Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^14 - x^12 + 117*x^10 - 343*x^8 + 384*x^6 - 181*x^4 + 30*x^2 - 1)
gp: K = bnfinit(y^16 - 7*y^14 - y^12 + 117*y^10 - 343*y^8 + 384*y^6 - 181*y^4 + 30*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^14 - x^12 + 117*x^10 - 343*x^8 + 384*x^6 - 181*x^4 + 30*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^14 - x^12 + 117*x^10 - 343*x^8 + 384*x^6 - 181*x^4 + 30*x^2 - 1)
\( x^{16} - 7x^{14} - x^{12} + 117x^{10} - 343x^{8} + 384x^{6} - 181x^{4} + 30x^{2} - 1 \)
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: | $[14, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-434911971099673600000000\)
\(\medspace = -\,2^{16}\cdot 5^{8}\cdot 13^{4}\cdot 29^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.02\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(5\), \(13\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{83}a^{14}-\frac{38}{83}a^{12}+\frac{15}{83}a^{10}-\frac{16}{83}a^{8}-\frac{13}{83}a^{6}+\frac{40}{83}a^{4}-\frac{10}{83}a^{2}+\frac{8}{83}$, $\frac{1}{83}a^{15}-\frac{38}{83}a^{13}+\frac{15}{83}a^{11}-\frac{16}{83}a^{9}-\frac{13}{83}a^{7}+\frac{40}{83}a^{5}-\frac{10}{83}a^{3}+\frac{8}{83}a$
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 $1$ (assuming GRH)
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: | $14$ | 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: |
$\frac{44}{83}a^{14}-\frac{261}{83}a^{12}-\frac{336}{83}a^{10}+\frac{4857}{83}a^{8}-\frac{9785}{83}a^{6}+\frac{5163}{83}a^{4}+\frac{58}{83}a^{2}-\frac{146}{83}$, $\frac{46}{83}a^{14}-\frac{337}{83}a^{12}-\frac{57}{83}a^{10}+\frac{5821}{83}a^{8}-\frac{15870}{83}a^{6}+\frac{14373}{83}a^{4}-\frac{4527}{83}a^{2}+\frac{285}{83}$, $a$, $\frac{90}{83}a^{14}-\frac{598}{83}a^{12}-\frac{393}{83}a^{10}+\frac{10678}{83}a^{8}-\frac{25655}{83}a^{6}+\frac{19536}{83}a^{4}-\frac{4469}{83}a^{2}+\frac{139}{83}$, $\frac{58}{83}a^{15}-\frac{378}{83}a^{13}-\frac{209}{83}a^{11}+\frac{6542}{83}a^{9}-\frac{17105}{83}a^{7}+\frac{16762}{83}a^{5}-\frac{6722}{83}a^{3}+\frac{713}{83}a$, $\frac{63}{83}a^{15}-\frac{319}{83}a^{13}-\frac{715}{83}a^{11}+\frac{6296}{83}a^{9}-\frac{9534}{83}a^{7}+\frac{362}{83}a^{5}+\frac{4516}{83}a^{3}-\frac{1073}{83}a$, $\frac{139}{83}a^{15}-\frac{883}{83}a^{13}-\frac{737}{83}a^{11}+\frac{15870}{83}a^{9}-\frac{36999}{83}a^{7}+\frac{27721}{83}a^{5}-\frac{5623}{83}a^{3}-\frac{299}{83}a+1$, $\frac{93}{83}a^{15}-\frac{46}{83}a^{14}-\frac{546}{83}a^{13}+\frac{337}{83}a^{12}-\frac{680}{83}a^{11}+\frac{57}{83}a^{10}+\frac{10049}{83}a^{9}-\frac{5821}{83}a^{8}-\frac{21129}{83}a^{7}+\frac{15870}{83}a^{6}+\frac{13348}{83}a^{5}-\frac{14373}{83}a^{4}-\frac{1096}{83}a^{3}+\frac{4527}{83}a^{2}-\frac{584}{83}a-\frac{285}{83}$, $a+1$, $\frac{126}{83}a^{15}+\frac{46}{83}a^{14}-\frac{804}{83}a^{13}-\frac{337}{83}a^{12}-\frac{517}{83}a^{11}-\frac{57}{83}a^{10}+\frac{14003}{83}a^{9}+\frac{5821}{83}a^{8}-\frac{36000}{83}a^{7}-\frac{15870}{83}a^{6}+\frac{34339}{83}a^{5}+\frac{14373}{83}a^{4}-\frac{12050}{83}a^{3}-\frac{4527}{83}a^{2}+\frac{759}{83}a+\frac{202}{83}$, $\frac{126}{83}a^{15}-\frac{46}{83}a^{14}-\frac{804}{83}a^{13}+\frac{337}{83}a^{12}-\frac{517}{83}a^{11}+\frac{57}{83}a^{10}+\frac{14003}{83}a^{9}-\frac{5821}{83}a^{8}-\frac{36000}{83}a^{7}+\frac{15870}{83}a^{6}+\frac{34339}{83}a^{5}-\frac{14373}{83}a^{4}-\frac{12050}{83}a^{3}+\frac{4527}{83}a^{2}+\frac{759}{83}a-\frac{202}{83}$, $\frac{44}{83}a^{14}-\frac{261}{83}a^{12}-\frac{336}{83}a^{10}+\frac{4857}{83}a^{8}-\frac{9785}{83}a^{6}+\frac{5163}{83}a^{4}+\frac{58}{83}a^{2}+a-\frac{63}{83}$, $\frac{213}{83}a^{15}+\frac{164}{83}a^{14}-\frac{1371}{83}a^{13}-\frac{1003}{83}a^{12}-\frac{955}{83}a^{11}-\frac{1026}{83}a^{10}+\frac{24231}{83}a^{9}+\frac{18209}{83}a^{8}-\frac{59707}{83}a^{7}-\frac{40561}{83}a^{6}+\frac{51016}{83}a^{5}+\frac{28638}{83}a^{4}-\frac{15078}{83}a^{3}-\frac{5292}{83}a^{2}+\frac{1040}{83}a+\frac{67}{83}$, $\frac{63}{83}a^{15}+\frac{30}{83}a^{14}-\frac{485}{83}a^{13}-\frac{227}{83}a^{12}+\frac{198}{83}a^{11}+\frac{35}{83}a^{10}+\frac{7707}{83}a^{9}+\frac{3753}{83}a^{8}-\frac{26466}{83}a^{7}-\frac{11595}{83}a^{6}+\frac{33977}{83}a^{5}+\frac{12986}{83}a^{4}-\frac{16566}{83}a^{3}-\frac{5612}{83}a^{2}+\frac{1832}{83}a+\frac{489}{83}$
| 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: | \( 2674022.50951 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 2674022.50951 \cdot 1}{2\cdot\sqrt{434911971099673600000000}}\cr\approx \mathstrut & 0.208705642075 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^14 - x^12 + 117*x^10 - 343*x^8 + 384*x^6 - 181*x^4 + 30*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 - 7*x^14 - x^12 + 117*x^10 - 343*x^8 + 384*x^6 - 181*x^4 + 30*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 - 7*x^14 - x^12 + 117*x^10 - 343*x^8 + 384*x^6 - 181*x^4 + 30*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 - 7*x^14 - x^12 + 117*x^10 - 343*x^8 + 384*x^6 - 181*x^4 + 30*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
$C_2\wr D_4.D_4$ (as 16T1775):
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 16384 |
| The 136 conjugacy class representatives for $C_2\wr D_4.D_4$ |
| Character table for $C_2\wr D_4.D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.2576088125.1 |
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: | 16.2.434911971099673600000000.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $16$ | $16$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $2$ | $8$ | $16$ | |||
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.4.3.3 | $x^{4} + 58$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.3.3 | $x^{4} + 58$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |