Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 + 11*y^14 - 24*y^13 + 46*y^12 - 84*y^11 + 143*y^10 - 208*y^9 + 259*y^8 - 284*y^7 + 278*y^6 - 234*y^5 + 160*y^4 - 84*y^3 + 32*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*x + 1)
\( x^{16} - 4 x^{15} + 11 x^{14} - 24 x^{13} + 46 x^{12} - 84 x^{11} + 143 x^{10} - 208 x^{9} + 259 x^{8} + \cdots + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(5415791883780096\)
\(\medspace = 2^{16}\cdot 3^{10}\cdot 7^{2}\cdot 13^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.62\) | 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: | $2\cdot 3^{3/4}7^{1/2}13^{1/2}\approx 43.49022283071465$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{26911}a^{15}+\frac{13101}{26911}a^{14}-\frac{3564}{26911}a^{13}+\frac{11252}{26911}a^{12}+\frac{12137}{26911}a^{11}+\frac{11291}{26911}a^{10}+\frac{12020}{26911}a^{9}+\frac{11809}{26911}a^{8}-\frac{7957}{26911}a^{7}+\frac{3356}{26911}a^{6}+\frac{8084}{26911}a^{5}-\frac{8021}{26911}a^{4}-\frac{679}{26911}a^{3}+\frac{9162}{26911}a^{2}-\frac{520}{1583}a+\frac{3647}{26911}$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{47932}{26911} a^{15} - \frac{226341}{26911} a^{14} + \frac{593422}{26911} a^{13} - \frac{1311126}{26911} a^{12} + \frac{2464498}{26911} a^{11} - \frac{4474135}{26911} a^{10} + \frac{7647765}{26911} a^{9} - \frac{11104318}{26911} a^{8} + \frac{13362535}{26911} a^{7} - \frac{14303907}{26911} a^{6} + \frac{13634676}{26911} a^{5} - \frac{11153180}{26911} a^{4} + \frac{6932609}{26911} a^{3} - \frac{3102390}{26911} a^{2} + \frac{51934}{1583} a - \frac{140407}{26911} \)
(order $12$)
| 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: |
$\frac{190008}{26911}a^{15}-\frac{726200}{26911}a^{14}+\frac{1937484}{26911}a^{13}-\frac{4145584}{26911}a^{12}+\frac{7820052}{26911}a^{11}-\frac{14200511}{26911}a^{10}+\frac{23964202}{26911}a^{9}-\frac{34023301}{26911}a^{8}+\frac{41140154}{26911}a^{7}-\frac{43961871}{26911}a^{6}+\frac{41872130}{26911}a^{5}-\frac{33803721}{26911}a^{4}+\frac{21470880}{26911}a^{3}-\frac{9949453}{26911}a^{2}+\frac{180830}{1583}a-\frac{483472}{26911}$, $\frac{330451}{26911}a^{15}-\frac{1118103}{26911}a^{14}+\frac{2938939}{26911}a^{13}-\frac{6086282}{26911}a^{12}+\frac{11359344}{26911}a^{11}-\frac{20567180}{26911}a^{10}+\frac{34215123}{26911}a^{9}-\frac{47001035}{26911}a^{8}+\frac{55511863}{26911}a^{7}-\frac{58028870}{26911}a^{6}+\frac{54217312}{26911}a^{5}-\frac{41983508}{26911}a^{4}+\frac{25142563}{26911}a^{3}-\frac{10821304}{26911}a^{2}+\frac{182175}{1583}a-\frac{458003}{26911}$, $\frac{7255}{26911}a^{15}+\frac{51925}{26911}a^{14}-\frac{129904}{26911}a^{13}+\frac{362040}{26911}a^{12}-\frac{698543}{26911}a^{11}+\frac{1263938}{26911}a^{10}-\frac{2354708}{26911}a^{9}+\frac{3891766}{26911}a^{8}-\frac{4821009}{26911}a^{7}+\frac{5429347}{26911}a^{6}-\frac{5371849}{26911}a^{5}+\frac{4725563}{26911}a^{4}-\frac{3096197}{26911}a^{3}+\frac{1426423}{26911}a^{2}-\frac{24056}{1583}a+\frac{32383}{26911}$, $\frac{228015}{26911}a^{15}-\frac{811459}{26911}a^{14}+\frac{2138887}{26911}a^{13}-\frac{4510275}{26911}a^{12}+\frac{8448513}{26911}a^{11}-\frac{15343453}{26911}a^{10}+\frac{25689510}{26911}a^{9}-\frac{35823333}{26911}a^{8}+\frac{42866577}{26911}a^{7}-\frac{45475535}{26911}a^{6}+\frac{42927360}{26911}a^{5}-\frac{34079170}{26911}a^{4}+\frac{21202666}{26911}a^{3}-\frac{9715460}{26911}a^{2}+\frac{176196}{1583}a-\frac{490504}{26911}$, $\frac{195184}{26911}a^{15}-\frac{596589}{26911}a^{14}+\frac{1547501}{26911}a^{13}-\frac{3144929}{26911}a^{12}+\frac{5812565}{26911}a^{11}-\frac{10549091}{26911}a^{10}+\frac{17314473}{26911}a^{9}-\frac{23088932}{26911}a^{8}+\frac{26973366}{26911}a^{7}-\frac{27990587}{26911}a^{6}+\frac{25721709}{26911}a^{5}-\frac{19372448}{26911}a^{4}+\frac{11309359}{26911}a^{3}-\frac{4831033}{26911}a^{2}+\frac{80681}{1583}a-\frac{202101}{26911}$, $\frac{76625}{26911}a^{15}-\frac{158374}{26911}a^{14}+\frac{378082}{26911}a^{13}-\frac{635982}{26911}a^{12}+\frac{1083727}{26911}a^{11}-\frac{1926456}{26911}a^{10}+\frac{2802269}{26911}a^{9}-\frac{2520473}{26911}a^{8}+\frac{1928083}{26911}a^{7}-\frac{1003723}{26911}a^{6}-\frac{27809}{26911}a^{5}+\frac{1598953}{26911}a^{4}-\frac{2269936}{26911}a^{3}+\frac{1787119}{26911}a^{2}-\frac{46797}{1583}a+\frac{195928}{26911}$, $\frac{169657}{26911}a^{15}-\frac{602819}{26911}a^{14}+\frac{1593460}{26911}a^{13}-\frac{3342007}{26911}a^{12}+\frac{6275196}{26911}a^{11}-\frac{11364968}{26911}a^{10}+\frac{19041459}{26911}a^{9}-\frac{26524861}{26911}a^{8}+\frac{31649991}{26911}a^{7}-\frac{33464419}{26911}a^{6}+\frac{31608498}{26911}a^{5}-\frac{24929846}{26911}a^{4}+\frac{15455802}{26911}a^{3}-\frac{6951875}{26911}a^{2}+\frac{125590}{1583}a-\frac{348476}{26911}$
| 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: | \( 63.1744988972 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 63.1744988972 \cdot 1}{12\cdot\sqrt{5415791883780096}}\cr\approx \mathstrut & 0.173767488034 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*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 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*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 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*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 - 4*x^15 + 11*x^14 - 24*x^13 + 46*x^12 - 84*x^11 + 143*x^10 - 208*x^9 + 259*x^8 - 284*x^7 + 278*x^6 - 234*x^5 + 160*x^4 - 84*x^3 + 32*x^2 - 8*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
$D_4^2:C_2^2$ (as 16T595):
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 256 |
| The 40 conjugacy class representatives for $D_4^2:C_2^2$ |
| Character table for $D_4^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), 4.0.1872.1, 4.0.117.1, \(\Q(\zeta_{12})\), 8.0.3504384.2 |
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: | 16.0.9329547737293056.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 $16$ | $2$ | $8$ | $16$ | |||
|
\(3\)
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(13\)
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |