Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 19*x^12 - 19*x^10 + 36*x^8 - 81*x^6 + 69*x^4 - 17*x^2 + 1)
gp: K = bnfinit(y^16 - 8*y^14 + 19*y^12 - 19*y^10 + 36*y^8 - 81*y^6 + 69*y^4 - 17*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 19*x^12 - 19*x^10 + 36*x^8 - 81*x^6 + 69*x^4 - 17*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 19*x^12 - 19*x^10 + 36*x^8 - 81*x^6 + 69*x^4 - 17*x^2 + 1)
\( x^{16} - 8x^{14} + 19x^{12} - 19x^{10} + 36x^{8} - 81x^{6} + 69x^{4} - 17x^{2} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(781528990105600000000\)
\(\medspace = 2^{20}\cdot 5^{8}\cdot 11^{4}\cdot 19^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.22\) | 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: | $2^{3/2}5^{1/2}11^{1/2}19^{1/2}\approx 91.43303560529968$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\), \(19\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2092}a^{14}-\frac{1}{4}a^{13}-\frac{39}{1046}a^{12}+\frac{1}{4}a^{11}+\frac{193}{523}a^{10}+\frac{1}{4}a^{9}+\frac{214}{523}a^{8}+\frac{1}{4}a^{7}-\frac{785}{2092}a^{6}-\frac{1}{2}a^{5}+\frac{250}{523}a^{4}-\frac{1}{2}a^{3}+\frac{151}{2092}a^{2}-\frac{1}{4}a+\frac{919}{2092}$, $\frac{1}{2092}a^{15}+\frac{445}{2092}a^{13}-\frac{1}{4}a^{12}+\frac{249}{2092}a^{11}+\frac{1}{4}a^{10}+\frac{333}{2092}a^{9}+\frac{1}{4}a^{8}+\frac{196}{523}a^{7}+\frac{1}{4}a^{6}-\frac{23}{1046}a^{5}-\frac{1}{2}a^{4}-\frac{895}{2092}a^{3}-\frac{1}{2}a^{2}-\frac{325}{1046}a-\frac{1}{4}$
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 $1$ (assuming GRH)
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: | $13$ | 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: |
$\frac{213}{523}a^{14}-\frac{1447}{523}a^{12}+\frac{2306}{523}a^{10}-\frac{1245}{523}a^{8}+\frac{5908}{523}a^{6}-\frac{9798}{523}a^{4}+\frac{2875}{523}a^{2}+\frac{668}{523}$, $a$, $\frac{12}{523}a^{14}-\frac{303}{1046}a^{12}+\frac{1269}{1046}a^{10}-\frac{1945}{1046}a^{8}+\frac{1557}{1046}a^{6}-\frac{2644}{523}a^{4}+\frac{4427}{523}a^{2}-\frac{1479}{1046}$, $\frac{661}{523}a^{15}+\frac{627}{2092}a^{14}-\frac{20567}{2092}a^{13}-\frac{2487}{1046}a^{12}+\frac{45923}{2092}a^{11}+\frac{2813}{523}a^{10}-\frac{41597}{2092}a^{9}-\frac{2325}{523}a^{8}+\frac{88111}{2092}a^{7}+\frac{20345}{2092}a^{6}-\frac{97945}{1046}a^{5}-\frac{12179}{523}a^{4}+\frac{72533}{1046}a^{3}+\frac{31917}{2092}a^{2}-\frac{20419}{2092}a-\frac{1179}{2092}$, $\frac{373}{523}a^{15}-\frac{6411}{1046}a^{13}+\frac{16825}{1046}a^{11}-\frac{16743}{1046}a^{9}+\frac{26823}{1046}a^{7}-\frac{35986}{523}a^{5}+\frac{31219}{523}a^{3}-\frac{7401}{1046}a$, $\frac{1189}{1046}a^{15}-\frac{4531}{523}a^{13}+\frac{9697}{523}a^{11}-\frac{8355}{523}a^{9}+\frac{38369}{1046}a^{7}-\frac{41468}{523}a^{5}+\frac{57157}{1046}a^{3}-\frac{5609}{1046}a$, $\frac{157}{1046}a^{15}+\frac{627}{2092}a^{14}-\frac{3049}{2092}a^{13}-\frac{2487}{1046}a^{12}+\frac{9673}{2092}a^{11}+\frac{2813}{523}a^{10}-\frac{12067}{2092}a^{9}-\frac{2325}{523}a^{8}+\frac{14487}{2092}a^{7}+\frac{20345}{2092}a^{6}-\frac{20297}{1046}a^{5}-\frac{12179}{523}a^{4}+\frac{12115}{523}a^{3}+\frac{31917}{2092}a^{2}-\frac{12159}{2092}a-\frac{3271}{2092}$, $\frac{267}{2092}a^{15}+\frac{559}{1046}a^{14}-\frac{1475}{2092}a^{13}-\frac{8231}{2092}a^{12}+\frac{585}{2092}a^{11}+\frac{16359}{2092}a^{10}+\frac{2093}{2092}a^{9}-\frac{13157}{2092}a^{8}+\frac{2679}{1046}a^{7}+\frac{35005}{2092}a^{6}-\frac{911}{1046}a^{5}-\frac{34605}{1046}a^{4}-\frac{13029}{2092}a^{3}+\frac{10040}{523}a^{2}+\frac{1329}{523}a-\frac{2345}{2092}$, $\frac{157}{1046}a^{15}+\frac{627}{2092}a^{14}-\frac{3049}{2092}a^{13}-\frac{2487}{1046}a^{12}+\frac{9673}{2092}a^{11}+\frac{2813}{523}a^{10}-\frac{12067}{2092}a^{9}-\frac{2325}{523}a^{8}+\frac{14487}{2092}a^{7}+\frac{20345}{2092}a^{6}-\frac{20297}{1046}a^{5}-\frac{12179}{523}a^{4}+\frac{12115}{523}a^{3}+\frac{31917}{2092}a^{2}-\frac{12159}{2092}a-\frac{1179}{2092}$, $\frac{1793}{2092}a^{15}-\frac{623}{1046}a^{14}-\frac{13811}{2092}a^{13}+\frac{8801}{2092}a^{12}+\frac{30149}{2092}a^{11}-\frac{15805}{2092}a^{10}-\frac{26347}{2092}a^{9}+\frac{11327}{2092}a^{8}+\frac{14616}{523}a^{7}-\frac{38079}{2092}a^{6}-\frac{64251}{1046}a^{5}+\frac{34411}{1046}a^{4}+\frac{91877}{2092}a^{3}-\frac{8596}{523}a^{2}-\frac{6379}{1046}a+\frac{2911}{2092}$, $\frac{145}{523}a^{15}-\frac{1373}{523}a^{13}+\frac{4202}{523}a^{11}-\frac{5061}{523}a^{9}+\frac{6465}{523}a^{7}-\frac{17653}{523}a^{5}+\frac{19803}{523}a^{3}-\frac{5340}{523}a-1$, $\frac{3005}{2092}a^{15}+\frac{585}{2092}a^{14}-\frac{11549}{1046}a^{13}-\frac{4313}{2092}a^{12}+\frac{12510}{523}a^{11}+\frac{8639}{2092}a^{10}-\frac{10680}{523}a^{9}-\frac{7073}{2092}a^{8}+\frac{97083}{2092}a^{7}+\frac{9137}{1046}a^{6}-\frac{53647}{523}a^{5}-\frac{18685}{1046}a^{4}+\frac{146231}{2092}a^{3}+\frac{24529}{2092}a^{2}-\frac{14489}{2092}a-\frac{661}{523}$, $\frac{661}{523}a^{15}-\frac{1513}{2092}a^{14}-\frac{20567}{2092}a^{13}+\frac{2569}{523}a^{12}+\frac{45923}{2092}a^{11}-\frac{8195}{1046}a^{10}-\frac{41597}{2092}a^{9}+\frac{4617}{1046}a^{8}+\frac{88111}{2092}a^{7}-\frac{43437}{2092}a^{6}-\frac{97945}{1046}a^{5}+\frac{17661}{523}a^{4}+\frac{72533}{1046}a^{3}-\frac{21355}{2092}a^{2}-\frac{20419}{2092}a-\frac{313}{2092}$
| 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: | \( 65873.4365255 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 65873.4365255 \cdot 1}{2\cdot\sqrt{781528990105600000000}}\cr\approx \mathstrut & 0.190514249120 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 19*x^12 - 19*x^10 + 36*x^8 - 81*x^6 + 69*x^4 - 17*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))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^16 - 8*x^14 + 19*x^12 - 19*x^10 + 36*x^8 - 81*x^6 + 69*x^4 - 17*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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^14 + 19*x^12 - 19*x^10 + 36*x^8 - 81*x^6 + 69*x^4 - 17*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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 19*x^12 - 19*x^10 + 36*x^8 - 81*x^6 + 69*x^4 - 17*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))))
Galois group
$D_4^2:D_4$ (as 16T919):
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 512 |
| The 65 conjugacy class representatives for $D_4^2:D_4$ |
| Character table for $D_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.5225.1, 4.4.4400.1, 4.4.7600.1, 8.6.436810000.1, 8.6.6988960000.1, 8.8.6988960000.1 |
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 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.0.135306265600000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.14 | $x^{8} + 8 x^{7} + 28 x^{6} + 58 x^{5} + 95 x^{4} + 130 x^{3} + 58 x^{2} - 58 x + 13$ | $4$ | $2$ | $12$ | $D_4$ | $[2, 2]^{2}$ |
| 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(19\)
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |