Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 9*x^12 - 14*x^10 - 9*x^8 + 18*x^4 + 16*x^2 + 4)
gp: K = bnfinit(y^16 - 2*y^14 + 9*y^12 - 14*y^10 - 9*y^8 + 18*y^4 + 16*y^2 + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^14 + 9*x^12 - 14*x^10 - 9*x^8 + 18*x^4 + 16*x^2 + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^14 + 9*x^12 - 14*x^10 - 9*x^8 + 18*x^4 + 16*x^2 + 4)
\( x^{16} - 2x^{14} + 9x^{12} - 14x^{10} - 9x^{8} + 18x^{4} + 16x^{2} + 4 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1584616432828678144\)
\(\medspace = 2^{38}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.72\) | 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}$, $\frac{1}{11684}a^{14}-\frac{1}{4}a^{13}-\frac{119}{5842}a^{12}-\frac{1}{4}a^{11}+\frac{339}{5842}a^{10}-\frac{1}{4}a^{9}+\frac{633}{11684}a^{8}+\frac{1354}{2921}a^{6}-\frac{1155}{2921}a^{4}-\frac{23}{127}a^{2}-\frac{755}{2921}$, $\frac{1}{11684}a^{15}+\frac{2683}{11684}a^{13}-\frac{1}{4}a^{12}-\frac{2243}{11684}a^{11}-\frac{1}{4}a^{10}-\frac{572}{2921}a^{9}-\frac{1}{4}a^{8}-\frac{213}{5842}a^{7}-\frac{1155}{2921}a^{5}-\frac{23}{127}a^{3}-\frac{755}{2921}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $7$ | 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: |
$\frac{671}{5842}a^{14}-\frac{982}{2921}a^{12}+\frac{8025}{5842}a^{10}-\frac{8164}{2921}a^{8}+\frac{9175}{5842}a^{6}-\frac{1880}{2921}a^{4}+\frac{122}{127}a^{2}+\frac{377}{2921}$, $\frac{1115}{5842}a^{14}-\frac{1240}{2921}a^{12}+\frac{11115}{5842}a^{10}-\frac{18613}{5842}a^{8}-\frac{894}{2921}a^{6}-\frac{1577}{5842}a^{4}+\frac{272}{127}a^{2}+\frac{4688}{2921}$, $\frac{533}{2921}a^{14}-\frac{1251}{2921}a^{12}+\frac{5012}{2921}a^{10}-\frac{17499}{5842}a^{8}-\frac{7203}{5842}a^{6}+\frac{8649}{5842}a^{4}+\frac{494}{127}a^{2}+\frac{2732}{2921}$, $\frac{4825}{11684}a^{15}+\frac{41}{2921}a^{14}-\frac{12081}{11684}a^{13}-\frac{1059}{11684}a^{12}+\frac{49487}{11684}a^{11}+\frac{3115}{11684}a^{10}-\frac{22923}{2921}a^{9}-\frac{10107}{11684}a^{8}+\frac{467}{5842}a^{7}+\frac{2981}{2921}a^{6}+\frac{3707}{5842}a^{5}-\frac{2031}{5842}a^{4}+\frac{785}{127}a^{3}+\frac{165}{127}a^{2}+\frac{8375}{2921}a-\frac{1138}{2921}$, $\frac{2189}{5842}a^{15}+\frac{1515}{11684}a^{14}-\frac{10851}{11684}a^{13}-\frac{1052}{2921}a^{12}+\frac{44363}{11684}a^{11}+\frac{4126}{2921}a^{10}-\frac{82543}{11684}a^{9}-\frac{34145}{11684}a^{8}-\frac{715}{5842}a^{7}+\frac{4457}{5842}a^{6}-\frac{339}{2921}a^{5}-\frac{3213}{5842}a^{4}+\frac{906}{127}a^{3}+\frac{334}{127}a^{2}+\frac{7024}{2921}a+\frac{4128}{2921}$, $\frac{145}{508}a^{15}-\frac{1197}{5842}a^{14}-\frac{347}{508}a^{13}+\frac{6019}{11684}a^{12}+\frac{1409}{508}a^{11}-\frac{25345}{11684}a^{10}-\frac{644}{127}a^{9}+\frac{47333}{11684}a^{8}-\frac{139}{127}a^{7}-\frac{2087}{2921}a^{6}+\frac{38}{127}a^{5}+\frac{1804}{2921}a^{4}+\frac{765}{127}a^{3}-\frac{437}{127}a^{2}+\frac{380}{127}a-\frac{3550}{2921}$, $\frac{41}{2921}a^{15}+\frac{1783}{11684}a^{14}-\frac{1059}{11684}a^{13}-\frac{1865}{5842}a^{12}+\frac{3115}{11684}a^{11}+\frac{8553}{5842}a^{10}-\frac{10107}{11684}a^{9}-\frac{28077}{11684}a^{8}+\frac{2981}{2921}a^{7}-\frac{1485}{2921}a^{6}-\frac{2031}{5842}a^{5}-\frac{2981}{2921}a^{4}+\frac{165}{127}a^{3}+\frac{393}{127}a^{2}-\frac{1138}{2921}a+\frac{3337}{2921}$
| 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: | \( 349.923818839 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 349.923818839 \cdot 1}{2\cdot\sqrt{1584616432828678144}}\cr\approx \mathstrut & 0.337613946695 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 9*x^12 - 14*x^10 - 9*x^8 + 18*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 - 2*x^14 + 9*x^12 - 14*x^10 - 9*x^8 + 18*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 - 2*x^14 + 9*x^12 - 14*x^10 - 9*x^8 + 18*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 - 2*x^14 + 9*x^12 - 14*x^10 - 9*x^8 + 18*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
$C_4\wr C_2$ (as 16T42):
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 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), 4.0.392.1 x2, 4.2.448.1 x2, \(\Q(\sqrt{2}, \sqrt{-7})\), 8.0.314703872.4, 8.0.78675968.2, 8.0.9834496.2 |
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
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.0.78675968.2, 8.0.314703872.4 |
| Degree 16 sibling: | 16.4.33115249535031967744.3 |
| Minimal sibling: | 8.0.78675968.2 |
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 | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.22.6 | $x^{8} + 16 x^{7} + 76 x^{6} + 88 x^{5} + 8 x^{4} + 192 x^{3} + 280 x^{2} + 48 x + 252$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ |
| 2.8.16.3 | $x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{5} + 36 x^{4} - 32 x^{3} + 88 x^{2} - 32 x + 124$ | $4$ | $2$ | $16$ | $C_4\times C_2$ | $[2, 3]^{2}$ | |
|
\(7\)
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |