Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^12 + 20*x^8 - 16*x^6 - 80*x^4 - 32*x^2 + 2)
gp: K = bnfinit(y^16 - 8*y^12 + 20*y^8 - 16*y^6 - 80*y^4 - 32*y^2 + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^12 + 20*x^8 - 16*x^6 - 80*x^4 - 32*x^2 + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^12 + 20*x^8 - 16*x^6 - 80*x^4 - 32*x^2 + 2)
\( x^{16} - 8x^{12} + 20x^{8} - 16x^{6} - 80x^{4} - 32x^{2} + 2 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(604462909807314587353088\)
\(\medspace = 2^{79}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.64\) | 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: | $2^{2849/512}\approx 47.32245861429085$ | ||
| 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: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{184193}a^{14}-\frac{68601}{184193}a^{12}-\frac{33957}{184193}a^{10}-\frac{4714}{184193}a^{8}-\frac{57774}{184193}a^{6}+\frac{73377}{184193}a^{4}+\frac{74840}{184193}a^{2}-\frac{87383}{184193}$, $\frac{1}{184193}a^{15}-\frac{68601}{184193}a^{13}-\frac{33957}{184193}a^{11}-\frac{4714}{184193}a^{9}-\frac{57774}{184193}a^{7}+\frac{73377}{184193}a^{5}+\frac{74840}{184193}a^{3}-\frac{87383}{184193}a$
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 $1$
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: | $9$ | 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: |
$\frac{3136}{184193}a^{14}-\frac{4688}{184193}a^{12}+\frac{25598}{184193}a^{10}+\frac{47664}{184193}a^{8}-\frac{66648}{184193}a^{6}-\frac{53215}{184193}a^{4}+\frac{332028}{184193}a^{2}+\frac{138097}{184193}$, $\frac{11524}{184193}a^{14}+\frac{1568}{184193}a^{12}+\frac{94536}{184193}a^{10}-\frac{12799}{184193}a^{8}-\frac{254312}{184193}a^{6}+\frac{217708}{184193}a^{4}+\frac{1040624}{184193}a^{2}+\frac{386947}{184193}$, $\frac{17568}{184193}a^{14}+\frac{7569}{184193}a^{12}+\frac{139642}{184193}a^{10}-\frac{71298}{184193}a^{8}-\frac{298184}{184193}a^{6}+\frac{448057}{184193}a^{4}+\frac{1085672}{184193}a^{2}+\frac{264275}{184193}$, $\frac{11524}{184193}a^{14}-\frac{1568}{184193}a^{12}-\frac{94536}{184193}a^{10}+\frac{12799}{184193}a^{8}+\frac{254312}{184193}a^{6}-\frac{217708}{184193}a^{4}-\frac{856431}{184193}a^{2}-\frac{386947}{184193}$, $\frac{2387}{184193}a^{14}+\frac{3010}{184193}a^{12}+\frac{10439}{184193}a^{10}+\frac{16545}{184193}a^{8}-\frac{54019}{184193}a^{6}+\frac{16644}{184193}a^{4}+\frac{208323}{184193}a^{2}+\frac{76745}{184193}$, $\frac{27752}{184193}a^{15}-\frac{7032}{184193}a^{14}+\frac{3896}{184193}a^{13}+\frac{765}{184193}a^{12}-\frac{227469}{184193}a^{11}+\frac{71496}{184193}a^{10}-\frac{45898}{184193}a^{9}-\frac{5892}{184193}a^{8}+\frac{608596}{184193}a^{7}-\frac{247183}{184193}a^{6}-\frac{263497}{184193}a^{5}+\frac{121722}{184193}a^{4}-\frac{2395097}{184193}a^{3}+\frac{701293}{184193}a^{2}-\frac{1257329}{184193}a+\frac{193601}{184193}$, $\frac{39107}{184193}a^{15}+\frac{2908}{184193}a^{14}-\frac{8262}{184193}a^{13}-\frac{10689}{184193}a^{12}-\frac{293255}{184193}a^{11}-\frac{19508}{184193}a^{10}+\frac{26795}{184193}a^{9}+\frac{106163}{184193}a^{8}+\frac{680292}{184193}a^{7}-\frac{22776}{184193}a^{6}-\frac{540987}{184193}a^{5}-\frac{283564}{184193}a^{4}-\frac{3005978}{184193}a^{3}+\frac{286980}{184193}a^{2}-\frac{1243603}{184193}a+\frac{260769}{184193}$, $\frac{172669}{184193}a^{15}+\frac{42200}{184193}a^{14}-\frac{1568}{184193}a^{13}-\frac{819}{184193}a^{12}+\frac{1379008}{184193}a^{11}-\frac{332246}{184193}a^{10}+\frac{12799}{184193}a^{9}-\frac{2360}{184193}a^{8}-\frac{3429548}{184193}a^{7}+\frac{836713}{184193}a^{6}+\frac{2729380}{184193}a^{5}-\frac{695895}{184193}a^{4}+\frac{13694816}{184193}a^{3}-\frac{3424845}{184193}a^{2}+\frac{5691422}{184193}a-\frac{1308091}{184193}$, $\frac{40262}{184193}a^{15}-\frac{2261}{184193}a^{14}-\frac{39427}{184193}a^{13}+\frac{16355}{184193}a^{12}-\frac{280481}{184193}a^{11}-\frac{31704}{184193}a^{10}+\frac{292108}{184193}a^{9}-\frac{24840}{184193}a^{8}+\frac{444995}{184193}a^{7}+\frac{218370}{184193}a^{6}-\frac{1071911}{184193}a^{5}-\frac{315890}{184193}a^{4}-\frac{1847137}{184193}a^{3}+\frac{60127}{184193}a^{2}+\frac{56147}{184193}a+\frac{118067}{184193}$
| 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: | \( 597015.9991799914 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 597015.9991799914 \cdot 1}{2\cdot\sqrt{604462909807314587353088}}\cr\approx \mathstrut & 0.377981320719396 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^12 + 20*x^8 - 16*x^6 - 80*x^4 - 32*x^2 + 2)
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^12 + 20*x^8 - 16*x^6 - 80*x^4 - 32*x^2 + 2, 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^12 + 20*x^8 - 16*x^6 - 80*x^4 - 32*x^2 + 2);
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^12 + 20*x^8 - 16*x^6 - 80*x^4 - 32*x^2 + 2);
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
$C_2^7.C_8$ (as 16T1155):
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 1024 |
| The 40 conjugacy class representatives for $C_2^7.C_8$ |
| Character table for $C_2^7.C_8$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.2147483648.2 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $16$ | $16$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ | $16$ | $16$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | $16$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.79.2515 | $x^{16} + 48 x^{14} + 40 x^{12} + 32 x^{11} + 32 x^{9} + 52 x^{8} + 32 x^{7} + 48 x^{6} + 32 x^{5} + 32 x^{4} + 34$ | $16$ | $1$ | $79$ | 16T1155 | $[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$ |