Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 + 135*x^8 + 462*x^6 + 513*x^4 - 324*x^2 + 144)
gp: K = bnfinit(y^12 + 18*y^10 + 135*y^8 + 462*y^6 + 513*y^4 - 324*y^2 + 144, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 18*x^10 + 135*x^8 + 462*x^6 + 513*x^4 - 324*x^2 + 144);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 18*x^10 + 135*x^8 + 462*x^6 + 513*x^4 - 324*x^2 + 144)
\( x^{12} + 18x^{10} + 135x^{8} + 462x^{6} + 513x^{4} - 324x^{2} + 144 \)
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: |
\(32905425960566784\)
\(\medspace = 2^{20}\cdot 3^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.79\) | 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: | $2^{2}3^{13/6}\approx 43.2337303863361$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{18}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{1}{3}$, $\frac{1}{18}a^{7}-\frac{1}{6}a$, $\frac{1}{36}a^{8}-\frac{1}{36}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{5}{12}a^{2}-\frac{1}{6}a$, $\frac{1}{36}a^{9}-\frac{1}{36}a^{7}-\frac{1}{36}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{3}a^{3}+\frac{1}{4}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{1764}a^{10}-\frac{11}{1764}a^{8}-\frac{1}{36}a^{7}+\frac{13}{1764}a^{6}-\frac{1}{4}a^{5}+\frac{55}{294}a^{4}-\frac{1}{4}a^{3}+\frac{17}{147}a^{2}-\frac{1}{6}a+\frac{68}{147}$, $\frac{1}{10584}a^{11}-\frac{5}{882}a^{9}-\frac{61}{3528}a^{7}-\frac{1}{36}a^{6}-\frac{23}{441}a^{5}+\frac{1}{4}a^{4}-\frac{451}{1176}a^{3}+\frac{1}{4}a^{2}-\frac{59}{294}a+\frac{1}{3}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
\( \frac{2}{1323} a^{11} + \frac{1}{49} a^{9} + \frac{11}{98} a^{7} + \frac{73}{441} a^{5} - \frac{23}{49} a^{3} - \frac{135}{98} a \)
(order $12$)
| 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: |
$\frac{1}{216}a^{11}-\frac{1}{196}a^{10}+\frac{1}{12}a^{9}-\frac{73}{882}a^{8}+\frac{5}{8}a^{7}-\frac{111}{196}a^{6}+\frac{79}{36}a^{5}-\frac{165}{98}a^{4}+\frac{23}{8}a^{3}-\frac{955}{588}a^{2}+\frac{41}{49}$, $\frac{265}{10584}a^{11}-\frac{1}{126}a^{10}+\frac{65}{147}a^{9}-\frac{17}{126}a^{8}+\frac{11471}{3528}a^{7}-\frac{229}{252}a^{6}+\frac{9419}{882}a^{5}-\frac{199}{84}a^{4}+\frac{3935}{392}a^{3}+\frac{25}{84}a^{2}-\frac{3581}{294}a+\frac{151}{21}$, $\frac{29}{5292}a^{11}+\frac{1}{147}a^{10}+\frac{155}{1764}a^{9}+\frac{211}{1764}a^{8}+\frac{877}{1764}a^{7}+\frac{1577}{1764}a^{6}+\frac{419}{882}a^{5}+\frac{685}{196}a^{4}-\frac{734}{147}a^{3}+\frac{1037}{147}a^{2}-\frac{3863}{294}a+\frac{865}{147}$, $\frac{23}{2646}a^{11}+\frac{11}{441}a^{10}+\frac{305}{1764}a^{9}+\frac{395}{882}a^{8}+\frac{2633}{1764}a^{7}+\frac{6011}{1764}a^{6}+\frac{11381}{1764}a^{5}+\frac{7339}{588}a^{4}+\frac{1894}{147}a^{3}+\frac{10685}{588}a^{2}+\frac{2363}{294}a+\frac{52}{147}$, $\frac{37}{1764}a^{10}+\frac{20}{49}a^{8}+\frac{1957}{588}a^{6}+\frac{3799}{294}a^{4}+\frac{3795}{196}a^{2}-\frac{174}{49}$
| 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: | \( 17485.2782015 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 17485.2782015 \cdot 1}{12\cdot\sqrt{32905425960566784}}\cr\approx \mathstrut & 0.494238703990 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 + 135*x^8 + 462*x^6 + 513*x^4 - 324*x^2 + 144)
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 + 18*x^10 + 135*x^8 + 462*x^6 + 513*x^4 - 324*x^2 + 144, 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 + 18*x^10 + 135*x^8 + 462*x^6 + 513*x^4 - 324*x^2 + 144);
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 + 18*x^10 + 135*x^8 + 462*x^6 + 513*x^4 - 324*x^2 + 144);
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
$S_3\times D_6$ (as 12T37):
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 72 |
| The 18 conjugacy class representatives for $S_3\times D_6$ |
| Character table for $S_3\times D_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 6.0.11337408.2 |
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
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| 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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
|
\(3\)
| 3.12.22.25 | $x^{12} - 30 x^{9} - 18 x^{8} + 36 x^{7} + 573 x^{6} + 594 x^{5} - 378 x^{4} - 360 x^{3} + 270 x^{2} + 432 x + 225$ | $6$ | $2$ | $22$ | $C_6\times S_3$ | $[2, 5/2]_{2}^{2}$ |