Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - 16*x^9 + 3*x^8 + 42*x^7 + 28*x^6 - 6*x^5 - 24*x^4 - 36*x^3 - 36*x^2 - 18*x - 3)
gp: K = bnfinit(y^12 - 6*y^10 - 16*y^9 + 3*y^8 + 42*y^7 + 28*y^6 - 6*y^5 - 24*y^4 - 36*y^3 - 36*y^2 - 18*y - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^10 - 16*x^9 + 3*x^8 + 42*x^7 + 28*x^6 - 6*x^5 - 24*x^4 - 36*x^3 - 36*x^2 - 18*x - 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^10 - 16*x^9 + 3*x^8 + 42*x^7 + 28*x^6 - 6*x^5 - 24*x^4 - 36*x^3 - 36*x^2 - 18*x - 3)
\( x^{12} - 6x^{10} - 16x^{9} + 3x^{8} + 42x^{7} + 28x^{6} - 6x^{5} - 24x^{4} - 36x^{3} - 36x^{2} - 18x - 3 \)
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: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(101559956668416\)
\(\medspace = 2^{18}\cdot 3^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.70\) | 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: | $2^{7/4}3^{3/2}\approx 17.477703781463642$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13}a^{10}+\frac{3}{13}a^{9}+\frac{4}{13}a^{8}-\frac{1}{13}a^{7}+\frac{4}{13}a^{6}+\frac{1}{13}a^{5}-\frac{4}{13}a^{4}-\frac{4}{13}a^{3}-\frac{1}{13}a^{2}-\frac{4}{13}a+\frac{3}{13}$, $\frac{1}{2287753}a^{11}-\frac{42125}{2287753}a^{10}-\frac{780709}{2287753}a^{9}+\frac{917234}{2287753}a^{8}-\frac{621830}{2287753}a^{7}-\frac{183058}{2287753}a^{6}-\frac{697085}{2287753}a^{5}-\frac{891889}{2287753}a^{4}-\frac{943418}{2287753}a^{3}+\frac{925851}{2287753}a^{2}+\frac{139733}{2287753}a+\frac{135826}{2287753}$
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: | $7$ | 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: |
$\frac{19583}{175981}a^{11}-\frac{2632}{175981}a^{10}-\frac{126065}{175981}a^{9}-\frac{282007}{175981}a^{8}+\frac{128052}{175981}a^{7}+\frac{869283}{175981}a^{6}+\frac{290873}{175981}a^{5}-\frac{181221}{175981}a^{4}-\frac{198574}{175981}a^{3}-\frac{842555}{175981}a^{2}-\frac{722376}{175981}a-\frac{275312}{175981}$, $\frac{220283}{2287753}a^{11}-\frac{647169}{2287753}a^{10}-\frac{720264}{2287753}a^{9}-\frac{120080}{2287753}a^{8}+\frac{7848206}{2287753}a^{7}+\frac{248869}{2287753}a^{6}-\frac{13784422}{2287753}a^{5}-\frac{1212358}{2287753}a^{4}-\frac{344679}{2287753}a^{3}+\frac{3271104}{2287753}a^{2}+\frac{7358931}{2287753}a+\frac{6732397}{2287753}$, $\frac{1136817}{2287753}a^{11}-\frac{642386}{2287753}a^{10}-\frac{6205098}{2287753}a^{9}-\frac{14774932}{2287753}a^{8}+\frac{10521700}{2287753}a^{7}+\frac{38433326}{2287753}a^{6}+\frac{10937686}{2287753}a^{5}-\frac{7437015}{2287753}a^{4}-\frac{22291161}{2287753}a^{3}-\frac{26771916}{2287753}a^{2}-\frac{26556749}{2287753}a-\frac{7862323}{2287753}$, $\frac{75005}{175981}a^{11}-\frac{4874}{13537}a^{10}-\frac{402533}{175981}a^{9}-\frac{66341}{13537}a^{8}+\frac{981777}{175981}a^{7}+\frac{2425582}{175981}a^{6}+\frac{109950}{175981}a^{5}-\frac{666433}{175981}a^{4}-\frac{1408280}{175981}a^{3}-\frac{1599411}{175981}a^{2}-\frac{1296175}{175981}a-\frac{208774}{175981}$, $\frac{754782}{2287753}a^{11}-\frac{176537}{2287753}a^{10}-\frac{4512665}{2287753}a^{9}-\frac{11091549}{2287753}a^{8}+\frac{4914895}{2287753}a^{7}+\frac{31283697}{2287753}a^{6}+\frac{15213772}{2287753}a^{5}-\frac{7451271}{2287753}a^{4}-\frac{17962961}{2287753}a^{3}-\frac{24063447}{2287753}a^{2}-\frac{21921453}{2287753}a-\frac{9446459}{2287753}$, $\frac{1710177}{2287753}a^{11}-\frac{392098}{2287753}a^{10}-\frac{10357510}{2287753}a^{9}-\frac{24659629}{2287753}a^{8}+\frac{11707318}{2287753}a^{7}+\frac{70211084}{2287753}a^{6}+\frac{27128336}{2287753}a^{5}-\frac{20427704}{2287753}a^{4}-\frac{29043789}{2287753}a^{3}-\frac{52870320}{2287753}a^{2}-\frac{51871174}{2287753}a-\frac{18097753}{2287753}$, $\frac{225146}{2287753}a^{11}-\frac{131217}{2287753}a^{10}-\frac{1221980}{2287753}a^{9}-\frac{2841260}{2287753}a^{8}+\frac{1963066}{2287753}a^{7}+\frac{7012972}{2287753}a^{6}+\frac{2227497}{2287753}a^{5}+\frac{1222895}{2287753}a^{4}-\frac{1417629}{2287753}a^{3}-\frac{12099715}{2287753}a^{2}-\frac{6484630}{2287753}a-\frac{65717}{2287753}$
| 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: | \( 332.146241747 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 332.146241747 \cdot 1}{2\cdot\sqrt{101559956668416}}\cr\approx \mathstrut & 0.410939179735 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - 16*x^9 + 3*x^8 + 42*x^7 + 28*x^6 - 6*x^5 - 24*x^4 - 36*x^3 - 36*x^2 - 18*x - 3)
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 - 16*x^9 + 3*x^8 + 42*x^7 + 28*x^6 - 6*x^5 - 24*x^4 - 36*x^3 - 36*x^2 - 18*x - 3, 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 - 16*x^9 + 3*x^8 + 42*x^7 + 28*x^6 - 6*x^5 - 24*x^4 - 36*x^3 - 36*x^2 - 18*x - 3);
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 - 16*x^9 + 3*x^8 + 42*x^7 + 28*x^6 - 6*x^5 - 24*x^4 - 36*x^3 - 36*x^2 - 18*x - 3);
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
$C_2\times A_4$ (as 12T7):
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 8 conjugacy class representatives for $A_4 \times C_2$ |
| Character table for $A_4 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), 6.2.1259712.1, \(\Q(\zeta_{36})^+\), 6.2.419904.1 |
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: | deg 24 |
| Degree 6 sibling: | 6.2.1259712.1 |
| Degree 8 sibling: | 8.0.967458816.1 |
| Degree 12 sibling: | 12.0.101559956668416.13 |
| Minimal sibling: | 6.2.1259712.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.49 | $x^{12} + 2 x^{11} - 4 x^{10} + 12 x^{9} + 78 x^{8} + 96 x^{7} + 16 x^{6} + 72 x^{5} + 180 x^{4} + 248 x^{3} + 248$ | $4$ | $3$ | $18$ | $A_4 \times C_2$ | $[2, 2, 2]^{3}$ |
|
\(3\)
| 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |
| 3.6.9.3 | $x^{6} + 3 x^{4} + 24$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ |