Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 + 8*y^14 - 10*y^13 + 11*y^12 - 12*y^11 + 10*y^10 - 4*y^9 + y^8 - 4*y^7 + 10*y^6 - 12*y^5 + 11*y^4 - 10*y^3 + 8*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 8*x^14 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*x + 1)
\( x^{16} - 4 x^{15} + 8 x^{14} - 10 x^{13} + 11 x^{12} - 12 x^{11} + 10 x^{10} - 4 x^{9} + x^{8} + \cdots + 1 \)
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: |
\(6298009600000000\)
\(\medspace = 2^{24}\cdot 5^{8}\cdot 31^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.72\) | 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: | $2^{3/2}5^{1/2}31^{1/2}\approx 35.21363372331802$ | ||
| Ramified primes: |
\(2\), \(5\), \(31\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}+\frac{4}{13}a^{13}-\frac{1}{13}a^{11}+\frac{3}{13}a^{10}-\frac{6}{13}a^{8}-\frac{6}{13}a^{6}+\frac{3}{13}a^{4}-\frac{1}{13}a^{3}+\frac{4}{13}a+\frac{1}{13}$, $\frac{1}{143}a^{15}-\frac{5}{143}a^{14}+\frac{68}{143}a^{13}-\frac{1}{143}a^{12}+\frac{12}{143}a^{11}+\frac{64}{143}a^{10}-\frac{32}{143}a^{9}+\frac{28}{143}a^{8}-\frac{71}{143}a^{7}+\frac{67}{143}a^{6}+\frac{42}{143}a^{5}-\frac{54}{143}a^{4}-\frac{56}{143}a^{3}-\frac{9}{143}a^{2}+\frac{17}{143}a+\frac{56}{143}$
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: |
\( \frac{21}{143} a^{15} - \frac{25}{11} a^{14} + \frac{834}{143} a^{13} - \frac{1165}{143} a^{12} + \frac{1044}{143} a^{11} - \frac{1318}{143} a^{10} + \frac{1330}{143} a^{9} - \frac{523}{143} a^{8} - \frac{204}{143} a^{7} - \frac{419}{143} a^{6} + \frac{1025}{143} a^{5} - \frac{116}{11} a^{4} + \frac{1046}{143} a^{3} - \frac{1190}{143} a^{2} + \frac{1050}{143} a - \frac{474}{143} \)
(order $4$)
| 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: |
$a$, $\frac{112}{143}a^{15}-\frac{16}{11}a^{14}+\frac{158}{143}a^{13}+\frac{31}{143}a^{12}+\frac{134}{143}a^{11}-\frac{70}{143}a^{10}-\frac{295}{143}a^{9}+\frac{309}{143}a^{8}+\frac{199}{143}a^{7}-\frac{328}{143}a^{6}+\frac{271}{143}a^{5}+\frac{12}{11}a^{4}+\frac{97}{143}a^{3}-\frac{150}{143}a^{2}-\frac{263}{143}a+\frac{189}{143}$, $\frac{54}{143}a^{15}-\frac{347}{143}a^{14}+\frac{790}{143}a^{13}-\frac{1055}{143}a^{12}+\frac{1011}{143}a^{11}-\frac{1208}{143}a^{10}+\frac{1132}{143}a^{9}-\frac{457}{143}a^{8}-\frac{116}{143}a^{7}-\frac{353}{143}a^{6}+\frac{981}{143}a^{5}-\frac{1288}{143}a^{4}+\frac{1057}{143}a^{3}-\frac{1058}{143}a^{2}+\frac{896}{143}a-\frac{485}{143}$, $\frac{19}{143}a^{15}-\frac{51}{143}a^{14}+\frac{38}{143}a^{13}-\frac{19}{143}a^{12}+\frac{41}{143}a^{11}-\frac{82}{143}a^{10}-\frac{36}{143}a^{9}-\frac{18}{143}a^{8}+\frac{81}{143}a^{7}-\frac{135}{143}a^{6}+\frac{83}{143}a^{5}-\frac{36}{143}a^{4}-\frac{107}{143}a^{3}-\frac{171}{143}a^{2}-\frac{73}{143}a-\frac{36}{143}$, $\frac{304}{143}a^{15}-\frac{1179}{143}a^{14}+\frac{2159}{143}a^{13}-\frac{2449}{143}a^{12}+\frac{2592}{143}a^{11}-\frac{2973}{143}a^{10}+\frac{2141}{143}a^{9}-\frac{398}{143}a^{8}+\frac{9}{143}a^{7}-\frac{1269}{143}a^{6}+\frac{2615}{143}a^{5}-\frac{2809}{143}a^{4}+\frac{2369}{143}a^{3}-\frac{2450}{143}a^{2}+\frac{1670}{143}a-\frac{510}{143}$, $\frac{710}{143}a^{15}-\frac{2615}{143}a^{14}+\frac{4687}{143}a^{13}-\frac{5143}{143}a^{12}+\frac{5440}{143}a^{11}-\frac{6095}{143}a^{10}+\frac{4307}{143}a^{9}-\frac{602}{143}a^{8}+\frac{69}{143}a^{7}-\frac{2942}{143}a^{6}+\frac{6082}{143}a^{5}-\frac{6077}{143}a^{4}+\frac{4922}{143}a^{3}-\frac{4817}{143}a^{2}+\frac{3369}{143}a-\frac{1061}{143}$, $\frac{25}{11}a^{15}-\frac{1020}{143}a^{14}+\frac{1640}{143}a^{13}-\frac{124}{11}a^{12}+\frac{1865}{143}a^{11}-\frac{1981}{143}a^{10}+\frac{80}{11}a^{9}+\frac{179}{143}a^{8}+\frac{18}{11}a^{7}-\frac{1160}{143}a^{6}+\frac{159}{11}a^{5}-\frac{1864}{143}a^{4}+\frac{1787}{143}a^{3}-\frac{115}{11}a^{2}+\frac{795}{143}a-\frac{71}{143}$
| 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: | \( 22.3523537457 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22.3523537457 \cdot 1}{4\cdot\sqrt{6298009600000000}}\cr\approx \mathstrut & 0.171041130482 \end{aligned}\]
# 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 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*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 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*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 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*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 - 10*x^13 + 11*x^12 - 12*x^11 + 10*x^10 - 4*x^9 + x^8 - 4*x^7 + 10*x^6 - 12*x^5 + 11*x^4 - 10*x^3 + 8*x^2 - 4*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
$C_2\wr D_4$ (as 16T396):
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 20 conjugacy class representatives for $C_2\wr D_4$ |
| Character table for $C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.0.320.1 x2, 4.2.400.1 x2, \(\Q(i, \sqrt{5})\), 8.2.4960000.1, 8.2.79360000.2, 8.0.2560000.1 |
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
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.2.4960000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $24$ | |||
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |