Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - 9*x^8 - 30*x^7 + 62*x^6 + 90*x^5 - 33*x^4 + 210*x^3 + 693*x^2 + 510*x + 151)
gp: K = bnfinit(y^12 - 6*y^10 - 9*y^8 - 30*y^7 + 62*y^6 + 90*y^5 - 33*y^4 + 210*y^3 + 693*y^2 + 510*y + 151, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^10 - 9*x^8 - 30*x^7 + 62*x^6 + 90*x^5 - 33*x^4 + 210*x^3 + 693*x^2 + 510*x + 151);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 6*x^10 - 9*x^8 - 30*x^7 + 62*x^6 + 90*x^5 - 33*x^4 + 210*x^3 + 693*x^2 + 510*x + 151)
\( x^{12} - 6x^{10} - 9x^{8} - 30x^{7} + 62x^{6} + 90x^{5} - 33x^{4} + 210x^{3} + 693x^{2} + 510x + 151 \)
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: |
\(1586874322944000000\)
\(\medspace = 2^{18}\cdot 3^{18}\cdot 5^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.86\) | 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: | $2^{3/2}3^{31/18}5^{1/2}\approx 41.95066245002997$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}-\frac{2}{11}a^{9}-\frac{5}{11}a^{8}+\frac{5}{11}a^{7}-\frac{4}{11}a^{6}-\frac{4}{11}a^{5}+\frac{5}{11}a^{4}+\frac{4}{11}a^{3}-\frac{1}{11}a^{2}+\frac{2}{11}a-\frac{1}{11}$, $\frac{1}{135680484049}a^{11}-\frac{135725696}{12334589459}a^{10}+\frac{31270877266}{135680484049}a^{9}-\frac{24929020627}{135680484049}a^{8}-\frac{66312541916}{135680484049}a^{7}-\frac{25874197834}{135680484049}a^{6}-\frac{21237479585}{135680484049}a^{5}+\frac{53821003052}{135680484049}a^{4}+\frac{53199844815}{135680484049}a^{3}+\frac{3511063813}{12334589459}a^{2}-\frac{32984610557}{135680484049}a+\frac{881379981}{135680484049}$
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
$C_{4}$, 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: |
$\frac{20102310}{4678637381}a^{11}-\frac{24460479}{4678637381}a^{10}-\frac{117764798}{4678637381}a^{9}+\frac{183057192}{4678637381}a^{8}-\frac{262500570}{4678637381}a^{7}-\frac{475949607}{4678637381}a^{6}+\frac{2194984764}{4678637381}a^{5}-\frac{683948907}{4678637381}a^{4}-\frac{2471055846}{4678637381}a^{3}+\frac{6570504027}{4678637381}a^{2}+\frac{7012235124}{4678637381}a-\frac{4984041970}{4678637381}$, $\frac{313631838}{135680484049}a^{11}-\frac{671076746}{135680484049}a^{10}-\frac{343840548}{135680484049}a^{9}-\frac{227118142}{135680484049}a^{8}-\frac{7750676317}{135680484049}a^{7}+\frac{16957490752}{135680484049}a^{6}+\frac{5405708757}{135680484049}a^{5}+\frac{1943971472}{135680484049}a^{4}+\frac{86229464797}{135680484049}a^{3}-\frac{170323386311}{135680484049}a^{2}+\frac{8329619374}{12334589459}a+\frac{78228899797}{135680484049}$, $\frac{309217673}{135680484049}a^{11}-\frac{1356582534}{135680484049}a^{10}-\frac{482495190}{135680484049}a^{9}+\frac{2811086892}{135680484049}a^{8}-\frac{4522151846}{135680484049}a^{7}+\frac{13257126191}{135680484049}a^{6}+\frac{2512186792}{12334589459}a^{5}-\frac{22269553745}{135680484049}a^{4}-\frac{9936979556}{135680484049}a^{3}+\frac{64287927692}{135680484049}a^{2}+\frac{63569266825}{135680484049}a+\frac{23764689958}{135680484049}$, $\frac{4426458173}{135680484049}a^{11}-\frac{230120962}{135680484049}a^{10}-\frac{27861595951}{135680484049}a^{9}+\frac{514074317}{12334589459}a^{8}-\frac{34179776208}{135680484049}a^{7}-\frac{145123667069}{135680484049}a^{6}+\frac{313696015418}{135680484049}a^{5}+\frac{353707622016}{135680484049}a^{4}-\frac{313803641070}{135680484049}a^{3}+\frac{1015740550236}{135680484049}a^{2}+\frac{2875428659724}{135680484049}a+\frac{1617006602118}{135680484049}$, $\frac{318925197}{135680484049}a^{11}+\frac{962217640}{135680484049}a^{10}-\frac{3378230893}{135680484049}a^{9}-\frac{5605455583}{135680484049}a^{8}+\frac{8009211792}{135680484049}a^{7}-\frac{19188934340}{135680484049}a^{6}-\frac{1948467750}{135680484049}a^{5}+\frac{128592076962}{135680484049}a^{4}-\frac{40515560264}{135680484049}a^{3}-\frac{154954055046}{135680484049}a^{2}+\frac{38360426289}{12334589459}a+\frac{658823299471}{135680484049}$
| 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: | \( 4457.650182835134 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 4457.650182835134 \cdot 4}{2\cdot\sqrt{1586874322944000000}}\cr\approx \mathstrut & 0.435455643399551 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - 9*x^8 - 30*x^7 + 62*x^6 + 90*x^5 - 33*x^4 + 210*x^3 + 693*x^2 + 510*x + 151)
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 - 9*x^8 - 30*x^7 + 62*x^6 + 90*x^5 - 33*x^4 + 210*x^3 + 693*x^2 + 510*x + 151, 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 - 9*x^8 - 30*x^7 + 62*x^6 + 90*x^5 - 33*x^4 + 210*x^3 + 693*x^2 + 510*x + 151);
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 - 9*x^8 - 30*x^7 + 62*x^6 + 90*x^5 - 33*x^4 + 210*x^3 + 693*x^2 + 510*x + 151);
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
| \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-6})\), 6.0.1259712000.3 |
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: | deg 18, 18.0.1295354998363672018944000000.1 |
| 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 | R | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
|
\(3\)
| 3.12.18.31 | $x^{12} - 12 x^{11} + 84 x^{10} - 72 x^{9} + 108 x^{8} + 42 x^{6} - 36 x^{5} + 36 x^{4} + 9$ | $6$ | $2$ | $18$ | $C_6\times S_3$ | $[3/2, 2]_{2}^{2}$ |
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.120.2t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{-30}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.72.6t1.c.a | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.10077696.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.72.6t1.c.b | $1$ | $ 2^{3} \cdot 3^{2}$ | 6.0.10077696.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.360.6t1.b.a | $1$ | $ 2^{3} \cdot 3^{2} \cdot 5 $ | 6.0.1259712000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.360.6t1.b.b | $1$ | $ 2^{3} \cdot 3^{2} \cdot 5 $ | 6.0.1259712000.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.1080.3t2.a.a | $2$ | $ 2^{3} \cdot 3^{3} \cdot 5 $ | 3.1.1080.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.1080.6t3.c.a | $2$ | $ 2^{3} \cdot 3^{3} \cdot 5 $ | 6.0.27993600.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.3240.12t18.b.a | $2$ | $ 2^{3} \cdot 3^{4} \cdot 5 $ | 12.0.1586874322944000000.9 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.3240.6t5.c.a | $2$ | $ 2^{3} \cdot 3^{4} \cdot 5 $ | 6.0.1259712000.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.3240.12t18.b.b | $2$ | $ 2^{3} \cdot 3^{4} \cdot 5 $ | 12.0.1586874322944000000.9 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
| * | 2.3240.6t5.c.b | $2$ | $ 2^{3} \cdot 3^{4} \cdot 5 $ | 6.0.1259712000.3 | $S_3\times C_3$ (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 *.