Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 17*x^10 - 28*x^9 + 24*x^8 - 20*x^6 + 18*x^5 - 6*x^4 - 4*x^3 + 8*x^2 - 4*x + 1)
gp: K = bnfinit(y^12 - 6*y^11 + 17*y^10 - 28*y^9 + 24*y^8 - 20*y^6 + 18*y^5 - 6*y^4 - 4*y^3 + 8*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 17*x^10 - 28*x^9 + 24*x^8 - 20*x^6 + 18*x^5 - 6*x^4 - 4*x^3 + 8*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 17*x^10 - 28*x^9 + 24*x^8 - 20*x^6 + 18*x^5 - 6*x^4 - 4*x^3 + 8*x^2 - 4*x + 1)
\( x^{12} - 6x^{11} + 17x^{10} - 28x^{9} + 24x^{8} - 20x^{6} + 18x^{5} - 6x^{4} - 4x^{3} + 8x^{2} - 4x + 1 \)
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: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(458838245376\)
\(\medspace = 2^{18}\cdot 3^{6}\cdot 7^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.37\) | 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: | $2^{3/2}3^{1/2}7^{2/3}\approx 17.926863604818365$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2387}a^{11}-\frac{9}{341}a^{10}-\frac{106}{217}a^{9}-\frac{402}{2387}a^{8}-\frac{932}{2387}a^{7}+\frac{610}{2387}a^{6}+\frac{145}{341}a^{5}-\frac{549}{2387}a^{4}+\frac{256}{2387}a^{3}-\frac{274}{2387}a^{2}-\frac{1083}{2387}a-\frac{335}{2387}$
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: |
\( -\frac{234}{341} a^{11} + \frac{1443}{341} a^{10} - \frac{368}{31} a^{9} + \frac{6431}{341} a^{8} - \frac{4926}{341} a^{7} - \frac{1225}{341} a^{6} + \frac{5282}{341} a^{5} - \frac{3160}{341} a^{4} + \frac{112}{341} a^{3} + \frac{1372}{341} a^{2} - \frac{1646}{341} a + \frac{642}{341} \)
(order $6$)
| 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: |
$a-1$, $\frac{2683}{2387}a^{11}-\frac{2323}{341}a^{10}+\frac{4212}{217}a^{9}-\frac{76026}{2387}a^{8}+\frac{63082}{2387}a^{7}+\frac{6309}{2387}a^{6}-\frac{8912}{341}a^{5}+\frac{49939}{2387}a^{4}-\frac{7769}{2387}a^{3}-\frac{16655}{2387}a^{2}+\frac{20773}{2387}a-\frac{8454}{2387}$, $\frac{39}{341}a^{11}-\frac{70}{341}a^{10}-\frac{11}{31}a^{9}+\frac{690}{341}a^{8}-\frac{1225}{341}a^{7}+\frac{602}{341}a^{6}+\frac{1052}{341}a^{5}-\frac{1292}{341}a^{4}+\frac{436}{341}a^{3}+\frac{226}{341}a^{2}-\frac{294}{341}a+\frac{234}{341}$, $\frac{524}{2387}a^{11}-\frac{283}{341}a^{10}+\frac{225}{217}a^{9}+\frac{1795}{2387}a^{8}-\frac{10968}{2387}a^{7}+\frac{14104}{2387}a^{6}-\frac{63}{341}a^{5}-\frac{10784}{2387}a^{4}+\frac{7633}{2387}a^{3}-\frac{2743}{2387}a^{2}-\frac{1773}{2387}a+\frac{3485}{2387}$, $\frac{236}{2387}a^{11}-\frac{78}{341}a^{10}-\frac{61}{217}a^{9}+\frac{5382}{2387}a^{8}-\frac{12283}{2387}a^{7}+\frac{12675}{2387}a^{6}-\frac{221}{341}a^{5}-\frac{7827}{2387}a^{4}+\frac{7902}{2387}a^{3}-\frac{2602}{2387}a^{2}-\frac{2566}{2387}a+\frac{2098}{2387}$
| 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: | \( 16.1633727166 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 16.1633727166 \cdot 1}{6\cdot\sqrt{458838245376}}\cr\approx \mathstrut & 0.244697843624 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 17*x^10 - 28*x^9 + 24*x^8 - 20*x^6 + 18*x^5 - 6*x^4 - 4*x^3 + 8*x^2 - 4*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 - 6*x^11 + 17*x^10 - 28*x^9 + 24*x^8 - 20*x^6 + 18*x^5 - 6*x^4 - 4*x^3 + 8*x^2 - 4*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 - 6*x^11 + 17*x^10 - 28*x^9 + 24*x^8 - 20*x^6 + 18*x^5 - 6*x^4 - 4*x^3 + 8*x^2 - 4*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 - 6*x^11 + 17*x^10 - 28*x^9 + 24*x^8 - 20*x^6 + 18*x^5 - 6*x^4 - 4*x^3 + 8*x^2 - 4*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
$C_6\times S_3$ (as 12T18):
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 36 |
| The 18 conjugacy class representatives for $C_6\times S_3$ |
| Character table for $C_6\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), 6.0.677376.1 |
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 36 |
| Degree 18 siblings: | 18.0.71418001746108874752.2, 18.6.1354296922000286810112.1 |
| 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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(7\)
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |