Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - x^8 - 8*x^6 + 468*x^4 - 1690*x^2 + 2197)
gp: K = bnfinit(y^12 - 6*y^10 - y^8 - 8*y^6 + 468*y^4 - 1690*y^2 + 2197, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^10 - x^8 - 8*x^6 + 468*x^4 - 1690*x^2 + 2197);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^10 - x^8 - 8*x^6 + 468*x^4 - 1690*x^2 + 2197)
\( x^{12} - 6x^{10} - x^{8} - 8x^{6} + 468x^{4} - 1690x^{2} + 2197 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1052745423388672\)
\(\medspace = 2^{24}\cdot 13^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.86\) | 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: | $2^{87/32}13^{5/6}\approx 55.80992355327924$ | ||
| Ramified primes: |
\(2\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13}a^{8}-\frac{6}{13}a^{6}-\frac{1}{13}a^{4}+\frac{5}{13}a^{2}$, $\frac{1}{13}a^{9}-\frac{6}{13}a^{7}-\frac{1}{13}a^{5}+\frac{5}{13}a^{3}$, $\frac{1}{4772729}a^{10}+\frac{168604}{4772729}a^{8}+\frac{1946515}{4772729}a^{6}+\frac{411728}{4772729}a^{4}+\frac{162711}{367133}a^{2}-\frac{3547}{28241}$, $\frac{1}{4772729}a^{11}+\frac{168604}{4772729}a^{9}+\frac{1946515}{4772729}a^{7}+\frac{411728}{4772729}a^{5}+\frac{162711}{367133}a^{3}-\frac{3547}{28241}a$
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: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{67}{28241} a^{10} + \frac{68}{28241} a^{8} - \frac{433}{28241} a^{6} - \frac{5681}{28241} a^{4} + \frac{7943}{28241} a^{2} - \frac{3979}{28241} \)
(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: |
$\frac{50}{4772729}a^{10}-\frac{13859}{4772729}a^{8}+\frac{35505}{4772729}a^{6}+\frac{394085}{4772729}a^{4}+\frac{115106}{367133}a^{2}-\frac{36145}{28241}$, $\frac{6957}{4772729}a^{10}-\frac{11907}{4772729}a^{8}-\frac{162983}{4772729}a^{6}-\frac{347103}{4772729}a^{4}+\frac{165870}{367133}a^{2}+\frac{6155}{28241}$, $\frac{17228}{4772729}a^{11}+\frac{18430}{4772729}a^{10}-\frac{46584}{4772729}a^{9}-\frac{41992}{4772729}a^{8}-\frac{102066}{4772729}a^{7}-\frac{129645}{4772729}a^{6}-\frac{860975}{4772729}a^{5}-\frac{124937}{4772729}a^{4}+\frac{463545}{367133}a^{3}+\frac{247314}{367133}a^{2}-\frac{50674}{28241}a-\frac{49777}{28241}$, $\frac{24080}{4772729}a^{11}+\frac{18020}{4772729}a^{10}-\frac{139527}{4772729}a^{9}-\frac{148628}{4772729}a^{8}-\frac{156043}{4772729}a^{7}-\frac{53653}{4772729}a^{6}-\frac{16425}{4772729}a^{5}+\frac{682029}{4772729}a^{4}+\frac{969457}{367133}a^{3}+\frac{834107}{367133}a^{2}-\frac{208663}{28241}a-\frac{261726}{28241}$, $\frac{47214}{4772729}a^{11}-\frac{110145}{4772729}a^{10}+\frac{291550}{4772729}a^{9}+\frac{168092}{4772729}a^{8}-\frac{543281}{4772729}a^{7}-\frac{565335}{4772729}a^{6}-\frac{5506420}{4772729}a^{5}+\frac{323265}{4772729}a^{4}-\frac{472727}{367133}a^{3}-\frac{2267293}{367133}a^{2}+\frac{424687}{28241}a+\frac{817310}{28241}$
| 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: | \( 560.767266895 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 560.767266895 \cdot 1}{4\cdot\sqrt{1052745423388672}}\cr\approx \mathstrut & 0.265852159764 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - x^8 - 8*x^6 + 468*x^4 - 1690*x^2 + 2197)
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^12 - 6*x^10 - x^8 - 8*x^6 + 468*x^4 - 1690*x^2 + 2197, 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^12 - 6*x^10 - x^8 - 8*x^6 + 468*x^4 - 1690*x^2 + 2197);
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^12 - 6*x^10 - x^8 - 8*x^6 + 468*x^4 - 1690*x^2 + 2197);
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
$A_4^2:D_4$ (as 12T208):
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 1152 |
| The 44 conjugacy class representatives for $A_4^2:D_4$ |
| Character table for $A_4^2:D_4$ 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 12 sibling: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.6229262860288.4 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.24.112 | $x^{12} + 12 x^{11} + 58 x^{10} + 148 x^{9} + 210 x^{8} + 160 x^{7} + 256 x^{6} + 928 x^{5} + 2436 x^{4} + 3440 x^{3} + 2920 x^{2} + 912 x + 72$ | $4$ | $3$ | $24$ | 12T134 | $[2, 2, 2, 2, 3, 3]^{6}$ |
|
\(13\)
| 13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 13.6.4.1 | $x^{6} + 130 x^{3} - 1521$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |