Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + 4*y^14 - 3*y^13 + 11*y^10 - 24*y^9 + 29*y^8 - 24*y^7 + 11*y^6 - 3*y^3 + 4*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*x + 1)
\( x^{16} - 3 x^{15} + 4 x^{14} - 3 x^{13} + 11 x^{10} - 24 x^{9} + 29 x^{8} - 24 x^{7} + 11 x^{6} + \cdots + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(4540317078515625\)
\(\medspace = 3^{8}\cdot 5^{8}\cdot 11^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.52\) | 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: | $3^{1/2}5^{1/2}11^{3/4}\approx 23.393227447460962$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{163}a^{14}-\frac{43}{163}a^{13}-\frac{70}{163}a^{12}+\frac{69}{163}a^{11}+\frac{81}{163}a^{10}-\frac{49}{163}a^{9}-\frac{66}{163}a^{8}+\frac{57}{163}a^{7}-\frac{66}{163}a^{6}-\frac{49}{163}a^{5}+\frac{81}{163}a^{4}+\frac{69}{163}a^{3}-\frac{70}{163}a^{2}-\frac{43}{163}a+\frac{1}{163}$, $\frac{1}{1793}a^{15}-\frac{4}{1793}a^{14}-\frac{443}{1793}a^{13}+\frac{10}{163}a^{12}-\frac{74}{163}a^{11}+\frac{16}{163}a^{10}-\frac{76}{163}a^{9}-\frac{398}{1793}a^{8}-\frac{288}{1793}a^{7}-\frac{31}{163}a^{6}-\frac{33}{163}a^{5}-\frac{77}{163}a^{4}+\frac{16}{163}a^{3}-\frac{817}{1793}a^{2}+\frac{117}{1793}a-\frac{450}{1793}$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{2135}{1793} a^{15} - \frac{4745}{1793} a^{14} + \frac{4463}{1793} a^{13} - \frac{192}{163} a^{12} - \frac{199}{163} a^{11} - \frac{161}{163} a^{10} + \frac{2091}{163} a^{9} - \frac{33365}{1793} a^{8} + \frac{31755}{1793} a^{7} - \frac{1913}{163} a^{6} + \frac{334}{163} a^{5} + \frac{307}{163} a^{4} + \frac{589}{163} a^{3} - \frac{5371}{1793} a^{2} + \frac{4132}{1793} a - \frac{1286}{1793} \)
(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: |
$\frac{2345}{1793}a^{15}-\frac{4859}{1793}a^{14}+\frac{3932}{1793}a^{13}-\frac{104}{163}a^{12}-\frac{264}{163}a^{11}-\frac{257}{163}a^{10}+\frac{2294}{163}a^{9}-\frac{32179}{1793}a^{8}+\frac{27001}{1793}a^{7}-\frac{1369}{163}a^{6}+\frac{113}{163}a^{5}+\frac{241}{163}a^{4}+\frac{516}{163}a^{3}-\frac{5429}{1793}a^{2}+\frac{2863}{1793}a-\frac{31}{1793}$, $\frac{3079}{1793}a^{15}-\frac{7102}{1793}a^{14}+\frac{7571}{1793}a^{13}-\frac{434}{163}a^{12}-\frac{192}{163}a^{11}-\frac{199}{163}a^{10}+\frac{2918}{163}a^{9}-\frac{50895}{1793}a^{8}+\frac{55926}{1793}a^{7}-\frac{3831}{163}a^{6}+\frac{1166}{163}a^{5}+\frac{334}{163}a^{4}+\frac{307}{163}a^{3}-\frac{2758}{1793}a^{2}+\frac{6945}{1793}a-\frac{5105}{1793}$, $\frac{2560}{1793}a^{15}-\frac{7281}{1793}a^{14}+\frac{8127}{1793}a^{13}-\frac{402}{163}a^{12}-\frac{218}{163}a^{11}-\frac{6}{163}a^{10}+\frac{2692}{163}a^{9}-\frac{55896}{1793}a^{8}+\frac{55347}{1793}a^{7}-\frac{3552}{163}a^{6}+\frac{954}{163}a^{5}+\frac{546}{163}a^{4}+\frac{189}{163}a^{3}-\frac{10782}{1793}a^{2}+\frac{7327}{1793}a-\frac{5107}{1793}$, $\frac{3991}{1793}a^{15}-\frac{10046}{1793}a^{14}+\frac{10775}{1793}a^{13}-\frac{521}{163}a^{12}-\frac{346}{163}a^{11}-\frac{146}{163}a^{10}+\frac{3983}{163}a^{9}-\frac{74840}{1793}a^{8}+\frac{75453}{1793}a^{7}-\frac{4705}{163}a^{6}+\frac{1186}{163}a^{5}+\frac{820}{163}a^{4}+\frac{407}{163}a^{3}-\frac{11808}{1793}a^{2}+\frac{11657}{1793}a-\frac{5997}{1793}$, $\frac{1885}{1793}a^{15}-\frac{4438}{1793}a^{14}+\frac{3365}{1793}a^{13}-\frac{75}{163}a^{12}-\frac{227}{163}a^{11}-\frac{136}{163}a^{10}+\frac{2010}{163}a^{9}-\frac{29774}{1793}a^{8}+\frac{21222}{1793}a^{7}-\frac{1252}{163}a^{6}-\frac{65}{163}a^{5}+\frac{273}{163}a^{4}+\frac{555}{163}a^{3}-\frac{5424}{1793}a^{2}+\frac{1095}{1793}a-\frac{2438}{1793}$, $\frac{3079}{1793}a^{15}-\frac{7102}{1793}a^{14}+\frac{7571}{1793}a^{13}-\frac{434}{163}a^{12}-\frac{192}{163}a^{11}-\frac{199}{163}a^{10}+\frac{2918}{163}a^{9}-\frac{50895}{1793}a^{8}+\frac{55926}{1793}a^{7}-\frac{3831}{163}a^{6}+\frac{1166}{163}a^{5}+\frac{334}{163}a^{4}+\frac{307}{163}a^{3}-\frac{4551}{1793}a^{2}+\frac{6945}{1793}a-\frac{5105}{1793}$, $\frac{3009}{1793}a^{15}-\frac{7526}{1793}a^{14}+\frac{7891}{1793}a^{13}-\frac{403}{163}a^{12}-\frac{243}{163}a^{11}-\frac{146}{163}a^{10}+\frac{3061}{163}a^{9}-\frac{55463}{1793}a^{8}+\frac{55681}{1793}a^{7}-\frac{3631}{163}a^{6}+\frac{907}{163}a^{5}+\frac{214}{163}a^{4}+\frac{639}{163}a^{3}-\frac{9247}{1793}a^{2}+\frac{7511}{1793}a-\frac{4790}{1793}$
| 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: | \( 27.6507723749 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 27.6507723749 \cdot 1}{6\cdot\sqrt{4540317078515625}}\cr\approx \mathstrut & 0.166131456202 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*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))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 4*x^14 - 3*x^13 + 11*x^10 - 24*x^9 + 29*x^8 - 24*x^7 + 11*x^6 - 3*x^3 + 4*x^2 - 3*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))))
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 64 |
| The 16 conjugacy class representatives for $D_4:D_4$ |
| Character table for $D_4:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.2.2475.1, 4.2.275.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.2.67381875.1 x2, 8.2.7486875.1 x2, 8.0.6125625.1 |
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 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.2.7486875.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(11\)
| 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 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.8.6.2 | $x^{8} + 28 x^{7} + 302 x^{6} + 1540 x^{5} + 3623 x^{4} + 3388 x^{3} + 4178 x^{2} + 13468 x + 22324$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |