Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 + x^12 + 3*x^10 + 2*x^8 - x^6 - 9*x^4 - 4*x^2 - 1)
gp: K = bnfinit(y^16 - 3*y^14 + y^12 + 3*y^10 + 2*y^8 - y^6 - 9*y^4 - 4*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^14 + x^12 + 3*x^10 + 2*x^8 - x^6 - 9*x^4 - 4*x^2 - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^14 + x^12 + 3*x^10 + 2*x^8 - x^6 - 9*x^4 - 4*x^2 - 1)
\( x^{16} - 3x^{14} + x^{12} + 3x^{10} + 2x^{8} - x^{6} - 9x^{4} - 4x^{2} - 1 \)
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: |
\(-67969704100000000\)
\(\medspace = -\,2^{8}\cdot 5^{8}\cdot 29^{4}\cdot 31^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.27\) | 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: | $2^{15/8}5^{1/2}29^{1/2}31^{1/2}\approx 245.92126826496147$ | ||
| Ramified primes: |
\(2\), \(5\), \(29\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{14}a^{12}+\frac{3}{14}a^{10}+\frac{1}{7}a^{8}+\frac{3}{7}a^{6}-\frac{1}{2}a^{5}-\frac{3}{14}a^{4}-\frac{1}{2}a^{3}-\frac{1}{7}a^{2}-\frac{1}{2}a-\frac{5}{14}$, $\frac{1}{14}a^{13}+\frac{3}{14}a^{11}+\frac{1}{7}a^{9}+\frac{3}{7}a^{7}-\frac{1}{2}a^{6}-\frac{3}{14}a^{5}-\frac{1}{2}a^{4}-\frac{1}{7}a^{3}-\frac{1}{2}a^{2}-\frac{5}{14}a$, $\frac{1}{14}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{3}{7}a^{2}+\frac{1}{14}$, $\frac{1}{14}a^{15}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}+\frac{1}{14}a^{3}-\frac{1}{2}a^{2}+\frac{1}{14}a-\frac{1}{2}$
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: |
$a$, $\frac{5}{14}a^{14}-\frac{15}{14}a^{12}+\frac{2}{7}a^{10}+\frac{19}{14}a^{8}+\frac{4}{7}a^{6}-\frac{2}{7}a^{4}-4a^{2}-\frac{11}{14}$, $\frac{11}{14}a^{15}+\frac{1}{7}a^{14}-\frac{37}{14}a^{13}-\frac{2}{7}a^{12}+\frac{11}{7}a^{11}-\frac{5}{14}a^{10}+\frac{31}{14}a^{9}+\frac{13}{14}a^{8}+\frac{9}{14}a^{7}+\frac{2}{7}a^{6}-\frac{11}{7}a^{5}-\frac{1}{7}a^{4}-\frac{45}{7}a^{3}-\frac{9}{7}a^{2}-a-\frac{13}{14}$, $\frac{1}{2}a^{15}-\frac{1}{14}a^{14}-\frac{3}{2}a^{13}+\frac{2}{7}a^{12}+\frac{1}{2}a^{11}-\frac{1}{7}a^{10}+\frac{3}{2}a^{9}-\frac{3}{7}a^{8}+a^{7}+\frac{3}{14}a^{6}-\frac{1}{2}a^{5}+\frac{1}{7}a^{4}-5a^{3}+\frac{5}{14}a^{2}-\frac{3}{2}a+\frac{1}{2}$, $\frac{11}{14}a^{15}-\frac{19}{7}a^{13}+\frac{13}{7}a^{11}+\frac{29}{14}a^{9}+\frac{3}{14}a^{7}-\frac{19}{14}a^{5}-\frac{81}{14}a^{3}-\frac{1}{7}a$, $\frac{11}{14}a^{15}-\frac{1}{7}a^{14}-\frac{37}{14}a^{13}+\frac{2}{7}a^{12}+\frac{11}{7}a^{11}+\frac{5}{14}a^{10}+\frac{31}{14}a^{9}-\frac{13}{14}a^{8}+\frac{9}{14}a^{7}-\frac{2}{7}a^{6}-\frac{11}{7}a^{5}+\frac{1}{7}a^{4}-\frac{45}{7}a^{3}+\frac{9}{7}a^{2}-a+\frac{13}{14}$, $\frac{2}{7}a^{15}-\frac{1}{7}a^{14}-\frac{5}{7}a^{13}+\frac{1}{2}a^{12}-\frac{1}{7}a^{11}-\frac{1}{2}a^{10}+\frac{15}{14}a^{9}+\frac{5}{7}a^{7}+\frac{1}{7}a^{5}-\frac{39}{14}a^{3}+\frac{5}{14}a^{2}-\frac{15}{7}a-\frac{1}{7}$, $\frac{5}{7}a^{15}+\frac{1}{14}a^{14}-\frac{33}{14}a^{13}-\frac{5}{14}a^{12}+\frac{10}{7}a^{11}+\frac{3}{7}a^{10}+\frac{25}{14}a^{9}+\frac{2}{7}a^{8}+\frac{6}{7}a^{7}-\frac{9}{14}a^{6}-\frac{13}{14}a^{5}+\frac{1}{14}a^{4}-\frac{46}{7}a^{3}-\frac{5}{7}a^{2}-a+\frac{5}{14}$
| 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: | \( 62.5995169928 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 62.5995169928 \cdot 1}{2\cdot\sqrt{67969704100000000}}\cr\approx \mathstrut & 0.185653169009 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^14 + x^12 + 3*x^10 + 2*x^8 - x^6 - 9*x^4 - 4*x^2 - 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 - 3*x^14 + x^12 + 3*x^10 + 2*x^8 - x^6 - 9*x^4 - 4*x^2 - 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 - 3*x^14 + x^12 + 3*x^10 + 2*x^8 - x^6 - 9*x^4 - 4*x^2 - 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 - 3*x^14 + x^12 + 3*x^10 + 2*x^8 - x^6 - 9*x^4 - 4*x^2 - 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
$C_4^4.C_2\wr D_4$ (as 16T1823):
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 32768 |
| The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$ |
| Character table for $C_4^4.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.2.16294375.1 |
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 | $16$ | R | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.10 | $x^{8} - 6 x^{7} + 36 x^{6} - 60 x^{5} + 88 x^{4} + 136 x^{3} + 336 x^{2} + 400 x + 144$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(29\)
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(31\)
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |