Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511)
gp: K = bnfinit(y^18 - 36*y^16 + 540*y^14 - 4368*y^12 + 20592*y^10 - 5*y^9 - 57024*y^8 + 90*y^7 + 88704*y^6 - 540*y^5 - 69120*y^4 + 1200*y^3 + 20736*y^2 - 720*y - 1511, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511)
\( x^{18} - 36 x^{16} + 540 x^{14} - 4368 x^{12} + 20592 x^{10} - 5 x^{9} - 57024 x^{8} + 90 x^{7} + \cdots - 1511 \)
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: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(119217173915258668597396055301\)
\(\medspace = 3^{45}\cdot 7^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.24\) | 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: | $3^{5/2}7^{1/2}\approx 41.24318125460256$ | ||
| Ramified primes: |
\(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(189=3^{3}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{189}(64,·)$, $\chi_{189}(1,·)$, $\chi_{189}(146,·)$, $\chi_{189}(83,·)$, $\chi_{189}(20,·)$, $\chi_{189}(85,·)$, $\chi_{189}(22,·)$, $\chi_{189}(167,·)$, $\chi_{189}(104,·)$, $\chi_{189}(41,·)$, $\chi_{189}(106,·)$, $\chi_{189}(43,·)$, $\chi_{189}(169,·)$, $\chi_{189}(148,·)$, $\chi_{189}(188,·)$, $\chi_{189}(125,·)$, $\chi_{189}(62,·)$, $\chi_{189}(127,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17}a^{9}-\frac{1}{17}a^{7}+\frac{6}{17}a^{5}-\frac{2}{17}a^{3}+\frac{8}{17}a+\frac{6}{17}$, $\frac{1}{17}a^{10}-\frac{1}{17}a^{8}+\frac{6}{17}a^{6}-\frac{2}{17}a^{4}+\frac{8}{17}a^{2}+\frac{6}{17}a$, $\frac{1}{17}a^{11}+\frac{5}{17}a^{7}+\frac{4}{17}a^{5}+\frac{6}{17}a^{3}+\frac{6}{17}a^{2}+\frac{8}{17}a+\frac{6}{17}$, $\frac{1}{17}a^{12}+\frac{5}{17}a^{8}+\frac{4}{17}a^{6}+\frac{6}{17}a^{4}+\frac{6}{17}a^{3}+\frac{8}{17}a^{2}+\frac{6}{17}a$, $\frac{1}{17}a^{13}-\frac{8}{17}a^{7}-\frac{7}{17}a^{5}+\frac{6}{17}a^{4}+\frac{1}{17}a^{3}+\frac{6}{17}a^{2}-\frac{6}{17}a+\frac{4}{17}$, $\frac{1}{8279}a^{14}+\frac{91}{8279}a^{13}-\frac{28}{8279}a^{12}+\frac{69}{8279}a^{11}-\frac{179}{8279}a^{10}-\frac{203}{8279}a^{9}-\frac{1193}{8279}a^{8}-\frac{3019}{8279}a^{7}+\frac{1782}{8279}a^{6}-\frac{910}{8279}a^{5}+\frac{2981}{8279}a^{4}-\frac{3960}{8279}a^{3}-\frac{2708}{8279}a^{2}+\frac{42}{487}a+\frac{3640}{8279}$, $\frac{1}{8279}a^{15}-\frac{30}{8279}a^{13}+\frac{182}{8279}a^{12}-\frac{127}{8279}a^{11}+\frac{15}{8279}a^{10}+\frac{235}{8279}a^{9}+\frac{1813}{8279}a^{8}+\frac{2330}{8279}a^{7}+\frac{3969}{8279}a^{6}+\frac{566}{8279}a^{5}-\frac{1050}{8279}a^{4}+\frac{1168}{8279}a^{3}-\frac{3663}{8279}a^{2}-\frac{1433}{8279}a+\frac{1868}{8279}$, $\frac{1}{8279}a^{16}-\frac{10}{8279}a^{13}+\frac{7}{8279}a^{12}+\frac{137}{8279}a^{11}+\frac{222}{8279}a^{10}+\frac{106}{8279}a^{9}-\frac{831}{8279}a^{8}-\frac{2837}{8279}a^{7}-\frac{1005}{8279}a^{6}+\frac{2331}{8279}a^{5}+\frac{1964}{8279}a^{4}+\frac{44}{487}a^{3}-\frac{3292}{8279}a^{2}-\frac{1062}{8279}a-\frac{3784}{8279}$, $\frac{1}{8279}a^{17}-\frac{57}{8279}a^{13}-\frac{143}{8279}a^{12}-\frac{62}{8279}a^{11}-\frac{223}{8279}a^{10}+\frac{61}{8279}a^{9}+\frac{330}{8279}a^{8}+\frac{113}{487}a^{7}+\frac{240}{487}a^{6}-\frac{3240}{8279}a^{5}-\frac{3045}{8279}a^{4}+\frac{2399}{8279}a^{3}-\frac{3305}{8279}a^{2}+\frac{434}{8279}a+\frac{2797}{8279}$
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 $1$ (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: | $17$ | 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: |
$\frac{40}{8279}a^{15}-\frac{1200}{8279}a^{13}-\frac{25}{8279}a^{12}+\frac{14400}{8279}a^{11}+\frac{600}{8279}a^{10}-\frac{88000}{8279}a^{9}-\frac{5400}{8279}a^{8}+\frac{288000}{8279}a^{7}+\frac{1289}{487}a^{6}-\frac{483840}{8279}a^{5}-\frac{36156}{8279}a^{4}+\frac{355965}{8279}a^{3}+\frac{11268}{8279}a^{2}-\frac{62190}{8279}a+\frac{4592}{8279}$, $\frac{5}{8279}a^{15}-\frac{150}{8279}a^{13}-\frac{64}{8279}a^{12}+\frac{1800}{8279}a^{11}+\frac{1536}{8279}a^{10}-\frac{11000}{8279}a^{9}-\frac{13824}{8279}a^{8}+\frac{36000}{8279}a^{7}+\frac{57831}{8279}a^{6}-\frac{60480}{8279}a^{5}-\frac{113364}{8279}a^{4}+\frac{47235}{8279}a^{3}+\frac{91260}{8279}a^{2}-\frac{24210}{8279}a-\frac{7705}{8279}$, $\frac{6}{8279}a^{17}-\frac{12}{487}a^{15}+\frac{168}{487}a^{13}-\frac{1248}{487}a^{11}-\frac{15}{8279}a^{10}+\frac{5280}{487}a^{9}-\frac{11}{487}a^{8}-\frac{12672}{487}a^{7}+\frac{5692}{8279}a^{6}+\frac{16128}{487}a^{5}-\frac{32960}{8279}a^{4}-\frac{9216}{487}a^{3}+\frac{56336}{8279}a^{2}+\frac{24651}{8279}a-\frac{14624}{8279}$, $\frac{6}{8279}a^{17}-\frac{5}{8279}a^{16}-\frac{164}{8279}a^{15}+\frac{145}{8279}a^{14}+\frac{1752}{8279}a^{13}-\frac{1685}{8279}a^{12}-\frac{9184}{8279}a^{11}+\frac{10045}{8279}a^{10}+\frac{23904}{8279}a^{9}-\frac{33187}{8279}a^{8}-\frac{25191}{8279}a^{7}+\frac{64565}{8279}a^{6}-\frac{3603}{8279}a^{5}-\frac{82069}{8279}a^{4}+\frac{19719}{8279}a^{3}+\frac{61037}{8279}a^{2}-\frac{4671}{8279}a-\frac{10213}{8279}$, $\frac{11}{8279}a^{17}+\frac{28}{8279}a^{16}-\frac{329}{8279}a^{15}-\frac{880}{8279}a^{14}+\frac{3881}{8279}a^{13}+\frac{11111}{8279}a^{12}-\frac{22601}{8279}a^{11}-\frac{72055}{8279}a^{10}+\frac{65030}{8279}a^{9}+\frac{254996}{8279}a^{8}-\frac{70377}{8279}a^{7}-\frac{28532}{487}a^{6}-\frac{40021}{8279}a^{5}+\frac{460153}{8279}a^{4}+\frac{112390}{8279}a^{3}-\frac{184585}{8279}a^{2}-\frac{28420}{8279}a+\frac{17653}{8279}$, $\frac{5}{8279}a^{17}-\frac{5}{8279}a^{16}-\frac{175}{8279}a^{15}+\frac{175}{8279}a^{14}+\frac{2434}{8279}a^{13}-\frac{2436}{8279}a^{12}-\frac{16856}{8279}a^{11}+\frac{16908}{8279}a^{10}+\frac{58024}{8279}a^{9}-\frac{58569}{8279}a^{8}-\frac{73671}{8279}a^{7}+\frac{76617}{8279}a^{6}-\frac{61161}{8279}a^{5}+\frac{52637}{8279}a^{4}+\frac{172515}{8279}a^{3}-\frac{160691}{8279}a^{2}-\frac{21005}{8279}a+\frac{15741}{8279}$, $\frac{16}{8279}a^{14}-\frac{5}{8279}a^{13}-\frac{448}{8279}a^{12}+\frac{130}{8279}a^{11}+\frac{4928}{8279}a^{10}-\frac{1300}{8279}a^{9}-\frac{26880}{8279}a^{8}+\frac{6240}{8279}a^{7}+\frac{75264}{8279}a^{6}-\frac{13099}{8279}a^{5}-\frac{100839}{8279}a^{4}-\frac{50}{8279}a^{3}+\frac{54072}{8279}a^{2}+\frac{25060}{8279}a-\frac{7992}{8279}$, $\frac{28}{8279}a^{16}-\frac{896}{8279}a^{14}+\frac{11648}{8279}a^{12}-\frac{35}{8279}a^{11}-\frac{78848}{8279}a^{10}+\frac{770}{8279}a^{9}+\frac{295680}{8279}a^{8}-\frac{5673}{8279}a^{7}-\frac{602112}{8279}a^{6}+\frac{14742}{8279}a^{5}+\frac{602112}{8279}a^{4}-\frac{3528}{8279}a^{3}-\frac{232785}{8279}a^{2}-\frac{14952}{8279}a+\frac{27972}{8279}$, $\frac{12}{8279}a^{16}-\frac{385}{8279}a^{14}-\frac{91}{8279}a^{13}+\frac{5020}{8279}a^{12}+\frac{2351}{8279}a^{11}-\frac{34100}{8279}a^{10}-\frac{23330}{8279}a^{9}+\frac{128400}{8279}a^{8}+\frac{109954}{8279}a^{7}-\frac{15456}{487}a^{6}-\frac{242116}{8279}a^{5}+\frac{264320}{8279}a^{4}+\frac{197248}{8279}a^{3}-\frac{6053}{487}a^{2}-\frac{15888}{8279}a+\frac{3965}{8279}$, $\frac{11}{8279}a^{17}-\frac{22}{487}a^{15}+\frac{80}{8279}a^{14}+\frac{5211}{8279}a^{13}-\frac{2240}{8279}a^{12}-\frac{38246}{8279}a^{11}+\frac{24369}{8279}a^{10}+\frac{158060}{8279}a^{9}-\frac{128980}{8279}a^{8}-\frac{363744}{8279}a^{7}+\frac{338380}{8279}a^{6}+\frac{428882}{8279}a^{5}-\frac{395795}{8279}a^{4}-\frac{204692}{8279}a^{3}+\frac{161960}{8279}a^{2}+\frac{7592}{8279}a-\frac{14337}{8279}$, $\frac{28}{8279}a^{17}-\frac{61}{8279}a^{16}-\frac{891}{8279}a^{15}+\frac{1846}{8279}a^{14}+\frac{11592}{8279}a^{13}-\frac{22507}{8279}a^{12}-\frac{79294}{8279}a^{11}+\frac{141434}{8279}a^{10}+\frac{304764}{8279}a^{9}-\frac{485577}{8279}a^{8}-\frac{650037}{8279}a^{7}+\frac{885693}{8279}a^{6}+\frac{41905}{487}a^{5}-\frac{44901}{487}a^{4}-\frac{352244}{8279}a^{3}+\frac{241656}{8279}a^{2}+\frac{73149}{8279}a-\frac{32255}{8279}$, $\frac{5}{8279}a^{17}-\frac{165}{8279}a^{15}+\frac{2230}{8279}a^{13}-\frac{64}{8279}a^{12}-\frac{15880}{8279}a^{11}+\frac{1280}{8279}a^{10}+\frac{63800}{8279}a^{9}-\frac{8217}{8279}a^{8}-\frac{143520}{8279}a^{7}+\frac{14199}{8279}a^{6}+\frac{168000}{8279}a^{5}+\frac{27996}{8279}a^{4}-\frac{83325}{8279}a^{3}-\frac{73476}{8279}a^{2}-\frac{9268}{8279}a+\frac{15984}{8279}$, $\frac{10}{8279}a^{16}+\frac{40}{8279}a^{15}-\frac{320}{8279}a^{14}-\frac{1200}{8279}a^{13}+\frac{4135}{8279}a^{12}+\frac{832}{487}a^{11}-\frac{27560}{8279}a^{10}-\frac{82368}{8279}a^{9}+\frac{100200}{8279}a^{8}+\frac{243918}{8279}a^{7}-\frac{193127}{8279}a^{6}-\frac{339780}{8279}a^{5}+\frac{178884}{8279}a^{4}+\frac{185229}{8279}a^{3}-\frac{69191}{8279}a^{2}-\frac{1566}{487}a+\frac{3868}{8279}$, $\frac{5}{8279}a^{17}-\frac{10}{487}a^{15}+\frac{90}{8279}a^{14}+\frac{2291}{8279}a^{13}-\frac{2520}{8279}a^{12}-\frac{15366}{8279}a^{11}+\frac{27464}{8279}a^{10}+\frac{51660}{8279}a^{9}-\frac{145593}{8279}a^{8}-\frac{68448}{8279}a^{7}+\frac{379728}{8279}a^{6}-\frac{30688}{8279}a^{5}-\frac{423120}{8279}a^{4}+\frac{128608}{8279}a^{3}+\frac{6912}{487}a^{2}-\frac{50827}{8279}a+\frac{649}{8279}$, $\frac{6}{8279}a^{17}-\frac{174}{8279}a^{15}+\frac{1}{8279}a^{14}+\frac{2047}{8279}a^{13}+\frac{75}{8279}a^{12}-\frac{12782}{8279}a^{11}-\frac{2179}{8279}a^{10}+\frac{47420}{8279}a^{9}+\frac{20381}{8279}a^{8}-\frac{112992}{8279}a^{7}-\frac{83353}{8279}a^{6}+\frac{176288}{8279}a^{5}+\frac{151340}{8279}a^{4}-\frac{151890}{8279}a^{3}-\frac{111780}{8279}a^{2}+\frac{45198}{8279}a+\frac{21680}{8279}$, $\frac{45}{8279}a^{15}-\frac{96}{8279}a^{14}-\frac{1320}{8279}a^{13}+\frac{2599}{8279}a^{12}+\frac{15420}{8279}a^{11}-\frac{27432}{8279}a^{10}-\frac{91200}{8279}a^{9}+\frac{142056}{8279}a^{8}+\frac{286560}{8279}a^{7}-\frac{371840}{8279}a^{6}-\frac{457447}{8279}a^{5}+\frac{455514}{8279}a^{4}+\frac{320710}{8279}a^{3}-\frac{221904}{8279}a^{2}-\frac{71180}{8279}a+\frac{36560}{8279}$, $\frac{45}{8279}a^{15}+\frac{91}{8279}a^{14}-\frac{1348}{8279}a^{13}-\frac{2637}{8279}a^{12}+\frac{16148}{8279}a^{11}+\frac{30164}{8279}a^{10}-\frac{98480}{8279}a^{9}-\frac{172104}{8279}a^{8}+\frac{18912}{487}a^{7}+\frac{507808}{8279}a^{6}-\frac{538496}{8279}a^{5}-\frac{720272}{8279}a^{4}+\frac{397376}{8279}a^{3}+\frac{387904}{8279}a^{2}-\frac{84736}{8279}a-\frac{34688}{8279}$
| 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: | \( 550518949.413 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 550518949.413 \cdot 1}{2\cdot\sqrt{119217173915258668597396055301}}\cr\approx \mathstrut & 0.208983879715 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511)
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 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511, 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 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511);
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 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511);
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
| \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), 6.6.6751269.1, \(\Q(\zeta_{27})^+\) |
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 | $18$ | R | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | R | $18$ | $18$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | $18$ | $18$ | $18$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 $18$ | $18$ | $1$ | $45$ | |||
|
\(7\)
| 7.18.9.2 | $x^{18} + 63 x^{16} + 1764 x^{14} + 12 x^{13} + 28814 x^{12} - 504 x^{11} + 302370 x^{10} - 17044 x^{9} + 2112804 x^{8} - 150180 x^{7} + 9908221 x^{6} - 209592 x^{5} + 29960739 x^{4} + 1787108 x^{3} + 51556212 x^{2} + 7225224 x + 40408804$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |