Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 18*x^10 - 32*x^9 + 45*x^8 - 78*x^7 + 146*x^6 - 180*x^5 + 99*x^4 + 30*x^3 - 72*x^2 + 36*x - 6)
gp: K = bnfinit(y^12 - 6*y^11 + 18*y^10 - 32*y^9 + 45*y^8 - 78*y^7 + 146*y^6 - 180*y^5 + 99*y^4 + 30*y^3 - 72*y^2 + 36*y - 6, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 18*x^10 - 32*x^9 + 45*x^8 - 78*x^7 + 146*x^6 - 180*x^5 + 99*x^4 + 30*x^3 - 72*x^2 + 36*x - 6);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 18*x^10 - 32*x^9 + 45*x^8 - 78*x^7 + 146*x^6 - 180*x^5 + 99*x^4 + 30*x^3 - 72*x^2 + 36*x - 6)
\( x^{12} - 6 x^{11} + 18 x^{10} - 32 x^{9} + 45 x^{8} - 78 x^{7} + 146 x^{6} - 180 x^{5} + 99 x^{4} + \cdots - 6 \)
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: | $[2, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-171382426877952\)
\(\medspace = -\,2^{14}\cdot 3^{21}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.35\) | 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: | $2^{3/2}3^{11/6}\approx 21.196653174008656$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{136}a^{11}-\frac{1}{34}a^{10}-\frac{7}{136}a^{9}+\frac{5}{136}a^{8}+\frac{21}{136}a^{7}-\frac{19}{136}a^{6}-\frac{7}{34}a^{5}-\frac{49}{136}a^{4}+\frac{13}{34}a^{3}+\frac{33}{68}a^{2}+\frac{15}{34}a-\frac{7}{68}$
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
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: | $6$ | 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: |
$a-1$, $\frac{15}{17}a^{11}-\frac{633}{136}a^{10}+\frac{1727}{136}a^{9}-\frac{1349}{68}a^{8}+\frac{3693}{136}a^{7}-\frac{1743}{34}a^{6}+\frac{12875}{136}a^{5}-\frac{13241}{136}a^{4}+\frac{965}{34}a^{3}+\frac{684}{17}a^{2}-\frac{2605}{68}a+\frac{673}{68}$, $\frac{5}{68}a^{11}-\frac{91}{136}a^{10}+\frac{321}{136}a^{9}-\frac{83}{17}a^{8}+\frac{907}{136}a^{7}-\frac{707}{68}a^{6}+\frac{2831}{136}a^{5}-\frac{4145}{136}a^{4}+\frac{354}{17}a^{3}+\frac{131}{34}a^{2}-\frac{941}{68}a+\frac{389}{68}$, $\frac{28}{17}a^{11}-\frac{635}{68}a^{10}+\frac{450}{17}a^{9}-\frac{2959}{68}a^{8}+\frac{996}{17}a^{7}-\frac{7313}{68}a^{6}+\frac{3449}{17}a^{5}-\frac{7657}{34}a^{4}+\frac{1371}{17}a^{3}+\frac{2795}{34}a^{2}-\frac{1516}{17}a+\frac{407}{17}$, $\frac{253}{136}a^{11}-\frac{1335}{136}a^{10}+\frac{1809}{68}a^{9}-\frac{5603}{136}a^{8}+\frac{951}{17}a^{7}-\frac{14565}{136}a^{6}+\frac{26933}{136}a^{5}-\frac{13653}{68}a^{4}+\frac{1793}{34}a^{3}+\frac{5799}{68}a^{2}-\frac{4905}{68}a+\frac{517}{34}$, $\frac{231}{136}a^{11}-\frac{1281}{136}a^{10}+\frac{446}{17}a^{9}-\frac{5713}{136}a^{8}+\frac{3811}{68}a^{7}-\frac{14283}{136}a^{6}+\frac{26937}{136}a^{5}-\frac{14389}{68}a^{4}+\frac{1051}{17}a^{3}+\frac{6025}{68}a^{2}-\frac{5497}{68}a+\frac{611}{34}$
| 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: | \( 635.742109669 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 635.742109669 \cdot 1}{2\cdot\sqrt{171382426877952}}\cr\approx \mathstrut & 0.951102168826 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 18*x^10 - 32*x^9 + 45*x^8 - 78*x^7 + 146*x^6 - 180*x^5 + 99*x^4 + 30*x^3 - 72*x^2 + 36*x - 6)
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^11 + 18*x^10 - 32*x^9 + 45*x^8 - 78*x^7 + 146*x^6 - 180*x^5 + 99*x^4 + 30*x^3 - 72*x^2 + 36*x - 6, 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^11 + 18*x^10 - 32*x^9 + 45*x^8 - 78*x^7 + 146*x^6 - 180*x^5 + 99*x^4 + 30*x^3 - 72*x^2 + 36*x - 6);
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^11 + 18*x^10 - 32*x^9 + 45*x^8 - 78*x^7 + 146*x^6 - 180*x^5 + 99*x^4 + 30*x^3 - 72*x^2 + 36*x - 6);
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
$C_2\times S_4$ (as 12T22):
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 48 |
| The 10 conjugacy class representatives for $C_2 \times S_4$ |
| Character table for $C_2 \times S_4$ |
Intermediate fields
| 3.1.243.1, 6.2.3779136.1 |
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 6 siblings: | 6.2.2834352.1, 6.0.944784.1 |
| Degree 8 siblings: | 8.4.2176782336.1, 8.0.241864704.2 |
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Minimal sibling: | 6.0.944784.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
| 2.4.6.8 | $x^{4} + 2 x^{3} + 2$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ | |
|
\(3\)
| 3.6.11.1 | $x^{6} + 18 x^{2} + 15$ | $6$ | $1$ | $11$ | $D_{6}$ | $[5/2]_{2}^{2}$ |
| 3.6.10.2 | $x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ |