sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 + 28*x^4 - 19*x^3 + 10*x^2 - 4*x + 1)
gp: K = bnfinit(y^12 - 4*y^11 + 10*y^10 - 19*y^9 + 28*y^8 - 34*y^7 + 37*y^6 - 34*y^5 + 28*y^4 - 19*y^3 + 10*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 + 28*x^4 - 19*x^3 + 10*x^2 - 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 + 28*x^4 - 19*x^3 + 10*x^2 - 4*x + 1)
\( x^{12} - 4 x^{11} + 10 x^{10} - 19 x^{9} + 28 x^{8} - 34 x^{7} + 37 x^{6} - 34 x^{5} + 28 x^{4} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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 :
\(629763392149\)
\(\medspace = 229^{5}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \(9.62\)
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: OK = ring_of_integers(K);
(1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant : $229^{1/2}\approx 15.132745950421556$
Ramified primes :
\(229\)
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant(OK))
Discriminant root field : \(\Q(\sqrt{229}) \)
$\Aut(K/\Q)$ :
$C_2$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphism_group(K)
This field is not Galois over $\Q$. This is not a CM field . This field has no CM subfields.
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : Trivial group, which has order $1$
sage: K.narrow_class_group().invariants()
gp: bnfnarrow(K)
magma: NarrowClassGroup(K);
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 :
$a^{10}-3a^{9}+6a^{8}-10a^{7}+12a^{6}-12a^{5}+13a^{4}-9a^{3}+6a^{2}-3a$, $a^{9}-3a^{8}+6a^{7}-9a^{6}+10a^{5}-9a^{4}+9a^{3}-6a^{2}+4a-1$, $a^{10}-4a^{9}+9a^{8}-16a^{7}+22a^{6}-24a^{5}+25a^{4}-22a^{3}+15a^{2}-9a+3$, $a^{11}-5a^{10}+13a^{9}-25a^{8}+37a^{7}-44a^{6}+46a^{5}-43a^{4}+34a^{3}-23a^{2}+10a-4$, $a^{11}-3a^{10}+6a^{9}-9a^{8}+10a^{7}-9a^{6}+9a^{5}-6a^{4}+4a^{3}-a$
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 : \( 5.78414476876 \)
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 5.78414476876 \cdot 1}{2\cdot\sqrt{629763392149}}\cr\approx \mathstrut & 0.224233069938
\end{aligned}\]
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 + 28*x^4 - 19*x^3 + 10*x^2 - 4*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))))
gp: \\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 + 28*x^4 - 19*x^3 + 10*x^2 - 4*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)))]
magma: /* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 + 28*x^4 - 19*x^3 + 10*x^2 - 4*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)));
oscar: # self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 10*x^10 - 19*x^9 + 28*x^8 - 34*x^7 + 37*x^6 - 34*x^5 + 28*x^4 - 19*x^3 + 10*x^2 - 4*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))))
$S_4$ (as 12T8 ):
sage: K.galois_group()
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K);
degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
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(L)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
$p$
$2$
$3$
$5$
$7$
$11$
$13$
$17$
$19$
$23$
$29$
$31$
$37$
$41$
$43$
$47$
$53$
$59$
Cycle type
${\href{/padicField/2.4.0.1}{4} }^{3}$
${\href{/padicField/3.3.0.1}{3} }^{4}$
${\href{/padicField/5.3.0.1}{3} }^{4}$
${\href{/padicField/7.4.0.1}{4} }^{3}$
${\href{/padicField/11.3.0.1}{3} }^{4}$
${\href{/padicField/13.4.0.1}{4} }^{3}$
${\href{/padicField/17.3.0.1}{3} }^{4}$
${\href{/padicField/19.3.0.1}{3} }^{4}$
${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$
${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$
${\href{/padicField/31.4.0.1}{4} }^{3}$
${\href{/padicField/37.2.0.1}{2} }^{6}$
${\href{/padicField/41.4.0.1}{4} }^{3}$
${\href{/padicField/43.3.0.1}{3} }^{4}$
${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$
${\href{/padicField/53.2.0.1}{2} }^{6}$
${\href{/padicField/59.4.0.1}{4} }^{3}$
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
sage: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: \\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
magma: // to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
oscar: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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]
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
