Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 12*x^12 - 8*x^10 + 42*x^8 - 72*x^6 + 36*x^4 - 8*x^2 + 1)
gp: K = bnfinit(y^16 - 8*y^14 + 12*y^12 - 8*y^10 + 42*y^8 - 72*y^6 + 36*y^4 - 8*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 12*x^12 - 8*x^10 + 42*x^8 - 72*x^6 + 36*x^4 - 8*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 12*x^12 - 8*x^10 + 42*x^8 - 72*x^6 + 36*x^4 - 8*x^2 + 1)
\( x^{16} - 8x^{14} + 12x^{12} - 8x^{10} + 42x^{8} - 72x^{6} + 36x^{4} - 8x^{2} + 1 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(73786976294838206464\)
\(\medspace = 2^{66}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.45\) | 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: | $2^{67/16}\approx 18.220618156107065$ | ||
| Ramified primes: |
\(2\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{1028}a^{14}-\frac{25}{1028}a^{12}-\frac{77}{1028}a^{10}-\frac{241}{1028}a^{8}-\frac{487}{1028}a^{6}-\frac{17}{1028}a^{4}-\frac{189}{1028}a^{2}-\frac{393}{1028}$, $\frac{1}{1028}a^{15}-\frac{25}{1028}a^{13}-\frac{77}{1028}a^{11}+\frac{4}{257}a^{9}-\frac{1}{4}a^{8}+\frac{27}{1028}a^{7}-\frac{1}{2}a^{6}+\frac{497}{1028}a^{5}-\frac{1}{2}a^{4}+\frac{325}{1028}a^{3}-\frac{1}{2}a^{2}+\frac{189}{514}a+\frac{1}{4}$
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: |
$\frac{111}{514}a^{14}-\frac{1695}{1028}a^{12}+\frac{481}{257}a^{10}-\frac{303}{1028}a^{8}+\frac{4539}{514}a^{6}-\frac{12255}{1028}a^{4}-\frac{81}{257}a^{2}-\frac{123}{1028}$, $\frac{99}{1028}a^{15}-\frac{933}{1028}a^{13}+\frac{2143}{1028}a^{11}-\frac{1243}{1028}a^{9}+\frac{4215}{1028}a^{7}-\frac{12477}{1028}a^{5}+\frac{7503}{1028}a^{3}+\frac{157}{1028}a$, $a$, $\frac{122}{257}a^{15}-\frac{994}{257}a^{13}+\frac{6371}{1028}a^{11}-\frac{4271}{1028}a^{9}+\frac{5093}{257}a^{7}-\frac{9527}{257}a^{5}+\frac{19563}{1028}a^{3}-\frac{2375}{1028}a$, $\frac{99}{1028}a^{15}+\frac{51}{1028}a^{14}-\frac{933}{1028}a^{13}-\frac{247}{1028}a^{12}+\frac{2143}{1028}a^{11}-\frac{293}{514}a^{10}-\frac{1243}{1028}a^{9}+\frac{204}{257}a^{8}+\frac{4215}{1028}a^{7}+\frac{1891}{1028}a^{6}-\frac{12477}{1028}a^{5}+\frac{2217}{1028}a^{4}+\frac{7503}{1028}a^{3}-\frac{3149}{514}a^{2}+\frac{157}{1028}a+\frac{65}{257}$, $a^{15}+\frac{383}{1028}a^{14}-8a^{13}-\frac{2893}{1028}a^{12}+12a^{11}+\frac{787}{257}a^{10}-8a^{9}-\frac{277}{514}a^{8}+42a^{7}+\frac{14967}{1028}a^{6}-72a^{5}-\frac{20389}{1028}a^{4}+36a^{3}-\frac{171}{257}a^{2}-8a+\frac{941}{514}$, $\frac{333}{1028}a^{15}+\frac{573}{1028}a^{14}-\frac{1207}{514}a^{13}-\frac{2151}{514}a^{12}+\frac{2115}{1028}a^{11}+\frac{4709}{1028}a^{10}-\frac{583}{1028}a^{9}-\frac{1883}{1028}a^{8}+\frac{13103}{1028}a^{7}+\frac{22667}{1028}a^{6}-\frac{3407}{257}a^{5}-\frac{7511}{257}a^{4}-\frac{1771}{1028}a^{3}+\frac{4783}{1028}a^{2}-\frac{1341}{1028}a-\frac{1085}{1028}$, $\frac{32}{257}a^{15}-\frac{105}{514}a^{14}-\frac{286}{257}a^{13}+\frac{413}{257}a^{12}+\frac{620}{257}a^{11}-\frac{1167}{514}a^{10}-\frac{2321}{1028}a^{9}+\frac{1523}{1028}a^{8}+\frac{3013}{514}a^{7}-\frac{2060}{257}a^{6}-\frac{6999}{514}a^{5}+\frac{6925}{514}a^{4}+\frac{6151}{514}a^{3}-\frac{1514}{257}a^{2}-\frac{1731}{1028}a+\frac{1061}{1028}$, $\frac{461}{1028}a^{15}+\frac{783}{1028}a^{14}-\frac{1779}{514}a^{13}-\frac{2977}{514}a^{12}+\frac{4595}{1028}a^{11}+\frac{7043}{1028}a^{10}-\frac{726}{257}a^{9}-\frac{1703}{514}a^{8}+\frac{19129}{1028}a^{7}+\frac{30907}{1028}a^{6}-\frac{13813}{514}a^{5}-\frac{21947}{514}a^{4}+\frac{10531}{1028}a^{3}+\frac{10839}{1028}a^{2}-\frac{768}{257}a-\frac{559}{514}$, $\frac{1621}{1028}a^{15}-\frac{95}{514}a^{14}-\frac{12769}{1028}a^{13}+\frac{1409}{1028}a^{12}+\frac{8909}{514}a^{11}-\frac{1561}{1028}a^{10}-\frac{10301}{1028}a^{9}+\frac{1329}{1028}a^{8}+\frac{66383}{1028}a^{7}-\frac{1925}{257}a^{6}-\frac{108255}{1028}a^{5}+\frac{9655}{1028}a^{4}+\frac{10595}{257}a^{3}-\frac{5467}{1028}a^{2}-\frac{5347}{1028}a+\frac{1939}{1028}$, $\frac{1621}{1028}a^{15}+\frac{95}{514}a^{14}-\frac{12769}{1028}a^{13}-\frac{1409}{1028}a^{12}+\frac{8909}{514}a^{11}+\frac{1561}{1028}a^{10}-\frac{10301}{1028}a^{9}-\frac{1329}{1028}a^{8}+\frac{66383}{1028}a^{7}+\frac{1925}{257}a^{6}-\frac{108255}{1028}a^{5}-\frac{9655}{1028}a^{4}+\frac{10595}{257}a^{3}+\frac{5467}{1028}a^{2}-\frac{5347}{1028}a-\frac{1939}{1028}$
| 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: | \( 13514.1847612 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 13514.1847612 \cdot 1}{2\cdot\sqrt{73786976294838206464}}\cr\approx \mathstrut & 0.313855279363 \end{aligned}\] (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 + 12*x^12 - 8*x^10 + 42*x^8 - 72*x^6 + 36*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 - 8*x^14 + 12*x^12 - 8*x^10 + 42*x^8 - 72*x^6 + 36*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 - 8*x^14 + 12*x^12 - 8*x^10 + 42*x^8 - 72*x^6 + 36*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 - 8*x^14 + 12*x^12 - 8*x^10 + 42*x^8 - 72*x^6 + 36*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
$C_2^2:C_8$ (as 16T24):
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 $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.2.1024.1, 4.2.2048.1, 8.4.2147483648.1, \(\Q(\zeta_{32})^+\), 8.4.67108864.1 |
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
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.0.73786976294838206464.5 |
| 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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.66.2 | $x^{16} + 4 x^{12} + 16 x^{9} + 8 x^{8} + 16 x^{7} + 8 x^{6} + 24 x^{4} + 16 x^{3} + 2$ | $16$ | $1$ | $66$ | $C_2^2 : C_8$ | $[2, 3, 7/2, 4, 5]$ |