Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*x + 1)
gp: K = bnfinit(y^18 - 4*y^17 + 14*y^16 - 31*y^15 + 64*y^14 - 100*y^13 + 146*y^12 - 176*y^11 + 202*y^10 - 205*y^9 + 202*y^8 - 176*y^7 + 146*y^6 - 100*y^5 + 64*y^4 - 31*y^3 + 14*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*x + 1)
\( x^{18} - 4 x^{17} + 14 x^{16} - 31 x^{15} + 64 x^{14} - 100 x^{13} + 146 x^{12} - 176 x^{11} + 202 x^{10} + \cdots + 1 \)
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: |
\(-1115906277282951168\)
\(\medspace = -\,2^{12}\cdot 3^{9}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.06\) | 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: | $2^{2/3}3^{1/2}7^{2/3}\approx 10.061112020813587$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{15}+\frac{4}{9}a^{13}+\frac{1}{3}a^{12}+\frac{1}{9}a^{11}+\frac{4}{9}a^{10}-\frac{4}{9}a^{9}-\frac{2}{9}a^{8}-\frac{4}{9}a^{7}+\frac{4}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{3}a^{4}+\frac{4}{9}a^{3}+\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{369}a^{17}-\frac{5}{369}a^{16}+\frac{20}{123}a^{15}-\frac{50}{369}a^{14}-\frac{44}{123}a^{13}-\frac{173}{369}a^{12}-\frac{50}{369}a^{11}+\frac{161}{369}a^{10}+\frac{2}{9}a^{9}-\frac{2}{9}a^{8}+\frac{79}{369}a^{7}-\frac{50}{369}a^{6}-\frac{44}{123}a^{5}+\frac{73}{369}a^{4}-\frac{1}{41}a^{3}+\frac{19}{369}a^{2}-\frac{128}{369}a+\frac{55}{123}$
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: |
\( \frac{796}{369} a^{17} - \frac{616}{123} a^{16} + \frac{6104}{369} a^{15} - \frac{8066}{369} a^{14} + \frac{16616}{369} a^{13} - \frac{12617}{369} a^{12} + \frac{2074}{41} a^{11} - \frac{7226}{369} a^{10} + \frac{256}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1466}{369} a^{7} + \frac{3598}{123} a^{6} - \frac{6754}{369} a^{5} + \frac{14566}{369} a^{4} - \frac{5770}{369} a^{3} + \frac{7006}{369} a^{2} - \frac{616}{123} a + \frac{1370}{369} \)
(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: |
$\frac{572}{369}a^{17}-\frac{1138}{369}a^{16}+\frac{1231}{123}a^{15}-\frac{3877}{369}a^{14}+\frac{2794}{123}a^{13}-\frac{3508}{369}a^{12}+\frac{6578}{369}a^{11}+\frac{457}{369}a^{10}+\frac{64}{9}a^{9}+\frac{32}{9}a^{8}+\frac{170}{369}a^{7}+\frac{6332}{369}a^{6}-\frac{650}{123}a^{5}+\frac{5102}{369}a^{4}+\frac{47}{123}a^{3}+\frac{167}{369}a^{2}+\frac{215}{369}a-\frac{23}{41}$, $\frac{737}{369}a^{17}-\frac{914}{123}a^{16}+\frac{8878}{369}a^{15}-\frac{17908}{369}a^{14}+\frac{34285}{369}a^{13}-\frac{48658}{369}a^{12}+\frac{21610}{123}a^{11}-\frac{71788}{369}a^{10}+\frac{1880}{9}a^{9}-\frac{1838}{9}a^{8}+\frac{70810}{369}a^{7}-\frac{6313}{41}a^{6}+\frac{42157}{369}a^{5}-\frac{24919}{369}a^{4}+\frac{14113}{369}a^{3}-\frac{6415}{369}a^{2}+\frac{685}{123}a-\frac{575}{369}$, $a$, $\frac{499}{369}a^{17}-\frac{367}{123}a^{16}+\frac{3659}{369}a^{15}-\frac{4163}{369}a^{14}+\frac{8342}{369}a^{13}-\frac{2564}{369}a^{12}+\frac{362}{41}a^{11}+\frac{9778}{369}a^{10}-\frac{245}{9}a^{9}+\frac{479}{9}a^{8}-\frac{17446}{369}a^{7}+\frac{8261}{123}a^{6}-\frac{19825}{369}a^{5}+\frac{22774}{369}a^{4}-\frac{11584}{369}a^{3}+\frac{8620}{369}a^{2}-\frac{218}{41}a+\frac{827}{369}$, $\frac{43}{369}a^{17}+\frac{605}{369}a^{16}-\frac{2381}{369}a^{15}+\frac{8428}{369}a^{14}-\frac{17279}{369}a^{13}+\frac{34135}{369}a^{12}-\frac{48562}{369}a^{11}+\frac{21769}{123}a^{10}-\frac{575}{3}a^{9}+\frac{613}{3}a^{8}-\frac{23741}{123}a^{7}+\frac{67673}{369}a^{6}-\frac{52580}{369}a^{5}+\frac{38809}{369}a^{4}-\frac{20354}{369}a^{3}+\frac{10903}{369}a^{2}-\frac{3454}{369}a+\frac{1519}{369}$, $\frac{62}{123}a^{17}-\frac{1094}{369}a^{16}+\frac{3862}{369}a^{15}-\frac{1088}{41}a^{14}+\frac{19933}{369}a^{13}-\frac{11300}{123}a^{12}+\frac{48715}{369}a^{11}-\frac{60992}{369}a^{10}+\frac{1678}{9}a^{9}-\frac{1720}{9}a^{8}+\frac{68609}{369}a^{7}-\frac{60755}{369}a^{6}+\frac{48838}{369}a^{5}-\frac{11300}{123}a^{4}+\frac{20056}{369}a^{3}-\frac{2963}{123}a^{2}+\frac{3580}{369}a-\frac{593}{369}$, $\frac{392}{369}a^{17}-\frac{2206}{369}a^{16}+\frac{2387}{123}a^{15}-\frac{17263}{369}a^{14}+\frac{10796}{123}a^{13}-\frac{52564}{369}a^{12}+\frac{68468}{369}a^{11}-\frac{81905}{369}a^{10}+\frac{2065}{9}a^{9}-\frac{2080}{9}a^{8}+\frac{78692}{369}a^{7}-\frac{67570}{369}a^{6}+\frac{15962}{123}a^{5}-\frac{30670}{369}a^{4}+\frac{4810}{123}a^{3}-\frac{6328}{369}a^{2}+\frac{1730}{369}a-\frac{2}{41}$, $\frac{509}{123}a^{17}-\frac{5462}{369}a^{16}+\frac{17656}{369}a^{15}-\frac{11633}{123}a^{14}+\frac{66658}{369}a^{13}-\frac{10260}{41}a^{12}+\frac{121393}{369}a^{11}-\frac{130820}{369}a^{10}+\frac{3331}{9}a^{9}-\frac{3163}{9}a^{8}+\frac{119444}{369}a^{7}-\frac{91766}{369}a^{6}+\frac{65674}{369}a^{5}-\frac{3741}{41}a^{4}+\frac{15613}{369}a^{3}-\frac{603}{41}a^{2}+\frac{1057}{369}a-\frac{359}{369}$
| 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: | \( 64.801283476 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 64.801283476 \cdot 1}{6\cdot\sqrt{1115906277282951168}}\cr\approx \mathstrut & 0.15604049246 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*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 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*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 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*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 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*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
$C_3\times S_3$ (as 18T3):
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 18 |
| The 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.588.1 x3, \(\Q(\zeta_{7})^+\), 6.0.1037232.1, 6.0.21168.1 x2, 6.0.64827.1, 9.3.203297472.1 x3 |
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 6 sibling: | 6.0.21168.1 |
| Degree 9 sibling: | 9.3.203297472.1 |
| Minimal sibling: | 6.0.21168.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 | $x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |