sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 + 28*x^12 + 56*x^10 + 72*x^8 + 64*x^6 + 40*x^4 + 16*x^2 + 1)
gp: K = bnfinit(y^16 + 8*y^14 + 28*y^12 + 56*y^10 + 72*y^8 + 64*y^6 + 40*y^4 + 16*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 8*x^14 + 28*x^12 + 56*x^10 + 72*x^8 + 64*x^6 + 40*x^4 + 16*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 8*x^14 + 28*x^12 + 56*x^10 + 72*x^8 + 64*x^6 + 40*x^4 + 16*x^2 + 1)
\( x^{16} + 8x^{14} + 28x^{12} + 56x^{10} + 72x^{8} + 64x^{6} + 40x^{4} + 16x^{2} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : $16$
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : $[0, 8]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
\(7745861623526935561764864\)
\(\medspace = 2^{70}\cdot 3^{8}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \(35.94\)
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 : $2^{2645/512}3^{1/2}\approx 62.185093055303284$
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 : \(\Q\)
$\Aut(K/\Q)$ :
$C_2$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is not Galois over $\Q$. This is not a CM field . This field has no CM subfields.
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : $C_{2}$, which has order $2$ (assuming GRH )
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : $C_{2}$, which has order $2$ (assuming GRH )
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 : $7$
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 :
$a$, $a^{3}+2a$, $a^{4}+2a^{2}+2$, $a^{14}+a^{13}+7a^{12}+6a^{11}+20a^{10}+15a^{9}+30a^{8}+20a^{7}+27a^{6}+17a^{5}+18a^{4}+10a^{3}+8a^{2}+a$, $a^{13}+8a^{11}+28a^{9}+53a^{7}+55a^{5}+26a^{3}+2a$, $2a^{14}-a^{13}+12a^{12}-4a^{11}+30a^{10}-8a^{9}+44a^{8}-12a^{7}+45a^{6}-13a^{5}+34a^{4}-11a^{3}+17a^{2}-4a+2$, $a^{15}-9a^{13}+4a^{12}-30a^{11}+26a^{10}-41a^{9}+78a^{8}-6a^{7}+136a^{6}+45a^{5}+135a^{4}+45a^{3}+66a^{2}+5a+4$
(assuming GRH )
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 : \( 1050872.0756821379 \)
(assuming GRH )
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1050872.0756821379 \cdot 2}{2\cdot\sqrt{7745861623526935561764864}}\cr\approx \mathstrut & 0.917178639294228
\end{aligned}\]
(assuming GRH )
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 + 28*x^12 + 56*x^10 + 72*x^8 + 64*x^6 + 40*x^4 + 16*x^2 + 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^16 + 8*x^14 + 28*x^12 + 56*x^10 + 72*x^8 + 64*x^6 + 40*x^4 + 16*x^2 + 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^16 + 8*x^14 + 28*x^12 + 56*x^10 + 72*x^8 + 64*x^6 + 40*x^4 + 16*x^2 + 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 = PolynomialRing(QQ); K, a = NumberField(x^16 + 8*x^14 + 28*x^12 + 56*x^10 + 72*x^8 + 64*x^6 + 40*x^4 + 16*x^2 + 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))))
$C_2^5.C_2\wr C_2^2$ (as 16T1455 ):
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)
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]
Degree 16
siblings :
16.6.3872930811763467780882432.30 , 16.6.3872930811763467780882432.32 , 16.2.3872930811763467780882432.51 , 16.2.3872930811763467780882432.59 , 16.4.7745861623526935561764864.54 , 16.4.7745861623526935561764864.53 , 16.0.7745861623526935561764864.13
Degree 32
siblings :
deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32 , deg 32
Minimal sibling : 16.6.3872930811763467780882432.30
$p$
$2$
$3$
$5$
$7$
$11$
$13$
$17$
$19$
$23$
$29$
$31$
$37$
$41$
$43$
$47$
$53$
$59$
Cycle type
R
R
${\href{/padicField/5.8.0.1}{8} }^{2}$
${\href{/padicField/7.8.0.1}{8} }^{2}$
${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$
${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$
${\href{/padicField/17.8.0.1}{8} }^{2}$
${\href{/padicField/19.4.0.1}{4} }^{4}$
${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$
${\href{/padicField/29.8.0.1}{8} }^{2}$
${\href{/padicField/31.8.0.1}{8} }^{2}$
${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$
${\href{/padicField/41.8.0.1}{8} }^{2}$
${\href{/padicField/43.4.0.1}{4} }^{4}$
${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$
${\href{/padicField/53.8.0.1}{8} }^{2}$
${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$
In the table, R denotes a ramified prime.
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]
$p$ Label Polynomial
$e$ $f$ $c$ Galois group Slope content
\(2\)
2.1.16.70b1.15374 $x^{16} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{11} + 8 x^{10} + 18 x^{8} + 16 x^{7} + 32 x^{5} + 8 x^{4} + 16 x^{2} + 6$ $16$ $1$ $70$ 16T1455 $$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}, \frac{43}{8}]^{2}$$
\(3\)
3.2.2.2a1.2 $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ $2$ $2$ $2$ $C_2^2$ $$[\ ]_{2}^{2}$$ 3.2.2.2a1.2 $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ $2$ $2$ $2$ $C_2^2$ $$[\ ]_{2}^{2}$$ 3.4.2.4a1.2 $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ $2$ $4$ $4$ $C_4\times C_2$ $$[\ ]_{2}^{4}$$
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
