Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 + 19*x^12 + 12*x^10 - 29*x^8 - 3*x^6 + 3*x^4 - 18*x^2 + 9)
gp: K = bnfinit(y^16 + 7*y^14 + 19*y^12 + 12*y^10 - 29*y^8 - 3*y^6 + 3*y^4 - 18*y^2 + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 7*x^14 + 19*x^12 + 12*x^10 - 29*x^8 - 3*x^6 + 3*x^4 - 18*x^2 + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 7*x^14 + 19*x^12 + 12*x^10 - 29*x^8 - 3*x^6 + 3*x^4 - 18*x^2 + 9)
\( x^{16} + 7x^{14} + 19x^{12} + 12x^{10} - 29x^{8} - 3x^{6} + 3x^{4} - 18x^{2} + 9 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3703660408794134937600\)
\(\medspace = 2^{16}\cdot 3^{6}\cdot 5^{2}\cdot 1327^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.29\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(1327\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{8}+\frac{1}{5}a^{4}-\frac{2}{5}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{9}+\frac{1}{5}a^{5}-\frac{2}{5}a$, $\frac{1}{45}a^{12}+\frac{4}{45}a^{10}+\frac{19}{45}a^{8}+\frac{7}{15}a^{6}-\frac{8}{45}a^{4}+\frac{1}{15}a^{2}+\frac{2}{15}$, $\frac{1}{45}a^{13}+\frac{4}{45}a^{11}+\frac{19}{45}a^{9}+\frac{7}{15}a^{7}-\frac{8}{45}a^{5}+\frac{1}{15}a^{3}+\frac{2}{15}a$, $\frac{1}{2745}a^{14}-\frac{7}{2745}a^{12}+\frac{56}{2745}a^{10}-\frac{1016}{2745}a^{8}-\frac{689}{2745}a^{6}+\frac{127}{2745}a^{4}-\frac{143}{305}a^{2}+\frac{449}{915}$, $\frac{1}{2745}a^{15}-\frac{7}{2745}a^{13}+\frac{56}{2745}a^{11}-\frac{1016}{2745}a^{9}-\frac{689}{2745}a^{7}+\frac{127}{2745}a^{5}-\frac{143}{305}a^{3}+\frac{449}{915}a$
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 $1$ (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: | $9$ | 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: |
$\frac{18}{305}a^{14}+\frac{1184}{2745}a^{12}+\frac{3521}{2745}a^{10}+\frac{3524}{2745}a^{8}-\frac{170}{183}a^{6}-\frac{1813}{2745}a^{4}-\frac{260}{183}a^{2}-\frac{584}{915}$, $\frac{86}{2745}a^{14}+\frac{496}{2745}a^{12}+\frac{973}{2745}a^{10}-\frac{634}{2745}a^{8}-\frac{3256}{2745}a^{6}+\frac{2138}{2745}a^{4}-\frac{37}{305}a^{2}+\frac{110}{183}$, $\frac{22}{915}a^{14}+\frac{331}{2745}a^{12}+\frac{56}{549}a^{10}-\frac{406}{549}a^{8}-\frac{1372}{915}a^{6}+\frac{737}{549}a^{4}+\frac{844}{915}a^{2}-\frac{988}{915}$, $\frac{352}{2745}a^{14}+\frac{157}{183}a^{12}+\frac{672}{305}a^{10}+\frac{2939}{2745}a^{8}-\frac{9569}{2745}a^{6}+\frac{3407}{2745}a^{4}+\frac{211}{915}a^{2}-\frac{674}{915}$, $\frac{53}{2745}a^{14}+\frac{544}{2745}a^{12}+\frac{2236}{2745}a^{10}+\frac{4163}{2745}a^{8}+\frac{1913}{2745}a^{6}-\frac{2236}{2745}a^{4}+\frac{46}{305}a^{2}-\frac{145}{183}$, $\frac{59}{305}a^{15}-\frac{43}{549}a^{14}+\frac{806}{549}a^{13}-\frac{192}{305}a^{12}+\frac{12412}{2745}a^{11}-\frac{1919}{915}a^{10}+\frac{13408}{2745}a^{9}-\frac{7687}{2745}a^{8}-\frac{2759}{915}a^{7}+\frac{637}{2745}a^{6}-\frac{7166}{2745}a^{5}+\frac{4049}{2745}a^{4}-\frac{452}{915}a^{3}+\frac{1162}{915}a^{2}-\frac{3019}{915}a+\frac{1309}{915}$, $\frac{16}{2745}a^{15}+\frac{14}{549}a^{14}+\frac{44}{915}a^{13}+\frac{364}{2745}a^{12}+\frac{5}{61}a^{11}+\frac{199}{2745}a^{10}-\frac{1189}{2745}a^{9}-\frac{1096}{915}a^{8}-\frac{1180}{549}a^{7}-\frac{8336}{2745}a^{6}-\frac{7057}{2745}a^{5}+\frac{137}{915}a^{4}+\frac{323}{183}a^{3}+\frac{2849}{915}a^{2}+\frac{473}{183}a+\frac{432}{305}$, $\frac{262}{915}a^{15}-\frac{493}{2745}a^{14}+\frac{1948}{915}a^{13}-\frac{3686}{2745}a^{12}+\frac{5827}{915}a^{11}-\frac{11138}{2745}a^{10}+\frac{379}{61}a^{9}-\frac{11329}{2745}a^{8}-\frac{5021}{915}a^{7}+\frac{1726}{549}a^{6}-\frac{177}{61}a^{5}+\frac{6563}{2745}a^{4}-\frac{36}{305}a^{3}+\frac{82}{61}a^{2}-\frac{1556}{305}a+\frac{2452}{915}$, $\frac{203}{2745}a^{15}+\frac{34}{2745}a^{14}+\frac{181}{305}a^{13}+\frac{311}{2745}a^{12}+\frac{1817}{915}a^{11}+\frac{271}{549}a^{10}+\frac{7618}{2745}a^{9}+\frac{3337}{2745}a^{8}+\frac{1043}{2745}a^{7}+\frac{4573}{2745}a^{6}-\frac{266}{2745}a^{5}+\frac{2671}{2745}a^{4}+\frac{143}{915}a^{3}-\frac{104}{305}a^{2}-\frac{278}{183}a-\frac{106}{915}$
| 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: | \( 35087.261258 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 35087.261258 \cdot 1}{2\cdot\sqrt{3703660408794134937600}}\cr\approx \mathstrut & 0.28379392086 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 + 19*x^12 + 12*x^10 - 29*x^8 - 3*x^6 + 3*x^4 - 18*x^2 + 9)
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 + 7*x^14 + 19*x^12 + 12*x^10 - 29*x^8 - 3*x^6 + 3*x^4 - 18*x^2 + 9, 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 + 7*x^14 + 19*x^12 + 12*x^10 - 29*x^8 - 3*x^6 + 3*x^4 - 18*x^2 + 9);
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 + 7*x^14 + 19*x^12 + 12*x^10 - 29*x^8 - 3*x^6 + 3*x^4 - 18*x^2 + 9);
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 $C_2^8.S_4$ are not computed |
| Character table for $C_2^8.S_4$ is not computed |
Intermediate fields
| 4.4.3981.1, 8.4.12171541248.5, 8.4.60857706240.2, 8.4.79241805.1 |
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 | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $2$ | $8$ | $16$ | |||
|
\(3\)
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.8.6.2 | $x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(1327\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |