Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 9*x^10 + 24*x^9 + 27*x^8 - 194*x^7 + 409*x^6 - 490*x^5 + 333*x^4 - 160*x^3 + 97*x^2 - 18*x + 1)
gp: K = bnfinit(y^12 - 2*y^11 - 9*y^10 + 24*y^9 + 27*y^8 - 194*y^7 + 409*y^6 - 490*y^5 + 333*y^4 - 160*y^3 + 97*y^2 - 18*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 9*x^10 + 24*x^9 + 27*x^8 - 194*x^7 + 409*x^6 - 490*x^5 + 333*x^4 - 160*x^3 + 97*x^2 - 18*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 9*x^10 + 24*x^9 + 27*x^8 - 194*x^7 + 409*x^6 - 490*x^5 + 333*x^4 - 160*x^3 + 97*x^2 - 18*x + 1)
\( x^{12} - 2 x^{11} - 9 x^{10} + 24 x^{9} + 27 x^{8} - 194 x^{7} + 409 x^{6} - 490 x^{5} + 333 x^{4} + \cdots + 1 \)
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: |
\(7169347584000000\)
\(\medspace = 2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 7^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.96\) | 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: | $2^{3/2}3^{1/2}5^{1/2}7^{2/3}\approx 40.085685643740796$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{61184186}a^{11}-\frac{485755}{4370299}a^{10}-\frac{5414732}{30592093}a^{9}+\frac{9705083}{30592093}a^{8}-\frac{7126838}{30592093}a^{7}-\frac{3452957}{61184186}a^{6}+\frac{6582487}{61184186}a^{5}+\frac{12511781}{30592093}a^{4}+\frac{17195659}{61184186}a^{3}+\frac{15644073}{61184186}a^{2}+\frac{1426869}{8740598}a-\frac{4004193}{61184186}$
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
$C_{2}$, which has order $2$
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: |
$\frac{1475691}{4370299}a^{11}-\frac{5367649}{8740598}a^{10}-\frac{13392437}{4370299}a^{9}+\frac{65567911}{8740598}a^{8}+\frac{83918353}{8740598}a^{7}-\frac{275221662}{4370299}a^{6}+\frac{569965253}{4370299}a^{5}-\frac{670228958}{4370299}a^{4}+\frac{891905331}{8740598}a^{3}-\frac{208072675}{4370299}a^{2}+\frac{234304577}{8740598}a-\frac{24658361}{8740598}$, $\frac{4148377}{4370299}a^{11}-\frac{7998728}{4370299}a^{10}-\frac{76365235}{8740598}a^{9}+\frac{193958071}{8740598}a^{8}+\frac{121718640}{4370299}a^{7}-\frac{1597365747}{8740598}a^{6}+\frac{3255837815}{8740598}a^{5}-\frac{3756428941}{8740598}a^{4}+\frac{2377026183}{8740598}a^{3}-\frac{535989483}{4370299}a^{2}+\frac{350588503}{4370299}a-\frac{71789129}{8740598}$, $\frac{4891094}{30592093}a^{11}-\frac{2959543}{8740598}a^{10}-\frac{42363184}{30592093}a^{9}+\frac{243707083}{61184186}a^{8}+\frac{227823475}{61184186}a^{7}-\frac{953574075}{30592093}a^{6}+\frac{2126562693}{30592093}a^{5}-\frac{2729677105}{30592093}a^{4}+\frac{4186220459}{61184186}a^{3}-\frac{1137412249}{30592093}a^{2}+\frac{175342441}{8740598}a-\frac{234086851}{61184186}$, $\frac{48215809}{30592093}a^{11}-\frac{12903179}{4370299}a^{10}-\frac{446184938}{30592093}a^{9}+\frac{1100090854}{30592093}a^{8}+\frac{1450009021}{30592093}a^{7}-\frac{9166800047}{30592093}a^{6}+\frac{18511509866}{30592093}a^{5}-\frac{21229133458}{30592093}a^{4}+\frac{13357124792}{30592093}a^{3}-\frac{6037539789}{30592093}a^{2}+\frac{562385383}{4370299}a-\frac{388529252}{30592093}$, $a$
| 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: | \( 522.1707580630209 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 522.1707580630209 \cdot 2}{2\cdot\sqrt{7169347584000000}}\cr\approx \mathstrut & 0.379447712776125 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 9*x^10 + 24*x^9 + 27*x^8 - 194*x^7 + 409*x^6 - 490*x^5 + 333*x^4 - 160*x^3 + 97*x^2 - 18*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 - 2*x^11 - 9*x^10 + 24*x^9 + 27*x^8 - 194*x^7 + 409*x^6 - 490*x^5 + 333*x^4 - 160*x^3 + 97*x^2 - 18*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 - 2*x^11 - 9*x^10 + 24*x^9 + 27*x^8 - 194*x^7 + 409*x^6 - 490*x^5 + 333*x^4 - 160*x^3 + 97*x^2 - 18*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 - 2*x^11 - 9*x^10 + 24*x^9 + 27*x^8 - 194*x^7 + 409*x^6 - 490*x^5 + 333*x^4 - 160*x^3 + 97*x^2 - 18*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
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{6}, \sqrt{-10})\), 6.0.3136000.1 |
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: | 18.6.571344013968870998016000000.1, 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 | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.9.7 | $x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.6.9.7 | $x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
|
\(5\)
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.168.6t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | 6.6.33191424.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.105.6t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | 6.0.8103375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.105.6t1.a.b | $1$ | $ 3 \cdot 5 \cdot 7 $ | 6.0.8103375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.168.6t1.b.b | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | 6.6.33191424.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.280.6t1.d.a | $1$ | $ 2^{3} \cdot 5 \cdot 7 $ | 6.0.153664000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.280.6t1.d.b | $1$ | $ 2^{3} \cdot 5 \cdot 7 $ | 6.0.153664000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 2.1960.3t2.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 7^{2}$ | 3.1.1960.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.17640.6t3.h.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ | 6.0.518616000.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.280.6t5.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 7 $ | 6.0.3136000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2520.12t18.b.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $ | 12.0.7169347584000000.1 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.280.6t5.a.b | $2$ | $ 2^{3} \cdot 5 \cdot 7 $ | 6.0.3136000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.2520.12t18.b.b | $2$ | $ 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $ | 12.0.7169347584000000.1 | $C_6\times S_3$ (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 *.