Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 85*x^14 + 2709*x^12 + 41896*x^10 + 344677*x^8 + 1544032*x^6 + 3659235*x^4 + 4140999*x^2 + 1652141)
gp: K = bnfinit(y^16 + 85*y^14 + 2709*y^12 + 41896*y^10 + 344677*y^8 + 1544032*y^6 + 3659235*y^4 + 4140999*y^2 + 1652141, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 85*x^14 + 2709*x^12 + 41896*x^10 + 344677*x^8 + 1544032*x^6 + 3659235*x^4 + 4140999*x^2 + 1652141);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 85*x^14 + 2709*x^12 + 41896*x^10 + 344677*x^8 + 1544032*x^6 + 3659235*x^4 + 4140999*x^2 + 1652141)
\( x^{16} + 85 x^{14} + 2709 x^{12} + 41896 x^{10} + 344677 x^{8} + 1544032 x^{6} + 3659235 x^{4} + \cdots + 1652141 \)
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: |
\(115446638528641004057600000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 1652141^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(65.52\) | 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: |
\(2\), \(5\), \(1652141\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1652141}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{16\!\cdots\!05}a^{14}-\frac{17\!\cdots\!69}{16\!\cdots\!05}a^{12}-\frac{14\!\cdots\!13}{33\!\cdots\!61}a^{10}-\frac{15\!\cdots\!29}{16\!\cdots\!05}a^{8}+\frac{22\!\cdots\!68}{16\!\cdots\!05}a^{6}+\frac{65\!\cdots\!02}{33\!\cdots\!61}a^{4}+\frac{13\!\cdots\!98}{33\!\cdots\!61}a^{2}+\frac{31\!\cdots\!09}{16\!\cdots\!05}$, $\frac{1}{16\!\cdots\!05}a^{15}-\frac{17\!\cdots\!69}{16\!\cdots\!05}a^{13}-\frac{14\!\cdots\!13}{33\!\cdots\!61}a^{11}-\frac{15\!\cdots\!29}{16\!\cdots\!05}a^{9}+\frac{22\!\cdots\!68}{16\!\cdots\!05}a^{7}+\frac{65\!\cdots\!02}{33\!\cdots\!61}a^{5}+\frac{13\!\cdots\!98}{33\!\cdots\!61}a^{3}+\frac{31\!\cdots\!09}{16\!\cdots\!05}a$
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
$C_{2}\times C_{5530}$, which has order $11060$ (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: | $7$ | 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: |
$\frac{97576214}{13600451765645}a^{14}+\frac{7945151054}{13600451765645}a^{12}+\frac{47348100121}{2720090353129}a^{10}+\frac{3305937614269}{13600451765645}a^{8}+\frac{23620639467682}{13600451765645}a^{6}+\frac{17704729568449}{2720090353129}a^{4}+\frac{32853308773438}{2720090353129}a^{2}+\frac{95899959260976}{13600451765645}$, $\frac{77\!\cdots\!13}{16\!\cdots\!05}a^{14}+\frac{63\!\cdots\!13}{16\!\cdots\!05}a^{12}+\frac{37\!\cdots\!72}{33\!\cdots\!61}a^{10}+\frac{25\!\cdots\!63}{16\!\cdots\!05}a^{8}+\frac{17\!\cdots\!24}{16\!\cdots\!05}a^{6}+\frac{12\!\cdots\!85}{33\!\cdots\!61}a^{4}+\frac{16\!\cdots\!60}{33\!\cdots\!61}a^{2}+\frac{47\!\cdots\!22}{16\!\cdots\!05}$, $\frac{77\!\cdots\!13}{16\!\cdots\!05}a^{14}+\frac{63\!\cdots\!13}{16\!\cdots\!05}a^{12}+\frac{37\!\cdots\!72}{33\!\cdots\!61}a^{10}+\frac{25\!\cdots\!63}{16\!\cdots\!05}a^{8}+\frac{17\!\cdots\!24}{16\!\cdots\!05}a^{6}+\frac{12\!\cdots\!85}{33\!\cdots\!61}a^{4}+\frac{16\!\cdots\!60}{33\!\cdots\!61}a^{2}+\frac{21\!\cdots\!27}{16\!\cdots\!05}$, $\frac{43\!\cdots\!13}{16\!\cdots\!05}a^{14}+\frac{35\!\cdots\!73}{16\!\cdots\!05}a^{12}+\frac{21\!\cdots\!17}{33\!\cdots\!61}a^{10}+\frac{15\!\cdots\!58}{16\!\cdots\!05}a^{8}+\frac{11\!\cdots\!14}{16\!\cdots\!05}a^{6}+\frac{97\!\cdots\!56}{33\!\cdots\!61}a^{4}+\frac{24\!\cdots\!82}{33\!\cdots\!61}a^{2}+\frac{13\!\cdots\!67}{16\!\cdots\!05}$, $\frac{43\!\cdots\!13}{16\!\cdots\!05}a^{14}+\frac{35\!\cdots\!73}{16\!\cdots\!05}a^{12}+\frac{21\!\cdots\!17}{33\!\cdots\!61}a^{10}+\frac{15\!\cdots\!58}{16\!\cdots\!05}a^{8}+\frac{11\!\cdots\!14}{16\!\cdots\!05}a^{6}+\frac{97\!\cdots\!56}{33\!\cdots\!61}a^{4}+\frac{24\!\cdots\!82}{33\!\cdots\!61}a^{2}+\frac{11\!\cdots\!62}{16\!\cdots\!05}$, $\frac{10\!\cdots\!41}{16\!\cdots\!05}a^{14}+\frac{97\!\cdots\!41}{16\!\cdots\!05}a^{12}+\frac{68\!\cdots\!84}{33\!\cdots\!61}a^{10}+\frac{58\!\cdots\!81}{16\!\cdots\!05}a^{8}+\frac{53\!\cdots\!23}{16\!\cdots\!05}a^{6}+\frac{49\!\cdots\!30}{33\!\cdots\!61}a^{4}+\frac{10\!\cdots\!81}{33\!\cdots\!61}a^{2}+\frac{34\!\cdots\!89}{16\!\cdots\!05}$, $\frac{91\!\cdots\!72}{33\!\cdots\!61}a^{14}+\frac{74\!\cdots\!67}{33\!\cdots\!61}a^{12}+\frac{21\!\cdots\!01}{33\!\cdots\!61}a^{10}+\frac{29\!\cdots\!83}{33\!\cdots\!61}a^{8}+\frac{20\!\cdots\!35}{33\!\cdots\!61}a^{6}+\frac{65\!\cdots\!41}{33\!\cdots\!61}a^{4}+\frac{96\!\cdots\!73}{33\!\cdots\!61}a^{2}+\frac{47\!\cdots\!51}{33\!\cdots\!61}$
| 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: | \( 6960.86418224 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6960.86418224 \cdot 11060}{2\cdot\sqrt{115446638528641004057600000000}}\cr\approx \mathstrut & 0.275192499105 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 85*x^14 + 2709*x^12 + 41896*x^10 + 344677*x^8 + 1544032*x^6 + 3659235*x^4 + 4140999*x^2 + 1652141)
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 + 85*x^14 + 2709*x^12 + 41896*x^10 + 344677*x^8 + 1544032*x^6 + 3659235*x^4 + 4140999*x^2 + 1652141, 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 + 85*x^14 + 2709*x^12 + 41896*x^10 + 344677*x^8 + 1544032*x^6 + 3659235*x^4 + 4140999*x^2 + 1652141);
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 + 85*x^14 + 2709*x^12 + 41896*x^10 + 344677*x^8 + 1544032*x^6 + 3659235*x^4 + 4140999*x^2 + 1652141);
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
$C_2^6.S_4\wr C_2$ (as 16T1872):
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 73728 |
| The 77 conjugacy class representatives for $C_2^6.S_4\wr C_2$ |
| Character table for $C_2^6.S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 8.8.1032588125.1 |
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 | R | $16$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $16$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $2$ | $8$ | $16$ | |||
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(1652141\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |