Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + x^14 + 9*x^13 - 21*x^12 + 24*x^11 - 12*x^10 - 8*x^9 + 17*x^8 - 8*x^7 - 12*x^6 + 24*x^5 - 21*x^4 + 9*x^3 + x^2 - 3*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + y^14 + 9*y^13 - 21*y^12 + 24*y^11 - 12*y^10 - 8*y^9 + 17*y^8 - 8*y^7 - 12*y^6 + 24*y^5 - 21*y^4 + 9*y^3 + y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + x^14 + 9*x^13 - 21*x^12 + 24*x^11 - 12*x^10 - 8*x^9 + 17*x^8 - 8*x^7 - 12*x^6 + 24*x^5 - 21*x^4 + 9*x^3 + x^2 - 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + x^14 + 9*x^13 - 21*x^12 + 24*x^11 - 12*x^10 - 8*x^9 + 17*x^8 - 8*x^7 - 12*x^6 + 24*x^5 - 21*x^4 + 9*x^3 + x^2 - 3*x + 1)
\( x^{16} - 3 x^{15} + x^{14} + 9 x^{13} - 21 x^{12} + 24 x^{11} - 12 x^{10} - 8 x^{9} + 17 x^{8} + \cdots + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(499382553081640625\)
\(\medspace = 5^{8}\cdot 89\cdot 119851^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.77\) | 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: | $5^{1/2}89^{1/2}119851^{1/2}\approx 7302.99219498419$ | ||
| Ramified primes: |
\(5\), \(89\), \(119851\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{89}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{191}a^{14}+\frac{8}{191}a^{13}+\frac{88}{191}a^{12}+\frac{14}{191}a^{11}+\frac{45}{191}a^{10}-\frac{68}{191}a^{9}-\frac{41}{191}a^{8}-\frac{9}{191}a^{7}-\frac{41}{191}a^{6}-\frac{68}{191}a^{5}+\frac{45}{191}a^{4}+\frac{14}{191}a^{3}+\frac{88}{191}a^{2}+\frac{8}{191}a+\frac{1}{191}$, $\frac{1}{191}a^{15}+\frac{24}{191}a^{13}+\frac{74}{191}a^{12}-\frac{67}{191}a^{11}-\frac{46}{191}a^{10}-\frac{70}{191}a^{9}-\frac{63}{191}a^{8}+\frac{31}{191}a^{7}+\frac{69}{191}a^{6}+\frac{16}{191}a^{5}+\frac{36}{191}a^{4}-\frac{24}{191}a^{3}+\frac{68}{191}a^{2}-\frac{63}{191}a-\frac{8}{191}$
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: | $9$ | 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: |
$a$, $\frac{434}{191}a^{15}-\frac{923}{191}a^{14}-\frac{406}{191}a^{13}+\frac{3608}{191}a^{12}-\frac{5901}{191}a^{11}+\frac{4969}{191}a^{10}-\frac{468}{191}a^{9}-\frac{4015}{191}a^{8}+\frac{3425}{191}a^{7}+\frac{175}{191}a^{6}-\frac{5164}{191}a^{5}+\frac{5413}{191}a^{4}-\frac{3665}{191}a^{3}+\frac{622}{191}a^{2}+\frac{609}{191}a-\frac{384}{191}$, $\frac{31}{191}a^{15}-\frac{216}{191}a^{14}+\frac{162}{191}a^{13}+\frac{667}{191}a^{12}-\frac{1472}{191}a^{11}+\frac{1269}{191}a^{10}-\frac{279}{191}a^{9}-\frac{1119}{191}a^{8}+\frac{1568}{191}a^{7}-\frac{83}{191}a^{6}-\frac{1051}{191}a^{5}+\frac{1710}{191}a^{4}-\frac{903}{191}a^{3}+\frac{99}{191}a^{2}+\frac{521}{191}a-\frac{273}{191}$, $\frac{540}{191}a^{15}-\frac{1181}{191}a^{14}-\frac{499}{191}a^{13}+\frac{4601}{191}a^{12}-\frac{7447}{191}a^{11}+\frac{6246}{191}a^{10}-\frac{658}{191}a^{9}-\frac{5272}{191}a^{8}+\frac{4831}{191}a^{7}+\frac{113}{191}a^{6}-\frac{6743}{191}a^{5}+\frac{7360}{191}a^{4}-\frac{4664}{191}a^{3}+\frac{406}{191}a^{2}+\frac{1417}{191}a-\frac{535}{191}$, $\frac{156}{191}a^{15}-\frac{225}{191}a^{14}-\frac{348}{191}a^{13}+\frac{1103}{191}a^{12}-\frac{1187}{191}a^{11}+\frac{653}{191}a^{10}+\frac{369}{191}a^{9}-\frac{985}{191}a^{8}+\frac{176}{191}a^{7}+\frac{507}{191}a^{6}-\frac{1304}{191}a^{5}+\frac{648}{191}a^{4}-\frac{591}{191}a^{3}-\frac{24}{191}a^{2}+\frac{23}{191}a+\frac{55}{191}$, $\frac{550}{191}a^{15}-\frac{1287}{191}a^{14}-\frac{343}{191}a^{13}+\frac{4799}{191}a^{12}-\frac{8264}{191}a^{11}+\frac{7319}{191}a^{10}-\frac{1408}{191}a^{9}-\frac{5185}{191}a^{8}+\frac{5331}{191}a^{7}-\frac{390}{191}a^{6}-\frac{6633}{191}a^{5}+\frac{8107}{191}a^{4}-\frac{5433}{191}a^{3}+\frac{1117}{191}a^{2}+\frac{1085}{191}a-\frac{530}{191}$, $a^{15}-\frac{708}{191}a^{14}+\frac{257}{191}a^{13}+\frac{2254}{191}a^{12}-\frac{4755}{191}a^{11}+\frac{4621}{191}a^{10}-\frac{1516}{191}a^{9}-\frac{2869}{191}a^{8}+\frac{4080}{191}a^{7}-\frac{768}{191}a^{6}-\frac{3235}{191}a^{5}+\frac{5576}{191}a^{4}-\frac{3418}{191}a^{3}+\frac{917}{191}a^{2}+\frac{1021}{191}a-\frac{708}{191}$, $\frac{496}{191}a^{15}-\frac{1165}{191}a^{14}-\frac{281}{191}a^{13}+\frac{4281}{191}a^{12}-\frac{7522}{191}a^{11}+\frac{6889}{191}a^{10}-\frac{1531}{191}a^{9}-\frac{4684}{191}a^{8}+\frac{4851}{191}a^{7}-\frac{332}{191}a^{6}-\frac{6052}{191}a^{5}+\frac{7642}{191}a^{4}-\frac{5294}{191}a^{3}+\frac{1114}{191}a^{2}+\frac{879}{191}a-\frac{549}{191}$, $\frac{100}{191}a^{15}-\frac{246}{191}a^{14}+\frac{50}{191}a^{13}+\frac{841}{191}a^{12}-\frac{1931}{191}a^{11}+\frac{1902}{191}a^{10}-\frac{395}{191}a^{9}-\frac{1180}{191}a^{8}+\frac{1494}{191}a^{7}-\frac{395}{191}a^{6}-\frac{1727}{191}a^{5}+\frac{1889}{191}a^{4}-\frac{1451}{191}a^{3}+\frac{50}{191}a^{2}+\frac{327}{191}a-\frac{282}{191}$
| 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: | \( 251.294326231 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 251.294326231 \cdot 1}{2\cdot\sqrt{499382553081640625}}\cr\approx \mathstrut & 0.175039150763 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + x^14 + 9*x^13 - 21*x^12 + 24*x^11 - 12*x^10 - 8*x^9 + 17*x^8 - 8*x^7 - 12*x^6 + 24*x^5 - 21*x^4 + 9*x^3 + x^2 - 3*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 - 3*x^15 + x^14 + 9*x^13 - 21*x^12 + 24*x^11 - 12*x^10 - 8*x^9 + 17*x^8 - 8*x^7 - 12*x^6 + 24*x^5 - 21*x^4 + 9*x^3 + x^2 - 3*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 - 3*x^15 + x^14 + 9*x^13 - 21*x^12 + 24*x^11 - 12*x^10 - 8*x^9 + 17*x^8 - 8*x^7 - 12*x^6 + 24*x^5 - 21*x^4 + 9*x^3 + x^2 - 3*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 - 3*x^15 + x^14 + 9*x^13 - 21*x^12 + 24*x^11 - 12*x^10 - 8*x^9 + 17*x^8 - 8*x^7 - 12*x^6 + 24*x^5 - 21*x^4 + 9*x^3 + x^2 - 3*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
| \(\Q(\sqrt{5}) \), 8.6.74906875.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | R | $16$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $16$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(89\)
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(119851\)
| $\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{119851}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |