Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 98*x^12 + 42*x^11 + 213*x^10 - 163*x^9 - 270*x^8 + 232*x^7 + 208*x^6 - 162*x^5 - 88*x^4 + 49*x^3 + 16*x^2 - 4*x - 1)
gp: K = bnfinit(y^16 - 8*y^15 + 18*y^14 + 14*y^13 - 98*y^12 + 42*y^11 + 213*y^10 - 163*y^9 - 270*y^8 + 232*y^7 + 208*y^6 - 162*y^5 - 88*y^4 + 49*y^3 + 16*y^2 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 98*x^12 + 42*x^11 + 213*x^10 - 163*x^9 - 270*x^8 + 232*x^7 + 208*x^6 - 162*x^5 - 88*x^4 + 49*x^3 + 16*x^2 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 98*x^12 + 42*x^11 + 213*x^10 - 163*x^9 - 270*x^8 + 232*x^7 + 208*x^6 - 162*x^5 - 88*x^4 + 49*x^3 + 16*x^2 - 4*x - 1)
\( x^{16} - 8 x^{15} + 18 x^{14} + 14 x^{13} - 98 x^{12} + 42 x^{11} + 213 x^{10} - 163 x^{9} - 270 x^{8} + \cdots - 1 \)
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: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(284052722662986805169\)
\(\medspace = 43^{2}\cdot 521^{2}\cdot 4337^{2}\cdot 30089\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.98\) | 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: | $43^{1/2}521^{1/2}4337^{1/2}30089^{1/2}\approx 1709825.0586475213$ | ||
| Ramified primes: |
\(43\), \(521\), \(4337\), \(30089\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{30089}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: | $13$ | 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: |
$a^{2}-a-1$, $a^{14}-7a^{13}+12a^{12}+19a^{11}-67a^{10}-6a^{9}+140a^{8}-29a^{7}-159a^{6}+44a^{5}+93a^{4}-25a^{3}-21a^{2}+5a+1$, $2a^{12}-12a^{11}+14a^{10}+40a^{9}-79a^{8}-56a^{7}+148a^{6}+52a^{5}-136a^{4}-26a^{3}+50a^{2}+3a-2$, $a$, $a-1$, $a^{14}-7a^{13}+14a^{12}+7a^{11}-52a^{10}+29a^{9}+64a^{8}-67a^{7}-29a^{6}+66a^{5}-17a^{4}-22a^{3}+16a^{2}-3a-1$, $a^{15}-7a^{14}+13a^{13}+12a^{12}-53a^{11}+2a^{10}+83a^{9}+3a^{8}-77a^{7}-41a^{6}+33a^{5}+70a^{4}-7a^{3}-39a^{2}+4a+4$, $2a^{15}-15a^{14}+31a^{13}+25a^{12}-147a^{11}+48a^{10}+267a^{9}-163a^{8}-260a^{7}+191a^{6}+105a^{5}-106a^{4}+12a^{3}+30a^{2}-11a-4$, $2a^{14}-14a^{13}+24a^{12}+38a^{11}-133a^{10}-17a^{9}+283a^{8}-40a^{7}-337a^{6}+61a^{5}+214a^{4}-30a^{3}-57a^{2}+6a+4$, $a^{14}-7a^{13}+13a^{12}+14a^{11}-65a^{10}+16a^{9}+123a^{8}-76a^{7}-133a^{6}+107a^{5}+85a^{4}-67a^{3}-31a^{2}+13a+4$, $a^{15}-8a^{14}+20a^{13}+a^{12}-78a^{11}+75a^{10}+114a^{9}-180a^{8}-92a^{7}+210a^{6}+38a^{5}-116a^{4}-12a^{3}+22a^{2}+2a-2$, $2a^{14}-14a^{13}+25a^{12}+33a^{11}-132a^{10}+10a^{9}+263a^{8}-105a^{7}-292a^{6}+151a^{5}+179a^{4}-94a^{3}-53a^{2}+20a+6$, $2a^{15}-14a^{14}+22a^{13}+51a^{12}-153a^{11}-50a^{10}+382a^{9}-23a^{8}-514a^{7}+80a^{6}+381a^{5}-65a^{4}-128a^{3}+22a^{2}+11a-1$
| 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: | \( 30667.3160594 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 30667.3160594 \cdot 1}{2\cdot\sqrt{284052722662986805169}}\cr\approx \mathstrut & 0.147118056801 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 98*x^12 + 42*x^11 + 213*x^10 - 163*x^9 - 270*x^8 + 232*x^7 + 208*x^6 - 162*x^5 - 88*x^4 + 49*x^3 + 16*x^2 - 4*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 - 8*x^15 + 18*x^14 + 14*x^13 - 98*x^12 + 42*x^11 + 213*x^10 - 163*x^9 - 270*x^8 + 232*x^7 + 208*x^6 - 162*x^5 - 88*x^4 + 49*x^3 + 16*x^2 - 4*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 - 8*x^15 + 18*x^14 + 14*x^13 - 98*x^12 + 42*x^11 + 213*x^10 - 163*x^9 - 270*x^8 + 232*x^7 + 208*x^6 - 162*x^5 - 88*x^4 + 49*x^3 + 16*x^2 - 4*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 - 8*x^15 + 18*x^14 + 14*x^13 - 98*x^12 + 42*x^11 + 213*x^10 - 163*x^9 - 270*x^8 + 232*x^7 + 208*x^6 - 162*x^5 - 88*x^4 + 49*x^3 + 16*x^2 - 4*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
$C_2^8.S_8$ (as 16T1948):
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 non-solvable group of order 10321920 |
| The 185 conjugacy class representatives for $C_2^8.S_8$ |
| Character table for $C_2^8.S_8$ |
Intermediate fields
| 8.6.97161811.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $16$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $16$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(43\)
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
|
\(521\)
| $\Q_{521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{521}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(4337\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
|
\(30089\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |