Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 + 432*x^12 + 3200*x^10 + 14144*x^8 + 37888*x^6 + 59376*x^4 + 49024*x^2 + 16129)
gp: K = bnfinit(y^16 + 32*y^14 + 432*y^12 + 3200*y^10 + 14144*y^8 + 37888*y^6 + 59376*y^4 + 49024*y^2 + 16129, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 32*x^14 + 432*x^12 + 3200*x^10 + 14144*x^8 + 37888*x^6 + 59376*x^4 + 49024*x^2 + 16129);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 32*x^14 + 432*x^12 + 3200*x^10 + 14144*x^8 + 37888*x^6 + 59376*x^4 + 49024*x^2 + 16129)
\( x^{16} + 32x^{14} + 432x^{12} + 3200x^{10} + 14144x^{8} + 37888x^{6} + 59376x^{4} + 49024x^{2} + 16129 \)
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: |
\(1540690511780295460519936\)
\(\medspace = 2^{64}\cdot 17^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.49\) | 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: | $2^{4}17^{1/2}\approx 65.96969000988257$ | ||
| Ramified primes: |
\(2\), \(17\)
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{3}{7}a^{10}-\frac{1}{7}a^{8}-\frac{3}{7}a^{2}-\frac{2}{7}$, $\frac{1}{7}a^{13}+\frac{3}{7}a^{11}-\frac{1}{7}a^{9}-\frac{3}{7}a^{3}-\frac{2}{7}a$, $\frac{1}{7}a^{14}-\frac{3}{7}a^{10}+\frac{3}{7}a^{8}-\frac{3}{7}a^{4}-\frac{1}{7}$, $\frac{1}{889}a^{15}+\frac{32}{889}a^{13}+\frac{51}{889}a^{11}+\frac{279}{889}a^{9}+\frac{43}{127}a^{7}-\frac{339}{889}a^{5}-\frac{187}{889}a^{3}+\frac{383}{889}a$
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
$C_{2}\times C_{10}$, which has order $20$ (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: |
\( -1 \)
(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: |
$\frac{13}{7}a^{14}+52a^{12}+\frac{4161}{7}a^{10}+\frac{24980}{7}a^{8}+12025a^{6}+\frac{156852}{7}a^{4}+20966a^{2}+\frac{52648}{7}$, $a^{12}+24a^{10}+224a^{8}+1024a^{6}+2367a^{4}+2552a^{2}+999$, $\frac{6}{7}a^{14}+25a^{12}+\frac{2089}{7}a^{10}+\frac{13108}{7}a^{8}+6585a^{6}+\frac{89197}{7}a^{4}+12270a^{2}+\frac{31431}{7}$, $\frac{80}{127}a^{15}+\frac{15507}{889}a^{13}+\frac{175118}{889}a^{11}+\frac{1037874}{889}a^{9}+\frac{493345}{127}a^{7}+\frac{909251}{127}a^{5}+\frac{5911270}{889}a^{3}+\frac{2112876}{889}a$, $\frac{1322}{889}a^{15}+\frac{36843}{889}a^{13}+\frac{419085}{889}a^{11}+\frac{2503454}{889}a^{9}+\frac{1199592}{127}a^{7}+\frac{15588513}{889}a^{5}+\frac{14549683}{889}a^{3}+\frac{5210279}{889}a$, $\frac{151}{889}a^{15}+\frac{4070}{889}a^{13}+\frac{44531}{889}a^{11}+\frac{254473}{889}a^{9}+\frac{116094}{127}a^{7}+\frac{1431663}{889}a^{5}+\frac{1267544}{889}a^{3}+\frac{432737}{889}a$, $\frac{1137}{889}a^{15}+\frac{31685}{889}a^{13}+\frac{51482}{127}a^{11}+\frac{2152373}{889}a^{9}+\frac{1031109}{127}a^{7}+\frac{13394946}{889}a^{5}+\frac{12499065}{889}a^{3}+\frac{4476484}{889}a$
| 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: | \( 12675.9605024 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 12675.9605024 \cdot 20}{2\cdot\sqrt{1540690511780295460519936}}\cr\approx \mathstrut & 0.248063073409 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 + 432*x^12 + 3200*x^10 + 14144*x^8 + 37888*x^6 + 59376*x^4 + 49024*x^2 + 16129)
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 + 32*x^14 + 432*x^12 + 3200*x^10 + 14144*x^8 + 37888*x^6 + 59376*x^4 + 49024*x^2 + 16129, 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 + 32*x^14 + 432*x^12 + 3200*x^10 + 14144*x^8 + 37888*x^6 + 59376*x^4 + 49024*x^2 + 16129);
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 + 32*x^14 + 432*x^12 + 3200*x^10 + 14144*x^8 + 37888*x^6 + 59376*x^4 + 49024*x^2 + 16129);
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
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.34816.1, 4.4.4352.1, 8.0.620622774272.25, 8.0.2147483648.1, 8.8.4848615424.1 |
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: | deg 16 |
| 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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.64.2 | $x^{16} + 16 x^{13} + 20 x^{12} + 22 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 2$ | $16$ | $1$ | $64$ | $C_8\times C_2$ | $[2, 3, 4, 5]$ |
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |