Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 15*x^14 + 56*x^10 - 23*x^8 - 57*x^6 + 45*x^4 - 6*x^2 - 1)
gp: K = bnfinit(y^18 + y^16 - 15*y^14 + 56*y^10 - 23*y^8 - 57*y^6 + 45*y^4 - 6*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + x^16 - 15*x^14 + 56*x^10 - 23*x^8 - 57*x^6 + 45*x^4 - 6*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + x^16 - 15*x^14 + 56*x^10 - 23*x^8 - 57*x^6 + 45*x^4 - 6*x^2 - 1)
\( x^{18} + x^{16} - 15x^{14} + 56x^{10} - 23x^{8} - 57x^{6} + 45x^{4} - 6x^{2} - 1 \)
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: | $[10, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(180040394398004750319616\)
\(\medspace = 2^{18}\cdot 37^{6}\cdot 16361^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.59\) | 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: |
\(2\), \(37\), \(16361\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{316}a^{16}+\frac{73}{316}a^{14}-\frac{1}{4}a^{13}-\frac{13}{79}a^{12}-\frac{31}{316}a^{10}-\frac{1}{4}a^{9}-\frac{43}{316}a^{8}-\frac{19}{158}a^{6}-\frac{107}{316}a^{4}-\frac{1}{2}a^{3}-\frac{75}{316}a^{2}-\frac{1}{4}a+\frac{45}{316}$, $\frac{1}{316}a^{17}-\frac{3}{158}a^{15}-\frac{13}{79}a^{13}-\frac{1}{4}a^{12}+\frac{12}{79}a^{11}-\frac{43}{316}a^{9}+\frac{1}{4}a^{8}+\frac{30}{79}a^{7}-\frac{107}{316}a^{5}-\frac{1}{2}a^{4}+\frac{1}{79}a^{3}+\frac{45}{316}a+\frac{1}{4}$
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 $1$
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: | $13$ | 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: |
$a$, $\frac{21}{79}a^{17}+\frac{143}{158}a^{15}-\frac{223}{79}a^{13}-\frac{1381}{158}a^{11}+\frac{598}{79}a^{9}+\frac{1967}{79}a^{7}-\frac{351}{79}a^{5}-\frac{2755}{158}a^{3}+\frac{313}{79}a$, $\frac{11}{158}a^{16}+\frac{13}{158}a^{14}-\frac{177}{158}a^{12}-\frac{25}{158}a^{10}+\frac{435}{79}a^{8}-\frac{130}{79}a^{6}-\frac{1493}{158}a^{4}+\frac{439}{158}a^{2}+\frac{129}{79}$, $\frac{515}{316}a^{17}+\frac{3}{316}a^{16}+\frac{215}{79}a^{15}+\frac{61}{316}a^{14}-\frac{7109}{316}a^{13}+\frac{81}{316}a^{12}-\frac{1167}{79}a^{11}-\frac{725}{316}a^{10}+\frac{6294}{79}a^{9}-\frac{341}{158}a^{8}+\frac{1072}{79}a^{7}+\frac{564}{79}a^{6}-\frac{24295}{316}a^{5}+\frac{1259}{316}a^{4}+\frac{2016}{79}a^{3}-\frac{1805}{316}a^{2}-\frac{223}{158}a+\frac{107}{158}$, $\frac{515}{316}a^{17}-\frac{3}{316}a^{16}+\frac{215}{79}a^{15}-\frac{61}{316}a^{14}-\frac{7109}{316}a^{13}-\frac{81}{316}a^{12}-\frac{1167}{79}a^{11}+\frac{725}{316}a^{10}+\frac{6294}{79}a^{9}+\frac{341}{158}a^{8}+\frac{1072}{79}a^{7}-\frac{564}{79}a^{6}-\frac{24295}{316}a^{5}-\frac{1259}{316}a^{4}+\frac{2016}{79}a^{3}+\frac{1805}{316}a^{2}-\frac{223}{158}a-\frac{107}{158}$, $\frac{15}{79}a^{16}+\frac{57}{158}a^{14}-\frac{375}{158}a^{12}-\frac{377}{158}a^{10}+\frac{1001}{158}a^{8}+\frac{536}{79}a^{6}-\frac{262}{79}a^{4}-\frac{907}{158}a^{2}+\frac{165}{158}$, $\frac{53}{158}a^{17}+\frac{78}{79}a^{15}-\frac{623}{158}a^{13}-\frac{703}{79}a^{11}+\frac{1033}{79}a^{9}+\frac{1837}{79}a^{7}-\frac{2195}{158}a^{5}-\frac{1158}{79}a^{3}+\frac{521}{79}a$, $\frac{131}{158}a^{17}-\frac{39}{79}a^{16}+\frac{403}{316}a^{15}-\frac{249}{316}a^{14}-\frac{1835}{158}a^{13}+\frac{2187}{316}a^{12}-\frac{1881}{316}a^{11}+\frac{1281}{316}a^{10}+\frac{6533}{158}a^{9}-\frac{7907}{316}a^{8}+\frac{157}{158}a^{7}-\frac{433}{158}a^{6}-\frac{3098}{79}a^{5}+\frac{4001}{158}a^{4}+\frac{6025}{316}a^{3}-\frac{2599}{316}a^{2}-\frac{173}{79}a-\frac{463}{316}$, $\frac{67}{158}a^{17}-\frac{119}{316}a^{16}+\frac{151}{158}a^{15}-\frac{155}{316}a^{14}-\frac{1675}{316}a^{13}+\frac{441}{79}a^{12}-\frac{1129}{158}a^{11}+\frac{529}{316}a^{10}+\frac{5219}{316}a^{9}-\frac{6891}{316}a^{8}+\frac{1255}{79}a^{7}+\frac{523}{158}a^{6}-\frac{1955}{158}a^{5}+\frac{7993}{316}a^{4}-\frac{498}{79}a^{3}-\frac{4031}{316}a^{2}+\frac{263}{316}a-\frac{299}{316}$, $\frac{3}{316}a^{17}+\frac{361}{316}a^{16}+\frac{61}{316}a^{15}+\frac{130}{79}a^{14}+\frac{81}{316}a^{13}-\frac{1296}{79}a^{12}-\frac{725}{316}a^{11}-\frac{566}{79}a^{10}-\frac{341}{158}a^{9}+\frac{19237}{316}a^{8}+\frac{564}{79}a^{7}+\frac{7}{79}a^{6}+\frac{1259}{316}a^{5}-\frac{20615}{316}a^{4}-\frac{1805}{316}a^{3}+\frac{1862}{79}a^{2}+\frac{107}{158}a+\frac{919}{316}$, $\frac{67}{158}a^{17}+\frac{119}{316}a^{16}+\frac{151}{158}a^{15}+\frac{155}{316}a^{14}-\frac{1675}{316}a^{13}-\frac{441}{79}a^{12}-\frac{1129}{158}a^{11}-\frac{529}{316}a^{10}+\frac{5219}{316}a^{9}+\frac{6891}{316}a^{8}+\frac{1255}{79}a^{7}-\frac{523}{158}a^{6}-\frac{1955}{158}a^{5}-\frac{7993}{316}a^{4}-\frac{498}{79}a^{3}+\frac{4031}{316}a^{2}+\frac{263}{316}a+\frac{299}{316}$, $\frac{41}{158}a^{17}-\frac{19}{158}a^{16}+\frac{61}{316}a^{15}-\frac{9}{316}a^{14}-\frac{631}{158}a^{13}+\frac{633}{316}a^{12}+\frac{223}{316}a^{11}-\frac{323}{316}a^{10}+\frac{2345}{158}a^{9}-\frac{2553}{316}a^{8}-\frac{1005}{158}a^{7}+\frac{361}{79}a^{6}-\frac{1285}{79}a^{5}+\frac{819}{79}a^{4}+\frac{2777}{316}a^{3}-\frac{1337}{316}a^{2}+\frac{265}{158}a-\frac{367}{316}$, $\frac{51}{158}a^{17}-\frac{93}{316}a^{16}+\frac{99}{316}a^{15}-\frac{58}{79}a^{14}-\frac{1591}{316}a^{13}+\frac{1123}{316}a^{12}-\frac{81}{316}a^{11}+\frac{464}{79}a^{10}+\frac{6437}{316}a^{9}-\frac{837}{79}a^{8}-\frac{416}{79}a^{7}-\frac{1052}{79}a^{6}-\frac{4035}{158}a^{5}+\frac{1893}{316}a^{4}+\frac{3331}{316}a^{3}+\frac{381}{79}a^{2}+\frac{1193}{316}a+\frac{159}{158}$
| 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: | \( 100975.232776 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 100975.232776 \cdot 1}{2\cdot\sqrt{180040394398004750319616}}\cr\approx \mathstrut & 0.189897531619 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 15*x^14 + 56*x^10 - 23*x^8 - 57*x^6 + 45*x^4 - 6*x^2 - 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 + x^16 - 15*x^14 + 56*x^10 - 23*x^8 - 57*x^6 + 45*x^4 - 6*x^2 - 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 + x^16 - 15*x^14 + 56*x^10 - 23*x^8 - 57*x^6 + 45*x^4 - 6*x^2 - 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 + x^16 - 15*x^14 + 56*x^10 - 23*x^8 - 57*x^6 + 45*x^4 - 6*x^2 - 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
$S_4^3.S_4$ (as 18T883):
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 331776 |
| The 165 conjugacy class representatives for $S_4^3.S_4$ |
| Character table for $S_4^3.S_4$ |
Intermediate fields
| 3.3.148.1, 9.9.53038958912.1 |
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 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.10.149205548140250764484108310806528.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.119 | $x^{18} + 6 x^{14} - 4 x^{13} + 6 x^{12} + 16 x^{10} - 4 x^{9} + 28 x^{8} - 16 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 32 x^{2} - 16 x + 8$ | $6$ | $3$ | $18$ | 18T269 | $[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$ |
|
\(37\)
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(16361\)
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{16361}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |