Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 3*x^13 - 11*x^12 + 16*x^11 - 17*x^10 + 16*x^9 - 13*x^8 + 16*x^7 - 17*x^6 + 16*x^5 - 11*x^4 + 3*x^3 + x^2 - 2*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + y^14 + 3*y^13 - 11*y^12 + 16*y^11 - 17*y^10 + 16*y^9 - 13*y^8 + 16*y^7 - 17*y^6 + 16*y^5 - 11*y^4 + 3*y^3 + y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + x^14 + 3*x^13 - 11*x^12 + 16*x^11 - 17*x^10 + 16*x^9 - 13*x^8 + 16*x^7 - 17*x^6 + 16*x^5 - 11*x^4 + 3*x^3 + x^2 - 2*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + x^14 + 3*x^13 - 11*x^12 + 16*x^11 - 17*x^10 + 16*x^9 - 13*x^8 + 16*x^7 - 17*x^6 + 16*x^5 - 11*x^4 + 3*x^3 + x^2 - 2*x + 1)
\( x^{16} - 2 x^{15} + x^{14} + 3 x^{13} - 11 x^{12} + 16 x^{11} - 17 x^{10} + 16 x^{9} - 13 x^{8} + \cdots + 1 \)
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: | $[2, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-36286162879296875\)
\(\medspace = -\,5^{8}\cdot 31^{2}\cdot 131\cdot 859^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.84\) | 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: | $5^{1/2}31^{1/2}131^{1/2}859^{1/2}\approx 4176.361454663616$ | ||
| Ramified primes: |
\(5\), \(31\), \(131\), \(859\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-131}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{43}a^{14}-\frac{7}{43}a^{13}-\frac{8}{43}a^{12}+\frac{7}{43}a^{11}+\frac{5}{43}a^{10}-\frac{16}{43}a^{9}+\frac{15}{43}a^{8}+\frac{15}{43}a^{6}-\frac{16}{43}a^{5}+\frac{5}{43}a^{4}+\frac{7}{43}a^{3}-\frac{8}{43}a^{2}-\frac{7}{43}a+\frac{1}{43}$, $\frac{1}{43}a^{15}-\frac{14}{43}a^{13}-\frac{6}{43}a^{12}+\frac{11}{43}a^{11}+\frac{19}{43}a^{10}-\frac{11}{43}a^{9}+\frac{19}{43}a^{8}+\frac{15}{43}a^{7}+\frac{3}{43}a^{6}-\frac{21}{43}a^{5}-\frac{1}{43}a^{4}-\frac{2}{43}a^{3}-\frac{20}{43}a^{2}-\frac{5}{43}a+\frac{7}{43}$
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: | $8$ | 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: |
$\frac{8}{43}a^{15}-\frac{60}{43}a^{14}+\frac{93}{43}a^{13}+\frac{45}{43}a^{12}-\frac{289}{43}a^{11}+\frac{540}{43}a^{10}-\frac{590}{43}a^{9}+\frac{413}{43}a^{8}-\frac{439}{43}a^{7}+\frac{457}{43}a^{6}-\frac{541}{43}a^{5}+\frac{638}{43}a^{4}-\frac{350}{43}a^{3}+\frac{148}{43}a^{2}+\frac{79}{43}a-\frac{90}{43}$, $a$, $\frac{6}{43}a^{15}-\frac{4}{43}a^{14}+\frac{30}{43}a^{13}-\frac{4}{43}a^{12}-\frac{91}{43}a^{11}+\frac{137}{43}a^{10}-\frac{217}{43}a^{9}+\frac{97}{43}a^{8}-\frac{39}{43}a^{7}+\frac{130}{43}a^{6}-\frac{62}{43}a^{5}+\frac{232}{43}a^{4}-\frac{83}{43}a^{3}-\frac{2}{43}a^{2}+\frac{41}{43}a-\frac{91}{43}$, $\frac{41}{43}a^{15}-\frac{70}{43}a^{14}-\frac{41}{43}a^{13}+\frac{185}{43}a^{12}-\frac{340}{43}a^{11}+\frac{343}{43}a^{10}-\frac{148}{43}a^{9}+\frac{202}{43}a^{8}-\frac{202}{43}a^{7}+\frac{234}{43}a^{6}-\frac{343}{43}a^{5}+\frac{82}{43}a^{4}+\frac{30}{43}a^{3}-\frac{88}{43}a^{2}+\frac{113}{43}a+\frac{2}{43}$, $\frac{6}{43}a^{15}-\frac{4}{43}a^{14}+\frac{30}{43}a^{13}-\frac{4}{43}a^{12}-\frac{91}{43}a^{11}+\frac{137}{43}a^{10}-\frac{217}{43}a^{9}+\frac{97}{43}a^{8}-\frac{39}{43}a^{7}+\frac{130}{43}a^{6}-\frac{62}{43}a^{5}+\frac{232}{43}a^{4}-\frac{83}{43}a^{3}-\frac{2}{43}a^{2}+\frac{84}{43}a-\frac{91}{43}$, $\frac{41}{43}a^{14}+\frac{14}{43}a^{13}-\frac{70}{43}a^{12}+\frac{72}{43}a^{11}-\frac{139}{43}a^{10}-\frac{97}{43}a^{9}+\frac{13}{43}a^{8}+a^{7}+\frac{99}{43}a^{6}+\frac{204}{43}a^{5}+\frac{162}{43}a^{4}-\frac{57}{43}a^{3}+\frac{102}{43}a^{2}-\frac{72}{43}a-\frac{45}{43}$, $\frac{14}{43}a^{15}-\frac{21}{43}a^{14}-\frac{6}{43}a^{13}+\frac{41}{43}a^{12}-\frac{122}{43}a^{11}+\frac{161}{43}a^{10}-\frac{119}{43}a^{9}+\frac{123}{43}a^{8}-\frac{48}{43}a^{7}+\frac{71}{43}a^{6}-\frac{130}{43}a^{5}+\frac{53}{43}a^{4}-\frac{46}{43}a^{3}-\frac{26}{43}a^{2}+\frac{77}{43}a-\frac{52}{43}$, $\frac{85}{43}a^{15}-\frac{71}{43}a^{14}-\frac{91}{43}a^{13}+\frac{273}{43}a^{12}-\frac{551}{43}a^{11}+\frac{357}{43}a^{10}-\frac{272}{43}a^{9}+\frac{378}{43}a^{8}-\frac{230}{43}a^{7}+\frac{609}{43}a^{6}-\frac{348}{43}a^{5}+\frac{205}{43}a^{4}-\frac{22}{43}a^{3}-\frac{143}{43}a^{2}-\frac{14}{43}a+\frac{51}{43}$
| 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: | \( 40.8815236048 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 40.8815236048 \cdot 1}{2\cdot\sqrt{36286162879296875}}\cr\approx \mathstrut & 0.165937913275 \end{aligned}\]
# 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 + 3*x^13 - 11*x^12 + 16*x^11 - 17*x^10 + 16*x^9 - 13*x^8 + 16*x^7 - 17*x^6 + 16*x^5 - 11*x^4 + 3*x^3 + x^2 - 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 + 3*x^13 - 11*x^12 + 16*x^11 - 17*x^10 + 16*x^9 - 13*x^8 + 16*x^7 - 17*x^6 + 16*x^5 - 11*x^4 + 3*x^3 + x^2 - 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 + 3*x^13 - 11*x^12 + 16*x^11 - 17*x^10 + 16*x^9 - 13*x^8 + 16*x^7 - 17*x^6 + 16*x^5 - 11*x^4 + 3*x^3 + x^2 - 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 + 3*x^13 - 11*x^12 + 16*x^11 - 17*x^10 + 16*x^9 - 13*x^8 + 16*x^7 - 17*x^6 + 16*x^5 - 11*x^4 + 3*x^3 + x^2 - 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
$C_2^6.S_4^2:D_4$ (as 16T1905):
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 294912 |
| The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$ |
| Character table for $C_2^6.S_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 8.4.16643125.1 |
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 | $16$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(131\)
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.3.0.1 | $x^{3} + 3 x + 129$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 131.3.0.1 | $x^{3} + 3 x + 129$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 131.3.0.1 | $x^{3} + 3 x + 129$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 131.3.0.1 | $x^{3} + 3 x + 129$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
|
\(859\)
| 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |