Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 4*x^12 - 4*x^9 + 13*x^6 - 26*x^3 + 13)
gp: K = bnfinit(y^18 + 4*y^12 - 4*y^9 + 13*y^6 - 26*y^3 + 13, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 4*x^12 - 4*x^9 + 13*x^6 - 26*x^3 + 13);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 4*x^12 - 4*x^9 + 13*x^6 - 26*x^3 + 13)
\( x^{18} + 4x^{12} - 4x^{9} + 13x^{6} - 26x^{3} + 13 \)
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: |
\(-82845576677855741607936\)
\(\medspace = -\,2^{18}\cdot 3^{18}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.76\) | 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: |
\(2\), \(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{8}+\frac{1}{3}a^{5}$, $\frac{1}{111}a^{15}-\frac{17}{111}a^{12}-\frac{1}{37}a^{9}-\frac{9}{37}a^{6}-\frac{3}{37}a^{3}+\frac{16}{111}$, $\frac{1}{111}a^{16}-\frac{17}{111}a^{13}-\frac{1}{37}a^{10}-\frac{9}{37}a^{7}-\frac{3}{37}a^{4}+\frac{16}{111}a$, $\frac{1}{111}a^{17}-\frac{17}{111}a^{14}-\frac{1}{37}a^{11}-\frac{9}{37}a^{8}-\frac{3}{37}a^{5}+\frac{16}{111}a^{2}$
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 $1$ (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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{1}{3} a^{15} + \frac{1}{3} a^{12} + \frac{5}{3} a^{9} + \frac{13}{3} a^{3} - 5 \)
(order $4$)
| 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: |
$\frac{11}{111}a^{15}-\frac{2}{111}a^{12}+\frac{41}{111}a^{9}-\frac{38}{111}a^{6}+\frac{41}{37}a^{3}-\frac{194}{111}$, $\frac{52}{111}a^{15}+\frac{41}{111}a^{12}+\frac{251}{111}a^{9}+\frac{2}{111}a^{6}+\frac{716}{111}a^{3}-\frac{253}{37}$, $\frac{7}{37}a^{16}-\frac{25}{111}a^{15}+\frac{13}{111}a^{13}-\frac{19}{111}a^{12}+\frac{85}{111}a^{10}-\frac{110}{111}a^{9}-\frac{49}{111}a^{7}+\frac{3}{37}a^{6}+\frac{85}{37}a^{4}-\frac{256}{111}a^{3}-\frac{404}{111}a+\frac{138}{37}$, $\frac{50}{111}a^{17}+\frac{10}{111}a^{15}+\frac{38}{111}a^{14}+\frac{5}{37}a^{12}+\frac{220}{111}a^{11}+\frac{44}{111}a^{9}-\frac{6}{37}a^{8}-\frac{11}{111}a^{6}+\frac{623}{111}a^{5}+\frac{44}{37}a^{3}-\frac{239}{37}a^{2}-a-\frac{33}{37}$, $\frac{7}{37}a^{16}-\frac{7}{37}a^{15}+\frac{13}{111}a^{13}-\frac{13}{111}a^{12}+\frac{85}{111}a^{10}-\frac{85}{111}a^{9}-\frac{49}{111}a^{7}+\frac{49}{111}a^{6}+\frac{85}{37}a^{4}-\frac{85}{37}a^{3}-\frac{404}{111}a+\frac{404}{111}$, $\frac{9}{37}a^{16}+\frac{25}{111}a^{15}+\frac{22}{111}a^{13}+\frac{19}{111}a^{12}+\frac{47}{37}a^{10}+\frac{110}{111}a^{9}+\frac{11}{111}a^{7}-\frac{3}{37}a^{6}+\frac{349}{111}a^{4}+\frac{367}{111}a^{3}-\frac{152}{37}a-\frac{101}{37}$, $\frac{11}{111}a^{16}+\frac{23}{111}a^{15}-\frac{2}{111}a^{13}+\frac{16}{111}a^{12}+\frac{41}{111}a^{10}+\frac{116}{111}a^{9}-\frac{38}{111}a^{7}-\frac{29}{111}a^{6}+\frac{41}{37}a^{4}+\frac{116}{37}a^{3}-\frac{194}{111}a-\frac{446}{111}$, $\frac{11}{111}a^{17}-\frac{22}{37}a^{16}+\frac{14}{111}a^{15}-\frac{2}{111}a^{14}-\frac{62}{111}a^{13}+\frac{7}{37}a^{12}+\frac{41}{111}a^{11}-\frac{320}{111}a^{10}+\frac{23}{37}a^{9}-\frac{38}{111}a^{8}-\frac{31}{111}a^{7}+\frac{29}{111}a^{6}+\frac{41}{37}a^{5}-\frac{283}{37}a^{4}+\frac{133}{111}a^{3}-\frac{305}{111}a^{2}+\frac{979}{111}a-\frac{109}{111}$
| 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: | \( 18508.8363905 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 18508.8363905 \cdot 1}{4\cdot\sqrt{82845576677855741607936}}\cr\approx \mathstrut & 0.245359725836 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 4*x^12 - 4*x^9 + 13*x^6 - 26*x^3 + 13)
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^12 - 4*x^9 + 13*x^6 - 26*x^3 + 13, 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^12 - 4*x^9 + 13*x^6 - 26*x^3 + 13);
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^12 - 4*x^9 + 13*x^6 - 26*x^3 + 13);
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
$\He_3^2:S_3^2$ (as 18T650):
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 26244 |
| The 109 conjugacy class representatives for $\He_3^2:S_3^2$ are not computed |
| Character table for $\He_3^2:S_3^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 6.0.10816.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | $18$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $18$ | $18$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | $18$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(3\)
| Deg $18$ | $3$ | $6$ | $18$ | |||
|
\(13\)
| 13.9.8.1 | $x^{9} + 26$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |