Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
gp: K = bnfinit(y^12 - 4*y^11 + 22*y^10 - 48*y^9 + 108*y^8 - 132*y^7 + 170*y^6 - 140*y^5 + 107*y^4 - 12*y^3 - 12*y^2 + 12*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
\( x^{12} - 4 x^{11} + 22 x^{10} - 48 x^{9} + 108 x^{8} - 132 x^{7} + 170 x^{6} - 140 x^{5} + 107 x^{4} + \cdots + 9 \)
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: |
\(12950250637492224\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 7^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.01\) | 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: | $2^{2}3^{4/3}7^{1/2}\approx 45.79000429827802$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{3}$, $\frac{1}{27}a^{10}-\frac{1}{27}a^{9}-\frac{4}{27}a^{8}-\frac{1}{27}a^{7}-\frac{1}{27}a^{6}-\frac{13}{27}a^{5}+\frac{10}{27}a^{4}-\frac{1}{3}a^{3}+\frac{1}{9}a^{2}-\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{2577231}a^{11}+\frac{8842}{859077}a^{10}-\frac{135620}{2577231}a^{9}+\frac{102187}{2577231}a^{8}+\frac{60565}{2577231}a^{7}+\frac{601687}{2577231}a^{6}-\frac{395084}{859077}a^{5}+\frac{6530}{48627}a^{4}+\frac{175183}{859077}a^{3}-\frac{95981}{286359}a^{2}+\frac{357608}{859077}a-\frac{6595}{286359}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{4}$, which has order $4$
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: |
\( -1 \)
(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: |
$\frac{37892}{859077}a^{11}-\frac{754318}{2577231}a^{10}+\frac{3597577}{2577231}a^{9}-\frac{11779595}{2577231}a^{8}+\frac{24961642}{2577231}a^{7}-\frac{45388631}{2577231}a^{6}+\frac{54093136}{2577231}a^{5}-\frac{1249376}{48627}a^{4}+\frac{1954781}{95453}a^{3}-\frac{11588239}{859077}a^{2}+\frac{3264223}{859077}a+\frac{185548}{286359}$, $\frac{218251}{2577231}a^{11}-\frac{375689}{2577231}a^{10}+\frac{1085486}{859077}a^{9}-\frac{377681}{859077}a^{8}+\frac{969622}{286359}a^{7}+\frac{2511215}{859077}a^{6}+\frac{10372775}{2577231}a^{5}+\frac{271054}{48627}a^{4}+\frac{1215766}{859077}a^{3}+\frac{4984738}{859077}a^{2}+\frac{3258124}{859077}a+\frac{256801}{286359}$, $\frac{242558}{2577231}a^{11}-\frac{321980}{859077}a^{10}+\frac{4928627}{2577231}a^{9}-\frac{10410541}{2577231}a^{8}+\frac{18921905}{2577231}a^{7}-\frac{21205006}{2577231}a^{6}+\frac{1628735}{286359}a^{5}-\frac{293681}{48627}a^{4}-\frac{1632973}{859077}a^{3}+\frac{1554550}{286359}a^{2}-\frac{3187016}{859077}a-\frac{354995}{286359}$, $\frac{22544}{859077}a^{11}-\frac{223609}{2577231}a^{10}+\frac{1449631}{2577231}a^{9}-\frac{2549606}{2577231}a^{8}+\frac{7402825}{2577231}a^{7}-\frac{5859503}{2577231}a^{6}+\frac{11729590}{2577231}a^{5}-\frac{18668}{48627}a^{4}+\frac{816754}{286359}a^{3}+\frac{637349}{859077}a^{2}+\frac{626740}{859077}a+\frac{18829}{286359}$, $\frac{32965}{286359}a^{11}-\frac{36932}{95453}a^{10}+\frac{687032}{286359}a^{9}-\frac{1278257}{286359}a^{8}+\frac{3466585}{286359}a^{7}-\frac{3766847}{286359}a^{6}+\frac{2240491}{95453}a^{5}-\frac{91324}{5403}a^{4}+\frac{5825918}{286359}a^{3}-\frac{576487}{95453}a^{2}+\frac{579485}{95453}a+\frac{208730}{95453}$
| 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: | \( 3833.8726435419103 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3833.8726435419103 \cdot 4}{2\cdot\sqrt{12950250637492224}}\cr\approx \mathstrut & 4.14579478575610 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
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 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9, 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 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9);
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 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9);
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
| \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{7})\), 6.0.14224896.1 |
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: | 18.6.12237489813936932340847607808.2, deg 18 |
| 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 | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.24.79 | $x^{12} - 8 x^{11} + 14 x^{10} + 76 x^{9} + 138 x^{8} + 432 x^{7} + 688 x^{6} + 992 x^{5} + 1748 x^{4} + 1728 x^{3} + 1848 x^{2} + 1648 x + 968$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
|
\(7\)
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.28.2t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\sqrt{7}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.56.2t1.b.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{-14}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.252.6t1.i.a | $1$ | $ 2^{2} \cdot 3^{2} \cdot 7 $ | 6.6.144027072.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.504.6t1.f.a | $1$ | $ 2^{3} \cdot 3^{2} \cdot 7 $ | 6.0.1152216576.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.504.6t1.f.b | $1$ | $ 2^{3} \cdot 3^{2} \cdot 7 $ | 6.0.1152216576.2 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.72.6t1.a.a | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.3359232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.252.6t1.i.b | $1$ | $ 2^{2} \cdot 3^{2} \cdot 7 $ | 6.6.144027072.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.72.6t1.a.b | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.3359232.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.4536.3t2.a.a | $2$ | $ 2^{3} \cdot 3^{4} \cdot 7 $ | 3.1.4536.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.18144.6t3.g.a | $2$ | $ 2^{5} \cdot 3^{4} \cdot 7 $ | 6.0.658409472.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.504.6t5.b.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 7 $ | 6.0.14224896.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2016.12t18.c.a | $2$ | $ 2^{5} \cdot 3^{2} \cdot 7 $ | 12.0.12950250637492224.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.2016.12t18.c.b | $2$ | $ 2^{5} \cdot 3^{2} \cdot 7 $ | 12.0.12950250637492224.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.504.6t5.b.b | $2$ | $ 2^{3} \cdot 3^{2} \cdot 7 $ | 6.0.14224896.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.