Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 3*x^10 + 6*x^8 + 7*x^6 - 30*x^4 - 33*x^2 + 49)
gp: K = bnfinit(y^12 + 3*y^10 + 6*y^8 + 7*y^6 - 30*y^4 - 33*y^2 + 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 3*x^10 + 6*x^8 + 7*x^6 - 30*x^4 - 33*x^2 + 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 3*x^10 + 6*x^8 + 7*x^6 - 30*x^4 - 33*x^2 + 49)
\( x^{12} + 3x^{10} + 6x^{8} + 7x^{6} - 30x^{4} - 33x^{2} + 49 \)
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: |
\(128536820158464\)
\(\medspace = 2^{12}\cdot 3^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.99\) | 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: | $2^{7/6}3^{11/6}\approx 16.82379477331321$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{14}a^{9}-\frac{1}{14}a^{7}-\frac{2}{7}a^{5}-\frac{1}{2}a^{4}-\frac{5}{14}a^{3}-\frac{1}{2}a^{2}-\frac{3}{14}a-\frac{1}{2}$, $\frac{1}{140}a^{10}-\frac{1}{28}a^{9}-\frac{3}{28}a^{8}-\frac{3}{14}a^{7}+\frac{31}{140}a^{6}+\frac{11}{28}a^{5}+\frac{9}{140}a^{4}+\frac{3}{7}a^{3}+\frac{53}{140}a^{2}-\frac{11}{28}a+\frac{9}{20}$, $\frac{1}{140}a^{11}-\frac{1}{4}a^{8}-\frac{19}{140}a^{7}+\frac{27}{70}a^{5}+\frac{1}{4}a^{4}-\frac{57}{140}a^{3}-\frac{1}{2}a^{2}+\frac{9}{70}a+\frac{1}{4}$
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: |
\( \frac{6}{35} a^{10} + \frac{3}{7} a^{8} + \frac{46}{35} a^{6} + \frac{54}{35} a^{4} - \frac{102}{35} a^{2} - \frac{6}{5} \)
(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: |
$\frac{2}{35}a^{11}+\frac{3}{70}a^{10}+\frac{3}{7}a^{9}+\frac{5}{14}a^{8}+\frac{69}{70}a^{7}+\frac{29}{35}a^{6}+\frac{83}{35}a^{5}+\frac{66}{35}a^{4}+\frac{3}{5}a^{3}+\frac{27}{35}a^{2}-\frac{403}{70}a-\frac{53}{10}$, $\frac{1}{28}a^{11}-\frac{1}{14}a^{9}+\frac{1}{4}a^{8}-\frac{3}{28}a^{7}+\frac{1}{2}a^{6}-\frac{11}{14}a^{5}+\frac{7}{4}a^{4}-\frac{47}{28}a^{3}+\frac{3}{2}a^{2}+\frac{33}{14}a-\frac{19}{4}$, $\frac{2}{5}a^{11}+\frac{23}{14}a^{9}+\frac{149}{35}a^{7}+\frac{527}{70}a^{5}-\frac{123}{35}a^{3}-\frac{1171}{70}a$, $\frac{2}{35}a^{11}-\frac{4}{7}a^{10}+\frac{3}{7}a^{9}-\frac{27}{14}a^{8}+\frac{69}{70}a^{7}-\frac{73}{14}a^{6}+\frac{83}{35}a^{5}-\frac{57}{7}a^{4}+\frac{3}{5}a^{3}+\frac{101}{14}a^{2}-\frac{403}{70}a+\frac{31}{2}$, $\frac{3}{70}a^{11}+\frac{1}{14}a^{9}+\frac{4}{35}a^{7}+\frac{1}{35}a^{5}-\frac{1}{2}a^{4}-\frac{4}{5}a^{3}-\frac{1}{2}a^{2}-\frac{31}{70}a+\frac{3}{2}$
| 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: | \( 944.675683171 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 944.675683171 \cdot 1}{6\cdot\sqrt{128536820158464}}\cr\approx \mathstrut & 0.854470198229 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 + 3*x^10 + 6*x^8 + 7*x^6 - 30*x^4 - 33*x^2 + 49)
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 + 3*x^10 + 6*x^8 + 7*x^6 - 30*x^4 - 33*x^2 + 49, 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 + 3*x^10 + 6*x^8 + 7*x^6 - 30*x^4 - 33*x^2 + 49);
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 + 3*x^10 + 6*x^8 + 7*x^6 - 30*x^4 - 33*x^2 + 49);
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
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 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 6.0.11337408.4, 6.2.3779136.3, 6.0.2834352.2 |
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
| Galois closure: | deg 24 |
| Degree 4 sibling: | 4.2.3888.1 |
| Degree 6 siblings: | 6.2.3779136.3, 6.0.11337408.4 |
| Degree 8 sibling: | 8.0.136048896.1 |
| Degree 12 sibling: | 12.2.171382426877952.3 |
| Minimal sibling: | 4.2.3888.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.28 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 130 x^{7} + 159 x^{6} + 132 x^{5} + 10 x^{4} - 100 x^{3} - 53 x^{2} + 22 x + 19$ | $6$ | $2$ | $12$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
|
\(3\)
| 3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ |
| 3.6.11.7 | $x^{6} + 9 x^{2} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ |