Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 - 3*x^12 - 2*x^11 + 19*x^10 - 4*x^9 - 37*x^8 + 28*x^7 + 19*x^6 - 30*x^5 + 6*x^4 + 10*x^3 - 6*x^2 + 1)
gp: K = bnfinit(y^16 - 2*y^15 + y^14 - 3*y^12 - 2*y^11 + 19*y^10 - 4*y^9 - 37*y^8 + 28*y^7 + 19*y^6 - 30*y^5 + 6*y^4 + 10*y^3 - 6*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + x^14 - 3*x^12 - 2*x^11 + 19*x^10 - 4*x^9 - 37*x^8 + 28*x^7 + 19*x^6 - 30*x^5 + 6*x^4 + 10*x^3 - 6*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + x^14 - 3*x^12 - 2*x^11 + 19*x^10 - 4*x^9 - 37*x^8 + 28*x^7 + 19*x^6 - 30*x^5 + 6*x^4 + 10*x^3 - 6*x^2 + 1)
\( x^{16} - 2 x^{15} + x^{14} - 3 x^{12} - 2 x^{11} + 19 x^{10} - 4 x^{9} - 37 x^{8} + 28 x^{7} + 19 x^{6} + \cdots + 1 \)
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: |
\(-80157081600000000\)
\(\medspace = -\,2^{24}\cdot 3^{4}\cdot 5^{8}\cdot 151\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.39\) | 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: |
\(2\), \(3\), \(5\), \(151\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-151}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{10}-\frac{2}{9}a^{9}+\frac{4}{9}a^{8}-\frac{4}{9}a^{7}+\frac{1}{9}a^{6}-\frac{4}{9}a^{5}-\frac{1}{3}a^{4}-\frac{4}{9}a^{3}+\frac{4}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{1251}a^{15}+\frac{55}{1251}a^{14}-\frac{61}{1251}a^{13}-\frac{47}{417}a^{12}+\frac{161}{1251}a^{11}-\frac{277}{1251}a^{10}-\frac{202}{1251}a^{9}+\frac{33}{139}a^{8}+\frac{39}{139}a^{7}-\frac{40}{417}a^{6}+\frac{407}{1251}a^{5}+\frac{95}{1251}a^{4}-\frac{43}{417}a^{2}-\frac{548}{1251}a-\frac{100}{1251}$
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: |
$\frac{1745}{1251}a^{15}-\frac{4661}{1251}a^{14}+\frac{6145}{1251}a^{13}-\frac{6131}{1251}a^{12}-\frac{28}{139}a^{11}-\frac{1457}{417}a^{10}+\frac{11171}{417}a^{9}-\frac{34399}{1251}a^{8}-\frac{19538}{1251}a^{7}+\frac{65681}{1251}a^{6}-\frac{46501}{1251}a^{5}-\frac{12284}{1251}a^{4}+\frac{265}{9}a^{3}-\frac{13825}{1251}a^{2}-\frac{364}{139}a+\frac{3698}{1251}$, $\frac{229}{139}a^{15}-\frac{4517}{1251}a^{14}+\frac{1183}{417}a^{13}-\frac{2593}{1251}a^{12}-\frac{3725}{1251}a^{11}-\frac{4472}{1251}a^{10}+\frac{39181}{1251}a^{9}-\frac{15746}{1251}a^{8}-\frac{59437}{1251}a^{7}+\frac{54727}{1251}a^{6}+\frac{8858}{1251}a^{5}-\frac{9934}{417}a^{4}+\frac{110}{9}a^{3}-\frac{1352}{1251}a^{2}-\frac{2833}{1251}a+\frac{1427}{1251}$, $\frac{998}{1251}a^{15}-\frac{1544}{1251}a^{14}+\frac{4}{1251}a^{13}+\frac{367}{1251}a^{12}-\frac{325}{139}a^{11}-\frac{337}{139}a^{10}+\frac{2049}{139}a^{9}+\frac{5618}{1251}a^{8}-\frac{38624}{1251}a^{7}+\frac{7703}{1251}a^{6}+\frac{28940}{1251}a^{5}-\frac{17780}{1251}a^{4}-\frac{23}{9}a^{3}+\frac{8312}{1251}a^{2}-\frac{860}{417}a+\frac{419}{1251}$, $\frac{343}{417}a^{15}-\frac{317}{417}a^{14}-\frac{212}{417}a^{13}+\frac{148}{417}a^{12}-\frac{1211}{417}a^{11}-\frac{488}{139}a^{10}+\frac{4940}{417}a^{9}+\frac{1709}{139}a^{8}-\frac{11101}{417}a^{7}-\frac{1684}{417}a^{6}+\frac{8107}{417}a^{5}-\frac{629}{139}a^{4}-\frac{8}{3}a^{3}+\frac{928}{417}a^{2}-\frac{12}{139}a+\frac{311}{417}$, $a$, $\frac{16}{9}a^{15}-5a^{14}+\frac{50}{9}a^{13}-\frac{41}{9}a^{12}-\frac{8}{9}a^{11}-\frac{26}{9}a^{10}+\frac{334}{9}a^{9}-\frac{316}{9}a^{8}-\frac{368}{9}a^{7}+\frac{650}{9}a^{6}-18a^{5}-\frac{220}{9}a^{4}+\frac{196}{9}a^{3}-\frac{52}{9}a^{2}-\frac{10}{9}a$, $\frac{96}{139}a^{15}-\frac{3076}{1251}a^{14}+\frac{399}{139}a^{13}-\frac{2006}{1251}a^{12}-\frac{730}{1251}a^{11}+\frac{725}{1251}a^{10}+\frac{20072}{1251}a^{9}-\frac{28342}{1251}a^{8}-\frac{26027}{1251}a^{7}+\frac{60062}{1251}a^{6}-\frac{11003}{1251}a^{5}-\frac{9892}{417}a^{4}+\frac{139}{9}a^{3}-\frac{256}{1251}a^{2}-\frac{4625}{1251}a+\frac{475}{1251}$, $\frac{5}{139}a^{15}-\frac{305}{1251}a^{14}+\frac{197}{417}a^{13}-\frac{229}{1251}a^{12}-\frac{122}{1251}a^{11}+\frac{601}{1251}a^{10}+\frac{1057}{1251}a^{9}-\frac{5261}{1251}a^{8}+\frac{227}{1251}a^{7}+\frac{10585}{1251}a^{6}-\frac{6844}{1251}a^{5}-\frac{2189}{417}a^{4}+\frac{71}{9}a^{3}-\frac{1913}{1251}a^{2}-\frac{3532}{1251}a+\frac{1616}{1251}$
| 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: | \( 69.1847935897 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 69.1847935897 \cdot 1}{2\cdot\sqrt{80157081600000000}}\cr\approx \mathstrut & 0.188942116240 \end{aligned}\]
# 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^12 - 2*x^11 + 19*x^10 - 4*x^9 - 37*x^8 + 28*x^7 + 19*x^6 - 30*x^5 + 6*x^4 + 10*x^3 - 6*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 - 2*x^15 + x^14 - 3*x^12 - 2*x^11 + 19*x^10 - 4*x^9 - 37*x^8 + 28*x^7 + 19*x^6 - 30*x^5 + 6*x^4 + 10*x^3 - 6*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 - 2*x^15 + x^14 - 3*x^12 - 2*x^11 + 19*x^10 - 4*x^9 - 37*x^8 + 28*x^7 + 19*x^6 - 30*x^5 + 6*x^4 + 10*x^3 - 6*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 - 2*x^15 + x^14 - 3*x^12 - 2*x^11 + 19*x^10 - 4*x^9 - 37*x^8 + 28*x^7 + 19*x^6 - 30*x^5 + 6*x^4 + 10*x^3 - 6*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
$C_2\wr D_4.C_2$ (as 16T1577):
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 4096 |
| The 88 conjugacy class representatives for $C_2\wr D_4.C_2$ |
| Character table for $C_2\wr D_4.C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.23040000.1 |
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: | 16.4.1344857702400000000.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 $16$ | $2$ | $8$ | $24$ | |||
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.8.0.1 | $x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(151\)
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |