Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 61*x^12 - 2*x^11 - 107*x^10 + 172*x^9 - 129*x^8 + 34*x^7 + 21*x^6 - 20*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - 1)
gp: K = bnfinit(y^16 - 8*y^15 + 28*y^14 - 56*y^13 + 61*y^12 - 2*y^11 - 107*y^10 + 172*y^9 - 129*y^8 + 34*y^7 + 21*y^6 - 20*y^5 + 8*y^4 - 6*y^3 + 3*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 61*x^12 - 2*x^11 - 107*x^10 + 172*x^9 - 129*x^8 + 34*x^7 + 21*x^6 - 20*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 61*x^12 - 2*x^11 - 107*x^10 + 172*x^9 - 129*x^8 + 34*x^7 + 21*x^6 - 20*x^5 + 8*x^4 - 6*x^3 + 3*x^2 - 1)
\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 61 x^{12} - 2 x^{11} - 107 x^{10} + 172 x^{9} - 129 x^{8} + \cdots - 1 \)
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: | $[6, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-561298201600000000\)
\(\medspace = -\,2^{16}\cdot 5^{8}\cdot 29^{4}\cdot 31\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.86\) | 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: |
\(2\), \(5\), \(29\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}-\frac{6}{13}a^{11}+\frac{2}{13}a^{10}+\frac{6}{13}a^{9}+\frac{1}{13}a^{8}-\frac{2}{13}a^{7}-\frac{5}{13}a^{6}+\frac{4}{13}a^{5}-\frac{4}{13}a^{4}+\frac{6}{13}a^{3}-\frac{3}{13}a^{2}+\frac{5}{13}$, $\frac{1}{13}a^{13}+\frac{5}{13}a^{11}+\frac{5}{13}a^{10}-\frac{2}{13}a^{9}+\frac{4}{13}a^{8}-\frac{4}{13}a^{7}-\frac{6}{13}a^{5}-\frac{5}{13}a^{4}-\frac{6}{13}a^{3}-\frac{5}{13}a^{2}+\frac{5}{13}a+\frac{4}{13}$, $\frac{1}{13}a^{14}-\frac{4}{13}a^{11}+\frac{1}{13}a^{10}+\frac{4}{13}a^{8}-\frac{3}{13}a^{7}+\frac{6}{13}a^{6}+\frac{1}{13}a^{5}+\frac{1}{13}a^{4}+\frac{4}{13}a^{3}-\frac{6}{13}a^{2}+\frac{4}{13}a+\frac{1}{13}$, $\frac{1}{533}a^{15}+\frac{1}{41}a^{14}+\frac{14}{533}a^{13}-\frac{8}{533}a^{12}-\frac{230}{533}a^{11}+\frac{88}{533}a^{10}-\frac{22}{533}a^{9}-\frac{3}{533}a^{8}+\frac{218}{533}a^{7}-\frac{226}{533}a^{6}+\frac{31}{533}a^{5}+\frac{262}{533}a^{4}+\frac{16}{533}a^{3}-\frac{80}{533}a^{2}+\frac{45}{533}a+\frac{166}{533}$
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: | $10$ | 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: |
$\frac{10}{13}a^{14}-\frac{70}{13}a^{13}+\frac{210}{13}a^{12}-\frac{350}{13}a^{11}+\frac{262}{13}a^{10}+\frac{230}{13}a^{9}-\frac{810}{13}a^{8}+\frac{870}{13}a^{7}-\frac{405}{13}a^{6}-\frac{17}{13}a^{5}+\frac{105}{13}a^{4}-\frac{35}{13}a^{3}+\frac{11}{13}a^{2}-\frac{11}{13}a$, $\frac{242}{533}a^{15}-\frac{2061}{533}a^{14}+\frac{7529}{533}a^{13}-\frac{15425}{533}a^{12}+\frac{1326}{41}a^{11}-\frac{1377}{533}a^{10}-\frac{28981}{533}a^{9}+\frac{47080}{533}a^{8}-\frac{34738}{533}a^{7}+\frac{9760}{533}a^{6}+\frac{3443}{533}a^{5}-\frac{3426}{533}a^{4}+\frac{920}{533}a^{3}-\frac{1402}{533}a^{2}+\frac{640}{533}a+\frac{74}{533}$, $\frac{242}{533}a^{15}-\frac{2061}{533}a^{14}+\frac{7529}{533}a^{13}-\frac{15425}{533}a^{12}+\frac{1326}{41}a^{11}-\frac{1377}{533}a^{10}-\frac{28981}{533}a^{9}+\frac{47080}{533}a^{8}-\frac{34738}{533}a^{7}+\frac{9760}{533}a^{6}+\frac{3443}{533}a^{5}-\frac{3426}{533}a^{4}+\frac{920}{533}a^{3}-\frac{1402}{533}a^{2}+\frac{1173}{533}a-\frac{459}{533}$, $\frac{242}{533}a^{15}-\frac{1569}{533}a^{14}+\frac{4085}{533}a^{13}-\frac{5011}{533}a^{12}-\frac{474}{533}a^{11}+\frac{12850}{533}a^{10}-\frac{1526}{41}a^{9}+\frac{8909}{533}a^{8}+\frac{8968}{533}a^{7}-\frac{14307}{533}a^{6}+\frac{8158}{533}a^{5}-\frac{2852}{533}a^{4}+\frac{1822}{533}a^{3}-\frac{254}{533}a^{2}-\frac{58}{41}a+\frac{484}{533}$, $\frac{3}{13}a^{12}-\frac{18}{13}a^{11}+\frac{45}{13}a^{10}-\frac{60}{13}a^{9}+\frac{16}{13}a^{8}+\frac{98}{13}a^{7}-\frac{171}{13}a^{6}+\frac{116}{13}a^{5}+\frac{1}{13}a^{4}-\frac{60}{13}a^{3}+\frac{30}{13}a^{2}+\frac{2}{13}$, $\frac{316}{533}a^{15}-\frac{2329}{533}a^{14}+\frac{7745}{533}a^{13}-\frac{15402}{533}a^{12}+\frac{1354}{41}a^{11}-\frac{3762}{533}a^{10}-\frac{23680}{533}a^{9}+\frac{3431}{41}a^{8}-\frac{42181}{533}a^{7}+\frac{21490}{533}a^{6}-\frac{2504}{533}a^{5}-\frac{3349}{533}a^{4}+\frac{2596}{533}a^{3}-\frac{885}{533}a^{2}+\frac{280}{533}a-\frac{270}{533}$, $\frac{558}{533}a^{15}-\frac{4390}{533}a^{14}+\frac{14864}{533}a^{13}-\frac{27957}{533}a^{12}+\frac{26230}{533}a^{11}+\frac{9211}{533}a^{10}-\frac{63403}{533}a^{9}+\frac{82253}{533}a^{8}-\frac{43709}{533}a^{7}-\frac{340}{41}a^{6}+\frac{17544}{533}a^{5}-\frac{6078}{533}a^{4}-\frac{789}{533}a^{3}-\frac{852}{533}a^{2}+\frac{469}{533}a+\frac{255}{533}$, $\frac{171}{533}a^{15}-\frac{1098}{533}a^{14}+\frac{3214}{533}a^{13}-\frac{5919}{533}a^{12}+\frac{6426}{533}a^{11}-\frac{1270}{533}a^{10}-\frac{630}{41}a^{9}+\frac{16912}{533}a^{8}-\frac{19425}{533}a^{7}+\frac{12153}{533}a^{6}-\frac{890}{533}a^{5}-\frac{4644}{533}a^{4}+\frac{4130}{533}a^{3}-\frac{1257}{533}a^{2}+\frac{643}{533}a-\frac{21}{41}$, $a-1$, $a$
| 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: | \( 359.922310966 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{5}\cdot 359.922310966 \cdot 1}{2\cdot\sqrt{561298201600000000}}\cr\approx \mathstrut & 0.150543241625 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 61*x^12 - 2*x^11 - 107*x^10 + 172*x^9 - 129*x^8 + 34*x^7 + 21*x^6 - 20*x^5 + 8*x^4 - 6*x^3 + 3*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 - 8*x^15 + 28*x^14 - 56*x^13 + 61*x^12 - 2*x^11 - 107*x^10 + 172*x^9 - 129*x^8 + 34*x^7 + 21*x^6 - 20*x^5 + 8*x^4 - 6*x^3 + 3*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 - 8*x^15 + 28*x^14 - 56*x^13 + 61*x^12 - 2*x^11 - 107*x^10 + 172*x^9 - 129*x^8 + 34*x^7 + 21*x^6 - 20*x^5 + 8*x^4 - 6*x^3 + 3*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 - 8*x^15 + 28*x^14 - 56*x^13 + 61*x^12 - 2*x^11 - 107*x^10 + 172*x^9 - 129*x^8 + 34*x^7 + 21*x^6 - 20*x^5 + 8*x^4 - 6*x^3 + 3*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, 8.6.134560000.2 |
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: | 16.2.67969704100000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | R | R | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 $16$ | $2$ | $8$ | $16$ | |||
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |