Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 56*x^12 - 184*x^10 + 344*x^8 - 352*x^6 + 264*x^4 - 96*x^2 + 16)
gp: K = bnfinit(y^16 - 8*y^14 + 56*y^12 - 184*y^10 + 344*y^8 - 352*y^6 + 264*y^4 - 96*y^2 + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 56*x^12 - 184*x^10 + 344*x^8 - 352*x^6 + 264*x^4 - 96*x^2 + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 56*x^12 - 184*x^10 + 344*x^8 - 352*x^6 + 264*x^4 - 96*x^2 + 16)
\( x^{16} - 8x^{14} + 56x^{12} - 184x^{10} + 344x^{8} - 352x^{6} + 264x^{4} - 96x^{2} + 16 \)
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: |
\(4294967296000000000000\)
\(\medspace = 2^{44}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.49\) | 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: | $2^{3}5^{3/4}\approx 26.74961219905688$ | ||
| Ramified primes: |
\(2\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{249608}a^{14}+\frac{2979}{124804}a^{12}-\frac{2957}{31201}a^{10}+\frac{5361}{62402}a^{8}+\frac{2787}{62402}a^{6}-\frac{2979}{62402}a^{4}+\frac{5961}{31201}a^{2}-\frac{5826}{31201}$, $\frac{1}{249608}a^{15}+\frac{2979}{124804}a^{13}-\frac{2957}{31201}a^{11}+\frac{5361}{62402}a^{9}+\frac{2787}{62402}a^{7}-\frac{2979}{62402}a^{5}+\frac{5961}{31201}a^{3}-\frac{5826}{31201}a$
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{1826}{31201} a^{14} + \frac{109929}{249608} a^{12} - \frac{191063}{62402} a^{10} + \frac{1156481}{124804} a^{8} - \frac{481211}{31201} a^{6} + \frac{385931}{31201} a^{4} - \frac{277106}{31201} a^{2} + \frac{52282}{31201} \)
(order $10$)
| 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: |
$\frac{1129}{31201}a^{14}-\frac{71471}{249608}a^{12}+\frac{62834}{31201}a^{10}-\frac{409013}{62402}a^{8}+\frac{386501}{31201}a^{6}-\frac{791091}{62402}a^{4}+\frac{298836}{31201}a^{2}-\frac{109149}{31201}$, $\frac{2445}{124804}a^{14}-\frac{38315}{249608}a^{12}+\frac{66269}{62402}a^{10}-\frac{211957}{62402}a^{8}+\frac{368005}{62402}a^{6}-\frac{370855}{62402}a^{4}+\frac{132360}{31201}a^{2}-\frac{33828}{31201}$, $\frac{4069}{124804}a^{14}-\frac{7819}{31201}a^{12}+\frac{54307}{31201}a^{10}-\frac{334391}{62402}a^{8}+\frac{559097}{62402}a^{6}-\frac{374337}{62402}a^{4}+\frac{86666}{31201}a^{2}-\frac{17669}{31201}$, $\frac{23117}{249608}a^{15}+\frac{1036}{31201}a^{14}-\frac{88667}{124804}a^{13}-\frac{73681}{249608}a^{12}+\frac{610507}{124804}a^{11}+\frac{126343}{62402}a^{10}-\frac{936165}{62402}a^{9}-\frac{901341}{124804}a^{8}+\frac{747331}{31201}a^{7}+\frac{410571}{31201}a^{6}-\frac{1034569}{62402}a^{5}-\frac{332591}{31201}a^{4}+\frac{235228}{31201}a^{3}+\frac{75987}{31201}a^{2}-\frac{47327}{31201}a-\frac{17941}{31201}$, $\frac{17}{761}a^{15}-\frac{6093}{249608}a^{14}-\frac{469}{3044}a^{13}+\frac{47071}{249608}a^{12}+\frac{1595}{1522}a^{11}-\frac{162311}{124804}a^{10}-\frac{8255}{3044}a^{9}+\frac{126241}{31201}a^{8}+\frac{2310}{761}a^{7}-\frac{206730}{31201}a^{6}+\frac{469}{1522}a^{5}+\frac{167638}{31201}a^{4}-\frac{530}{761}a^{3}-\frac{96012}{31201}a^{2}+\frac{626}{761}a+\frac{22281}{31201}$, $\frac{148665}{249608}a^{15}+\frac{90511}{249608}a^{14}-\frac{1142355}{249608}a^{13}-\frac{87496}{31201}a^{12}+\frac{3978865}{124804}a^{11}+\frac{1219141}{62402}a^{10}-\frac{12397355}{124804}a^{9}-\frac{7661207}{124804}a^{8}+\frac{5372210}{31201}a^{7}+\frac{6702287}{62402}a^{6}-\frac{4729972}{31201}a^{5}-\frac{6046221}{62402}a^{4}+\frac{3203765}{31201}a^{3}+\frac{2067645}{31201}a^{2}-\frac{606550}{31201}a-\frac{457000}{31201}$, $\frac{6690}{31201}a^{15}-\frac{2727}{62402}a^{14}-\frac{219437}{124804}a^{13}+\frac{47711}{124804}a^{12}+\frac{765689}{62402}a^{11}-\frac{169761}{62402}a^{10}-\frac{5155517}{124804}a^{9}+\frac{308453}{31201}a^{8}+\frac{4855615}{62402}a^{7}-\frac{660632}{31201}a^{6}-\frac{2495565}{31201}a^{5}+\frac{1574741}{62402}a^{4}+\frac{1749751}{31201}a^{3}-\frac{561322}{31201}a^{2}-\frac{608746}{31201}a+\frac{274380}{31201}$
| 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: | \( 73929.9567996 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 73929.9567996 \cdot 1}{10\cdot\sqrt{4294967296000000000000}}\cr\approx \mathstrut & 0.274018237543 \end{aligned}\] (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 + 56*x^12 - 184*x^10 + 344*x^8 - 352*x^6 + 264*x^4 - 96*x^2 + 16)
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 + 56*x^12 - 184*x^10 + 344*x^8 - 352*x^6 + 264*x^4 - 96*x^2 + 16, 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 + 56*x^12 - 184*x^10 + 344*x^8 - 352*x^6 + 264*x^4 - 96*x^2 + 16);
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 + 56*x^12 - 184*x^10 + 344*x^8 - 352*x^6 + 264*x^4 - 96*x^2 + 16);
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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), 4.0.8000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.2621440000.2, 8.0.64000000.2, 8.4.65536000000.1 |
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: | 16.8.68719476736000000000000.1 |
| 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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $44$ | |||
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |