Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 200*x^12 + 696*x^10 + 1046*x^8 + 696*x^6 + 200*x^4 + 24*x^2 + 1)
gp: K = bnfinit(y^16 + 24*y^14 + 200*y^12 + 696*y^10 + 1046*y^8 + 696*y^6 + 200*y^4 + 24*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 24*x^14 + 200*x^12 + 696*x^10 + 1046*x^8 + 696*x^6 + 200*x^4 + 24*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 24*x^14 + 200*x^12 + 696*x^10 + 1046*x^8 + 696*x^6 + 200*x^4 + 24*x^2 + 1)
\( x^{16} + 24x^{14} + 200x^{12} + 696x^{10} + 1046x^{8} + 696x^{6} + 200x^{4} + 24x^{2} + 1 \)
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: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1661590760669764013522944\)
\(\medspace = 2^{58}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.64\) | 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: | $2^{63/16}7^{1/2}\approx 40.53728216620807$ | ||
| 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 a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{12}a^{12}+\frac{1}{12}a^{8}-\frac{5}{12}a^{4}-\frac{5}{12}$, $\frac{1}{12}a^{13}+\frac{1}{12}a^{9}-\frac{5}{12}a^{5}-\frac{5}{12}a$, $\frac{1}{228}a^{14}+\frac{1}{38}a^{12}-\frac{41}{228}a^{10}-\frac{4}{19}a^{8}-\frac{47}{228}a^{6}+\frac{5}{19}a^{4}-\frac{101}{228}a^{2}+\frac{3}{38}$, $\frac{1}{228}a^{15}+\frac{1}{38}a^{13}+\frac{4}{57}a^{11}-\frac{1}{4}a^{10}+\frac{3}{76}a^{9}-\frac{1}{4}a^{8}-\frac{47}{228}a^{7}+\frac{5}{19}a^{5}+\frac{35}{114}a^{3}+\frac{1}{4}a^{2}-\frac{13}{76}a+\frac{1}{4}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{159}{76}a^{15}+\frac{11393}{228}a^{13}+\frac{31367}{76}a^{11}+\frac{321479}{228}a^{9}+\frac{154825}{76}a^{7}+\frac{286127}{228}a^{5}+\frac{23461}{76}a^{3}+\frac{5489}{228}a$, $a$, $\frac{153}{38}a^{14}+\frac{1827}{19}a^{12}+\frac{30169}{38}a^{10}+\frac{102913}{38}a^{8}+\frac{147735}{38}a^{6}+\frac{44281}{19}a^{4}+\frac{19735}{38}a^{2}+\frac{1329}{38}$, $\frac{107}{228}a^{15}+\frac{2485}{228}a^{13}+\frac{19439}{228}a^{11}+\frac{58609}{228}a^{9}+\frac{60179}{228}a^{7}+\frac{7693}{228}a^{5}-\frac{9553}{228}a^{3}-\frac{563}{228}a$, $\frac{1}{38}a^{15}+\frac{169}{228}a^{13}+\frac{301}{38}a^{11}+\frac{9307}{228}a^{9}+\frac{4095}{38}a^{7}+\frac{32071}{228}a^{5}+\frac{2901}{38}a^{3}+\frac{1381}{228}a$, $\frac{4231}{228}a^{15}+\frac{100987}{228}a^{13}+\frac{832921}{228}a^{11}+\frac{2835487}{228}a^{9}+\frac{4055167}{228}a^{7}+\frac{2419423}{228}a^{5}+\frac{536881}{228}a^{3}+\frac{33883}{228}a$, $\frac{515}{57}a^{15}+\frac{49163}{228}a^{13}+\frac{202699}{114}a^{11}+\frac{1379453}{228}a^{9}+\frac{492728}{57}a^{7}+\frac{1173977}{228}a^{5}+\frac{130297}{114}a^{3}+\frac{17111}{228}a$
| 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: | \( 22325.8162501 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22325.8162501 \cdot 20}{2\cdot\sqrt{1661590760669764013522944}}\cr\approx \mathstrut & 0.420711379006 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 200*x^12 + 696*x^10 + 1046*x^8 + 696*x^6 + 200*x^4 + 24*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 + 24*x^14 + 200*x^12 + 696*x^10 + 1046*x^8 + 696*x^6 + 200*x^4 + 24*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 + 24*x^14 + 200*x^12 + 696*x^10 + 1046*x^8 + 696*x^6 + 200*x^4 + 24*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 + 24*x^14 + 200*x^12 + 696*x^10 + 1046*x^8 + 696*x^6 + 200*x^4 + 24*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
$C_4\wr C_2$ (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
| \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), 4.4.25088.1 x2, 4.4.7168.1 x2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.0.322256764928.1, 8.0.322256764928.2, 8.8.10070523904.1 |
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 8 siblings: | 8.0.322256764928.2, 8.0.322256764928.1 |
| Degree 16 sibling: | 16.0.542560248381963759517696.3 |
| Minimal sibling: | 8.0.322256764928.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.58.41 | $x^{16} + 12 x^{14} + 8 x^{11} + 14 x^{8} + 28 x^{4} + 2$ | $16$ | $1$ | $58$ | 16T42 | $[2, 3, 7/2, 4, 9/2]$ |
|
\(7\)
| 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 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |