Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 30*x^16 - 50*x^15 + 552*x^14 - 814*x^13 + 6983*x^12 - 8936*x^11 + 65641*x^10 - 72082*x^9 + 465597*x^8 - 424984*x^7 + 2471666*x^6 - 1778572*x^5 + 9429355*x^4 - 4814010*x^3 + 23461605*x^2 - 6474360*x + 29134601)
gp: K = bnfinit(y^18 - 2*y^17 + 30*y^16 - 50*y^15 + 552*y^14 - 814*y^13 + 6983*y^12 - 8936*y^11 + 65641*y^10 - 72082*y^9 + 465597*y^8 - 424984*y^7 + 2471666*y^6 - 1778572*y^5 + 9429355*y^4 - 4814010*y^3 + 23461605*y^2 - 6474360*y + 29134601, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 30*x^16 - 50*x^15 + 552*x^14 - 814*x^13 + 6983*x^12 - 8936*x^11 + 65641*x^10 - 72082*x^9 + 465597*x^8 - 424984*x^7 + 2471666*x^6 - 1778572*x^5 + 9429355*x^4 - 4814010*x^3 + 23461605*x^2 - 6474360*x + 29134601);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 30*x^16 - 50*x^15 + 552*x^14 - 814*x^13 + 6983*x^12 - 8936*x^11 + 65641*x^10 - 72082*x^9 + 465597*x^8 - 424984*x^7 + 2471666*x^6 - 1778572*x^5 + 9429355*x^4 - 4814010*x^3 + 23461605*x^2 - 6474360*x + 29134601)
\( x^{18} - 2 x^{17} + 30 x^{16} - 50 x^{15} + 552 x^{14} - 814 x^{13} + 6983 x^{12} - 8936 x^{11} + \cdots + 29134601 \)
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: |
\(-147682003746622037852672000000000\)
\(\medspace = -\,2^{18}\cdot 5^{9}\cdot 19^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(61.26\) | 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: | $2\cdot 5^{1/2}19^{8/9}\approx 61.26111052918685$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(380=2^{2}\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(321,·)$, $\chi_{380}(201,·)$, $\chi_{380}(139,·)$, $\chi_{380}(81,·)$, $\chi_{380}(339,·)$, $\chi_{380}(199,·)$, $\chi_{380}(159,·)$, $\chi_{380}(161,·)$, $\chi_{380}(99,·)$, $\chi_{380}(101,·)$, $\chi_{380}(39,·)$, $\chi_{380}(359,·)$, $\chi_{380}(301,·)$, $\chi_{380}(239,·)$, $\chi_{380}(119,·)$, $\chi_{380}(121,·)$, $\chi_{380}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{40\!\cdots\!61}a^{17}-\frac{38\!\cdots\!16}{40\!\cdots\!61}a^{16}+\frac{13\!\cdots\!79}{40\!\cdots\!61}a^{15}+\frac{16\!\cdots\!49}{40\!\cdots\!61}a^{14}-\frac{78\!\cdots\!26}{40\!\cdots\!61}a^{13}+\frac{62\!\cdots\!23}{40\!\cdots\!61}a^{12}-\frac{17\!\cdots\!24}{40\!\cdots\!61}a^{11}-\frac{14\!\cdots\!50}{40\!\cdots\!61}a^{10}+\frac{14\!\cdots\!54}{40\!\cdots\!61}a^{9}+\frac{39\!\cdots\!74}{40\!\cdots\!61}a^{8}-\frac{10\!\cdots\!99}{40\!\cdots\!61}a^{7}-\frac{23\!\cdots\!54}{40\!\cdots\!61}a^{6}-\frac{55\!\cdots\!45}{40\!\cdots\!61}a^{5}-\frac{15\!\cdots\!27}{40\!\cdots\!61}a^{4}+\frac{72\!\cdots\!44}{40\!\cdots\!61}a^{3}+\frac{68\!\cdots\!06}{40\!\cdots\!61}a^{2}+\frac{12\!\cdots\!22}{40\!\cdots\!61}a+\frac{16\!\cdots\!29}{40\!\cdots\!61}$
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
$C_{8582}$, which has order $8582$ (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: |
$\frac{33\!\cdots\!60}{40\!\cdots\!61}a^{17}-\frac{49\!\cdots\!57}{40\!\cdots\!61}a^{16}+\frac{70\!\cdots\!76}{40\!\cdots\!61}a^{15}-\frac{77\!\cdots\!83}{40\!\cdots\!61}a^{14}+\frac{10\!\cdots\!94}{40\!\cdots\!61}a^{13}-\frac{99\!\cdots\!16}{40\!\cdots\!61}a^{12}+\frac{10\!\cdots\!18}{40\!\cdots\!61}a^{11}-\frac{81\!\cdots\!60}{40\!\cdots\!61}a^{10}+\frac{79\!\cdots\!10}{40\!\cdots\!61}a^{9}-\frac{49\!\cdots\!47}{40\!\cdots\!61}a^{8}+\frac{45\!\cdots\!24}{40\!\cdots\!61}a^{7}-\frac{24\!\cdots\!92}{40\!\cdots\!61}a^{6}+\frac{18\!\cdots\!40}{40\!\cdots\!61}a^{5}-\frac{10\!\cdots\!85}{40\!\cdots\!61}a^{4}+\frac{54\!\cdots\!82}{40\!\cdots\!61}a^{3}-\frac{39\!\cdots\!67}{40\!\cdots\!61}a^{2}+\frac{86\!\cdots\!24}{40\!\cdots\!61}a-\frac{10\!\cdots\!31}{40\!\cdots\!61}$, $\frac{45\!\cdots\!38}{40\!\cdots\!61}a^{17}-\frac{12\!\cdots\!37}{40\!\cdots\!61}a^{16}+\frac{13\!\cdots\!34}{40\!\cdots\!61}a^{15}-\frac{24\!\cdots\!08}{40\!\cdots\!61}a^{14}+\frac{23\!\cdots\!46}{40\!\cdots\!61}a^{13}-\frac{36\!\cdots\!61}{40\!\cdots\!61}a^{12}+\frac{27\!\cdots\!24}{40\!\cdots\!61}a^{11}-\frac{33\!\cdots\!70}{40\!\cdots\!61}a^{10}+\frac{23\!\cdots\!68}{40\!\cdots\!61}a^{9}-\frac{22\!\cdots\!36}{40\!\cdots\!61}a^{8}+\frac{15\!\cdots\!66}{40\!\cdots\!61}a^{7}-\frac{98\!\cdots\!75}{40\!\cdots\!61}a^{6}+\frac{70\!\cdots\!10}{40\!\cdots\!61}a^{5}-\frac{23\!\cdots\!65}{40\!\cdots\!61}a^{4}+\frac{21\!\cdots\!30}{40\!\cdots\!61}a^{3}-\frac{49\!\cdots\!91}{40\!\cdots\!61}a^{2}+\frac{33\!\cdots\!48}{40\!\cdots\!61}a+\frac{16\!\cdots\!37}{40\!\cdots\!61}$, $\frac{20\!\cdots\!18}{40\!\cdots\!61}a^{17}-\frac{78\!\cdots\!20}{40\!\cdots\!61}a^{16}-\frac{13\!\cdots\!86}{40\!\cdots\!61}a^{15}-\frac{51\!\cdots\!75}{40\!\cdots\!61}a^{14}-\frac{30\!\cdots\!88}{40\!\cdots\!61}a^{13}-\frac{83\!\cdots\!88}{40\!\cdots\!61}a^{12}-\frac{50\!\cdots\!46}{40\!\cdots\!61}a^{11}-\frac{10\!\cdots\!35}{40\!\cdots\!61}a^{10}-\frac{52\!\cdots\!40}{40\!\cdots\!61}a^{9}-\frac{84\!\cdots\!62}{40\!\cdots\!61}a^{8}-\frac{40\!\cdots\!28}{40\!\cdots\!61}a^{7}-\frac{48\!\cdots\!04}{40\!\cdots\!61}a^{6}-\frac{20\!\cdots\!58}{40\!\cdots\!61}a^{5}-\frac{16\!\cdots\!80}{40\!\cdots\!61}a^{4}-\frac{72\!\cdots\!70}{40\!\cdots\!61}a^{3}-\frac{27\!\cdots\!05}{40\!\cdots\!61}a^{2}-\frac{12\!\cdots\!80}{40\!\cdots\!61}a+\frac{22\!\cdots\!66}{40\!\cdots\!61}$, $\frac{12\!\cdots\!60}{40\!\cdots\!61}a^{17}-\frac{23\!\cdots\!77}{40\!\cdots\!61}a^{16}+\frac{36\!\cdots\!40}{40\!\cdots\!61}a^{15}-\frac{59\!\cdots\!68}{40\!\cdots\!61}a^{14}+\frac{67\!\cdots\!80}{40\!\cdots\!61}a^{13}-\frac{97\!\cdots\!48}{40\!\cdots\!61}a^{12}+\frac{85\!\cdots\!00}{40\!\cdots\!61}a^{11}-\frac{10\!\cdots\!96}{40\!\cdots\!61}a^{10}+\frac{80\!\cdots\!60}{40\!\cdots\!61}a^{9}-\frac{86\!\cdots\!90}{40\!\cdots\!61}a^{8}+\frac{56\!\cdots\!60}{40\!\cdots\!61}a^{7}-\frac{50\!\cdots\!16}{40\!\cdots\!61}a^{6}+\frac{28\!\cdots\!00}{40\!\cdots\!61}a^{5}-\frac{21\!\cdots\!52}{40\!\cdots\!61}a^{4}+\frac{10\!\cdots\!04}{40\!\cdots\!61}a^{3}-\frac{57\!\cdots\!92}{40\!\cdots\!61}a^{2}+\frac{17\!\cdots\!68}{40\!\cdots\!61}a-\frac{11\!\cdots\!40}{40\!\cdots\!61}$, $\frac{52\!\cdots\!00}{40\!\cdots\!61}a^{17}-\frac{34\!\cdots\!00}{40\!\cdots\!61}a^{16}+\frac{19\!\cdots\!00}{40\!\cdots\!61}a^{15}-\frac{99\!\cdots\!05}{40\!\cdots\!61}a^{14}+\frac{39\!\cdots\!86}{40\!\cdots\!61}a^{13}-\frac{18\!\cdots\!72}{40\!\cdots\!61}a^{12}+\frac{53\!\cdots\!50}{40\!\cdots\!61}a^{11}-\frac{22\!\cdots\!65}{40\!\cdots\!61}a^{10}+\frac{52\!\cdots\!32}{40\!\cdots\!61}a^{9}-\frac{20\!\cdots\!49}{40\!\cdots\!61}a^{8}+\frac{38\!\cdots\!58}{40\!\cdots\!61}a^{7}-\frac{14\!\cdots\!61}{40\!\cdots\!61}a^{6}+\frac{19\!\cdots\!56}{40\!\cdots\!61}a^{5}-\frac{69\!\cdots\!07}{40\!\cdots\!61}a^{4}+\frac{68\!\cdots\!52}{40\!\cdots\!61}a^{3}-\frac{22\!\cdots\!94}{40\!\cdots\!61}a^{2}+\frac{11\!\cdots\!16}{40\!\cdots\!61}a-\frac{35\!\cdots\!47}{40\!\cdots\!61}$, $\frac{49\!\cdots\!22}{40\!\cdots\!61}a^{17}+\frac{14\!\cdots\!60}{40\!\cdots\!61}a^{16}+\frac{10\!\cdots\!30}{40\!\cdots\!61}a^{15}+\frac{49\!\cdots\!40}{40\!\cdots\!61}a^{14}+\frac{16\!\cdots\!88}{40\!\cdots\!61}a^{13}+\frac{99\!\cdots\!80}{40\!\cdots\!61}a^{12}+\frac{16\!\cdots\!86}{40\!\cdots\!61}a^{11}+\frac{13\!\cdots\!00}{40\!\cdots\!61}a^{10}+\frac{13\!\cdots\!70}{40\!\cdots\!61}a^{9}+\frac{13\!\cdots\!60}{40\!\cdots\!61}a^{8}+\frac{72\!\cdots\!46}{40\!\cdots\!61}a^{7}+\frac{99\!\cdots\!60}{40\!\cdots\!61}a^{6}+\frac{28\!\cdots\!16}{40\!\cdots\!61}a^{5}+\frac{51\!\cdots\!85}{40\!\cdots\!61}a^{4}+\frac{74\!\cdots\!62}{40\!\cdots\!61}a^{3}+\frac{17\!\cdots\!33}{40\!\cdots\!61}a^{2}+\frac{94\!\cdots\!54}{40\!\cdots\!61}a+\frac{29\!\cdots\!05}{40\!\cdots\!61}$, $\frac{37\!\cdots\!00}{40\!\cdots\!61}a^{17}-\frac{84\!\cdots\!00}{40\!\cdots\!61}a^{16}+\frac{10\!\cdots\!00}{40\!\cdots\!61}a^{15}-\frac{20\!\cdots\!05}{40\!\cdots\!61}a^{14}+\frac{19\!\cdots\!00}{40\!\cdots\!61}a^{13}-\frac{33\!\cdots\!05}{40\!\cdots\!61}a^{12}+\frac{23\!\cdots\!00}{40\!\cdots\!61}a^{11}-\frac{36\!\cdots\!10}{40\!\cdots\!61}a^{10}+\frac{20\!\cdots\!00}{40\!\cdots\!61}a^{9}-\frac{29\!\cdots\!00}{40\!\cdots\!61}a^{8}+\frac{13\!\cdots\!00}{40\!\cdots\!61}a^{7}-\frac{17\!\cdots\!95}{40\!\cdots\!61}a^{6}+\frac{59\!\cdots\!34}{40\!\cdots\!61}a^{5}-\frac{72\!\cdots\!95}{40\!\cdots\!61}a^{4}+\frac{18\!\cdots\!30}{40\!\cdots\!61}a^{3}-\frac{19\!\cdots\!95}{40\!\cdots\!61}a^{2}+\frac{27\!\cdots\!70}{40\!\cdots\!61}a-\frac{22\!\cdots\!54}{40\!\cdots\!61}$, $\frac{30\!\cdots\!00}{40\!\cdots\!61}a^{17}+\frac{10\!\cdots\!20}{40\!\cdots\!61}a^{16}+\frac{55\!\cdots\!60}{40\!\cdots\!61}a^{15}+\frac{19\!\cdots\!80}{40\!\cdots\!61}a^{14}+\frac{84\!\cdots\!40}{40\!\cdots\!61}a^{13}+\frac{48\!\cdots\!60}{40\!\cdots\!61}a^{12}+\frac{79\!\cdots\!28}{40\!\cdots\!61}a^{11}+\frac{52\!\cdots\!00}{40\!\cdots\!61}a^{10}+\frac{59\!\cdots\!72}{40\!\cdots\!61}a^{9}+\frac{50\!\cdots\!29}{40\!\cdots\!61}a^{8}+\frac{31\!\cdots\!12}{40\!\cdots\!61}a^{7}+\frac{30\!\cdots\!28}{40\!\cdots\!61}a^{6}+\frac{11\!\cdots\!04}{40\!\cdots\!61}a^{5}+\frac{13\!\cdots\!70}{40\!\cdots\!61}a^{4}+\frac{28\!\cdots\!40}{40\!\cdots\!61}a^{3}+\frac{38\!\cdots\!16}{40\!\cdots\!61}a^{2}+\frac{34\!\cdots\!72}{40\!\cdots\!61}a+\frac{93\!\cdots\!24}{40\!\cdots\!61}$
| 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: | \( 22305.8950792 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22305.8950792 \cdot 8582}{2\cdot\sqrt{147682003746622037852672000000000}}\cr\approx \mathstrut & 0.120207951622 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 30*x^16 - 50*x^15 + 552*x^14 - 814*x^13 + 6983*x^12 - 8936*x^11 + 65641*x^10 - 72082*x^9 + 465597*x^8 - 424984*x^7 + 2471666*x^6 - 1778572*x^5 + 9429355*x^4 - 4814010*x^3 + 23461605*x^2 - 6474360*x + 29134601)
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 - 2*x^17 + 30*x^16 - 50*x^15 + 552*x^14 - 814*x^13 + 6983*x^12 - 8936*x^11 + 65641*x^10 - 72082*x^9 + 465597*x^8 - 424984*x^7 + 2471666*x^6 - 1778572*x^5 + 9429355*x^4 - 4814010*x^3 + 23461605*x^2 - 6474360*x + 29134601, 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 - 2*x^17 + 30*x^16 - 50*x^15 + 552*x^14 - 814*x^13 + 6983*x^12 - 8936*x^11 + 65641*x^10 - 72082*x^9 + 465597*x^8 - 424984*x^7 + 2471666*x^6 - 1778572*x^5 + 9429355*x^4 - 4814010*x^3 + 23461605*x^2 - 6474360*x + 29134601);
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 - 2*x^17 + 30*x^16 - 50*x^15 + 552*x^14 - 814*x^13 + 6983*x^12 - 8936*x^11 + 65641*x^10 - 72082*x^9 + 465597*x^8 - 424984*x^7 + 2471666*x^6 - 1778572*x^5 + 9429355*x^4 - 4814010*x^3 + 23461605*x^2 - 6474360*x + 29134601);
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
| \(\Q(\sqrt{-5}) \), 3.3.361.1, 6.0.1042568000.1, \(\Q(\zeta_{19})^+\) |
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 | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | $18$ | $18$ | R | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | $18$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.118 | $x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
|
\(5\)
| 5.18.9.1 | $x^{18} + 180 x^{17} + 14445 x^{16} + 679200 x^{15} + 20664900 x^{14} + 423486000 x^{13} + 5887570504 x^{12} + 54397260480 x^{11} + 316143109712 x^{10} + 1034969211206 x^{9} + 1580754753720 x^{8} + 1360718216520 x^{7} + 746510415004 x^{6} + 357128191140 x^{5} + 552895560364 x^{4} + 1314509471572 x^{3} + 1121303668936 x^{2} + 1315877100296 x + 1500010785049$ | $2$ | $9$ | $9$ | $C_{18}$ | $[\ ]_{2}^{9}$ |
|
\(19\)
| 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |