sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 + 40*x^12 + 16*x^10 - 136*x^8 - 32*x^6 + 160*x^4 - 64*x^2 + 16)
gp: K = bnfinit(y^16 + 8*y^14 + 40*y^12 + 16*y^10 - 136*y^8 - 32*y^6 + 160*y^4 - 64*y^2 + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 8*x^14 + 40*x^12 + 16*x^10 - 136*x^8 - 32*x^6 + 160*x^4 - 64*x^2 + 16);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 8*x^14 + 40*x^12 + 16*x^10 - 136*x^8 - 32*x^6 + 160*x^4 - 64*x^2 + 16)
\( x^{16} + 8x^{14} + 40x^{12} + 16x^{10} - 136x^{8} - 32x^{6} + 160x^{4} - 64x^{2} + 16 \)
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 :
\(30257271966902092038144\)
\(\medspace = 2^{62}\cdot 3^{8}\)
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : \(25.41\)
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 : $2^{555/128}3^{3/4}\approx 46.035004279893066$
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 : \(\Q\)
$\Aut(K/\Q)$ :
$C_2^2$
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is not Galois over $\Q$. This is not a CM field . Maximal CM subfield : \(\Q(\sqrt{-2}) \)
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{8}-\frac{1}{2}$, $\frac{1}{16}a^{9}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{48}a^{10}-\frac{1}{12}a^{6}+\frac{1}{6}a^{4}-\frac{1}{4}a^{2}+\frac{1}{3}$, $\frac{1}{48}a^{11}-\frac{1}{12}a^{7}-\frac{1}{12}a^{5}-\frac{1}{4}a^{3}-\frac{1}{6}a$, $\frac{1}{96}a^{12}+\frac{1}{48}a^{8}+\frac{1}{12}a^{6}+\frac{1}{8}a^{4}+\frac{1}{6}a^{2}+\frac{1}{4}$, $\frac{1}{192}a^{13}+\frac{1}{96}a^{9}-\frac{1}{16}a^{8}+\frac{1}{24}a^{7}+\frac{1}{16}a^{5}-\frac{1}{4}a^{4}-\frac{5}{12}a^{3}-\frac{1}{2}a^{2}+\frac{1}{8}a-\frac{1}{4}$, $\frac{1}{192}a^{14}-\frac{1}{96}a^{10}+\frac{1}{24}a^{8}-\frac{5}{48}a^{6}-\frac{1}{12}a^{4}-\frac{1}{8}a^{2}-\frac{1}{3}$, $\frac{1}{192}a^{15}-\frac{1}{96}a^{11}-\frac{1}{48}a^{9}-\frac{5}{48}a^{7}-\frac{1}{12}a^{5}+\frac{3}{8}a^{3}-\frac{1}{12}a$
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 : $7$
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 :
$\frac{1}{96}a^{14}+\frac{11}{96}a^{12}+\frac{17}{24}a^{10}+\frac{29}{16}a^{8}+\frac{31}{24}a^{6}-\frac{23}{24}a^{4}-\frac{7}{6}a^{2}-\frac{1}{4}$, $\frac{5}{48}a^{14}+\frac{43}{48}a^{12}+\frac{75}{16}a^{10}+\frac{35}{8}a^{8}-12a^{6}-\frac{39}{4}a^{4}+\frac{157}{12}a^{2}+\frac{1}{6}$, $\frac{1}{12}a^{15}-\frac{35}{48}a^{13}-\frac{31}{8}a^{11}-\frac{17}{4}a^{9}+8a^{7}+7a^{5}-\frac{49}{6}a^{3}+\frac{13}{6}a$, $\frac{1}{32}a^{15}-\frac{7}{192}a^{14}-\frac{13}{48}a^{13}-\frac{7}{24}a^{12}-\frac{23}{16}a^{11}-\frac{47}{32}a^{10}-\frac{71}{48}a^{9}-\frac{5}{8}a^{8}+\frac{77}{24}a^{7}+\frac{77}{16}a^{6}+4a^{5}+\frac{9}{4}a^{4}-\frac{31}{12}a^{3}-\frac{115}{24}a^{2}-\frac{9}{4}a-\frac{1}{3}$, $\frac{5}{192}a^{15}-\frac{11}{192}a^{14}+\frac{5}{24}a^{13}-\frac{25}{48}a^{12}+\frac{101}{96}a^{11}-\frac{265}{96}a^{10}+\frac{7}{16}a^{9}-\frac{13}{4}a^{8}-\frac{169}{48}a^{7}+\frac{359}{48}a^{6}-\frac{31}{12}a^{5}+\frac{49}{6}a^{4}+\frac{107}{24}a^{3}-\frac{227}{24}a^{2}+\frac{7}{4}a+\frac{7}{6}$, $\frac{7}{64}a^{15}+\frac{5}{96}a^{14}+\frac{43}{48}a^{13}+\frac{41}{96}a^{12}+\frac{437}{96}a^{11}+\frac{13}{6}a^{10}+\frac{8}{3}a^{9}+\frac{61}{48}a^{8}-\frac{677}{48}a^{7}-\frac{161}{24}a^{6}-\frac{73}{12}a^{5}-\frac{49}{24}a^{4}+\frac{395}{24}a^{3}+\frac{25}{3}a^{2}-\frac{11}{3}a-\frac{11}{4}$, $\frac{23}{64}a^{15}+\frac{11}{64}a^{14}+\frac{287}{96}a^{13}+\frac{173}{96}a^{12}+\frac{1519}{96}a^{11}+\frac{983}{96}a^{10}+\frac{85}{6}a^{9}+\frac{959}{48}a^{8}-\frac{443}{16}a^{7}-\frac{689}{48}a^{6}-\frac{625}{24}a^{5}-\frac{1043}{24}a^{4}+\frac{449}{24}a^{3}+\frac{569}{24}a^{2}-\frac{29}{6}a-\frac{77}{12}$
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 : \( 387370.6549598008 \)
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
\[
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 387370.6549598008 \cdot 1}{2\cdot\sqrt{30257271966902092038144}}\cr\approx \mathstrut & 2.70471048524471
\end{aligned}\]
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 + 40*x^12 + 16*x^10 - 136*x^8 - 32*x^6 + 160*x^4 - 64*x^2 + 16)
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 + 40*x^12 + 16*x^10 - 136*x^8 - 32*x^6 + 160*x^4 - 64*x^2 + 16, 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 + 40*x^12 + 16*x^10 - 136*x^8 - 32*x^6 + 160*x^4 - 64*x^2 + 16);
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 + 40*x^12 + 16*x^10 - 136*x^8 - 32*x^6 + 160*x^4 - 64*x^2 + 16);
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))))
$C_2^6:D_4$ (as 16T969 ):
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 :
16.0.30257271966902092038144.4 , 16.0.484116351470433472610304.7 , 16.0.30257271966902092038144.5 , 16.0.30257271966902092038144.6 , 16.0.121029087867608368152576.17 , 16.0.121029087867608368152576.18 , 16.0.30257271966902092038144.11 , 16.0.484116351470433472610304.18 , 16.0.4357047163233901253492736.147 , 16.0.484116351470433472610304.266 , 16.0.484116351470433472610304.269 , 16.0.4357047163233901253492736.67 , 16.0.484116351470433472610304.274 , 16.0.272315447702118828343296.127 , 16.0.272315447702118828343296.130 , 16.0.484116351470433472610304.169 , 16.0.30257271966902092038144.56 , 16.0.484116351470433472610304.276 , 16.0.484116351470433472610304.277 , 16.0.484116351470433472610304.75 , 16.0.4357047163233901253492736.264 , 16.0.484116351470433472610304.181 , 16.0.1089261790808475313373184.138 , 16.0.4357047163233901253492736.184 , 16.0.1089261790808475313373184.69 , 16.0.4357047163233901253492736.78 , 16.0.4357047163233901253492736.186 , 16.0.4357047163233901253492736.79 , 16.0.272315447702118828343296.141 , 16.0.272315447702118828343296.160 , 16.0.121029087867608368152576.171 , 16.0.121029087867608368152576.172 , 16.0.484116351470433472610304.217 , 16.0.4357047163233901253492736.287 , 16.0.4357047163233901253492736.288 , 16.0.1089261790808475313373184.147 , 16.0.4357047163233901253492736.289 , 16.0.4357047163233901253492736.223 , 16.0.484116351470433472610304.234 , 16.0.484116351470433472610304.235 , 16.0.4357047163233901253492736.108 , 16.0.272315447702118828343296.179 , 16.0.272315447702118828343296.184 , 16.0.30257271966902092038144.73 , 16.0.1089261790808475313373184.95 , 16.0.121029087867608368152576.106 , 16.0.484116351470433472610304.116 , 16.0.30257271966902092038144.78 , 16.0.1089261790808475313373184.109 , 16.0.121029087867608368152576.123 , 16.0.484116351470433472610304.129 , 16.0.4357047163233901253492736.141 , 16.0.484116351470433472610304.150 , 16.0.1089261790808475313373184.18 , 16.0.121029087867608368152576.56 , 16.0.272315447702118828343296.21 , 16.0.1089261790808475313373184.64 , 16.0.121029087867608368152576.59 , 16.0.4357047163233901253492736.52 , 16.0.272315447702118828343296.39 , 16.0.1089261790808475313373184.55 , 16.0.4357047163233901253492736.304 , 16.0.4357047163233901253492736.249
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 , 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 , 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 , 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 : 16.0.30257271966902092038144.4
$p$
$2$
$3$
$5$
$7$
$11$
$13$
$17$
$19$
$23$
$29$
$31$
$37$
$41$
$43$
$47$
$53$
$59$
Cycle type
R
R
${\href{/padicField/5.4.0.1}{4} }^{4}$
${\href{/padicField/7.8.0.1}{8} }^{2}$
${\href{/padicField/11.4.0.1}{4} }^{4}$
${\href{/padicField/13.4.0.1}{4} }^{4}$
${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$
${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$
${\href{/padicField/23.4.0.1}{4} }^{4}$
${\href{/padicField/29.4.0.1}{4} }^{4}$
${\href{/padicField/31.8.0.1}{8} }^{2}$
${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$
${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$
${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$
${\href{/padicField/47.4.0.1}{4} }^{4}$
${\href{/padicField/53.4.0.1}{4} }^{4}$
${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$
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]
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
