Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^15 + 76*x^12 - 715*x^9 + 2632*x^6 - 1785*x^3 + 343)
gp: K = bnfinit(y^18 - 5*y^15 + 76*y^12 - 715*y^9 + 2632*y^6 - 1785*y^3 + 343, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^15 + 76*x^12 - 715*x^9 + 2632*x^6 - 1785*x^3 + 343);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^15 + 76*x^12 - 715*x^9 + 2632*x^6 - 1785*x^3 + 343)
\( x^{18} - 5x^{15} + 76x^{12} - 715x^{9} + 2632x^{6} - 1785x^{3} + 343 \)
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: |
\(-137858723094110501552832000000\)
\(\medspace = -\,2^{12}\cdot 3^{27}\cdot 5^{6}\cdot 7^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.58\) | 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: | $2^{2/3}3^{31/18}5^{1/2}7^{5/6}\approx 119.15955157499168$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{35}a^{12}-\frac{3}{7}a^{9}+\frac{9}{35}a^{6}+\frac{2}{5}$, $\frac{1}{35}a^{13}-\frac{3}{7}a^{10}+\frac{9}{35}a^{7}+\frac{2}{5}a$, $\frac{1}{35}a^{14}-\frac{3}{7}a^{11}+\frac{9}{35}a^{8}+\frac{2}{5}a^{2}$, $\frac{1}{19589255}a^{15}+\frac{246794}{19589255}a^{12}-\frac{6234866}{19589255}a^{9}-\frac{3125144}{19589255}a^{6}-\frac{358573}{2798465}a^{3}+\frac{400613}{2798465}$, $\frac{1}{19589255}a^{16}+\frac{246794}{19589255}a^{13}-\frac{6234866}{19589255}a^{10}-\frac{3125144}{19589255}a^{7}-\frac{358573}{2798465}a^{4}+\frac{400613}{2798465}a$, $\frac{1}{137124785}a^{17}+\frac{246794}{137124785}a^{14}-\frac{6234866}{137124785}a^{11}-\frac{22714399}{137124785}a^{8}+\frac{5238357}{19589255}a^{5}+\frac{8796008}{19589255}a^{2}$
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: |
\( -\frac{4488}{559693} a^{15} + \frac{20975}{559693} a^{12} - \frac{332620}{559693} a^{9} + \frac{3097850}{559693} a^{6} - \frac{10645810}{559693} a^{3} + \frac{4076334}{559693} \)
(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: |
$\frac{351978}{19589255}a^{15}-\frac{1653226}{19589255}a^{12}+\frac{26184982}{19589255}a^{9}-\frac{243397904}{19589255}a^{6}+\frac{121298101}{2798465}a^{3}-\frac{45188267}{2798465}$, $\frac{33897}{19589255}a^{15}-\frac{155053}{19589255}a^{12}+\frac{2420928}{19589255}a^{9}-\frac{3425551}{2798465}a^{6}+\frac{10378374}{2798465}a^{3}-\frac{3610556}{2798465}$, $\frac{2744162}{137124785}a^{17}+\frac{351978}{19589255}a^{16}+\frac{8976}{559693}a^{15}-\frac{12678636}{137124785}a^{14}-\frac{1653226}{19589255}a^{13}-\frac{41950}{559693}a^{12}+\frac{203928318}{137124785}a^{11}+\frac{26184982}{19589255}a^{10}+\frac{665240}{559693}a^{9}-\frac{1884306229}{137124785}a^{8}-\frac{243397904}{19589255}a^{7}-\frac{6195700}{559693}a^{6}+\frac{931758994}{19589255}a^{5}+\frac{121298101}{2798465}a^{4}+\frac{21291620}{559693}a^{3}-\frac{357984042}{19589255}a^{2}-\frac{47986732}{2798465}a-\frac{8152668}{559693}$, $\frac{572035}{27424957}a^{17}-\frac{2633}{559693}a^{16}-\frac{8976}{559693}a^{15}-\frac{13021422}{137124785}a^{14}+\frac{383678}{19589255}a^{13}+\frac{41950}{559693}a^{12}+\frac{42331987}{27424957}a^{11}-\frac{1351514}{3917851}a^{10}-\frac{665240}{559693}a^{9}-\frac{1950309028}{137124785}a^{8}+\frac{60223312}{19589255}a^{7}+\frac{6195700}{559693}a^{6}+\frac{190280205}{3917851}a^{5}-\frac{5592911}{559693}a^{4}-\frac{21291620}{559693}a^{3}-\frac{314918739}{19589255}a^{2}+\frac{2777116}{2798465}a+\frac{8152668}{559693}$, $\frac{2802308}{137124785}a^{17}+\frac{39226}{19589255}a^{16}+\frac{586102}{19589255}a^{15}-\frac{2736585}{27424957}a^{14}-\frac{268377}{19589255}a^{13}-\frac{2840704}{19589255}a^{12}+\frac{209752167}{137124785}a^{11}+\frac{2994959}{19589255}a^{10}+\frac{43723223}{19589255}a^{9}-\frac{395038519}{27424957}a^{8}-\frac{32601413}{19589255}a^{7}-\frac{58710353}{2798465}a^{6}+\frac{1006296141}{19589255}a^{5}+\frac{19289847}{2798465}a^{4}+\frac{208656999}{2798465}a^{3}-\frac{92605159}{3917851}a^{2}-\frac{6771139}{2798465}a-\frac{76766003}{2798465}$, $\frac{2802308}{137124785}a^{17}-\frac{7061}{3917851}a^{16}-\frac{420947}{19589255}a^{15}-\frac{2736585}{27424957}a^{14}+\frac{238454}{19589255}a^{13}+\frac{1901356}{19589255}a^{12}+\frac{209752167}{137124785}a^{11}-\frac{502861}{3917851}a^{10}-\frac{31440658}{19589255}a^{9}-\frac{395038519}{27424957}a^{8}+\frac{30031866}{19589255}a^{7}+\frac{40648247}{2798465}a^{6}+\frac{1006296141}{19589255}a^{5}-\frac{2965644}{559693}a^{4}-\frac{143299789}{2798465}a^{3}-\frac{92605159}{3917851}a^{2}+\frac{5956768}{2798465}a+\frac{43995522}{2798465}$, $\frac{21049498}{137124785}a^{17}-\frac{3443338}{19589255}a^{16}+\frac{1078384}{19589255}a^{15}-\frac{98341717}{137124785}a^{14}+\frac{15827564}{19589255}a^{13}-\frac{967077}{3917851}a^{12}+\frac{1567397292}{137124785}a^{11}-\frac{36472216}{2798465}a^{10}+\frac{79435616}{19589255}a^{9}-\frac{14535128763}{137124785}a^{8}+\frac{2358806641}{19589255}a^{7}-\frac{145984038}{3917851}a^{6}+\frac{7231515681}{19589255}a^{5}-\frac{1158882766}{2798465}a^{4}+\frac{351121933}{2798465}a^{3}-\frac{2982518024}{19589255}a^{2}+\frac{409402453}{2798465}a-\frac{18034927}{559693}$, $\frac{33418682}{137124785}a^{17}-\frac{1885393}{19589255}a^{16}-\frac{1078384}{19589255}a^{15}-\frac{153416397}{137124785}a^{14}+\frac{8849061}{19589255}a^{13}+\frac{967077}{3917851}a^{12}+\frac{2476940743}{137124785}a^{11}-\frac{140522147}{19589255}a^{10}-\frac{79435616}{19589255}a^{9}-\frac{22880504538}{137124785}a^{8}+\frac{1304864529}{19589255}a^{7}+\frac{145984038}{3917851}a^{6}+\frac{11225906189}{19589255}a^{5}-\frac{651394391}{2798465}a^{4}-\frac{351121933}{2798465}a^{3}-\frac{3916703674}{19589255}a^{2}+\frac{276521942}{2798465}a+\frac{18034927}{559693}$
| 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: | \( 13242502.889658673 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 13242502.889658673 \cdot 3}{6\cdot\sqrt{137858723094110501552832000000}}\cr\approx \mathstrut & 0.272170950608512 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^15 + 76*x^12 - 715*x^9 + 2632*x^6 - 1785*x^3 + 343)
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 - 5*x^15 + 76*x^12 - 715*x^9 + 2632*x^6 - 1785*x^3 + 343, 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 - 5*x^15 + 76*x^12 - 715*x^9 + 2632*x^6 - 1785*x^3 + 343);
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 - 5*x^15 + 76*x^12 - 715*x^9 + 2632*x^6 - 1785*x^3 + 343);
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 $C_3\times S_3^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.140.1, 6.0.964467.2, 6.0.529200.1 |
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: | 12.0.8933608089630756000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 $18$ | $6$ | $3$ | $27$ | |||
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.6.5.1 | $x^{6} + 21$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |