Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*x^2 + x + 1)
gp: K = bnfinit(y^16 - 2*y^15 - y^14 + y^13 + 11*y^12 - 11*y^11 - 10*y^10 + 5*y^9 + 22*y^8 - 10*y^7 - 15*y^6 + 3*y^5 + 11*y^4 - 2*y^3 - 4*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*x^2 + x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*x^2 + x + 1)
\( x^{16} - 2 x^{15} - x^{14} + x^{13} + 11 x^{12} - 11 x^{11} - 10 x^{10} + 5 x^{9} + 22 x^{8} - 10 x^{7} + \cdots + 1 \)
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: |
\(3207314697265625\)
\(\medspace = 5^{12}\cdot 181^{2}\cdot 401\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.31\) | 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: | $5^{3/4}181^{1/2}401^{1/2}\approx 900.8219853969824$ | ||
| Ramified primes: |
\(5\), \(181\), \(401\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{401}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{131}a^{15}+\frac{63}{131}a^{14}+\frac{33}{131}a^{13}+\frac{50}{131}a^{12}-\frac{14}{131}a^{11}-\frac{4}{131}a^{10}-\frac{8}{131}a^{9}+\frac{9}{131}a^{8}-\frac{48}{131}a^{7}+\frac{14}{131}a^{6}-\frac{22}{131}a^{5}+\frac{14}{131}a^{4}+\frac{4}{131}a^{3}-\frac{4}{131}a^{2}-\frac{2}{131}a+\frac{2}{131}$
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: |
\( \frac{54}{131} a^{15} - \frac{135}{131} a^{14} + \frac{79}{131} a^{13} + \frac{80}{131} a^{12} + \frac{423}{131} a^{11} - \frac{1133}{131} a^{10} + \frac{223}{131} a^{9} + \frac{879}{131} a^{8} + \frac{814}{131} a^{7} - \frac{1733}{131} a^{6} - \frac{271}{131} a^{5} + \frac{756}{131} a^{4} + \frac{609}{131} a^{3} - \frac{478}{131} a^{2} + \frac{23}{131} a + \frac{108}{131} \)
(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: |
$\frac{48}{131}a^{15}+\frac{11}{131}a^{14}-\frac{119}{131}a^{13}-\frac{220}{131}a^{12}+\frac{376}{131}a^{11}+\frac{463}{131}a^{10}-\frac{253}{131}a^{9}-\frac{1009}{131}a^{8}+\frac{316}{131}a^{7}+\frac{803}{131}a^{6}-\frac{8}{131}a^{5}-\frac{507}{131}a^{4}+\frac{61}{131}a^{3}+\frac{201}{131}a^{2}-\frac{227}{131}a-\frac{35}{131}$, $\frac{11}{131}a^{15}-\frac{93}{131}a^{14}-\frac{30}{131}a^{13}+\frac{157}{131}a^{12}+\frac{370}{131}a^{11}-\frac{437}{131}a^{10}-\frac{612}{131}a^{9}+\frac{230}{131}a^{8}+\frac{782}{131}a^{7}+\frac{154}{131}a^{6}-\frac{635}{131}a^{5}-\frac{239}{131}a^{4}+\frac{175}{131}a^{3}+\frac{218}{131}a^{2}-\frac{22}{131}a-\frac{109}{131}$, $\frac{215}{131}a^{15}-\frac{341}{131}a^{14}-\frac{372}{131}a^{13}+\frac{8}{131}a^{12}+\frac{2361}{131}a^{11}-\frac{1122}{131}a^{10}-\frac{2506}{131}a^{9}-\frac{423}{131}a^{8}+\frac{3828}{131}a^{7}+\frac{259}{131}a^{6}-\frac{2372}{131}a^{5}-\frac{658}{131}a^{4}+\frac{1122}{131}a^{3}+\frac{319}{131}a^{2}-\frac{430}{131}a-\frac{94}{131}$, $\frac{19}{131}a^{15}+\frac{18}{131}a^{14}-\frac{28}{131}a^{13}-\frac{229}{131}a^{12}+\frac{127}{131}a^{11}+\frac{448}{131}a^{10}+\frac{372}{131}a^{9}-\frac{1270}{131}a^{8}-\frac{388}{131}a^{7}+\frac{921}{131}a^{6}+\frac{1154}{131}a^{5}-\frac{913}{131}a^{4}-\frac{579}{131}a^{3}+\frac{186}{131}a^{2}+\frac{355}{131}a-\frac{93}{131}$, $\frac{84}{131}a^{15}-\frac{79}{131}a^{14}-\frac{110}{131}a^{13}-\frac{254}{131}a^{12}+\frac{789}{131}a^{11}-\frac{74}{131}a^{10}-\frac{148}{131}a^{9}-\frac{1209}{131}a^{8}+\frac{1077}{131}a^{7}+\frac{128}{131}a^{6}+\frac{248}{131}a^{5}-\frac{789}{131}a^{4}+\frac{336}{131}a^{3}+\frac{57}{131}a^{2}-\frac{37}{131}a-\frac{94}{131}$, $\frac{115}{131}a^{15}-\frac{91}{131}a^{14}-\frac{397}{131}a^{13}-\frac{145}{131}a^{12}+\frac{1403}{131}a^{11}+\frac{457}{131}a^{10}-\frac{2230}{131}a^{9}-\frac{1454}{131}a^{8}+\frac{2602}{131}a^{7}+\frac{1872}{131}a^{6}-\frac{1875}{131}a^{5}-\frac{1665}{131}a^{4}+\frac{984}{131}a^{3}+\frac{850}{131}a^{2}-\frac{492}{131}a-\frac{294}{131}$, $\frac{96}{131}a^{15}-\frac{240}{131}a^{14}+\frac{24}{131}a^{13}+\frac{84}{131}a^{12}+\frac{1014}{131}a^{11}-\frac{1563}{131}a^{10}-\frac{244}{131}a^{9}+\frac{602}{131}a^{8}+\frac{1942}{131}a^{7}-\frac{1669}{131}a^{6}-\frac{933}{131}a^{5}+\frac{427}{131}a^{4}+\frac{908}{131}a^{3}-\frac{122}{131}a^{2}-\frac{323}{131}a+\frac{61}{131}$
| 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: | \( 37.3668375913 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 37.3668375913 \cdot 1}{10\cdot\sqrt{3207314697265625}}\cr\approx \mathstrut & 0.160270836674 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*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 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*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 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*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 - 2*x^15 - x^14 + x^13 + 11*x^12 - 11*x^11 - 10*x^10 + 5*x^9 + 22*x^8 - 10*x^7 - 15*x^6 + 3*x^5 + 11*x^4 - 2*x^3 - 4*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
$C_4^4.C_2\wr C_4$ (as 16T1771):
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 16384 |
| The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$ |
| Character table for $C_4^4.C_2\wr C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.2828125.1 |
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: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | $16$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(181\)
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
|
\(401\)
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{401}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |