Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 19*x^16 + 152*x^14 - 665*x^12 + 1729*x^10 - 2717*x^8 + 2508*x^6 - 1254*x^4 + 285*x^2 - 19)
gp: K = bnfinit(y^18 - 19*y^16 + 152*y^14 - 665*y^12 + 1729*y^10 - 2717*y^8 + 2508*y^6 - 1254*y^4 + 285*y^2 - 19, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 19*x^16 + 152*x^14 - 665*x^12 + 1729*x^10 - 2717*x^8 + 2508*x^6 - 1254*x^4 + 285*x^2 - 19);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 19*x^16 + 152*x^14 - 665*x^12 + 1729*x^10 - 2717*x^8 + 2508*x^6 - 1254*x^4 + 285*x^2 - 19)
\( x^{18} - 19x^{16} + 152x^{14} - 665x^{12} + 1729x^{10} - 2717x^{8} + 2508x^{6} - 1254x^{4} + 285x^{2} - 19 \)
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: |
\(1436650532447139184230793216\)
\(\medspace = 2^{18}\cdot 19^{17}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.27\) | 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: | $2\cdot 19^{17/18}\approx 32.26574742904561$ | ||
| 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: | \(\Q(\sqrt{19}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(76=2^{2}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{76}(1,·)$, $\chi_{76}(67,·)$, $\chi_{76}(5,·)$, $\chi_{76}(71,·)$, $\chi_{76}(9,·)$, $\chi_{76}(75,·)$, $\chi_{76}(15,·)$, $\chi_{76}(17,·)$, $\chi_{76}(3,·)$, $\chi_{76}(25,·)$, $\chi_{76}(27,·)$, $\chi_{76}(31,·)$, $\chi_{76}(45,·)$, $\chi_{76}(49,·)$, $\chi_{76}(51,·)$, $\chi_{76}(73,·)$, $\chi_{76}(59,·)$, $\chi_{76}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: |
$a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{10}-10a^{8}+36a^{6}-56a^{4}+35a^{2}-6$, $a^{4}-5a^{2}+5$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{14}-15a^{12}+90a^{10}-274a^{8}+441a^{6}-351a^{4}+111a^{2}-9$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+295a^{6}-203a^{4}+62a^{2}-6$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-3$, $a^{16}-a^{15}-15a^{14}+14a^{13}+91a^{12}-78a^{11}-286a^{10}+220a^{9}+494a^{8}-329a^{7}-455a^{6}+246a^{5}+194a^{4}-73a^{3}-23a^{2}+2a-1$, $a^{12}-a^{11}-11a^{10}+10a^{9}+45a^{8}-36a^{7}-84a^{6}+56a^{5}+70a^{4}-35a^{3}-21a^{2}+6a+1$, $a^{12}+a^{11}-12a^{10}-11a^{9}+55a^{8}+44a^{7}-121a^{6}-78a^{5}+131a^{4}+60a^{3}-62a^{2}-16a+9$, $a^{9}-9a^{7}+27a^{5}-a^{4}-30a^{3}+4a^{2}+9a-2$, $a^{12}+a^{11}-12a^{10}-11a^{9}+54a^{8}+44a^{7}-112a^{6}-77a^{5}+105a^{4}+55a^{3}-36a^{2}-11a+2$, $a^{14}-a^{13}-14a^{12}+13a^{11}+77a^{10}-65a^{9}-210a^{8}+156a^{7}+294a^{6}-182a^{5}-196a^{4}+91a^{3}+49a^{2}-13a-2$, $a^{12}-a^{11}-13a^{10}+12a^{9}+64a^{8}-53a^{7}-147a^{6}+104a^{5}+156a^{4}-85a^{3}-66a^{2}+20a+9$, $a+1$, $a^{7}-7a^{5}+14a^{3}-7a+1$
| 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: | \( 39601653.8367 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 39601653.8367 \cdot 1}{2\cdot\sqrt{1436650532447139184230793216}}\cr\approx \mathstrut & 0.136945473021 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 19*x^16 + 152*x^14 - 665*x^12 + 1729*x^10 - 2717*x^8 + 2508*x^6 - 1254*x^4 + 285*x^2 - 19)
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 - 19*x^16 + 152*x^14 - 665*x^12 + 1729*x^10 - 2717*x^8 + 2508*x^6 - 1254*x^4 + 285*x^2 - 19, 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 - 19*x^16 + 152*x^14 - 665*x^12 + 1729*x^10 - 2717*x^8 + 2508*x^6 - 1254*x^4 + 285*x^2 - 19);
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 - 19*x^16 + 152*x^14 - 665*x^12 + 1729*x^10 - 2717*x^8 + 2508*x^6 - 1254*x^4 + 285*x^2 - 19);
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
| \(\Q(\sqrt{19}) \), 3.3.361.1, 6.6.158470336.1, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | $18$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | R | $18$ | $18$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | $18$ | $18$ | $18$ | $18$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.118 | $x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
|
\(19\)
| 19.18.17.1 | $x^{18} + 342$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |