Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 + 8*y^14 - 8*y^13 - 4*y^12 + 30*y^11 - 56*y^10 + 58*y^9 - 18*y^8 - 48*y^7 + 98*y^6 - 102*y^5 + 73*y^4 - 40*y^3 + 18*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*x + 1)
\( x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} - 4 x^{12} + 30 x^{11} - 56 x^{10} + 58 x^{9} - 18 x^{8} + \cdots + 1 \)
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: |
\(3143861048180736\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 13^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.30\) | 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: | $2^{7/4}3^{1/2}13^{1/2}\approx 21.005585720479342$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23711}a^{15}+\frac{7757}{23711}a^{14}-\frac{144}{23711}a^{13}-\frac{3175}{23711}a^{12}-\frac{5450}{23711}a^{11}+\frac{3004}{23711}a^{10}+\frac{6075}{23711}a^{9}+\frac{10665}{23711}a^{8}-\frac{4054}{23711}a^{7}+\frac{1355}{23711}a^{6}-\frac{11431}{23711}a^{5}+\frac{10469}{23711}a^{4}-\frac{7615}{23711}a^{3}+\frac{11468}{23711}a^{2}-\frac{7928}{23711}a+\frac{831}{23711}$
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: |
\( -\frac{101867}{23711} a^{15} + \frac{366132}{23711} a^{14} - \frac{648458}{23711} a^{13} + \frac{507616}{23711} a^{12} + \frac{669704}{23711} a^{11} - \frac{2792200}{23711} a^{10} + \frac{4425321}{23711} a^{9} - \frac{3791006}{23711} a^{8} - \frac{53091}{23711} a^{7} + \frac{5018678}{23711} a^{6} - \frac{7664186}{23711} a^{5} + \frac{6714237}{23711} a^{4} - \frac{4161296}{23711} a^{3} + \frac{2045649}{23711} a^{2} - \frac{801258}{23711} a + \frac{186481}{23711} \)
(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: |
$\frac{48145}{23711}a^{15}-\frac{177173}{23711}a^{14}+\frac{322686}{23711}a^{13}-\frac{256379}{23711}a^{12}-\frac{312567}{23711}a^{11}+\frac{1365718}{23711}a^{10}-\frac{2199433}{23711}a^{9}+\frac{1901600}{23711}a^{8}-\frac{14589}{23711}a^{7}-\frac{2458430}{23711}a^{6}+\frac{3827997}{23711}a^{5}-\frac{3314262}{23711}a^{4}+\frac{2081875}{23711}a^{3}-\frac{1027059}{23711}a^{2}+\frac{432916}{23711}a-\frac{86806}{23711}$, $\frac{7262}{23711}a^{15}+\frac{17709}{23711}a^{14}-\frac{73577}{23711}a^{13}+\frac{132508}{23711}a^{12}-\frac{99085}{23711}a^{11}-\frac{165049}{23711}a^{10}+\frac{583254}{23711}a^{9}-\frac{797070}{23711}a^{8}+\frac{554267}{23711}a^{7}+\frac{331899}{23711}a^{6}-\frac{1114128}{23711}a^{5}+\frac{1265095}{23711}a^{4}-\frac{788541}{23711}a^{3}+\frac{410671}{23711}a^{2}-\frac{168805}{23711}a+\frac{59550}{23711}$, $\frac{45487}{23711}a^{15}-\frac{142998}{23711}a^{14}+\frac{231218}{23711}a^{13}-\frac{139790}{23711}a^{12}-\frac{337599}{23711}a^{11}+\frac{1087161}{23711}a^{10}-\frac{1535684}{23711}a^{9}+\frac{1129923}{23711}a^{8}+\frac{375525}{23711}a^{7}-\frac{1981728}{23711}a^{6}+\frac{2604832}{23711}a^{5}-\frac{2047467}{23711}a^{4}+\frac{1196044}{23711}a^{3}-\frac{566148}{23711}a^{2}+\frac{236773}{23711}a-\frac{43059}{23711}$, $\frac{89192}{23711}a^{15}-\frac{332879}{23711}a^{14}+\frac{600489}{23711}a^{13}-\frac{478347}{23711}a^{12}-\frac{589964}{23711}a^{11}+\frac{2559256}{23711}a^{10}-\frac{4104375}{23711}a^{9}+\frac{3527721}{23711}a^{8}+\frac{8382}{23711}a^{7}-\frac{4647163}{23711}a^{6}+\frac{7132548}{23711}a^{5}-\frac{6176703}{23711}a^{4}+\frac{3798275}{23711}a^{3}-\frac{1864431}{23711}a^{2}+\frac{730307}{23711}a-\frac{144300}{23711}$, $\frac{468}{131}a^{15}-\frac{1568}{131}a^{14}+\frac{2824}{131}a^{13}-\frac{2194}{131}a^{12}-\frac{2912}{131}a^{11}+\frac{12032}{131}a^{10}-\frac{19250}{131}a^{9}+\frac{16757}{131}a^{8}+\frac{1}{131}a^{7}-\frac{21253}{131}a^{6}+\frac{33737}{131}a^{5}-\frac{29776}{131}a^{4}+\frac{18768}{131}a^{3}-\frac{8954}{131}a^{2}+\frac{3808}{131}a-\frac{948}{131}$, $\frac{67493}{23711}a^{15}-\frac{232789}{23711}a^{14}+\frac{405605}{23711}a^{13}-\frac{298500}{23711}a^{12}-\frac{458616}{23711}a^{11}+\frac{1774536}{23711}a^{10}-\frac{2741113}{23711}a^{9}+\frac{2270563}{23711}a^{8}+\frac{174295}{23711}a^{7}-\frac{3177586}{23711}a^{6}+\frac{4714524}{23711}a^{5}-\frac{4034453}{23711}a^{4}+\frac{2513807}{23711}a^{3}-\frac{1268843}{23711}a^{2}+\frac{499564}{23711}a-\frac{108387}{23711}$, $\frac{920}{23711}a^{15}-\frac{571}{23711}a^{14}-\frac{13925}{23711}a^{13}+\frac{42875}{23711}a^{12}-\frac{58401}{23711}a^{11}+\frac{13204}{23711}a^{10}+\frac{135470}{23711}a^{9}-\frac{312797}{23711}a^{8}+\frac{348612}{23711}a^{7}-\frac{128638}{23711}a^{6}-\frac{297079}{23711}a^{5}+\frac{597589}{23711}a^{4}-\frac{532697}{23711}a^{3}+\frac{283697}{23711}a^{2}-\frac{109327}{23711}a+\frac{53190}{23711}$
| 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: | \( 44.6663869612 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 44.6663869612 \cdot 1}{12\cdot\sqrt{3143861048180736}}\cr\approx \mathstrut & 0.161252647372 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*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 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*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 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*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 + 8*x^14 - 8*x^13 - 4*x^12 + 30*x^11 - 56*x^10 + 58*x^9 - 18*x^8 - 48*x^7 + 98*x^6 - 102*x^5 + 73*x^4 - 40*x^3 + 18*x^2 - 6*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
$C_2\wr C_2^2$ (as 16T149):
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 64 |
| The 16 conjugacy class representatives for $C_2\wr C_2^2$ |
| Character table for $C_2\wr C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.117.1, \(\Q(\zeta_{12})\), 4.0.1872.1, 8.0.3504384.1 x2, 8.0.3504384.2, 8.0.4313088.1 x2 |
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 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.0.3504384.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 $16$ | $4$ | $4$ | $24$ | |||
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |