Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 + 26*x^12 - 32*x^10 + 31*x^8 - 32*x^6 + 26*x^4 - 10*x^2 + 1)
gp: K = bnfinit(y^16 - 10*y^14 + 26*y^12 - 32*y^10 + 31*y^8 - 32*y^6 + 26*y^4 - 10*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 10*x^14 + 26*x^12 - 32*x^10 + 31*x^8 - 32*x^6 + 26*x^4 - 10*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 10*x^14 + 26*x^12 - 32*x^10 + 31*x^8 - 32*x^6 + 26*x^4 - 10*x^2 + 1)
\( x^{16} - 10x^{14} + 26x^{12} - 32x^{10} + 31x^{8} - 32x^{6} + 26x^{4} - 10x^{2} + 1 \)
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: |
\(29960650073923649536\)
\(\medspace = 2^{32}\cdot 17^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.49\) | 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: | $2^{2}17^{1/2}\approx 16.492422502470642$ | ||
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{6}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{7}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{104}a^{14}-\frac{1}{8}a^{13}+\frac{1}{52}a^{12}-\frac{1}{8}a^{11}-\frac{15}{104}a^{10}-\frac{17}{104}a^{8}-\frac{3}{8}a^{7}+\frac{35}{104}a^{6}+\frac{37}{104}a^{4}-\frac{1}{8}a^{3}-\frac{25}{52}a^{2}+\frac{3}{8}a+\frac{1}{104}$, $\frac{1}{104}a^{15}-\frac{11}{104}a^{13}-\frac{1}{8}a^{12}+\frac{3}{13}a^{11}-\frac{1}{8}a^{10}-\frac{17}{104}a^{9}+\frac{6}{13}a^{7}-\frac{3}{8}a^{6}+\frac{37}{104}a^{5}-\frac{11}{104}a^{3}-\frac{1}{8}a^{2}+\frac{5}{13}a+\frac{3}{8}$
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 $1$ (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: |
$\frac{67}{52}a^{15}-\frac{323}{26}a^{13}+\frac{1517}{52}a^{11}-\frac{1659}{52}a^{9}+\frac{1591}{52}a^{7}-\frac{1629}{52}a^{5}+\frac{313}{13}a^{3}-\frac{349}{52}a$, $\frac{6}{13}a^{14}-\frac{225}{52}a^{12}+\frac{485}{52}a^{10}-\frac{243}{26}a^{8}+\frac{515}{52}a^{6}-\frac{245}{26}a^{4}+\frac{321}{52}a^{2}-\frac{67}{52}$, $\frac{13}{8}a^{15}-\frac{3}{52}a^{14}-\frac{125}{8}a^{13}+\frac{27}{104}a^{12}+\frac{145}{4}a^{11}+\frac{129}{104}a^{10}-\frac{305}{8}a^{9}-\frac{131}{52}a^{8}+\frac{143}{4}a^{7}+\frac{115}{104}a^{6}-\frac{307}{8}a^{5}-\frac{85}{52}a^{4}+\frac{219}{8}a^{3}+\frac{131}{104}a^{2}-\frac{25}{4}a-\frac{71}{104}$, $a^{15}-9a^{13}+17a^{11}-15a^{9}+16a^{7}-16a^{5}+10a^{3}$, $\frac{33}{52}a^{15}-\frac{68}{13}a^{13}+\frac{337}{52}a^{11}-\frac{93}{52}a^{9}+\frac{271}{52}a^{7}-\frac{183}{52}a^{5}-\frac{19}{26}a^{3}+\frac{59}{52}a$, $\frac{7}{13}a^{15}-\frac{243}{52}a^{13}+\frac{373}{52}a^{11}-\frac{43}{26}a^{9}+\frac{83}{52}a^{7}-\frac{93}{26}a^{5}-\frac{87}{52}a^{3}+\frac{275}{52}a$, $\frac{63}{104}a^{15}+\frac{35}{104}a^{14}-\frac{72}{13}a^{13}-\frac{333}{104}a^{12}+\frac{1161}{104}a^{11}+\frac{92}{13}a^{10}-\frac{1175}{104}a^{9}-\frac{647}{104}a^{8}+\frac{1243}{104}a^{7}+\frac{67}{13}a^{6}-\frac{1101}{104}a^{5}-\frac{733}{104}a^{4}+\frac{207}{26}a^{3}+\frac{343}{104}a^{2}-\frac{119}{104}a+\frac{6}{13}$, $\frac{15}{104}a^{15}-\frac{7}{13}a^{14}-\frac{165}{104}a^{13}+\frac{525}{104}a^{12}+\frac{129}{26}a^{11}-\frac{1123}{104}a^{10}-\frac{671}{104}a^{9}+\frac{132}{13}a^{8}+\frac{167}{26}a^{7}-\frac{1089}{104}a^{6}-\frac{693}{104}a^{5}+\frac{157}{13}a^{4}+\frac{511}{104}a^{3}-\frac{723}{104}a^{2}-\frac{42}{13}a+\frac{217}{104}$, $\frac{49}{52}a^{15}+\frac{89}{104}a^{14}-\frac{909}{104}a^{13}-\frac{405}{52}a^{12}+\frac{1897}{104}a^{11}+\frac{1577}{104}a^{10}-\frac{911}{52}a^{9}-\frac{1357}{104}a^{8}+\frac{1779}{104}a^{7}+\frac{1451}{104}a^{6}-\frac{917}{52}a^{5}-\frac{1543}{104}a^{4}+\frac{1171}{104}a^{3}+\frac{427}{52}a^{2}-\frac{71}{104}a-\frac{15}{104}$, $\frac{83}{104}a^{15}-\frac{15}{104}a^{14}-\frac{106}{13}a^{13}+\frac{165}{104}a^{12}+\frac{2317}{104}a^{11}-\frac{129}{26}a^{10}-\frac{2919}{104}a^{9}+\frac{671}{104}a^{8}+\frac{2671}{104}a^{7}-\frac{167}{26}a^{6}-\frac{2805}{104}a^{5}+\frac{693}{104}a^{4}+\frac{581}{26}a^{3}-\frac{511}{104}a^{2}-\frac{775}{104}a+\frac{29}{13}$, $\frac{49}{52}a^{15}+\frac{113}{104}a^{14}-\frac{909}{104}a^{13}-\frac{537}{52}a^{12}+\frac{1897}{104}a^{11}+\frac{2413}{104}a^{10}-\frac{911}{52}a^{9}-\frac{2493}{104}a^{8}+\frac{1779}{104}a^{7}+\frac{2447}{104}a^{6}-\frac{917}{52}a^{5}-\frac{2527}{104}a^{4}+\frac{1171}{104}a^{3}+\frac{841}{52}a^{2}-\frac{71}{104}a-\frac{303}{104}$
| 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: | \( 6607.13613637 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 6607.13613637 \cdot 1}{2\cdot\sqrt{29960650073923649536}}\cr\approx \mathstrut & 0.240805877183 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 + 26*x^12 - 32*x^10 + 31*x^8 - 32*x^6 + 26*x^4 - 10*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 - 10*x^14 + 26*x^12 - 32*x^10 + 31*x^8 - 32*x^6 + 26*x^4 - 10*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 - 10*x^14 + 26*x^12 - 32*x^10 + 31*x^8 - 32*x^6 + 26*x^4 - 10*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 - 10*x^14 + 26*x^12 - 32*x^10 + 31*x^8 - 32*x^6 + 26*x^4 - 10*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
$C_2^2\wr C_2$ (as 16T39):
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 $C_2^2\wr C_2$ |
| Character table for $C_2^2\wr C_2$ |
Intermediate fields
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 | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
|
\(17\)
| 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 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 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 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |