Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + 10*x^13 - 13*x^12 + 6*x^11 + 12*x^10 - 33*x^9 + 41*x^8 - 33*x^7 + 12*x^6 + 6*x^5 - 13*x^4 + 10*x^3 - 3*x^2 - x + 1)
gp: K = bnfinit(y^16 - y^15 - 3*y^14 + 10*y^13 - 13*y^12 + 6*y^11 + 12*y^10 - 33*y^9 + 41*y^8 - 33*y^7 + 12*y^6 + 6*y^5 - 13*y^4 + 10*y^3 - 3*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 3*x^14 + 10*x^13 - 13*x^12 + 6*x^11 + 12*x^10 - 33*x^9 + 41*x^8 - 33*x^7 + 12*x^6 + 6*x^5 - 13*x^4 + 10*x^3 - 3*x^2 - x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 3*x^14 + 10*x^13 - 13*x^12 + 6*x^11 + 12*x^10 - 33*x^9 + 41*x^8 - 33*x^7 + 12*x^6 + 6*x^5 - 13*x^4 + 10*x^3 - 3*x^2 - x + 1)
\( x^{16} - x^{15} - 3 x^{14} + 10 x^{13} - 13 x^{12} + 6 x^{11} + 12 x^{10} - 33 x^{9} + 41 x^{8} + \cdots + 1 \)
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: | $[6, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-505241847543359375\)
\(\medspace = -\,5^{8}\cdot 71^{3}\cdot 1901^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.78\) | 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: | $5^{1/2}71^{3/4}1901^{1/2}\approx 2384.6235143470617$ | ||
| Ramified primes: |
\(5\), \(71\), \(1901\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{323}a^{14}-\frac{147}{323}a^{13}+\frac{140}{323}a^{12}+\frac{66}{323}a^{11}-\frac{99}{323}a^{10}-\frac{141}{323}a^{9}+\frac{25}{323}a^{8}+\frac{11}{323}a^{7}+\frac{25}{323}a^{6}-\frac{141}{323}a^{5}-\frac{99}{323}a^{4}+\frac{66}{323}a^{3}+\frac{140}{323}a^{2}-\frac{147}{323}a+\frac{1}{323}$, $\frac{1}{323}a^{15}-\frac{151}{323}a^{13}-\frac{26}{323}a^{12}-\frac{87}{323}a^{11}-\frac{159}{323}a^{10}-\frac{30}{323}a^{9}+\frac{7}{17}a^{8}+\frac{27}{323}a^{7}-\frac{1}{17}a^{6}-\frac{154}{323}a^{5}+\frac{48}{323}a^{4}+\frac{8}{17}a^{3}+\frac{84}{323}a^{2}+\frac{33}{323}a+\frac{147}{323}$
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: | $10$ | 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: |
$\frac{4}{17}a^{15}+\frac{155}{323}a^{14}-\frac{346}{323}a^{13}+\frac{21}{323}a^{12}+\frac{1034}{323}a^{11}-\frac{1589}{323}a^{10}+\frac{1059}{323}a^{9}+\frac{740}{323}a^{8}-\frac{178}{19}a^{7}+\frac{200}{19}a^{6}-\frac{2874}{323}a^{5}+\frac{577}{323}a^{4}+\frac{464}{323}a^{3}-\frac{58}{19}a^{2}+\frac{395}{323}a+\frac{22}{323}$, $\frac{300}{323}a^{15}-\frac{63}{323}a^{14}-\frac{1155}{323}a^{13}+\frac{2114}{323}a^{12}-\frac{1511}{323}a^{11}-\frac{45}{19}a^{10}+\frac{4082}{323}a^{9}-\frac{6572}{323}a^{8}+\frac{5146}{323}a^{7}-\frac{2107}{323}a^{6}-\frac{1787}{323}a^{5}+\frac{3518}{323}a^{4}-\frac{2163}{323}a^{3}+\frac{1199}{323}a^{2}+\frac{427}{323}a-\frac{537}{323}$, $a$, $\frac{108}{323}a^{15}-\frac{212}{323}a^{14}-\frac{325}{323}a^{13}+\frac{1427}{323}a^{12}-\frac{2070}{323}a^{11}+\frac{1232}{323}a^{10}+\frac{1458}{323}a^{9}-\frac{4825}{323}a^{8}+\frac{6398}{323}a^{7}-\frac{5091}{323}a^{6}+\frac{134}{19}a^{5}+\frac{655}{323}a^{4}-\frac{1452}{323}a^{3}+\frac{1033}{323}a^{2}-\frac{479}{323}a-\frac{163}{323}$, $\frac{124}{323}a^{15}-\frac{63}{323}a^{14}-\frac{419}{323}a^{13}+\frac{876}{323}a^{12}-\frac{1057}{323}a^{11}+\frac{733}{323}a^{10}+\frac{641}{323}a^{9}-\frac{2848}{323}a^{8}+\frac{3947}{323}a^{7}-\frac{3931}{323}a^{6}+\frac{2707}{323}a^{5}-\frac{43}{19}a^{4}-\frac{491}{323}a^{3}+\frac{67}{17}a^{2}-\frac{536}{323}a+\frac{77}{323}$, $\frac{94}{323}a^{15}+\frac{145}{323}a^{14}-\frac{302}{323}a^{13}+\frac{91}{323}a^{12}+\frac{423}{323}a^{11}-\frac{554}{323}a^{10}+\frac{637}{323}a^{9}-\frac{346}{323}a^{8}-\frac{712}{323}a^{7}+\frac{870}{323}a^{6}-\frac{1329}{323}a^{5}+\frac{10}{19}a^{4}-\frac{44}{323}a^{3}-\frac{29}{17}a^{2}+\frac{521}{323}a+\frac{74}{323}$, $\frac{143}{323}a^{15}-\frac{287}{323}a^{14}-\frac{21}{17}a^{13}+\frac{1968}{323}a^{12}-\frac{2959}{323}a^{11}+\frac{1477}{323}a^{10}+\frac{2262}{323}a^{9}-\frac{6567}{323}a^{8}+\frac{8456}{323}a^{7}-\frac{6662}{323}a^{6}+\frac{154}{19}a^{5}+\frac{1039}{323}a^{4}-\frac{2051}{323}a^{3}+\frac{1871}{323}a^{2}-\frac{573}{323}a+\frac{62}{323}$, $\frac{28}{323}a^{15}+\frac{135}{323}a^{14}-\frac{9}{17}a^{13}-\frac{239}{323}a^{12}+\frac{983}{323}a^{11}-\frac{1344}{323}a^{10}+\frac{797}{323}a^{9}+\frac{962}{323}a^{8}-\frac{2927}{323}a^{7}+\frac{3489}{323}a^{6}-\frac{3321}{323}a^{5}+\frac{1868}{323}a^{4}-\frac{400}{323}a^{3}-\frac{389}{323}a^{2}+\frac{27}{19}a-\frac{271}{323}$, $\frac{59}{323}a^{15}-\frac{156}{323}a^{14}-\frac{189}{323}a^{13}+\frac{851}{323}a^{12}-\frac{1540}{323}a^{11}+\frac{895}{323}a^{10}+\frac{1169}{323}a^{9}-\frac{3482}{323}a^{8}+\frac{4076}{323}a^{7}-\frac{3083}{323}a^{6}+\frac{1282}{323}a^{5}+\frac{1157}{323}a^{4}-\frac{1005}{323}a^{3}+\frac{881}{323}a^{2}+\frac{331}{323}a-\frac{12}{19}$, $\frac{39}{323}a^{15}-\frac{24}{323}a^{14}-\frac{100}{323}a^{13}+\frac{471}{323}a^{12}-\frac{455}{323}a^{11}-\frac{16}{19}a^{10}+\frac{1245}{323}a^{9}-\frac{1227}{323}a^{8}+\frac{466}{323}a^{7}+\frac{274}{323}a^{6}-\frac{53}{17}a^{5}+\frac{695}{323}a^{4}-\frac{178}{323}a^{3}-\frac{407}{323}a^{2}+\frac{293}{323}a-\frac{105}{323}$
| 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: | \( 335.766972694 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{5}\cdot 335.766972694 \cdot 1}{2\cdot\sqrt{505241847543359375}}\cr\approx \mathstrut & 0.148025875324 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + 10*x^13 - 13*x^12 + 6*x^11 + 12*x^10 - 33*x^9 + 41*x^8 - 33*x^7 + 12*x^6 + 6*x^5 - 13*x^4 + 10*x^3 - 3*x^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 - x^15 - 3*x^14 + 10*x^13 - 13*x^12 + 6*x^11 + 12*x^10 - 33*x^9 + 41*x^8 - 33*x^7 + 12*x^6 + 6*x^5 - 13*x^4 + 10*x^3 - 3*x^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 - x^15 - 3*x^14 + 10*x^13 - 13*x^12 + 6*x^11 + 12*x^10 - 33*x^9 + 41*x^8 - 33*x^7 + 12*x^6 + 6*x^5 - 13*x^4 + 10*x^3 - 3*x^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 - x^15 - 3*x^14 + 10*x^13 - 13*x^12 + 6*x^11 + 12*x^10 - 33*x^9 + 41*x^8 - 33*x^7 + 12*x^6 + 6*x^5 - 13*x^4 + 10*x^3 - 3*x^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
$C_2^6.S_4^2:D_4$ (as 16T1905):
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 294912 |
| The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$ |
| Character table for $C_2^6.S_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 8.6.84356875.1 |
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 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | $16$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | $16$ | $16$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(71\)
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.4.3.1 | $x^{4} + 71$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 71.8.0.1 | $x^{8} + 53 x^{3} + 22 x^{2} + 19 x + 7$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(1901\)
| $\Q_{1901}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1901}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1901}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1901}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1901}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1901}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |