Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 + 21*x^12 + 43*x^10 + 60*x^8 + 57*x^6 + 31*x^4 + 8*x^2 + 1)
gp: K = bnfinit(y^16 + 7*y^14 + 21*y^12 + 43*y^10 + 60*y^8 + 57*y^6 + 31*y^4 + 8*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 7*x^14 + 21*x^12 + 43*x^10 + 60*x^8 + 57*x^6 + 31*x^4 + 8*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 7*x^14 + 21*x^12 + 43*x^10 + 60*x^8 + 57*x^6 + 31*x^4 + 8*x^2 + 1)
\( x^{16} + 7x^{14} + 21x^{12} + 43x^{10} + 60x^{8} + 57x^{6} + 31x^{4} + 8x^{2} + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(23642137600000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 31^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.55\) | 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{164}a^{14}-\frac{7}{164}a^{12}-\frac{1}{2}a^{11}+\frac{39}{82}a^{10}+\frac{17}{164}a^{8}-\frac{1}{2}a^{7}-\frac{7}{82}a^{6}+\frac{12}{41}a^{4}-\frac{1}{2}a^{3}+\frac{14}{41}a^{2}-\frac{1}{2}a-\frac{79}{164}$, $\frac{1}{164}a^{15}-\frac{7}{164}a^{13}+\frac{39}{82}a^{11}+\frac{17}{164}a^{9}-\frac{1}{2}a^{8}-\frac{7}{82}a^{7}-\frac{1}{2}a^{6}+\frac{12}{41}a^{5}+\frac{14}{41}a^{3}-\frac{79}{164}a-\frac{1}{2}$
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: |
\( -1 \)
(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: |
$\frac{115}{82}a^{15}+\frac{397}{41}a^{13}+\frac{1164}{41}a^{11}+\frac{4661}{82}a^{9}+\frac{6303}{82}a^{7}+\frac{5643}{82}a^{5}+\frac{2709}{82}a^{3}+\frac{193}{41}a$, $\frac{151}{82}a^{15}+\frac{993}{82}a^{13}+\frac{1379}{41}a^{11}+\frac{5355}{82}a^{9}+\frac{3412}{41}a^{7}+\frac{2886}{41}a^{5}+\frac{1112}{41}a^{3}+\frac{207}{82}a$, $\frac{11}{82}a^{14}+\frac{87}{82}a^{12}+\frac{142}{41}a^{10}+\frac{597}{82}a^{8}+\frac{456}{41}a^{6}+\frac{428}{41}a^{4}+\frac{267}{41}a^{2}+\frac{115}{82}$, $\frac{11}{82}a^{15}+\frac{27}{82}a^{14}+\frac{133}{164}a^{13}+\frac{90}{41}a^{12}+\frac{161}{82}a^{11}+\frac{507}{82}a^{10}+\frac{269}{82}a^{9}+\frac{496}{41}a^{8}+\frac{471}{164}a^{7}+\frac{1303}{82}a^{6}+\frac{113}{164}a^{5}+\frac{1173}{82}a^{4}-\frac{367}{164}a^{3}+\frac{264}{41}a^{2}-\frac{385}{164}a+\frac{61}{41}$, $\frac{61}{41}a^{15}-\frac{11}{82}a^{14}+\frac{827}{82}a^{13}-\frac{133}{164}a^{12}+\frac{1191}{41}a^{11}-\frac{161}{82}a^{10}+\frac{4739}{82}a^{9}-\frac{269}{82}a^{8}+\frac{3164}{41}a^{7}-\frac{471}{164}a^{6}+\frac{5651}{82}a^{5}-\frac{113}{164}a^{4}+\frac{2609}{82}a^{3}+\frac{367}{164}a^{2}+\frac{224}{41}a+\frac{221}{164}$, $\frac{61}{41}a^{15}+\frac{11}{82}a^{14}+\frac{827}{82}a^{13}+\frac{133}{164}a^{12}+\frac{1191}{41}a^{11}+\frac{161}{82}a^{10}+\frac{4739}{82}a^{9}+\frac{269}{82}a^{8}+\frac{3164}{41}a^{7}+\frac{471}{164}a^{6}+\frac{5651}{82}a^{5}+\frac{113}{164}a^{4}+\frac{2609}{82}a^{3}-\frac{367}{164}a^{2}+\frac{224}{41}a-\frac{221}{164}$, $\frac{75}{164}a^{15}+\frac{133}{164}a^{14}+\frac{291}{82}a^{13}+\frac{873}{164}a^{12}+\frac{957}{82}a^{11}+\frac{605}{41}a^{10}+\frac{4063}{164}a^{9}+\frac{4721}{164}a^{8}+\frac{6043}{164}a^{7}+\frac{1523}{41}a^{6}+\frac{5937}{164}a^{5}+\frac{1309}{41}a^{4}+\frac{3503}{164}a^{3}+\frac{1141}{82}a^{2}+\frac{379}{82}a+\frac{399}{164}$
| 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: | \( 23.6552396581 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 23.6552396581 \cdot 1}{2\cdot\sqrt{23642137600000000}}\cr\approx \mathstrut & 0.186849927316 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 + 21*x^12 + 43*x^10 + 60*x^8 + 57*x^6 + 31*x^4 + 8*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 + 21*x^12 + 43*x^10 + 60*x^8 + 57*x^6 + 31*x^4 + 8*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 + 21*x^12 + 43*x^10 + 60*x^8 + 57*x^6 + 31*x^4 + 8*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 + 21*x^12 + 43*x^10 + 60*x^8 + 57*x^6 + 31*x^4 + 8*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
$C_2\wr D_4$ (as 16T388):
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
| \(\Q(\sqrt{5}) \), 4.2.775.1, 4.2.400.1, 4.0.12400.1, 8.2.9610000.1 x2, 8.2.4960000.1 x2, 8.0.153760000.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
| 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
|
\(5\)
| 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 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(31\)
| 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |