Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*x + 1)
gp: K = bnfinit(y^18 - 2*y^17 + 5*y^16 - 7*y^15 + 10*y^14 - 13*y^13 + 17*y^12 - 24*y^11 + 28*y^10 - 28*y^9 + 25*y^8 - 24*y^7 + 28*y^6 - 31*y^5 + 29*y^4 - 22*y^3 + 13*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*x + 1)
\( x^{18} - 2 x^{17} + 5 x^{16} - 7 x^{15} + 10 x^{14} - 13 x^{13} + 17 x^{12} - 24 x^{11} + 28 x^{10} + \cdots + 1 \)
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: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-5585027926119911424\)
\(\medspace = -\,2^{12}\cdot 3^{12}\cdot 37^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.00\) | 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: | $2^{3/2}3^{3/4}37^{2/3}\approx 71.59022882794918$ | ||
| Ramified primes: |
\(2\), \(3\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2507}a^{17}-\frac{1001}{2507}a^{16}-\frac{289}{2507}a^{15}+\frac{399}{2507}a^{14}+\frac{22}{2507}a^{13}+\frac{572}{2507}a^{12}+\frac{185}{2507}a^{11}+\frac{679}{2507}a^{10}+\frac{48}{109}a^{9}+\frac{156}{2507}a^{8}-\frac{385}{2507}a^{7}+\frac{1020}{2507}a^{6}-\frac{1110}{2507}a^{5}+\frac{765}{2507}a^{4}+\frac{429}{2507}a^{3}+\frac{104}{2507}a^{2}-\frac{1096}{2507}a-\frac{660}{2507}$
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{82}{23} a^{17} + \frac{179}{23} a^{16} - \frac{383}{23} a^{15} + \frac{586}{23} a^{14} - \frac{723}{23} a^{13} + \frac{1028}{23} a^{12} - \frac{1278}{23} a^{11} + \frac{1845}{23} a^{10} - 94 a^{9} + \frac{1951}{23} a^{8} - \frac{1780}{23} a^{7} + \frac{1690}{23} a^{6} - \frac{2107}{23} a^{5} + \frac{2337}{23} a^{4} - \frac{2012}{23} a^{3} + \frac{1477}{23} a^{2} - \frac{771}{23} a + \frac{231}{23} \)
(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: |
$\frac{2562}{2507}a^{17}+\frac{99}{2507}a^{16}+\frac{6668}{2507}a^{15}+\frac{1889}{2507}a^{14}+\frac{6224}{2507}a^{13}-\frac{1131}{2507}a^{12}+\frac{5161}{2507}a^{11}-\frac{10288}{2507}a^{10}-\frac{194}{109}a^{9}+\frac{3566}{2507}a^{8}-\frac{6133}{2507}a^{7}+\frac{946}{2507}a^{6}+\frac{9146}{2507}a^{5}+\frac{4470}{2507}a^{4}-\frac{8996}{2507}a^{3}+\frac{13241}{2507}a^{2}-\frac{12647}{2507}a+\frac{6319}{2507}$, $\frac{6319}{2507}a^{17}-\frac{15200}{2507}a^{16}+\frac{31496}{2507}a^{15}-\frac{50901}{2507}a^{14}+\frac{61301}{2507}a^{13}-\frac{88371}{2507}a^{12}+\frac{108554}{2507}a^{11}-\frac{156817}{2507}a^{10}+\frac{8140}{109}a^{9}-\frac{172470}{2507}a^{8}+\frac{154409}{2507}a^{7}-\frac{145523}{2507}a^{6}+\frac{175986}{2507}a^{5}-\frac{205035}{2507}a^{4}+\frac{178781}{2507}a^{3}-\frac{130022}{2507}a^{2}+\frac{68906}{2507}a-\frac{18948}{2507}$, $\frac{1093}{2507}a^{17}-\frac{6055}{2507}a^{16}+\frac{7526}{2507}a^{15}-\frac{20167}{2507}a^{14}+\frac{16525}{2507}a^{13}-\frac{31638}{2507}a^{12}+\frac{34236}{2507}a^{11}-\frac{50065}{2507}a^{10}+\frac{2978}{109}a^{9}-\frac{55122}{2507}a^{8}+\frac{55525}{2507}a^{7}-\frac{45881}{2507}a^{6}+\frac{57819}{2507}a^{5}-\frac{73896}{2507}a^{4}+\frac{57749}{2507}a^{3}-\frac{46776}{2507}a^{2}+\frac{22981}{2507}a-\frac{6885}{2507}$, $\frac{4101}{2507}a^{17}+\frac{6379}{2507}a^{16}+\frac{5636}{2507}a^{15}+\frac{24298}{2507}a^{14}-\frac{5044}{2507}a^{13}+\frac{29304}{2507}a^{12}-\frac{26006}{2507}a^{11}+\frac{29386}{2507}a^{10}-\frac{3494}{109}a^{9}+\frac{63146}{2507}a^{8}-\frac{69671}{2507}a^{7}+\frac{46470}{2507}a^{6}-\frac{42017}{2507}a^{5}+\frac{83739}{2507}a^{4}-\frac{78302}{2507}a^{3}+\frac{73017}{2507}a^{2}-\frac{47278}{2507}a+\frac{20956}{2507}$, $\frac{3339}{2507}a^{17}-\frac{5522}{2507}a^{16}+\frac{15266}{2507}a^{15}-\frac{19012}{2507}a^{14}+\frac{28332}{2507}a^{13}-\frac{35524}{2507}a^{12}+\frac{46119}{2507}a^{11}-\frac{66836}{2507}a^{10}+\frac{3203}{109}a^{9}-\frac{73275}{2507}a^{8}+\frac{63251}{2507}a^{7}-\frac{58894}{2507}a^{6}+\frac{74266}{2507}a^{5}-\frac{80522}{2507}a^{4}+\frac{76144}{2507}a^{3}-\frac{53864}{2507}a^{2}+\frac{25746}{2507}a-\frac{7608}{2507}$, $\frac{8321}{2507}a^{17}-\frac{6081}{2507}a^{16}+\frac{27021}{2507}a^{15}-\frac{16738}{2507}a^{14}+\frac{37656}{2507}a^{13}-\frac{38786}{2507}a^{12}+\frac{55241}{2507}a^{11}-\frac{86057}{2507}a^{10}+\frac{2866}{109}a^{9}-\frac{60718}{2507}a^{8}+\frac{50501}{2507}a^{7}-\frac{61450}{2507}a^{6}+\frac{92237}{2507}a^{5}-\frac{72411}{2507}a^{4}+\frac{54895}{2507}a^{3}-\frac{24601}{2507}a^{2}+\frac{3157}{2507}a+\frac{5991}{2507}$, $\frac{1971}{2507}a^{17}-\frac{4976}{2507}a^{16}+\frac{9498}{2507}a^{15}-\frac{15811}{2507}a^{14}+\frac{18292}{2507}a^{13}-\frac{25808}{2507}a^{12}+\frac{33711}{2507}a^{11}-\frac{45555}{2507}a^{10}+\frac{2503}{109}a^{9}-\frac{48518}{2507}a^{8}+\frac{43405}{2507}a^{7}-\frac{42813}{2507}a^{6}+\frac{50941}{2507}a^{5}-\frac{61567}{2507}a^{4}+\frac{50840}{2507}a^{3}-\frac{33181}{2507}a^{2}+\frac{18367}{2507}a-\frac{4741}{2507}$, $\frac{1548}{2507}a^{17}+\frac{4792}{2507}a^{16}-\frac{1126}{2507}a^{15}+\frac{18479}{2507}a^{14}-\frac{11070}{2507}a^{13}+\frac{25555}{2507}a^{12}-\frac{26995}{2507}a^{11}+\frac{33250}{2507}a^{10}-\frac{2868}{109}a^{9}+\frac{55970}{2507}a^{8}-\frac{56975}{2507}a^{7}+\frac{39662}{2507}a^{6}-\frac{41097}{2507}a^{5}+\frac{71112}{2507}a^{4}-\frac{65445}{2507}a^{3}+\frac{58205}{2507}a^{2}-\frac{34467}{2507}a+\frac{13711}{2507}$
| 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: | \( 172.576549312 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 172.576549312 \cdot 1}{6\cdot\sqrt{5585027926119911424}}\cr\approx \mathstrut & 0.185753357035 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*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 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*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 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*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 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*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
$C_3\wr D_4$ (as 18T189):
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 648 |
| The 54 conjugacy class representatives for $C_3\wr D_4$ are not computed |
| Character table for $C_3\wr D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 6.0.36963.1, 6.0.1774224.1 |
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 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.110370926592.1 |
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 | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.12.12.18 | $x^{12} - 10 x^{11} + 72 x^{10} - 332 x^{9} + 1316 x^{8} - 4160 x^{7} + 12128 x^{6} - 27904 x^{5} + 53744 x^{4} - 69600 x^{3} + 71680 x^{2} - 41536 x + 38848$ | $2$ | $6$ | $12$ | $D_4 \times C_3$ | $[2, 2]^{6}$ | |
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.12.9.2 | $x^{12} + 8 x^{10} + 4 x^{9} + 33 x^{8} + 24 x^{7} - 10 x^{6} - 96 x^{5} + 163 x^{4} + 12 x^{3} - 6 x^{2} + 68 x + 172$ | $4$ | $3$ | $9$ | $D_4 \times C_3$ | $[\ ]_{4}^{6}$ | |
|
\(37\)
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.0.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.9.6.1 | $x^{9} + 18 x^{7} + 216 x^{6} + 108 x^{5} + 594 x^{4} - 19197 x^{3} + 7776 x^{2} - 50544 x + 381240$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |