Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + 6*x^13 + x^12 - 10*x^11 + 8*x^10 + 5*x^9 - 13*x^8 + 5*x^7 + 8*x^6 - 10*x^5 + x^4 + 6*x^3 - 3*x^2 - x + 1)
gp: K = bnfinit(y^16 - y^15 - 3*y^14 + 6*y^13 + y^12 - 10*y^11 + 8*y^10 + 5*y^9 - 13*y^8 + 5*y^7 + 8*y^6 - 10*y^5 + y^4 + 6*y^3 - 3*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 3*x^14 + 6*x^13 + x^12 - 10*x^11 + 8*x^10 + 5*x^9 - 13*x^8 + 5*x^7 + 8*x^6 - 10*x^5 + x^4 + 6*x^3 - 3*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 3*x^14 + 6*x^13 + x^12 - 10*x^11 + 8*x^10 + 5*x^9 - 13*x^8 + 5*x^7 + 8*x^6 - 10*x^5 + x^4 + 6*x^3 - 3*x^2 - x + 1)
\( x^{16} - x^{15} - 3 x^{14} + 6 x^{13} + x^{12} - 10 x^{11} + 8 x^{10} + 5 x^{9} - 13 x^{8} + 5 x^{7} + \cdots + 1 \)
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: |
\(3006123291015625\)
\(\medspace = 5^{12}\cdot 11^{4}\cdot 29^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.28\) | 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: | $5^{3/4}11^{1/2}29^{1/2}\approx 59.72041882063591$ | ||
| Ramified primes: |
\(5\), \(11\), \(29\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a$
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: |
\( -\frac{4}{3} a^{15} + \frac{8}{3} a^{14} + \frac{4}{3} a^{13} - \frac{31}{3} a^{12} + \frac{29}{3} a^{11} + \frac{20}{3} a^{10} - 22 a^{9} + \frac{40}{3} a^{8} + \frac{35}{3} a^{7} - 23 a^{6} + 8 a^{5} + \frac{41}{3} a^{4} - \frac{49}{3} a^{3} + \frac{10}{3} a^{2} + 6 a - \frac{11}{3} \)
(order $10$)
| 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: |
$\frac{10}{3}a^{15}-4a^{14}-7a^{13}+\frac{58}{3}a^{12}-6a^{11}-\frac{62}{3}a^{10}+\frac{92}{3}a^{9}-\frac{13}{3}a^{8}-\frac{80}{3}a^{7}+25a^{6}+\frac{11}{3}a^{5}-\frac{71}{3}a^{4}+15a^{3}+\frac{11}{3}a^{2}-\frac{17}{3}a+\frac{7}{3}$, $a^{15}-2a^{14}-\frac{5}{3}a^{13}+\frac{25}{3}a^{12}-\frac{16}{3}a^{11}-\frac{25}{3}a^{10}+\frac{46}{3}a^{9}-5a^{8}-12a^{7}+15a^{6}-a^{5}-\frac{38}{3}a^{4}+\frac{29}{3}a^{3}+\frac{2}{3}a^{2}-\frac{14}{3}a+\frac{4}{3}$, $\frac{5}{3}a^{14}-2a^{13}-4a^{12}+10a^{11}-2a^{10}-\frac{38}{3}a^{9}+\frac{46}{3}a^{8}+\frac{2}{3}a^{7}-\frac{50}{3}a^{6}+\frac{34}{3}a^{5}+5a^{4}-14a^{3}+6a^{2}+4a-\frac{10}{3}$, $\frac{7}{3}a^{15}-\frac{7}{3}a^{14}-6a^{13}+\frac{38}{3}a^{12}-17a^{10}+17a^{9}+\frac{13}{3}a^{8}-\frac{61}{3}a^{7}+\frac{32}{3}a^{6}+\frac{26}{3}a^{5}-\frac{46}{3}a^{4}+5a^{3}+\frac{19}{3}a^{2}-\frac{8}{3}a-\frac{2}{3}$, $\frac{5}{3}a^{15}-2a^{14}-4a^{13}+\frac{31}{3}a^{12}-2a^{11}-13a^{10}+\frac{49}{3}a^{9}-17a^{7}+\frac{41}{3}a^{6}+\frac{14}{3}a^{5}-\frac{41}{3}a^{4}+7a^{3}+\frac{11}{3}a^{2}-\frac{13}{3}a+\frac{4}{3}$, $\frac{10}{3}a^{15}-\frac{16}{3}a^{14}-\frac{19}{3}a^{13}+\frac{70}{3}a^{12}-\frac{35}{3}a^{11}-\frac{71}{3}a^{10}+\frac{122}{3}a^{9}-\frac{32}{3}a^{8}-\frac{100}{3}a^{7}+\frac{110}{3}a^{6}+\frac{5}{3}a^{5}-\frac{92}{3}a^{4}+\frac{67}{3}a^{3}+\frac{11}{3}a^{2}-10a+3$, $\frac{2}{3}a^{15}-\frac{5}{3}a^{14}-\frac{2}{3}a^{13}+\frac{20}{3}a^{12}-\frac{19}{3}a^{11}-\frac{16}{3}a^{10}+\frac{46}{3}a^{9}-\frac{25}{3}a^{8}-\frac{29}{3}a^{7}+\frac{52}{3}a^{6}-\frac{14}{3}a^{5}-\frac{34}{3}a^{4}+\frac{38}{3}a^{3}-\frac{2}{3}a^{2}-6a+3$
| 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: | \( 35.8754367412 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 35.8754367412 \cdot 1}{10\cdot\sqrt{3006123291015625}}\cr\approx \mathstrut & 0.158939831927 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + 6*x^13 + x^12 - 10*x^11 + 8*x^10 + 5*x^9 - 13*x^8 + 5*x^7 + 8*x^6 - 10*x^5 + x^4 + 6*x^3 - 3*x^2 - 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 - x^15 - 3*x^14 + 6*x^13 + x^12 - 10*x^11 + 8*x^10 + 5*x^9 - 13*x^8 + 5*x^7 + 8*x^6 - 10*x^5 + x^4 + 6*x^3 - 3*x^2 - 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 - x^15 - 3*x^14 + 6*x^13 + x^12 - 10*x^11 + 8*x^10 + 5*x^9 - 13*x^8 + 5*x^7 + 8*x^6 - 10*x^5 + x^4 + 6*x^3 - 3*x^2 - 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 - x^15 - 3*x^14 + 6*x^13 + x^12 - 10*x^11 + 8*x^10 + 5*x^9 - 13*x^8 + 5*x^7 + 8*x^6 - 10*x^5 + x^4 + 6*x^3 - 3*x^2 - 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
$D_4^2:C_4$ (as 16T547):
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 256 |
| The 40 conjugacy class representatives for $D_4^2:C_4$ |
| Character table for $D_4^2:C_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.275.1, 4.2.1375.1, 8.0.2193125.1, 8.4.54828125.1, 8.0.1890625.1 |
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: | 16.4.2528149687744140625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(11\)
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |