Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*x^2 - 12*x + 9)
gp: K = bnfinit(x^12 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*x^2 - 12*x + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*x^2 - 12*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*x^2 - 12*x + 9)
\( x^{12} - 2x^{11} + 3x^{10} - 6x^{9} + 5x^{8} - 2x^{7} + 2x^{6} + 2x^{5} + 5x^{4} - 12x^{3} + 16x^{2} - 12x + 9 \)
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: |
\(24904730935296\)
\(\medspace = 2^{18}\cdot 3^{6}\cdot 19^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.07\) | 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))
| |
| Ramified primes: |
\(2\), \(3\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{6219}a^{11}-\frac{337}{6219}a^{10}+\frac{956}{6219}a^{9}+\frac{1049}{6219}a^{8}+\frac{3073}{6219}a^{7}-\frac{1249}{6219}a^{6}-\frac{329}{6219}a^{5}+\frac{116}{2073}a^{4}-\frac{487}{6219}a^{3}-\frac{2707}{6219}a^{2}+\frac{320}{2073}a-\frac{33}{691}$
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: |
\( \frac{302}{2073} a^{11} - \frac{296}{691} a^{10} + \frac{1256}{2073} a^{9} - \frac{1753}{2073} a^{8} + \frac{702}{691} a^{7} - \frac{201}{691} a^{6} - \frac{412}{691} a^{5} + \frac{755}{2073} a^{4} + \frac{2182}{2073} a^{3} - \frac{4207}{2073} a^{2} + \frac{6610}{2073} a - \frac{876}{691} \)
(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: |
$\frac{985}{6219}a^{11}-\frac{2338}{6219}a^{10}+\frac{2591}{6219}a^{9}-\frac{5308}{6219}a^{8}+\frac{4471}{6219}a^{7}+\frac{1097}{6219}a^{6}-\frac{677}{6219}a^{5}+\frac{245}{2073}a^{4}+\frac{5387}{6219}a^{3}-\frac{17101}{6219}a^{2}+\frac{4250}{2073}a-\frac{719}{691}$, $\frac{1039}{6219}a^{11}-\frac{3952}{6219}a^{10}+\frac{6536}{6219}a^{9}-\frac{8779}{6219}a^{8}+\frac{10792}{6219}a^{7}-\frac{6232}{6219}a^{6}-\frac{3932}{6219}a^{5}+\frac{1672}{2073}a^{4}+\frac{10184}{6219}a^{3}-\frac{24388}{6219}a^{2}+\frac{3952}{691}a-\frac{3192}{691}$, $\frac{425}{6219}a^{11}+\frac{3958}{6219}a^{10}-\frac{8300}{6219}a^{9}+\frac{6349}{6219}a^{8}-\frac{16549}{6219}a^{7}+\frac{14374}{6219}a^{6}+\frac{11504}{6219}a^{5}-\frac{381}{691}a^{4}-\frac{1748}{6219}a^{3}+\frac{20770}{6219}a^{2}-\frac{14638}{2073}a+\frac{4632}{691}$, $\frac{124}{2073}a^{11}+\frac{121}{691}a^{10}-\frac{333}{691}a^{9}+\frac{859}{2073}a^{8}-\frac{2453}{2073}a^{7}+\frac{2672}{2073}a^{6}+\frac{664}{2073}a^{5}-\frac{127}{691}a^{4}+\frac{140}{691}a^{3}+\frac{2231}{2073}a^{2}-\frac{6722}{2073}a+\frac{2235}{691}$, $\frac{574}{2073}a^{11}-\frac{677}{691}a^{10}+\frac{2854}{2073}a^{9}-\frac{3881}{2073}a^{8}+\frac{1769}{691}a^{7}-\frac{350}{691}a^{6}-\frac{989}{691}a^{5}+\frac{1435}{2073}a^{4}+\frac{2390}{2073}a^{3}-\frac{10124}{2073}a^{2}+\frac{15515}{2073}a-\frac{2928}{691}$
| 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: | \( 300.56279584783476 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} \approx\frac{2^{0}\cdot(2\pi)^{6}\cdot 300.56279584783476 \cdot 1}{6\cdot\sqrt{24904730935296}}\approx 0.617621285008397$
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*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 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*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 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*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 - 2*x^11 + 3*x^10 - 6*x^9 + 5*x^8 - 2*x^7 + 2*x^6 + 2*x^5 + 5*x^4 - 12*x^3 + 16*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
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.0.184832.1 |
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
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.3.2.1 | $x^{3} + 76$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.152.6t1.c.a | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.456.6t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 19 $ | 6.6.1801557504.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.57.6t1.a.a | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.152.6t1.c.b | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.57.6t1.a.b | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.456.6t1.b.b | $1$ | $ 2^{3} \cdot 3 \cdot 19 $ | 6.6.1801557504.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 2.2888.3t2.b.a | $2$ | $ 2^{3} \cdot 19^{2}$ | 3.1.2888.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.25992.6t3.b.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 19^{2}$ | 6.0.225194688.10 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.152.6t5.a.a | $2$ | $ 2^{3} \cdot 19 $ | 6.0.184832.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1368.12t18.b.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 19 $ | 12.0.24904730935296.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.152.6t5.a.b | $2$ | $ 2^{3} \cdot 19 $ | 6.0.184832.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1368.12t18.b.b | $2$ | $ 2^{3} \cdot 3^{2} \cdot 19 $ | 12.0.24904730935296.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
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 *.