Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^14 + 26*x^12 - 36*x^10 + 180*x^8 - 127*x^6 + 256*x^4 - 264*x^2 + 81)
gp: K = bnfinit(y^16 - 5*y^14 + 26*y^12 - 36*y^10 + 180*y^8 - 127*y^6 + 256*y^4 - 264*y^2 + 81, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^14 + 26*x^12 - 36*x^10 + 180*x^8 - 127*x^6 + 256*x^4 - 264*x^2 + 81);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^14 + 26*x^12 - 36*x^10 + 180*x^8 - 127*x^6 + 256*x^4 - 264*x^2 + 81)
\( x^{16} - 5x^{14} + 26x^{12} - 36x^{10} + 180x^{8} - 127x^{6} + 256x^{4} - 264x^{2} + 81 \)
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: |
\(81400900859025914658816\)
\(\medspace = 2^{16}\cdot 3^{2}\cdot 19^{4}\cdot 97^{2}\cdot 103^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.03\) | 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: | $2^{15/8}3^{1/2}19^{1/2}97^{1/2}103^{1/2}\approx 2768.0450193841634$ | ||
| Ramified primes: |
\(2\), \(3\), \(19\), \(97\), \(103\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{6}a^{10}+\frac{1}{6}a^{6}-\frac{1}{3}a^{4}-\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{11}+\frac{1}{6}a^{7}-\frac{1}{3}a^{5}-\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{12}a^{12}-\frac{1}{12}a^{10}-\frac{1}{12}a^{8}-\frac{1}{12}a^{6}+\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{12}a^{13}-\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{12}a^{7}+\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{3201384}a^{14}-\frac{12793}{1600692}a^{12}-\frac{20942}{400173}a^{10}-\frac{47654}{400173}a^{8}+\frac{43343}{400173}a^{6}+\frac{147703}{1067128}a^{4}+\frac{396107}{3201384}a^{2}-\frac{144357}{1067128}$, $\frac{1}{9604152}a^{15}-\frac{12793}{4802076}a^{13}-\frac{20942}{1200519}a^{11}+\frac{28579}{400173}a^{9}+\frac{103375}{400173}a^{7}-\frac{3825403}{9604152}a^{5}+\frac{2530363}{9604152}a^{3}-\frac{1211485}{3201384}a$
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
$C_{2}$, which has order $2$ (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: | $7$ | 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: |
$\frac{989}{533564}a^{14}-\frac{14183}{1600692}a^{12}+\frac{68945}{1600692}a^{10}-\frac{93055}{1600692}a^{8}+\frac{478229}{1600692}a^{6}-\frac{55354}{133391}a^{4}+\frac{113847}{533564}a^{2}-\frac{197266}{133391}$, $\frac{1781}{800346}a^{14}-\frac{31279}{1600692}a^{12}+\frac{53831}{533564}a^{10}-\frac{428177}{1600692}a^{8}+\frac{835205}{1600692}a^{6}-\frac{1395355}{1600692}a^{4}+\frac{82522}{133391}a^{2}-\frac{244111}{533564}$, $\frac{6529}{1600692}a^{14}-\frac{7577}{266782}a^{12}+\frac{115219}{800346}a^{10}-\frac{43436}{133391}a^{8}+\frac{656717}{800346}a^{6}-\frac{2059603}{1600692}a^{4}+\frac{443935}{533564}a^{2}-\frac{1033175}{533564}$, $\frac{44855}{3201384}a^{15}+\frac{21883}{1067128}a^{14}-\frac{28831}{400173}a^{13}-\frac{37828}{400173}a^{12}+\frac{611003}{1600692}a^{11}+\frac{825671}{1600692}a^{10}-\frac{306033}{533564}a^{9}-\frac{1007029}{1600692}a^{8}+\frac{4307663}{1600692}a^{7}+\frac{6247339}{1600692}a^{6}-\frac{6274355}{3201384}a^{5}-\frac{1771799}{1067128}a^{4}+\frac{14104513}{3201384}a^{3}+\frac{23730155}{3201384}a^{2}-\frac{2201569}{1067128}a-\frac{4264219}{1067128}$, $\frac{35096}{1200519}a^{15}-\frac{989}{533564}a^{14}-\frac{356261}{2401038}a^{13}+\frac{14183}{1600692}a^{12}+\frac{1797731}{2401038}a^{11}-\frac{68945}{1600692}a^{10}-\frac{800821}{800346}a^{9}+\frac{93055}{1600692}a^{8}+\frac{3823741}{800346}a^{7}-\frac{478229}{1600692}a^{6}-\frac{8209393}{2401038}a^{5}+\frac{55354}{133391}a^{4}+\frac{4830110}{1200519}a^{3}-\frac{113847}{533564}a^{2}-\frac{4994869}{800346}a+\frac{464048}{133391}$, $\frac{37397}{1600692}a^{15}-\frac{11933}{800346}a^{14}-\frac{26565}{266782}a^{13}+\frac{104869}{1600692}a^{12}+\frac{207956}{400173}a^{11}-\frac{514613}{1600692}a^{10}-\frac{53049}{133391}a^{9}+\frac{94325}{533564}a^{8}+\frac{487390}{133391}a^{7}-\frac{2882657}{1600692}a^{6}-\frac{483439}{1600692}a^{5}-\frac{401087}{1600692}a^{4}+\frac{7879607}{1600692}a^{3}-\frac{701701}{800346}a^{2}-\frac{1518869}{533564}a+\frac{401587}{533564}$, $\frac{127915}{4802076}a^{15}-\frac{21883}{1067128}a^{14}-\frac{618569}{4802076}a^{13}+\frac{37828}{400173}a^{12}+\frac{3298613}{4802076}a^{11}-\frac{825671}{1600692}a^{10}-\frac{1448207}{1600692}a^{9}+\frac{1007029}{1600692}a^{8}+\frac{2614173}{533564}a^{7}-\frac{6247339}{1600692}a^{6}-\frac{5643719}{2401038}a^{5}+\frac{1771799}{1067128}a^{4}+\frac{36271471}{4802076}a^{3}-\frac{23730155}{3201384}a^{2}-\frac{1528030}{400173}a+\frac{4264219}{1067128}$
| 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: | \( 101705.265084 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 101705.265084 \cdot 2}{2\cdot\sqrt{81400900859025914658816}}\cr\approx \mathstrut & 0.865900075155 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^14 + 26*x^12 - 36*x^10 + 180*x^8 - 127*x^6 + 256*x^4 - 264*x^2 + 81)
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 - 5*x^14 + 26*x^12 - 36*x^10 + 180*x^8 - 127*x^6 + 256*x^4 - 264*x^2 + 81, 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 - 5*x^14 + 26*x^12 - 36*x^10 + 180*x^8 - 127*x^6 + 256*x^4 - 264*x^2 + 81);
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 - 5*x^14 + 26*x^12 - 36*x^10 + 180*x^8 - 127*x^6 + 256*x^4 - 264*x^2 + 81);
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
$C_2^8.S_4$ (as 16T1664):
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 6144 |
| The 105 conjugacy class representatives for $C_2^8.S_4$ |
| Character table for $C_2^8.S_4$ |
Intermediate fields
| 4.4.1957.1, 8.0.2941324032.1, 8.8.285308431104.1, 8.0.371495353.1 |
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 | R | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# 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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.10 | $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.8.10 | $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
|
\(19\)
| 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 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(97\)
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(103\)
| 103.4.0.1 | $x^{4} + 2 x^{2} + 88 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 103.4.0.1 | $x^{4} + 2 x^{2} + 88 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |