Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 4*x^13 - 5*x^12 + 2*x^11 + 3*x^10 + x^8 + 3*x^6 - 2*x^5 - 5*x^4 - 4*x^3 + x^2 + 2*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + y^14 + 4*y^13 - 5*y^12 + 2*y^11 + 3*y^10 + y^8 + 3*y^6 - 2*y^5 - 5*y^4 - 4*y^3 + y^2 + 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + x^14 + 4*x^13 - 5*x^12 + 2*x^11 + 3*x^10 + x^8 + 3*x^6 - 2*x^5 - 5*x^4 - 4*x^3 + x^2 + 2*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + x^14 + 4*x^13 - 5*x^12 + 2*x^11 + 3*x^10 + x^8 + 3*x^6 - 2*x^5 - 5*x^4 - 4*x^3 + x^2 + 2*x + 1)
\( x^{16} - 2 x^{15} + x^{14} + 4 x^{13} - 5 x^{12} + 2 x^{11} + 3 x^{10} + x^{8} + 3 x^{6} - 2 x^{5} + \cdots + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(882697830400000000\)
\(\medspace = 2^{24}\cdot 5^{8}\cdot 367^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.23\) | 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: | $2^{3/2}5^{1/2}367^{1/2}\approx 121.16104984688768$ | ||
| Ramified primes: |
\(2\), \(5\), \(367\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29}a^{14}+\frac{8}{29}a^{13}-\frac{5}{29}a^{12}-\frac{9}{29}a^{11}-\frac{13}{29}a^{10}+\frac{8}{29}a^{9}+\frac{12}{29}a^{8}+\frac{12}{29}a^{7}-\frac{12}{29}a^{6}+\frac{8}{29}a^{5}+\frac{13}{29}a^{4}-\frac{9}{29}a^{3}+\frac{5}{29}a^{2}+\frac{8}{29}a-\frac{1}{29}$, $\frac{1}{493}a^{15}+\frac{2}{493}a^{14}+\frac{179}{493}a^{13}+\frac{108}{493}a^{12}+\frac{70}{493}a^{11}+\frac{231}{493}a^{10}+\frac{196}{493}a^{9}+\frac{172}{493}a^{8}-\frac{229}{493}a^{7}+\frac{138}{493}a^{6}-\frac{6}{493}a^{5}-\frac{5}{17}a^{4}+\frac{146}{493}a^{3}+\frac{36}{493}a^{2}+\frac{9}{493}a-\frac{81}{493}$
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: | $9$ | 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: |
$a$, $\frac{614}{493}a^{15}-\frac{1611}{493}a^{14}+\frac{1412}{493}a^{13}+\frac{2120}{493}a^{12}-\frac{4926}{493}a^{11}+\frac{3726}{493}a^{10}+\frac{1004}{493}a^{9}-\frac{1917}{493}a^{8}+\frac{1820}{493}a^{7}-\frac{999}{493}a^{6}+\frac{2198}{493}a^{5}-\frac{2194}{493}a^{4}-\frac{2632}{493}a^{3}+\frac{21}{493}a^{2}+\frac{1548}{493}a+\frac{433}{493}$, $\frac{332}{493}a^{15}-\frac{917}{493}a^{14}+\frac{931}{493}a^{13}+\frac{30}{17}a^{12}-\frac{2464}{493}a^{11}+\frac{2096}{493}a^{10}+\frac{166}{493}a^{9}-\frac{322}{493}a^{8}+\frac{149}{493}a^{7}+\frac{205}{493}a^{6}+\frac{1136}{493}a^{5}-\frac{1645}{493}a^{4}-\frac{403}{493}a^{3}+\frac{103}{493}a^{2}+\frac{693}{493}a+\frac{325}{493}$, $\frac{9}{17}a^{15}-\frac{787}{493}a^{14}+\frac{1244}{493}a^{13}-\frac{270}{493}a^{12}-\frac{1501}{493}a^{11}+\frac{3358}{493}a^{10}-\frac{2700}{493}a^{9}+\frac{1576}{493}a^{8}+\frac{938}{493}a^{7}-\frac{1025}{493}a^{6}+\frac{2259}{493}a^{5}-\frac{1618}{493}a^{4}+\frac{587}{493}a^{3}-\frac{600}{493}a^{2}-\frac{235}{493}a-\frac{605}{493}$, $\frac{54}{493}a^{15}+\frac{193}{493}a^{14}-\frac{500}{493}a^{13}+\frac{477}{493}a^{12}+\frac{1043}{493}a^{11}-\frac{1449}{493}a^{10}+\frac{911}{493}a^{9}+\frac{1434}{493}a^{8}-\frac{500}{493}a^{7}+\frac{1009}{493}a^{6}+\frac{849}{493}a^{5}+\frac{670}{493}a^{4}+\frac{217}{493}a^{3}-\frac{1575}{493}a^{2}-\frac{806}{493}a-\frac{22}{493}$, $\frac{30}{493}a^{15}+\frac{111}{493}a^{14}-\frac{138}{493}a^{13}+\frac{27}{493}a^{12}+\frac{655}{493}a^{11}-\frac{142}{493}a^{10}-\frac{121}{493}a^{9}+\frac{842}{493}a^{8}+\frac{644}{493}a^{7}+\frac{570}{493}a^{6}+\frac{228}{493}a^{5}+\frac{1243}{493}a^{4}+\frac{470}{493}a^{3}-\frac{1130}{493}a^{2}-\frac{801}{493}a-\frac{509}{493}$, $\frac{229}{493}a^{15}-\frac{171}{493}a^{14}-\frac{523}{493}a^{13}+\frac{1748}{493}a^{12}-\frac{494}{493}a^{11}-\frac{1535}{493}a^{10}+\frac{2877}{493}a^{9}-\frac{205}{493}a^{8}+\frac{157}{493}a^{7}+\frac{24}{17}a^{6}+\frac{989}{493}a^{5}+\frac{1016}{493}a^{4}-\frac{2317}{493}a^{3}-\frac{817}{493}a^{2}-\frac{13}{493}a+\frac{814}{493}$, $\frac{31}{493}a^{15}-\frac{363}{493}a^{14}+\frac{670}{493}a^{13}-\frac{443}{493}a^{12}-\frac{907}{493}a^{11}+\frac{1347}{493}a^{10}-\frac{1268}{493}a^{9}+\frac{8}{17}a^{8}-\frac{860}{493}a^{7}-\frac{482}{493}a^{6}+\frac{358}{493}a^{5}-\frac{1639}{493}a^{4}+\frac{463}{493}a^{3}+\frac{963}{493}a^{2}+\frac{1316}{493}a+\frac{379}{493}$, $\frac{678}{493}a^{15}-\frac{1806}{493}a^{14}+\frac{1410}{493}a^{13}+\frac{2759}{493}a^{12}-\frac{5920}{493}a^{11}+\frac{3482}{493}a^{10}+\frac{2583}{493}a^{9}-\frac{3659}{493}a^{8}+\frac{1529}{493}a^{7}-\frac{123}{493}a^{6}+\frac{709}{493}a^{5}-\frac{2362}{493}a^{4}-\frac{3692}{493}a^{3}+\frac{217}{493}a^{2}+\frac{2498}{493}a+\frac{995}{493}$
| 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: | \( 407.518266474 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 407.518266474 \cdot 1}{2\cdot\sqrt{882697830400000000}}\cr\approx \mathstrut & 0.213506080421 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 4*x^13 - 5*x^12 + 2*x^11 + 3*x^10 + x^8 + 3*x^6 - 2*x^5 - 5*x^4 - 4*x^3 + x^2 + 2*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 - 2*x^15 + x^14 + 4*x^13 - 5*x^12 + 2*x^11 + 3*x^10 + x^8 + 3*x^6 - 2*x^5 - 5*x^4 - 4*x^3 + x^2 + 2*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 - 2*x^15 + x^14 + 4*x^13 - 5*x^12 + 2*x^11 + 3*x^10 + x^8 + 3*x^6 - 2*x^5 - 5*x^4 - 4*x^3 + x^2 + 2*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 - 2*x^15 + x^14 + 4*x^13 - 5*x^12 + 2*x^11 + 3*x^10 + x^8 + 3*x^6 - 2*x^5 - 5*x^4 - 4*x^3 + x^2 + 2*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
$S_4^2:C_2^2$ (as 16T1494):
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 2304 |
| The 40 conjugacy class representatives for $S_4^2:C_2^2$ |
| Character table for $S_4^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.2.939520000.1 |
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 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.2.323950103756800.1 |
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 | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(367\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |