Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 20*x^12 - 56*x^10 + 160*x^8 + 32*x^6 - 40*x^4 - 16*x^2 + 4)
gp: K = bnfinit(y^16 - 8*y^14 + 20*y^12 - 56*y^10 + 160*y^8 + 32*y^6 - 40*y^4 - 16*y^2 + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 20*x^12 - 56*x^10 + 160*x^8 + 32*x^6 - 40*x^4 - 16*x^2 + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 20*x^12 - 56*x^10 + 160*x^8 + 32*x^6 - 40*x^4 - 16*x^2 + 4)
\( x^{16} - 8x^{14} + 20x^{12} - 56x^{10} + 160x^{8} + 32x^{6} - 40x^{4} - 16x^{2} + 4 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2393397489569403764736\)
\(\medspace = 2^{52}\cdot 3^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.69\) | 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: | $2^{13/4}3^{3/4}\approx 21.686448086636275$ | ||
| Ramified primes: |
\(2\), \(3\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{8}a^{9}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}+\frac{1}{4}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{40}a^{12}+\frac{1}{40}a^{10}+\frac{1}{20}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{3}{20}a^{4}+\frac{3}{20}a^{2}-\frac{1}{5}$, $\frac{1}{40}a^{13}+\frac{1}{40}a^{11}+\frac{1}{20}a^{9}+\frac{1}{4}a^{7}+\frac{3}{20}a^{5}+\frac{3}{20}a^{3}-\frac{1}{5}a$, $\frac{1}{7720}a^{14}-\frac{47}{3860}a^{12}-\frac{97}{1930}a^{10}-\frac{27}{772}a^{8}-\frac{1627}{3860}a^{6}-\frac{238}{965}a^{4}+\frac{203}{1930}a^{2}-\frac{57}{386}$, $\frac{1}{7720}a^{15}-\frac{47}{3860}a^{13}-\frac{97}{1930}a^{11}-\frac{27}{772}a^{9}-\frac{1627}{3860}a^{7}-\frac{238}{965}a^{5}+\frac{203}{1930}a^{3}-\frac{57}{386}a$
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: | $11$ | 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: |
$\frac{15}{772}a^{14}-\frac{122}{965}a^{12}+\frac{311}{1930}a^{10}-\frac{2301}{3860}a^{8}+\frac{685}{386}a^{6}+\frac{9081}{1930}a^{4}+\frac{2873}{965}a^{2}-\frac{2413}{1930}$, $\frac{697}{3860}a^{15}-\frac{10797}{7720}a^{13}+\frac{25197}{7720}a^{11}-\frac{18053}{1930}a^{9}+\frac{102967}{3860}a^{7}+\frac{46879}{3860}a^{5}-\frac{9369}{3860}a^{3}-\frac{5693}{1930}a$, $\frac{439}{7720}a^{15}-\frac{649}{1544}a^{13}+\frac{6649}{7720}a^{11}-\frac{2416}{965}a^{9}+\frac{6958}{965}a^{7}+\frac{5619}{772}a^{5}-\frac{4923}{3860}a^{3}-\frac{2367}{1930}a$, $\frac{373}{7720}a^{15}-\frac{378}{965}a^{13}+\frac{7939}{7720}a^{11}-\frac{5781}{1930}a^{9}+\frac{8231}{965}a^{7}-\frac{573}{965}a^{5}+\frac{7267}{3860}a^{3}-\frac{753}{965}a$, $\frac{11}{772}a^{15}-\frac{69}{772}a^{13}+\frac{149}{1544}a^{11}-\frac{67}{193}a^{9}+\frac{683}{772}a^{7}+\frac{1687}{386}a^{5}+\frac{247}{772}a^{3}+\frac{339}{193}a$, $\frac{8}{965}a^{15}-\frac{183}{7720}a^{14}-\frac{161}{1544}a^{13}+\frac{1569}{7720}a^{12}+\frac{3539}{7720}a^{11}-\frac{4459}{7720}a^{10}-\frac{2197}{1930}a^{9}+\frac{6177}{3860}a^{8}+\frac{12637}{3860}a^{7}-\frac{8907}{1930}a^{6}-\frac{4041}{772}a^{5}+\frac{5727}{3860}a^{4}-\frac{12423}{3860}a^{3}+\frac{393}{3860}a^{2}-\frac{677}{1930}a+\frac{698}{965}$, $\frac{423}{7720}a^{15}-\frac{223}{7720}a^{14}-\frac{657}{1544}a^{13}+\frac{256}{965}a^{12}+\frac{7453}{7720}a^{11}-\frac{1339}{1544}a^{10}-\frac{10013}{3860}a^{9}+\frac{9261}{3860}a^{8}+\frac{14387}{1930}a^{7}-\frac{13037}{1930}a^{6}+\frac{4265}{772}a^{5}+\frac{10227}{1930}a^{4}-\frac{23771}{3860}a^{3}+\frac{845}{772}a^{2}+\frac{228}{965}a-\frac{2837}{1930}$, $\frac{33}{3860}a^{15}-\frac{59}{965}a^{14}-\frac{221}{7720}a^{13}+\frac{1919}{3860}a^{12}-\frac{129}{772}a^{11}-\frac{1233}{965}a^{10}+\frac{949}{1930}a^{9}+\frac{677}{193}a^{8}-\frac{1273}{965}a^{7}-\frac{9699}{965}a^{6}+\frac{30387}{3860}a^{5}-\frac{2103}{1930}a^{4}-\frac{1219}{386}a^{3}+\frac{4202}{965}a^{2}-\frac{2791}{1930}a-\frac{58}{193}$, $\frac{323}{3860}a^{15}-\frac{223}{7720}a^{14}-\frac{5333}{7720}a^{13}+\frac{256}{965}a^{12}+\frac{3537}{1930}a^{11}-\frac{1339}{1544}a^{10}-\frac{9637}{1930}a^{9}+\frac{9261}{3860}a^{8}+\frac{13712}{965}a^{7}-\frac{13037}{1930}a^{6}+\frac{871}{3860}a^{5}+\frac{10227}{1930}a^{4}-\frac{6999}{965}a^{3}+\frac{845}{772}a^{2}+\frac{1363}{1930}a-\frac{907}{1930}$, $\frac{43}{3860}a^{15}+\frac{71}{1930}a^{14}-\frac{943}{7720}a^{13}-\frac{249}{965}a^{12}+\frac{461}{965}a^{11}+\frac{823}{1930}a^{10}-\frac{9903}{7720}a^{9}-\frac{4951}{3860}a^{8}+\frac{7239}{1930}a^{7}+\frac{7321}{1930}a^{6}-\frac{9477}{1930}a^{5}+\frac{14777}{1930}a^{4}-\frac{149}{965}a^{3}+\frac{69}{965}a^{2}-\frac{3279}{3860}a-\frac{2003}{1930}$, $\frac{15}{193}a^{15}+\frac{107}{3860}a^{14}-\frac{1169}{1930}a^{13}-\frac{1781}{7720}a^{12}+\frac{10959}{7720}a^{11}+\frac{4783}{7720}a^{10}-\frac{30567}{7720}a^{9}-\frac{1339}{772}a^{8}+\frac{8761}{772}a^{7}+\frac{19487}{3860}a^{6}+\frac{21117}{3860}a^{5}-\frac{2043}{3860}a^{4}-\frac{17143}{3860}a^{3}-\frac{931}{3860}a^{2}-\frac{9461}{3860}a-\frac{309}{193}$
| 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: | \( 79205.5923155 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 79205.5923155 \cdot 1}{2\cdot\sqrt{2393397489569403764736}}\cr\approx \mathstrut & 0.322981668366 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 20*x^12 - 56*x^10 + 160*x^8 + 32*x^6 - 40*x^4 - 16*x^2 + 4)
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 - 8*x^14 + 20*x^12 - 56*x^10 + 160*x^8 + 32*x^6 - 40*x^4 - 16*x^2 + 4, 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 - 8*x^14 + 20*x^12 - 56*x^10 + 160*x^8 + 32*x^6 - 40*x^4 - 16*x^2 + 4);
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 - 8*x^14 + 20*x^12 - 56*x^10 + 160*x^8 + 32*x^6 - 40*x^4 - 16*x^2 + 4);
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
$C_4^2:C_2$ (as 16T17):
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 32 |
| The 20 conjugacy class representatives for $C_4^2:C_2$ |
| Character table for $C_4^2:C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{16})^+\), 4.4.18432.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.4.12230590464.2, 8.4.764411904.3, \(\Q(\zeta_{48})^+\) |
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
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.0.2393397489569403764736.3 |
| 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 | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.52.1 | $x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 8 x^{5} + 30$ | $16$ | $1$ | $52$ | $C_4^2:C_2$ | $[2, 3, 3, 4]^{2}$ |
|
\(3\)
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 36 x^{12} + 120 x^{11} + 312 x^{10} + 352 x^{9} - 522 x^{8} - 2664 x^{7} - 3672 x^{6} - 1440 x^{5} + 4292 x^{4} + 7720 x^{3} + 6408 x^{2} + 2592 x + 433$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $[\ ]_{4}^{4}$ |