Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 16*x^9 + 21*x^8 + 66*x^7 - 18*x^6 - 78*x^5 + 12*x^4 + 20*x^3 - 6*x^2 + 6*x + 1)
gp: K = bnfinit(y^12 - 12*y^10 - 16*y^9 + 21*y^8 + 66*y^7 - 18*y^6 - 78*y^5 + 12*y^4 + 20*y^3 - 6*y^2 + 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 12*x^10 - 16*x^9 + 21*x^8 + 66*x^7 - 18*x^6 - 78*x^5 + 12*x^4 + 20*x^3 - 6*x^2 + 6*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 12*x^10 - 16*x^9 + 21*x^8 + 66*x^7 - 18*x^6 - 78*x^5 + 12*x^4 + 20*x^3 - 6*x^2 + 6*x + 1)
\( x^{12} - 12x^{10} - 16x^{9} + 21x^{8} + 66x^{7} - 18x^{6} - 78x^{5} + 12x^{4} + 20x^{3} - 6x^{2} + 6x + 1 \)
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: | $[6, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-304679870005248\)
\(\medspace = -\,2^{18}\cdot 3^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.11\) | 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: | $2^{3/2}3^{19/12}\approx 16.105973497991872$ | ||
| 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: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{97030909}a^{11}-\frac{9915098}{97030909}a^{10}-\frac{25845574}{97030909}a^{9}-\frac{45489125}{97030909}a^{8}-\frac{30119065}{97030909}a^{7}-\frac{2949499}{97030909}a^{6}+\frac{37848738}{97030909}a^{5}-\frac{451199}{2622457}a^{4}-\frac{43872331}{97030909}a^{3}-\frac{30537170}{97030909}a^{2}+\frac{543603}{97030909}a-\frac{4084956}{97030909}$
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: |
\( -1 \)
(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: |
$\frac{23189894}{97030909}a^{11}-\frac{1860490}{97030909}a^{10}-\frac{271834334}{97030909}a^{9}-\frac{352578356}{97030909}a^{8}+\frac{443772862}{97030909}a^{7}+\frac{1429518264}{97030909}a^{6}-\frac{390051168}{97030909}a^{5}-\frac{39054085}{2622457}a^{4}+\frac{185663880}{97030909}a^{3}+\frac{155631090}{97030909}a^{2}-\frac{159749198}{97030909}a+\frac{27327492}{97030909}$, $\frac{23189894}{97030909}a^{11}-\frac{1860490}{97030909}a^{10}-\frac{271834334}{97030909}a^{9}-\frac{352578356}{97030909}a^{8}+\frac{443772862}{97030909}a^{7}+\frac{1429518264}{97030909}a^{6}-\frac{390051168}{97030909}a^{5}-\frac{39054085}{2622457}a^{4}+\frac{185663880}{97030909}a^{3}+\frac{155631090}{97030909}a^{2}-\frac{159749198}{97030909}a+\frac{124358401}{97030909}$, $\frac{3790010}{97030909}a^{11}+\frac{3928358}{97030909}a^{10}-\frac{49569333}{97030909}a^{9}-\frac{107653686}{97030909}a^{8}+\frac{67228764}{97030909}a^{7}+\frac{397351809}{97030909}a^{6}+\frac{92754504}{97030909}a^{5}-\frac{17941086}{2622457}a^{4}-\frac{235221823}{97030909}a^{3}+\frac{444996229}{97030909}a^{2}+\frac{3515233}{97030909}a+\frac{33688662}{97030909}$, $\frac{24230075}{97030909}a^{11}+\frac{13935291}{97030909}a^{10}-\frac{283979325}{97030909}a^{9}-\frac{541459767}{97030909}a^{8}+\frac{213427290}{97030909}a^{7}+\frac{1631072434}{97030909}a^{6}+\frac{312227295}{97030909}a^{5}-\frac{44909013}{2622457}a^{4}-\frac{81563514}{97030909}a^{3}+\frac{495128105}{97030909}a^{2}-\frac{307405616}{97030909}a+\frac{111277384}{97030909}$, $\frac{325428}{2622457}a^{11}-\frac{398800}{2622457}a^{10}-\frac{3487336}{2622457}a^{9}-\frac{879996}{2622457}a^{8}+\frac{8545014}{2622457}a^{7}+\frac{11660740}{2622457}a^{6}-\frac{19896827}{2622457}a^{5}-\frac{2740771}{2622457}a^{4}+\frac{5609926}{2622457}a^{3}-\frac{9327137}{2622457}a^{2}+\frac{11045063}{2622457}a+\frac{1106530}{2622457}$, $\frac{1408414}{97030909}a^{11}+\frac{28557799}{97030909}a^{10}-\frac{25717377}{97030909}a^{9}-\frac{340026866}{97030909}a^{8}-\frac{334078108}{97030909}a^{7}+\frac{553776976}{97030909}a^{6}+\frac{1404189656}{97030909}a^{5}-\frac{19565345}{2622457}a^{4}-\frac{831249107}{97030909}a^{3}+\frac{263758597}{97030909}a^{2}-\frac{246889095}{97030909}a+\frac{41498462}{97030909}$, $\frac{14657011}{97030909}a^{11}+\frac{14760856}{97030909}a^{10}-\frac{199499687}{97030909}a^{9}-\frac{395670044}{97030909}a^{8}+\frac{334838954}{97030909}a^{7}+\frac{1476279279}{97030909}a^{6}+\frac{180969140}{97030909}a^{5}-\frac{66123638}{2622457}a^{4}+\frac{83176710}{97030909}a^{3}+\frac{1122621784}{97030909}a^{2}-\frac{486065538}{97030909}a+\frac{259557388}{97030909}$, $\frac{3790010}{97030909}a^{11}+\frac{3928358}{97030909}a^{10}-\frac{49569333}{97030909}a^{9}-\frac{107653686}{97030909}a^{8}+\frac{67228764}{97030909}a^{7}+\frac{397351809}{97030909}a^{6}+\frac{92754504}{97030909}a^{5}-\frac{17941086}{2622457}a^{4}-\frac{235221823}{97030909}a^{3}+\frac{444996229}{97030909}a^{2}+\frac{100546142}{97030909}a-\frac{63342247}{97030909}$
| 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: | \( 647.806861381 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 647.806861381 \cdot 1}{2\cdot\sqrt{304679870005248}}\cr\approx \mathstrut & 0.294586775142 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 12*x^10 - 16*x^9 + 21*x^8 + 66*x^7 - 18*x^6 - 78*x^5 + 12*x^4 + 20*x^3 - 6*x^2 + 6*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^12 - 12*x^10 - 16*x^9 + 21*x^8 + 66*x^7 - 18*x^6 - 78*x^5 + 12*x^4 + 20*x^3 - 6*x^2 + 6*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^12 - 12*x^10 - 16*x^9 + 21*x^8 + 66*x^7 - 18*x^6 - 78*x^5 + 12*x^4 + 20*x^3 - 6*x^2 + 6*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^12 - 12*x^10 - 16*x^9 + 21*x^8 + 66*x^7 - 18*x^6 - 78*x^5 + 12*x^4 + 20*x^3 - 6*x^2 + 6*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
$C_3\times D_4$ (as 12T14):
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 24 |
| The 15 conjugacy class representatives for $D_4 \times C_3$ |
| Character table for $D_4 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), 4.2.1728.1, \(\Q(\zeta_{36})^+\) |
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: | deg 24 |
| Degree 12 sibling: | 12.0.4760622968832.2 |
| Minimal sibling: | 12.0.4760622968832.2 |
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 | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.12.0.1}{12} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.65 | $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 10 x^{8} + 40 x^{7} + 40 x^{6} + 92 x^{4} + 184 x^{3} + 344$ | $4$ | $3$ | $18$ | $D_4 \times C_3$ | $[2, 2]^{6}$ |
|
\(3\)
| 3.12.19.24 | $x^{12} + 6 x^{11} + 6 x^{10} + 3 x^{9} + 3 x^{8} + 3 x^{6} + 24$ | $12$ | $1$ | $19$ | $D_4 \times C_3$ | $[2]_{4}^{2}$ |