!\cdots\!72}{15\!\cdots\!25}a^{12}-\frac{30\!\cdots\!14}{78\!\cdots\!25}a^{11}-\frac{59\!\cdots\!47}{78\!\cdots\!25}a^{10}+\frac{21\!\cdots\!76}{78\!\cdots\!25}a^{9}-\frac{33\!\cdots\!93}{78\!\cdots\!25}a^{8}+\frac{25\!\cdots\!02}{78\!\cdots\!25}a^{7}+\frac{15\!\cdots\!84}{78\!\cdots\!25}a^{6}-\frac{14\!\cdots\!52}{15\!\cdots\!25}a^{5}+\frac{12\!\cdots\!64}{78\!\cdots\!25}a^{4}-\frac{13\!\cdots\!33}{78\!\cdots\!25}a^{3}+\frac{11\!\cdots\!57}{78\!\cdots\!25}a^{2}-\frac{57\!\cdots\!31}{78\!\cdots\!25}a+\frac{11\!\cdots\!81}{62\!\cdots\!21}$, $\frac{21\!\cdots\!53}{78\!\cdots\!25}a^{17}-\frac{13\!\cdots\!68}{78\!\cdots\!25}a^{16}+\frac{34\!\cdots\!82}{78\!\cdots\!25}a^{15}-\frac{29\!\cdots\!66}{78\!\cdots\!25}a^{14}-\frac{20\!\cdots\!36}{78\!\cdots\!25}a^{13}+\frac{12\!\cdots\!73}{15\!\cdots\!25}a^{12}-\frac{83\!\cdots\!09}{78\!\cdots\!25}a^{11}+\frac{22\!\cdots\!53}{78\!\cdots\!25}a^{10}+\frac{16\!\cdots\!61}{78\!\cdots\!25}a^{9}-\frac{39\!\cdots\!03}{78\!\cdots\!25}a^{8}+\frac{45\!\cdots\!82}{78\!\cdots\!25}a^{7}-\frac{85\!\cdots\!66}{78\!\cdots\!25}a^{6}-\frac{10\!\cdots\!53}{15\!\cdots\!25}a^{5}+\frac{11\!\cdots\!84}{78\!\cdots\!25}a^{4}-\frac{13\!\cdots\!83}{78\!\cdots\!25}a^{3}+\frac{10\!\cdots\!57}{78\!\cdots\!25}a^{2}-\frac{48\!\cdots\!41}{78\!\cdots\!25}a+\frac{53\!\cdots\!21}{62\!\cdots\!21}$, $\frac{79\!\cdots\!04}{15\!\cdots\!25}a^{17}-\frac{68\!\cdots\!09}{15\!\cdots\!25}a^{16}+\frac{61\!\cdots\!99}{31\!\cdots\!05}a^{15}-\frac{16\!\cdots\!13}{31\!\cdots\!05}a^{14}+\frac{13\!\cdots\!99}{15\!\cdots\!25}a^{13}-\frac{11\!\cdots\!73}{15\!\cdots\!25}a^{12}-\frac{10\!\cdots\!98}{15\!\cdots\!25}a^{11}+\frac{53\!\cdots\!02}{15\!\cdots\!25}a^{10}-\frac{93\!\cdots\!63}{15\!\cdots\!25}a^{9}+\frac{86\!\cdots\!31}{15\!\cdots\!25}a^{8}+\frac{10\!\cdots\!01}{15\!\cdots\!25}a^{7}-\frac{17\!\cdots\!87}{15\!\cdots\!25}a^{6}+\frac{34\!\cdots\!49}{15\!\cdots\!25}a^{5}-\frac{43\!\cdots\!61}{15\!\cdots\!25}a^{4}+\frac{38\!\cdots\!39}{15\!\cdots\!25}a^{3}-\frac{24\!\cdots\!87}{15\!\cdots\!25}a^{2}+\frac{96\!\cdots\!48}{15\!\cdots\!25}a-\frac{66\!\cdots\!02}{62\!\cdots\!21}$, $\frac{46\!\cdots\!21}{78\!\cdots\!25}a^{17}-\frac{41\!\cdots\!71}{78\!\cdots\!25}a^{16}+\frac{18\!\cdots\!54}{78\!\cdots\!25}a^{15}-\frac{51\!\cdots\!82}{78\!\cdots\!25}a^{14}+\frac{90\!\cdots\!23}{78\!\cdots\!25}a^{13}-\frac{15\!\cdots\!86}{15\!\cdots\!25}a^{12}-\frac{57\!\cdots\!33}{78\!\cdots\!25}a^{11}+\frac{33\!\cdots\!41}{78\!\cdots\!25}a^{10}-\frac{61\!\cdots\!28}{78\!\cdots\!25}a^{9}+\frac{58\!\cdots\!04}{78\!\cdots\!25}a^{8}+\frac{20\!\cdots\!94}{78\!\cdots\!25}a^{7}-\frac{10\!\cdots\!77}{78\!\cdots\!25}a^{6}+\frac{45\!\cdots\!01}{15\!\cdots\!25}a^{5}-\frac{28\!\cdots\!42}{78\!\cdots\!25}a^{4}+\frac{25\!\cdots\!99}{78\!\cdots\!25}a^{3}-\frac{16\!\cdots\!71}{78\!\cdots\!25}a^{2}+\frac{65\!\cdots\!93}{78\!\cdots\!25}a-\frac{94\!\cdots\!23}{62\!\cdots\!21}$, $\frac{33\!\cdots\!84}{78\!\cdots\!25}a^{17}-\frac{32\!\cdots\!64}{78\!\cdots\!25}a^{16}+\frac{16\!\cdots\!86}{78\!\cdots\!25}a^{15}-\frac{47\!\cdots\!58}{78\!\cdots\!25}a^{14}+\frac{92\!\cdots\!42}{78\!\cdots\!25}a^{13}-\frac{20\!\cdots\!42}{15\!\cdots\!25}a^{12}-\frac{12\!\cdots\!37}{78\!\cdots\!25}a^{11}+\frac{29\!\cdots\!69}{78\!\cdots\!25}a^{10}-\frac{62\!\cdots\!57}{78\!\cdots\!25}a^{9}+\frac{71\!\cdots\!41}{78\!\cdots\!25}a^{8}-\frac{22\!\cdots\!69}{78\!\cdots\!25}a^{7}-\frac{90\!\cdots\!18}{78\!\cdots\!25}a^{6}+\frac{44\!\cdots\!32}{15\!\cdots\!25}a^{5}-\frac{31\!\cdots\!63}{78\!\cdots\!25}a^{4}+\frac{31\!\cdots\!66}{78\!\cdots\!25}a^{3}-\frac{22\!\cdots\!89}{78\!\cdots\!25}a^{2}+\frac{10\!\cdots\!67}{78\!\cdots\!25}a-\frac{20\!\cdots\!02}{62\!\cdots\!21}$, $\frac{19261326113178}{10\!\cdots\!25}a^{17}-\frac{16212970441333}{200623814692765}a^{16}+\frac{45868159711874}{10\!\cdots\!25}a^{15}+\frac{202137728546933}{200623814692765}a^{14}-\frac{42\!\cdots\!04}{10\!\cdots\!25}a^{13}+\frac{93\!\cdots\!66}{10\!\cdots\!25}a^{12}-\frac{10\!\cdots\!62}{10\!\cdots\!25}a^{11}-\frac{649955929409617}{10\!\cdots\!25}a^{10}+\frac{29\!\cdots\!42}{10\!\cdots\!25}a^{9}-\frac{60\!\cdots\!83}{10\!\cdots\!25}a^{8}+\frac{66\!\cdots\!74}{10\!\cdots\!25}a^{7}-\frac{84\!\cdots\!49}{10\!\cdots\!25}a^{6}-\frac{17\!\cdots\!46}{200623814692765}a^{5}+\frac{20\!\cdots\!03}{10\!\cdots\!25}a^{4}-\frac{25\!\cdots\!82}{10\!\cdots\!25}a^{3}+\frac{22\!\cdots\!47}{10\!\cdots\!25}a^{2}-\frac{12\!\cdots\!82}{10\!\cdots\!25}a+\frac{15\!\cdots\!69}{40124762938553}$
| 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: | \( 311633.43352525914 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 311633.43352525914 \cdot 3}{2\cdot\sqrt{54783205459591112303652864}}\cr\approx \mathstrut & 0.963896490388146 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125)
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 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125, 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 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125);
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 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125);
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
$C_3^2:D_6$ (as 18T52):
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 20 conjugacy class representatives for $C_3^2:D_6$ |
| Character table for $C_3^2:D_6$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 3.1.108.1, 6.0.15524784.3, 9.3.18443443392.1 |
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 siblings: | data not computed |
| Degree 36 siblings: | 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 | R | ${\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.12.1 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 3.9.9.6 | $x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ |
| 3.9.9.6 | $x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ | |
|
\(11\)
| 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 11.12.10.2 | $x^{12} - 198 x^{6} - 10043$ | $6$ | $2$ | $10$ | $C_6\times S_3$ | $[\ ]_{6}^{6}$ |