Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 45*x^12 + 112*x^9 + 63*x^6 + 12*x^3 - 1)
gp: K = bnfinit(y^18 - 12*y^15 + 45*y^12 + 112*y^9 + 63*y^6 + 12*y^3 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12*x^15 + 45*x^12 + 112*x^9 + 63*x^6 + 12*x^3 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 12*x^15 + 45*x^12 + 112*x^9 + 63*x^6 + 12*x^3 - 1)
\( x^{18} - 12x^{15} + 45x^{12} + 112x^{9} + 63x^{6} + 12x^{3} - 1 \)
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: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1200757082375992968000000000\)
\(\medspace = 2^{12}\cdot 3^{36}\cdot 5^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.95\) | 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: | $2^{2/3}3^{37/18}5^{1/2}\approx 33.956342994275154$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{2}$, $\frac{1}{2852}a^{15}+\frac{16}{713}a^{12}-\frac{41}{1426}a^{9}+\frac{505}{1426}a^{6}-\frac{181}{2852}a^{3}+\frac{129}{713}$, $\frac{1}{2852}a^{16}+\frac{16}{713}a^{13}-\frac{41}{1426}a^{10}+\frac{505}{1426}a^{7}-\frac{181}{2852}a^{4}+\frac{129}{713}a$, $\frac{1}{2852}a^{17}+\frac{16}{713}a^{14}-\frac{41}{1426}a^{11}+\frac{505}{1426}a^{8}-\frac{181}{2852}a^{5}+\frac{129}{713}a^{2}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: | $9$ | 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: |
$\frac{831}{1426}a^{17}-\frac{19833}{2852}a^{14}+\frac{36669}{1426}a^{11}+\frac{48182}{713}a^{8}+\frac{54933}{1426}a^{5}+\frac{21243}{2852}a^{2}$, $\frac{111}{2852}a^{15}-\frac{363}{713}a^{12}+\frac{1646}{713}a^{9}+\frac{1290}{713}a^{6}+\frac{1299}{2852}a^{3}+\frac{831}{1426}$, $\frac{111}{2852}a^{17}-\frac{111}{2852}a^{16}+\frac{111}{2852}a^{15}-\frac{363}{713}a^{14}+\frac{363}{713}a^{13}-\frac{363}{713}a^{12}+\frac{1646}{713}a^{11}-\frac{1646}{713}a^{10}+\frac{1646}{713}a^{9}+\frac{1290}{713}a^{8}-\frac{1290}{713}a^{7}+\frac{1290}{713}a^{6}+\frac{1299}{2852}a^{5}-\frac{1299}{2852}a^{4}+\frac{1299}{2852}a^{3}+\frac{3683}{1426}a^{2}-\frac{3683}{1426}a+\frac{2257}{1426}$, $\frac{111}{2852}a^{17}-\frac{111}{2852}a^{15}-\frac{363}{713}a^{14}+\frac{363}{713}a^{12}+\frac{1646}{713}a^{11}-\frac{1646}{713}a^{9}+\frac{1290}{713}a^{8}-\frac{1290}{713}a^{6}+\frac{1299}{2852}a^{5}-\frac{1299}{2852}a^{3}+\frac{3683}{1426}a^{2}+a-\frac{831}{1426}$, $\frac{305}{1426}a^{15}-\frac{7305}{2852}a^{12}+\frac{6746}{713}a^{9}+\frac{36397}{1426}a^{6}+\frac{5552}{713}a^{3}-\frac{1099}{2852}$, $\frac{321}{713}a^{17}+\frac{60}{713}a^{15}-\frac{3698}{713}a^{14}-\frac{1589}{1426}a^{12}+\frac{25073}{1426}a^{11}+\frac{3636}{713}a^{9}+\frac{87289}{1426}a^{8}+\frac{2847}{713}a^{6}+\frac{68465}{1426}a^{5}-\frac{2304}{713}a^{3}+\frac{18265}{1426}a^{2}-\frac{2963}{1426}$, $\frac{745}{713}a^{17}+\frac{829}{2852}a^{16}+\frac{111}{2852}a^{15}-\frac{36727}{2852}a^{14}-\frac{10401}{2852}a^{13}-\frac{363}{713}a^{12}+\frac{36591}{713}a^{11}+\frac{21625}{1426}a^{10}+\frac{1646}{713}a^{9}+\frac{140931}{1426}a^{8}+\frac{16456}{713}a^{7}+\frac{1290}{713}a^{6}+\frac{57577}{1426}a^{5}+\frac{31053}{2852}a^{4}+\frac{1299}{2852}a^{3}+\frac{22555}{2852}a^{2}+\frac{9233}{2852}a+\frac{2257}{1426}$, $\frac{831}{713}a^{17}-\frac{941}{2852}a^{16}-\frac{19833}{1426}a^{14}+\frac{2769}{713}a^{13}+\frac{36669}{713}a^{11}-\frac{19885}{1426}a^{10}+\frac{96364}{713}a^{8}-\frac{57387}{1426}a^{7}+\frac{54933}{713}a^{5}-\frac{86359}{2852}a^{4}+\frac{21243}{1426}a^{2}-\frac{4457}{713}a$, $\frac{58}{713}a^{17}+\frac{429}{1426}a^{16}-\frac{111}{2852}a^{15}-\frac{566}{713}a^{14}-\frac{11397}{2852}a^{13}+\frac{363}{713}a^{12}+\frac{948}{713}a^{11}+\frac{13070}{713}a^{10}-\frac{1646}{713}a^{9}+\frac{13661}{713}a^{8}+\frac{20463}{1426}a^{7}-\frac{1290}{713}a^{6}+\frac{12318}{713}a^{5}-\frac{11374}{713}a^{4}-\frac{1299}{2852}a^{3}+\frac{2834}{713}a^{2}-\frac{27139}{2852}a+\frac{4873}{1426}$
| 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: | \( 2550553.0510749845 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 2550553.0510749845 \cdot 2}{2\cdot\sqrt{1200757082375992968000000000}}\cr\approx \mathstrut & 0.715164125446342 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 45*x^12 + 112*x^9 + 63*x^6 + 12*x^3 - 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 - 12*x^15 + 45*x^12 + 112*x^9 + 63*x^6 + 12*x^3 - 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 - 12*x^15 + 45*x^12 + 112*x^9 + 63*x^6 + 12*x^3 - 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 - 12*x^15 + 45*x^12 + 112*x^9 + 63*x^6 + 12*x^3 - 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
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 36 |
| The 12 conjugacy class representatives for $C_6:S_3$ |
| Character table for $C_6:S_3$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.243.1, 3.1.972.2, 3.1.108.1, 3.1.972.1, 6.2.7381125.1, 6.2.1458000.2, 6.2.118098000.8, 6.2.118098000.10, 9.1.24794911296.1 |
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
| Galois closure: | data not computed |
| Degree 18 sibling: | 18.0.3602271247127978904000000000.3 |
| 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 | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| Deg $18$ | $9$ | $2$ | $36$ | |||
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 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 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |