Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 7*x^13 + x^12 + 11*x^11 - 24*x^10 + 35*x^9 - 12*x^8 - 10*x^7 + 14*x^6 - 14*x^5 + 32*x^4 - 36*x^3 + 22*x^2 - 7*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + 5*y^14 - 7*y^13 + y^12 + 11*y^11 - 24*y^10 + 35*y^9 - 12*y^8 - 10*y^7 + 14*y^6 - 14*y^5 + 32*y^4 - 36*y^3 + 22*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 5*x^14 - 7*x^13 + x^12 + 11*x^11 - 24*x^10 + 35*x^9 - 12*x^8 - 10*x^7 + 14*x^6 - 14*x^5 + 32*x^4 - 36*x^3 + 22*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 5*x^14 - 7*x^13 + x^12 + 11*x^11 - 24*x^10 + 35*x^9 - 12*x^8 - 10*x^7 + 14*x^6 - 14*x^5 + 32*x^4 - 36*x^3 + 22*x^2 - 7*x + 1)
\( x^{16} - 3 x^{15} + 5 x^{14} - 7 x^{13} + x^{12} + 11 x^{11} - 24 x^{10} + 35 x^{9} - 12 x^{8} + \cdots + 1 \)
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: |
\(169561224228515625\)
\(\medspace = 3^{4}\cdot 5^{10}\cdot 11^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.94\) | 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: | $3^{1/2}5^{3/4}11^{1/2}\approx 19.208102881010017$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9}a^{14}+\frac{2}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{4}{9}a^{10}-\frac{1}{3}a^{9}-\frac{4}{9}a^{8}-\frac{2}{9}a^{6}-\frac{2}{9}a^{5}-\frac{4}{9}a^{2}-\frac{2}{9}a+\frac{4}{9}$, $\frac{1}{73554651}a^{15}-\frac{97703}{73554651}a^{14}+\frac{10838303}{24518217}a^{13}-\frac{3823702}{73554651}a^{12}+\frac{5377063}{24518217}a^{11}+\frac{13823471}{73554651}a^{10}+\frac{32866721}{73554651}a^{9}+\frac{23202391}{73554651}a^{8}+\frac{31706674}{73554651}a^{7}+\frac{1341895}{8172739}a^{6}+\frac{10919639}{73554651}a^{5}-\frac{4065603}{8172739}a^{4}-\frac{18429970}{73554651}a^{3}-\frac{9787516}{73554651}a^{2}-\frac{6395404}{24518217}a+\frac{1668092}{73554651}$
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: |
\( -1 \)
(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: |
$\frac{72249637}{73554651}a^{15}-\frac{81433489}{24518217}a^{14}+\frac{416743318}{73554651}a^{13}-\frac{189221566}{24518217}a^{12}+\frac{150643985}{73554651}a^{11}+\frac{917894437}{73554651}a^{10}-\frac{2010186730}{73554651}a^{9}+\frac{2865791663}{73554651}a^{8}-\frac{1249855010}{73554651}a^{7}-\frac{1159685419}{73554651}a^{6}+\frac{1340498254}{73554651}a^{5}-\frac{106016314}{8172739}a^{4}+\frac{2381506526}{73554651}a^{3}-\frac{357947498}{8172739}a^{2}+\frac{1784023283}{73554651}a-\frac{409462937}{73554651}$, $\frac{66197597}{73554651}a^{15}-\frac{44419130}{24518217}a^{14}+\frac{167754554}{73554651}a^{13}-\frac{73054490}{24518217}a^{12}-\frac{263481620}{73554651}a^{11}+\frac{617982059}{73554651}a^{10}-\frac{912660281}{73554651}a^{9}+\frac{1072392652}{73554651}a^{8}+\frac{859435568}{73554651}a^{7}-\frac{583992356}{73554651}a^{6}+\frac{269393603}{73554651}a^{5}-\frac{31769981}{8172739}a^{4}+\frac{1553275906}{73554651}a^{3}-\frac{66075444}{8172739}a^{2}-\frac{66360683}{73554651}a+\frac{211661819}{73554651}$, $\frac{66197597}{73554651}a^{15}-\frac{44419130}{24518217}a^{14}+\frac{167754554}{73554651}a^{13}-\frac{73054490}{24518217}a^{12}-\frac{263481620}{73554651}a^{11}+\frac{617982059}{73554651}a^{10}-\frac{912660281}{73554651}a^{9}+\frac{1072392652}{73554651}a^{8}+\frac{859435568}{73554651}a^{7}-\frac{583992356}{73554651}a^{6}+\frac{269393603}{73554651}a^{5}-\frac{31769981}{8172739}a^{4}+\frac{1553275906}{73554651}a^{3}-\frac{66075444}{8172739}a^{2}-\frac{66360683}{73554651}a+\frac{138107168}{73554651}$, $\frac{1578176}{73554651}a^{15}-\frac{30153971}{73554651}a^{14}+\frac{80839423}{73554651}a^{13}-\frac{116541424}{73554651}a^{12}+\frac{141119519}{73554651}a^{11}+\frac{14291299}{24518217}a^{10}-\frac{362430275}{73554651}a^{9}+\frac{65167067}{8172739}a^{8}-\frac{737792629}{73554651}a^{7}+\frac{32866085}{73554651}a^{6}+\frac{58249301}{8172739}a^{5}-\frac{24009703}{8172739}a^{4}+\frac{347083465}{73554651}a^{3}-\frac{796105672}{73554651}a^{2}+\frac{672440081}{73554651}a-\frac{17555484}{8172739}$, $\frac{18357107}{24518217}a^{15}-\frac{216800620}{73554651}a^{14}+\frac{375573349}{73554651}a^{13}-\frac{502053338}{73554651}a^{12}+\frac{215946536}{73554651}a^{11}+\frac{824646218}{73554651}a^{10}-\frac{599846119}{24518217}a^{9}+\frac{2558894737}{73554651}a^{8}-\frac{491102188}{24518217}a^{7}-\frac{1281857599}{73554651}a^{6}+\frac{1191960371}{73554651}a^{5}-\frac{83112700}{8172739}a^{4}+\frac{747224206}{24518217}a^{3}-\frac{3116355416}{73554651}a^{2}+\frac{1557405746}{73554651}a-\frac{343274467}{73554651}$, $\frac{9431947}{8172739}a^{15}-\frac{28677274}{8172739}a^{14}+\frac{46618294}{8172739}a^{13}-\frac{63111902}{8172739}a^{12}+\frac{4900451}{8172739}a^{11}+\frac{113257554}{8172739}a^{10}-\frac{230087431}{8172739}a^{9}+\frac{320352178}{8172739}a^{8}-\frac{93453957}{8172739}a^{7}-\frac{136831241}{8172739}a^{6}+\frac{141755749}{8172739}a^{5}-\frac{112524202}{8172739}a^{4}+\frac{299282681}{8172739}a^{3}-\frac{338599410}{8172739}a^{2}+\frac{173745323}{8172739}a-\frac{33550210}{8172739}$, $\frac{15443447}{24518217}a^{15}-\frac{44546278}{24518217}a^{14}+\frac{22971533}{8172739}a^{13}-\frac{92337842}{24518217}a^{12}-\frac{2100623}{8172739}a^{11}+\frac{185532130}{24518217}a^{10}-\frac{345769892}{24518217}a^{9}+\frac{471149237}{24518217}a^{8}-\frac{82524652}{24518217}a^{7}-\frac{79346623}{8172739}a^{6}+\frac{177338161}{24518217}a^{5}-\frac{53449724}{8172739}a^{4}+\frac{472878601}{24518217}a^{3}-\frac{456996833}{24518217}a^{2}+\frac{75805940}{8172739}a-\frac{53099474}{24518217}$
| 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: | \( 73.2440423866 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 73.2440423866 \cdot 1}{2\cdot\sqrt{169561224228515625}}\cr\approx \mathstrut & 0.216031949475 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 7*x^13 + x^12 + 11*x^11 - 24*x^10 + 35*x^9 - 12*x^8 - 10*x^7 + 14*x^6 - 14*x^5 + 32*x^4 - 36*x^3 + 22*x^2 - 7*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 - 3*x^15 + 5*x^14 - 7*x^13 + x^12 + 11*x^11 - 24*x^10 + 35*x^9 - 12*x^8 - 10*x^7 + 14*x^6 - 14*x^5 + 32*x^4 - 36*x^3 + 22*x^2 - 7*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 - 3*x^15 + 5*x^14 - 7*x^13 + x^12 + 11*x^11 - 24*x^10 + 35*x^9 - 12*x^8 - 10*x^7 + 14*x^6 - 14*x^5 + 32*x^4 - 36*x^3 + 22*x^2 - 7*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 - 3*x^15 + 5*x^14 - 7*x^13 + x^12 + 11*x^11 - 24*x^10 + 35*x^9 - 12*x^8 - 10*x^7 + 14*x^6 - 14*x^5 + 32*x^4 - 36*x^3 + 22*x^2 - 7*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
$C_4\wr C_2$ (as 16T42):
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 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-55}) \), 4.2.275.1 x2, 4.0.605.1 x2, \(\Q(\sqrt{5}, \sqrt{-11})\), 8.0.411778125.1, 8.0.16471125.1, 8.0.9150625.1 |
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
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.0.16471125.1, 8.0.411778125.1 |
| Degree 16 sibling: | 16.4.2837698174072265625.1 |
| Minimal sibling: | 8.0.16471125.1 |
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 | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(5\)
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(11\)
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |