Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 19*x^16 - 14*x^15 + 117*x^14 + 104*x^13 - 418*x^12 - 444*x^11 + 890*x^10 + 1282*x^9 - 793*x^8 - 2154*x^7 - 769*x^6 + 932*x^5 + 1107*x^4 + 526*x^3 + 134*x^2 + 18*x + 1)
gp: K = bnfinit(y^18 - 19*y^16 - 14*y^15 + 117*y^14 + 104*y^13 - 418*y^12 - 444*y^11 + 890*y^10 + 1282*y^9 - 793*y^8 - 2154*y^7 - 769*y^6 + 932*y^5 + 1107*y^4 + 526*y^3 + 134*y^2 + 18*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 19*x^16 - 14*x^15 + 117*x^14 + 104*x^13 - 418*x^12 - 444*x^11 + 890*x^10 + 1282*x^9 - 793*x^8 - 2154*x^7 - 769*x^6 + 932*x^5 + 1107*x^4 + 526*x^3 + 134*x^2 + 18*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 19*x^16 - 14*x^15 + 117*x^14 + 104*x^13 - 418*x^12 - 444*x^11 + 890*x^10 + 1282*x^9 - 793*x^8 - 2154*x^7 - 769*x^6 + 932*x^5 + 1107*x^4 + 526*x^3 + 134*x^2 + 18*x + 1)
\( x^{18} - 19 x^{16} - 14 x^{15} + 117 x^{14} + 104 x^{13} - 418 x^{12} - 444 x^{11} + 890 x^{10} + \cdots + 1 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(9445660200192205743157319565312\)
\(\medspace = 2^{24}\cdot 3^{13}\cdot 13^{4}\cdot 17^{4}\cdot 23^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.58\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(17\), \(23\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11}a^{15}-\frac{3}{11}a^{14}+\frac{1}{11}a^{13}-\frac{5}{11}a^{12}-\frac{3}{11}a^{11}+\frac{4}{11}a^{10}+\frac{5}{11}a^{9}+\frac{3}{11}a^{7}+\frac{2}{11}a^{6}+\frac{4}{11}a^{5}+\frac{4}{11}a^{4}+\frac{2}{11}a^{3}-\frac{5}{11}a^{2}+\frac{4}{11}a-\frac{1}{11}$, $\frac{1}{11}a^{16}+\frac{3}{11}a^{14}-\frac{2}{11}a^{13}+\frac{4}{11}a^{12}-\frac{5}{11}a^{11}-\frac{5}{11}a^{10}+\frac{4}{11}a^{9}+\frac{3}{11}a^{8}-\frac{1}{11}a^{6}+\frac{5}{11}a^{5}+\frac{3}{11}a^{4}+\frac{1}{11}a^{3}-\frac{3}{11}$, $\frac{1}{11}a^{17}-\frac{4}{11}a^{14}+\frac{1}{11}a^{13}-\frac{1}{11}a^{12}+\frac{4}{11}a^{11}+\frac{3}{11}a^{10}-\frac{1}{11}a^{9}+\frac{1}{11}a^{7}-\frac{1}{11}a^{6}+\frac{2}{11}a^{5}+\frac{5}{11}a^{3}+\frac{4}{11}a^{2}-\frac{4}{11}a+\frac{3}{11}$
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: | $11$ | 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: |
$a$, $a+1$, $\frac{246}{11}a^{17}+\frac{72}{11}a^{16}-\frac{4746}{11}a^{15}-\frac{4779}{11}a^{14}+\frac{29137}{11}a^{13}+\frac{34376}{11}a^{12}-\frac{104103}{11}a^{11}-\frac{142675}{11}a^{10}+\frac{219632}{11}a^{9}+\frac{395952}{11}a^{8}-16182a^{7}-\frac{643905}{11}a^{6}-\frac{253037}{11}a^{5}+\frac{288198}{11}a^{4}+\frac{331941}{11}a^{3}+\frac{138707}{11}a^{2}+\frac{26892}{11}a+\frac{1990}{11}$, $\frac{268}{11}a^{17}+\frac{61}{11}a^{16}-\frac{5153}{11}a^{15}-\frac{4889}{11}a^{14}+\frac{31667}{11}a^{13}+\frac{35421}{11}a^{12}-\frac{113200}{11}a^{11}-\frac{147944}{11}a^{10}+\frac{239597}{11}a^{9}+\frac{413981}{11}a^{8}-18125a^{7}-\frac{678643}{11}a^{6}-\frac{258911}{11}a^{5}+\frac{306117}{11}a^{4}+\frac{348628}{11}a^{3}+\frac{145769}{11}a^{2}+\frac{28575}{11}a+\frac{2144}{11}$, $\frac{160}{11}a^{17}+\frac{283}{11}a^{16}-\frac{3251}{11}a^{15}-\frac{7506}{11}a^{14}+\frac{18706}{11}a^{13}+\frac{50645}{11}a^{12}-\frac{62588}{11}a^{11}-\frac{198288}{11}a^{10}+\frac{109655}{11}a^{9}+\frac{503120}{11}a^{8}+\frac{23473}{11}a^{7}-\frac{732010}{11}a^{6}-\frac{478142}{11}a^{5}+23683a^{4}+\frac{428806}{11}a^{3}+\frac{197174}{11}a^{2}+\frac{39651}{11}a+267$, $\frac{1608}{11}a^{17}-\frac{379}{11}a^{16}-\frac{30491}{11}a^{15}-\frac{15305}{11}a^{14}+\frac{192271}{11}a^{13}+\frac{121930}{11}a^{12}-\frac{704289}{11}a^{11}-\frac{548469}{11}a^{10}+\frac{1573148}{11}a^{9}+\frac{1694216}{11}a^{8}-\frac{1704313}{11}a^{7}-\frac{3077421}{11}a^{6}-\frac{472709}{11}a^{5}+149620a^{4}+\frac{1381937}{11}a^{3}+\frac{495531}{11}a^{2}+\frac{86720}{11}a+\frac{6015}{11}$, $\frac{5848}{11}a^{17}-\frac{2668}{11}a^{16}-\frac{109874}{11}a^{15}-\frac{31763}{11}a^{14}+\frac{698326}{11}a^{13}+\frac{289640}{11}a^{12}-\frac{2574166}{11}a^{11}-\frac{1422053}{11}a^{10}+\frac{5844448}{11}a^{9}+\frac{4829235}{11}a^{8}-619966a^{7}-\frac{9476315}{11}a^{6}-\frac{201120}{11}a^{5}+\frac{5519054}{11}a^{4}+\frac{3963900}{11}a^{3}+\frac{1284044}{11}a^{2}+\frac{205340}{11}a+\frac{13047}{11}$, $\frac{239}{11}a^{17}+14a^{16}-\frac{4713}{11}a^{15}-\frac{6144}{11}a^{14}+\frac{28977}{11}a^{13}+\frac{42939}{11}a^{12}-\frac{104035}{11}a^{11}-\frac{173609}{11}a^{10}+19864a^{9}+42358a^{8}-\frac{162136}{11}a^{7}-\frac{735126}{11}a^{6}-\frac{304891}{11}a^{5}+\frac{326504}{11}a^{4}+\frac{373942}{11}a^{3}+\frac{151428}{11}a^{2}+\frac{28108}{11}a+\frac{1976}{11}$, $\frac{2846}{11}a^{17}+\frac{87}{11}a^{16}-\frac{54532}{11}a^{15}-\frac{41171}{11}a^{14}+\frac{340290}{11}a^{13}+\frac{306462}{11}a^{12}-\frac{1235413}{11}a^{11}-\frac{1308562}{11}a^{10}+\frac{2698483}{11}a^{9}+\frac{3784701}{11}a^{8}-\frac{2619349}{11}a^{7}-\frac{6454993}{11}a^{6}-\frac{1773605}{11}a^{5}+\frac{3171937}{11}a^{4}+\frac{3093196}{11}a^{3}+\frac{1196868}{11}a^{2}+\frac{220872}{11}a+\frac{15795}{11}$, $142a^{17}-\frac{755}{11}a^{16}-\frac{29302}{11}a^{15}-\frac{7771}{11}a^{14}+\frac{186411}{11}a^{13}+\frac{73299}{11}a^{12}-\frac{688043}{11}a^{11}-\frac{366499}{11}a^{10}+\frac{1567581}{11}a^{9}+\frac{1263032}{11}a^{8}-\frac{1851437}{11}a^{7}-227884a^{6}+\frac{9510}{11}a^{5}+\frac{1494810}{11}a^{4}+\frac{1022986}{11}a^{3}+\frac{309519}{11}a^{2}+\frac{44030}{11}a+\frac{2219}{11}$, $25a^{17}+\frac{311}{11}a^{16}-\frac{5425}{11}a^{15}-\frac{9698}{11}a^{14}+\frac{31639}{11}a^{13}+6052a^{12}-\frac{107127}{11}a^{11}-\frac{265035}{11}a^{10}+\frac{198167}{11}a^{9}+\frac{688455}{11}a^{8}-\frac{29497}{11}a^{7}-\frac{1027528}{11}a^{6}-\frac{633439}{11}a^{5}+\frac{374232}{11}a^{4}+\frac{597475}{11}a^{3}+\frac{279344}{11}a^{2}+\frac{58644}{11}a+\frac{4613}{11}$
| 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: | \( 249959188.611 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 249959188.611 \cdot 3}{2\cdot\sqrt{9445660200192205743157319565312}}\cr\approx \mathstrut & 0.480400433206 \end{aligned}\] (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 - 14*x^15 + 117*x^14 + 104*x^13 - 418*x^12 - 444*x^11 + 890*x^10 + 1282*x^9 - 793*x^8 - 2154*x^7 - 769*x^6 + 932*x^5 + 1107*x^4 + 526*x^3 + 134*x^2 + 18*x + 1)
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 - 14*x^15 + 117*x^14 + 104*x^13 - 418*x^12 - 444*x^11 + 890*x^10 + 1282*x^9 - 793*x^8 - 2154*x^7 - 769*x^6 + 932*x^5 + 1107*x^4 + 526*x^3 + 134*x^2 + 18*x + 1, 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 - 14*x^15 + 117*x^14 + 104*x^13 - 418*x^12 - 444*x^11 + 890*x^10 + 1282*x^9 - 793*x^8 - 2154*x^7 - 769*x^6 + 932*x^5 + 1107*x^4 + 526*x^3 + 134*x^2 + 18*x + 1);
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 - 14*x^15 + 117*x^14 + 104*x^13 - 418*x^12 - 444*x^11 + 890*x^10 + 1282*x^9 - 793*x^8 - 2154*x^7 - 769*x^6 + 932*x^5 + 1107*x^4 + 526*x^3 + 134*x^2 + 18*x + 1);
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
$C_3^5:S_3^2$ (as 18T541):
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 8748 |
| The 68 conjugacy class representatives for $C_3^5:S_3^2$ |
| Character table for $C_3^5:S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 6.2.3656448.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | R | R | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.24.10 | $x^{18} + 6 x^{17} + 8 x^{16} - 2 x^{15} + 14 x^{14} + 24 x^{13} + 18 x^{12} + 120 x^{11} + 64 x^{10} + 24 x^{9} + 232 x^{8} - 16 x^{7} + 228 x^{6} + 264 x^{5} - 48 x^{4} + 216 x^{3} + 72 x^{2} + 216$ | $6$ | $3$ | $24$ | $S_3 \times C_6$ | $[2]_{3}^{6}$ |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.6.7.1 | $x^{6} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
|
\(13\)
| 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.3.2.3 | $x^{3} + 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.6.4.2 | $x^{6} + 204 x^{3} - 7225$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
| 17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(23\)
| 23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 23.6.0.1 | $x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 23.9.6.1 | $x^{9} + 6 x^{7} + 123 x^{6} + 12 x^{5} + 78 x^{4} - 6127 x^{3} + 492 x^{2} - 6888 x + 69105$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |