Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*x^2 + 1)
gp: K = bnfinit(y^16 - 5*y^15 + 11*y^14 - 10*y^13 - 10*y^12 + 45*y^11 - 56*y^10 - 10*y^9 + 159*y^8 - 315*y^7 + 374*y^6 - 300*y^5 + 160*y^4 - 50*y^3 + 6*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*x^2 + 1)
\( x^{16} - 5 x^{15} + 11 x^{14} - 10 x^{13} - 10 x^{12} + 45 x^{11} - 56 x^{10} - 10 x^{9} + 159 x^{8} + \cdots + 1 \)
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: |
\(5960322509765625\)
\(\medspace = 3^{8}\cdot 5^{12}\cdot 61^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.68\) | 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: | $3^{1/2}5^{3/4}61^{1/2}\approx 45.23275582376457$ | ||
| Ramified primes: |
\(3\), \(5\), \(61\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{673721}a^{15}-\frac{249772}{673721}a^{14}+\frac{259698}{673721}a^{13}-\frac{153659}{673721}a^{12}-\frac{243043}{673721}a^{11}-\frac{162237}{673721}a^{10}-\frac{174543}{673721}a^{9}-\frac{56997}{673721}a^{8}+\frac{245128}{673721}a^{7}+\frac{184105}{673721}a^{6}+\frac{126252}{673721}a^{5}-\frac{72179}{673721}a^{4}-\frac{167786}{673721}a^{3}-\frac{61551}{673721}a^{2}-\frac{230876}{673721}a+\frac{78060}{673721}$
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: |
\( -\frac{333300}{673721} a^{15} + \frac{1345956}{673721} a^{14} - \frac{2385367}{673721} a^{13} + \frac{969164}{673721} a^{12} + \frac{4756070}{673721} a^{11} - \frac{11382338}{673721} a^{10} + \frac{8131923}{673721} a^{9} + \frac{12989762}{673721} a^{8} - \frac{44829758}{673721} a^{7} + \frac{64989396}{673721} a^{6} - \frac{57117946}{673721} a^{5} + \frac{29001235}{673721} a^{4} - \frac{3853852}{673721} a^{3} - \frac{4572197}{673721} a^{2} + \frac{1926785}{673721} a + \frac{359578}{673721} \)
(order $30$)
| 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: |
$\frac{218781}{673721}a^{15}-\frac{1205064}{673721}a^{14}+\frac{2769929}{673721}a^{13}-\frac{2360384}{673721}a^{12}-\frac{3129263}{673721}a^{11}+\frac{12071045}{673721}a^{10}-\frac{13660223}{673721}a^{9}-\frac{5348436}{673721}a^{8}+\frac{43601791}{673721}a^{7}-\frac{77211895}{673721}a^{6}+\frac{81171774}{673721}a^{5}-\frac{54618681}{673721}a^{4}+\frac{20858891}{673721}a^{3}-\frac{1875146}{673721}a^{2}-\frac{397623}{673721}a-\frac{108769}{673721}$, $\frac{482684}{673721}a^{15}-\frac{1943703}{673721}a^{14}+\frac{3582498}{673721}a^{13}-\frac{2164471}{673721}a^{12}-\frac{5340613}{673721}a^{11}+\frac{15578189}{673721}a^{10}-\frac{14450503}{673721}a^{9}-\frac{11596170}{673721}a^{8}+\frac{59095259}{673721}a^{7}-\frac{100993951}{673721}a^{6}+\frac{109551278}{673721}a^{5}-\frac{81034604}{673721}a^{4}+\frac{39459604}{673721}a^{3}-\frac{10713762}{673721}a^{2}+\frac{713147}{673721}a+\frac{466115}{673721}$, $\frac{230251}{673721}a^{15}-\frac{754491}{673721}a^{14}+\frac{964285}{673721}a^{13}+\frac{319906}{673721}a^{12}-\frac{2974975}{673721}a^{11}+\frac{4619126}{673721}a^{10}-\frac{568922}{673721}a^{9}-\frac{9636982}{673721}a^{8}+\frac{20201983}{673721}a^{7}-\frac{23071479}{673721}a^{6}+\frac{18126011}{673721}a^{5}-\frac{10043116}{673721}a^{4}+\frac{5678785}{673721}a^{3}-\frac{3826671}{673721}a^{2}+\frac{2546792}{673721}a-\frac{135778}{673721}$, $\frac{454175}{673721}a^{15}-\frac{2424725}{673721}a^{14}+\frac{5393448}{673721}a^{13}-\frac{4728866}{673721}a^{12}-\frac{5648211}{673721}a^{11}+\frac{23109088}{673721}a^{10}-\frac{27308121}{673721}a^{9}-\frac{8315144}{673721}a^{8}+\frac{82829275}{673721}a^{7}-\frac{153907044}{673721}a^{6}+\frac{169885482}{673721}a^{5}-\frac{121924408}{673721}a^{4}+\frac{52252277}{673721}a^{3}-\frac{8978345}{673721}a^{2}-\frac{1518302}{673721}a-\frac{319683}{673721}$, $\frac{429524}{673721}a^{15}-\frac{1757651}{673721}a^{14}+\frac{3253829}{673721}a^{13}-\frac{1845435}{673721}a^{12}-\frac{5122350}{673721}a^{11}+\frac{14447146}{673721}a^{10}-\frac{12882793}{673721}a^{9}-\frac{12032708}{673721}a^{8}+\frac{55160035}{673721}a^{7}-\frac{90764969}{673721}a^{6}+\frac{94107977}{673721}a^{5}-\frac{64670755}{673721}a^{4}+\frac{28117788}{673721}a^{3}-\frac{6209452}{673721}a^{2}+\frac{232129}{673721}a-\frac{429567}{673721}$, $\frac{107622}{673721}a^{15}-\frac{168005}{673721}a^{14}-\frac{97529}{673721}a^{13}+\frac{740489}{673721}a^{12}-\frac{903363}{673721}a^{11}-\frac{116978}{673721}a^{10}+\frac{2717060}{673721}a^{9}-\frac{3943755}{673721}a^{8}+\frac{1619861}{673721}a^{7}+\frac{2982305}{673721}a^{6}-\frac{5502152}{673721}a^{5}+\frac{4670839}{673721}a^{4}-\frac{1068371}{673721}a^{3}-\frac{216850}{673721}a^{2}-\frac{506392}{673721}a+\frac{346171}{673721}$, $\frac{750447}{673721}a^{15}-\frac{3381232}{673721}a^{14}+\frac{6353662}{673721}a^{13}-\frac{3565260}{673721}a^{12}-\frac{10573195}{673721}a^{11}+\frac{28843137}{673721}a^{10}-\frac{24687857}{673721}a^{9}-\frac{26303930}{673721}a^{8}+\frac{109912015}{673721}a^{7}-\frac{172740553}{673721}a^{6}+\frac{168480664}{673721}a^{5}-\frac{106467252}{673721}a^{4}+\frac{39661771}{673721}a^{3}-\frac{7188747}{673721}a^{2}+\frac{1301719}{673721}a-\frac{821851}{673721}$
| 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: | \( 161.821613678 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 161.821613678 \cdot 1}{30\cdot\sqrt{5960322509765625}}\cr\approx \mathstrut & 0.169714652749 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*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 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*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 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*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 - 5*x^15 + 11*x^14 - 10*x^13 - 10*x^12 + 45*x^11 - 56*x^10 - 10*x^9 + 159*x^8 - 315*x^7 + 374*x^6 - 300*x^5 + 160*x^4 - 50*x^3 + 6*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
$C_2^5:C_4$ (as 16T261):
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 128 |
| The 26 conjugacy class representatives for $C_2^5:C_4$ |
| Character table for $C_2^5:C_4$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.4.77203125.1, 8.4.77203125.2, \(\Q(\zeta_{15})\) |
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: | 16.0.273806914306640625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |