Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 3*x^14 + 8*x^12 + 4*x^11 - x^10 + 10*x^9 + 15*x^8 + 10*x^7 - x^6 + 4*x^5 + 8*x^4 - 3*x^2 - 2*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 - 3*y^14 + 8*y^12 + 4*y^11 - y^10 + 10*y^9 + 15*y^8 + 10*y^7 - y^6 + 4*y^5 + 8*y^4 - 3*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 3*x^14 + 8*x^12 + 4*x^11 - x^10 + 10*x^9 + 15*x^8 + 10*x^7 - x^6 + 4*x^5 + 8*x^4 - 3*x^2 - 2*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 3*x^14 + 8*x^12 + 4*x^11 - x^10 + 10*x^9 + 15*x^8 + 10*x^7 - x^6 + 4*x^5 + 8*x^4 - 3*x^2 - 2*x + 1)
\( x^{16} - 2 x^{15} - 3 x^{14} + 8 x^{12} + 4 x^{11} - x^{10} + 10 x^{9} + 15 x^{8} + 10 x^{7} - x^{6} + \cdots + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(887213286400000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 7^{2}\cdot 29^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.24\) | 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: | $2^{15/8}5^{1/2}7^{1/2}29^{1/2}\approx 116.85956436561077$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1169}a^{14}+\frac{22}{167}a^{13}-\frac{529}{1169}a^{12}+\frac{321}{1169}a^{11}+\frac{346}{1169}a^{10}-\frac{115}{1169}a^{9}+\frac{417}{1169}a^{8}-\frac{41}{167}a^{7}+\frac{417}{1169}a^{6}-\frac{115}{1169}a^{5}+\frac{346}{1169}a^{4}+\frac{321}{1169}a^{3}-\frac{529}{1169}a^{2}+\frac{22}{167}a+\frac{1}{1169}$, $\frac{1}{1169}a^{15}+\frac{304}{1169}a^{13}-\frac{43}{1169}a^{12}+\frac{10}{1169}a^{11}+\frac{375}{1169}a^{10}-\frac{577}{1169}a^{9}-\frac{30}{167}a^{8}+\frac{193}{1169}a^{7}-\frac{38}{1169}a^{6}+\frac{521}{1169}a^{5}-\frac{358}{1169}a^{4}+\frac{304}{1169}a^{3}-\frac{30}{167}a^{2}-\frac{335}{1169}a-\frac{22}{167}$
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: |
$\frac{44}{167}a^{15}-\frac{906}{1169}a^{14}-\frac{43}{167}a^{13}+\frac{768}{1169}a^{12}+\frac{2166}{1169}a^{11}-\frac{415}{1169}a^{10}-\frac{1048}{1169}a^{9}+\frac{2907}{1169}a^{8}+\frac{47}{167}a^{7}-\frac{229}{1169}a^{6}-\frac{1874}{1169}a^{5}-\frac{562}{1169}a^{4}+\frac{367}{1169}a^{3}-\frac{2739}{1169}a^{2}+\frac{64}{167}a-\frac{409}{1169}$, $\frac{214}{167}a^{15}+\frac{2281}{1169}a^{14}+\frac{824}{167}a^{13}+\frac{2216}{1169}a^{12}-\frac{11066}{1169}a^{11}-\frac{11000}{1169}a^{10}-\frac{2342}{1169}a^{9}-\frac{16636}{1169}a^{8}-\frac{4396}{167}a^{7}-\frac{27633}{1169}a^{6}-\frac{10546}{1169}a^{5}-\frac{10657}{1169}a^{4}-\frac{18948}{1169}a^{3}-\frac{7136}{1169}a^{2}+\frac{296}{167}a+\frac{2680}{1169}$, $\frac{2127}{1169}a^{15}+\frac{3611}{1169}a^{14}+\frac{46}{7}a^{13}+\frac{1375}{1169}a^{12}-\frac{15942}{1169}a^{11}-\frac{12312}{1169}a^{10}-\frac{63}{167}a^{9}-\frac{24322}{1169}a^{8}-\frac{39392}{1169}a^{7}-\frac{4469}{167}a^{6}-\frac{10746}{1169}a^{5}-\frac{15005}{1169}a^{4}-\frac{137}{7}a^{3}-\frac{5806}{1169}a^{2}+\frac{2612}{1169}a+\frac{2680}{1169}$, $\frac{828}{1169}a^{15}+\frac{1338}{1169}a^{14}+\frac{3438}{1169}a^{13}-\frac{23}{1169}a^{12}-\frac{948}{167}a^{11}-\frac{5367}{1169}a^{10}+\frac{1242}{1169}a^{9}-\frac{9320}{1169}a^{8}-\frac{14253}{1169}a^{7}-\frac{9118}{1169}a^{6}-\frac{1927}{1169}a^{5}-\frac{3985}{1169}a^{4}-\frac{988}{167}a^{3}-\frac{858}{1169}a^{2}+\frac{1804}{1169}a+\frac{1429}{1169}$, $\frac{507}{1169}a^{15}+\frac{914}{1169}a^{14}+\frac{1825}{1169}a^{13}+\frac{50}{1169}a^{12}-\frac{3926}{1169}a^{11}-\frac{520}{167}a^{10}+\frac{389}{1169}a^{9}-\frac{4541}{1169}a^{8}-\frac{5962}{1169}a^{7}-\frac{5239}{1169}a^{6}+\frac{21}{167}a^{5}+\frac{925}{1169}a^{4}-\frac{3352}{1169}a^{3}+\frac{551}{1169}a^{2}+\frac{816}{1169}a+\frac{1838}{1169}$, $\frac{1299}{1169}a^{15}+\frac{2273}{1169}a^{14}+\frac{4244}{1169}a^{13}+\frac{1398}{1169}a^{12}-\frac{9306}{1169}a^{11}-\frac{6945}{1169}a^{10}-\frac{1683}{1169}a^{9}-\frac{15002}{1169}a^{8}-\frac{25139}{1169}a^{7}-\frac{22165}{1169}a^{6}-\frac{8819}{1169}a^{5}-\frac{11020}{1169}a^{4}-\frac{15963}{1169}a^{3}-\frac{4948}{1169}a^{2}+\frac{808}{1169}a+\frac{2420}{1169}$, $\frac{49}{167}a^{15}+\frac{303}{1169}a^{14}+\frac{287}{167}a^{13}+\frac{587}{1169}a^{12}-\frac{3194}{1169}a^{11}-\frac{3914}{1169}a^{10}+\frac{575}{1169}a^{9}-\frac{2687}{1169}a^{8}-\frac{1339}{167}a^{7}-\frac{7909}{1169}a^{6}-\frac{4297}{1169}a^{5}-\frac{2661}{1169}a^{4}-\frac{5840}{1169}a^{3}-\frac{4089}{1169}a^{2}-\frac{132}{167}a+\frac{520}{1169}$, $\frac{325}{1169}a^{15}+\frac{282}{1169}a^{14}-\frac{2728}{1169}a^{13}-\frac{3000}{1169}a^{12}+\frac{370}{167}a^{11}+\frac{7858}{1169}a^{10}+\frac{3324}{1169}a^{9}+\frac{3753}{1169}a^{8}+\frac{16861}{1169}a^{7}+\frac{18738}{1169}a^{6}+\frac{9474}{1169}a^{5}+\frac{5771}{1169}a^{4}+\frac{1328}{167}a^{3}+\frac{8189}{1169}a^{2}-\frac{1152}{1169}a-\frac{3008}{1169}$, $\frac{633}{1169}a^{15}+\frac{624}{1169}a^{14}+\frac{3029}{1169}a^{13}+\frac{2232}{1169}a^{12}-\frac{4756}{1169}a^{11}-\frac{7443}{1169}a^{10}-\frac{2276}{1169}a^{9}-\frac{6660}{1169}a^{8}-\frac{17190}{1169}a^{7}-\frac{18509}{1169}a^{6}-\frac{7600}{1169}a^{5}-\frac{5209}{1169}a^{4}-\frac{9663}{1169}a^{3}-\frac{5450}{1169}a^{2}+\frac{704}{1169}a+\frac{2248}{1169}$
| 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: | \( 355.4268966475079 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 355.4268966475079 \cdot 1}{2\cdot\sqrt{887213286400000000}}\cr\approx \mathstrut & 0.185740011574205 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 3*x^14 + 8*x^12 + 4*x^11 - x^10 + 10*x^9 + 15*x^8 + 10*x^7 - x^6 + 4*x^5 + 8*x^4 - 3*x^2 - 2*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 - 2*x^15 - 3*x^14 + 8*x^12 + 4*x^11 - x^10 + 10*x^9 + 15*x^8 + 10*x^7 - x^6 + 4*x^5 + 8*x^4 - 3*x^2 - 2*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 - 2*x^15 - 3*x^14 + 8*x^12 + 4*x^11 - x^10 + 10*x^9 + 15*x^8 + 10*x^7 - x^6 + 4*x^5 + 8*x^4 - 3*x^2 - 2*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 - 2*x^15 - 3*x^14 + 8*x^12 + 4*x^11 - x^10 + 10*x^9 + 15*x^8 + 10*x^7 - x^6 + 4*x^5 + 8*x^4 - 3*x^2 - 2*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
$C_2^7.C_2\wr D_4$ (as 16T1772):
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 16384 |
| The 148 conjugacy class representatives for $C_2^7.C_2\wr D_4$ |
| Character table for $C_2^7.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.6.134560000.2 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.8 | $x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.8.8 | $x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.8.0.1 | $x^{8} + 4 x^{3} + 6 x^{2} + 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(29\)
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 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 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |