Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 32*x^12 - 100*x^10 + 141*x^8 - 120*x^6 + 92*x^4 - 32*x^2 + 4)
gp: K = bnfinit(y^16 - 4*y^14 + 32*y^12 - 100*y^10 + 141*y^8 - 120*y^6 + 92*y^4 - 32*y^2 + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 32*x^12 - 100*x^10 + 141*x^8 - 120*x^6 + 92*x^4 - 32*x^2 + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 + 32*x^12 - 100*x^10 + 141*x^8 - 120*x^6 + 92*x^4 - 32*x^2 + 4)
\( x^{16} - 4x^{14} + 32x^{12} - 100x^{10} + 141x^{8} - 120x^{6} + 92x^{4} - 32x^{2} + 4 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1622647227216566419456\)
\(\medspace = 2^{48}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.17\) | 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: | $2^{3}7^{1/2}\approx 21.166010488516726$ | ||
| Ramified primes: |
\(2\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{56}a^{12}-\frac{1}{2}a^{10}-\frac{1}{4}a^{8}+\frac{3}{14}a^{6}-\frac{1}{8}a^{4}+\frac{11}{28}$, $\frac{1}{56}a^{13}-\frac{1}{2}a^{11}-\frac{1}{4}a^{9}+\frac{3}{14}a^{7}-\frac{1}{8}a^{5}+\frac{11}{28}a$, $\frac{1}{33544}a^{14}+\frac{1}{16772}a^{12}+\frac{431}{2396}a^{10}+\frac{320}{4193}a^{8}-\frac{1271}{33544}a^{6}-\frac{211}{2396}a^{4}+\frac{3763}{16772}a^{2}+\frac{2895}{8386}$, $\frac{1}{67088}a^{15}+\frac{1}{33544}a^{13}-\frac{1965}{4792}a^{11}+\frac{160}{4193}a^{9}-\frac{1271}{67088}a^{7}-\frac{211}{4792}a^{5}-\frac{13009}{33544}a^{3}-\frac{5491}{16772}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $7$ |
Class group and class number
$C_{2}$, which has order $2$
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: |
$\frac{613}{4193}a^{14}-\frac{19543}{33544}a^{12}+\frac{5567}{1198}a^{10}-\frac{242995}{16772}a^{8}+\frac{165077}{8386}a^{6}-\frac{75417}{4792}a^{4}+\frac{43068}{4193}a^{2}-\frac{38573}{16772}$, $\frac{811}{4792}a^{14}-\frac{2624}{4193}a^{12}+\frac{12451}{2396}a^{10}-\frac{18263}{1198}a^{8}+\frac{614661}{33544}a^{6}-\frac{14599}{1198}a^{4}+\frac{20853}{2396}a^{2}-\frac{5914}{4193}$, $\frac{75}{2396}a^{14}-\frac{673}{8386}a^{12}+\frac{1051}{1198}a^{10}-\frac{1118}{599}a^{8}+\frac{25169}{16772}a^{6}-\frac{1757}{1198}a^{4}+\frac{695}{1198}a^{2}+\frac{1410}{4193}$, $\frac{4847}{67088}a^{15}+\frac{104}{599}a^{14}-\frac{11925}{33544}a^{13}-\frac{12745}{16772}a^{12}+\frac{11725}{4792}a^{11}+\frac{3378}{599}a^{10}-\frac{37922}{4193}a^{9}-\frac{22793}{1198}a^{8}+\frac{883703}{67088}a^{7}+\frac{109185}{4193}a^{6}-\frac{52337}{4792}a^{5}-\frac{48235}{2396}a^{4}+\frac{276449}{33544}a^{3}+\frac{9395}{599}a^{2}-\frac{31257}{16772}a-\frac{31919}{8386}$, $\frac{2687}{8386}a^{15}-\frac{166}{4193}a^{14}-\frac{3171}{2396}a^{13}+\frac{6329}{33544}a^{12}+\frac{6220}{599}a^{11}-\frac{1659}{1198}a^{10}-\frac{278731}{8386}a^{9}+\frac{82185}{16772}a^{8}+\frac{57721}{1198}a^{7}-\frac{71005}{8386}a^{6}-\frac{97651}{2396}a^{5}+\frac{38687}{4792}a^{4}+\frac{123455}{4193}a^{3}-\frac{24960}{4193}a^{2}-\frac{11751}{1198}a+\frac{57747}{16772}$, $\frac{2647}{67088}a^{15}+\frac{4793}{33544}a^{14}-\frac{8135}{33544}a^{13}-\frac{19765}{33544}a^{12}+\frac{7549}{4792}a^{11}+\frac{11213}{2396}a^{10}-\frac{54455}{8386}a^{9}-\frac{250899}{16772}a^{8}+\frac{871791}{67088}a^{7}+\frac{767845}{33544}a^{6}-\frac{58951}{4792}a^{5}-\frac{105247}{4792}a^{4}+\frac{182585}{33544}a^{3}+\frac{257739}{16772}a^{2}-\frac{11323}{16772}a-\frac{43927}{16772}$, $\frac{10299}{67088}a^{15}+\frac{271}{4792}a^{14}-\frac{3565}{8386}a^{13}-\frac{7587}{33544}a^{12}+\frac{20645}{4792}a^{11}+\frac{4165}{2396}a^{10}-\frac{163563}{16772}a^{9}-\frac{13119}{2396}a^{8}+\frac{476123}{67088}a^{7}+\frac{202949}{33544}a^{6}-\frac{5649}{2396}a^{5}-\frac{12849}{4792}a^{4}+\frac{28589}{33544}a^{3}+\frac{8661}{2396}a^{2}+\frac{34333}{8386}a+\frac{23727}{16772}$
| 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: | \( 31437.5544959 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 31437.5544959 \cdot 2}{2\cdot\sqrt{1622647227216566419456}}\cr\approx \mathstrut & 1.89572634749 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 32*x^12 - 100*x^10 + 141*x^8 - 120*x^6 + 92*x^4 - 32*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 - 4*x^14 + 32*x^12 - 100*x^10 + 141*x^8 - 120*x^6 + 92*x^4 - 32*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 - 4*x^14 + 32*x^12 - 100*x^10 + 141*x^8 - 120*x^6 + 92*x^4 - 32*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 - 4*x^14 + 32*x^12 - 100*x^10 + 141*x^8 - 120*x^6 + 92*x^4 - 32*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
$\SD_{16}$ (as 16T12):
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 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.2.1792.1 x2, 4.0.1568.1 x2, 8.0.157351936.3, 8.2.5754585088.2 x4 |
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 sibling: | 8.2.5754585088.2 |
| Minimal sibling: | 8.2.5754585088.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.24.4 | $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $Q_8$ | $[2, 3, 4]$ |
| 2.8.24.4 | $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $Q_8$ | $[2, 3, 4]$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |