Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 18*x^12 + 4*x^10 - 11*x^8 - 8*x^6 + 60*x^4 + 8*x^2 + 4)
gp: K = bnfinit(y^16 - 4*y^14 + 18*y^12 + 4*y^10 - 11*y^8 - 8*y^6 + 60*y^4 + 8*y^2 + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 18*x^12 + 4*x^10 - 11*x^8 - 8*x^6 + 60*x^4 + 8*x^2 + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 + 18*x^12 + 4*x^10 - 11*x^8 - 8*x^6 + 60*x^4 + 8*x^2 + 4)
\( x^{16} - 4x^{14} + 18x^{12} + 4x^{10} - 11x^{8} - 8x^{6} + 60x^{4} + 8x^{2} + 4 \)
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: |
\(32088482764780732416\)
\(\medspace = 2^{36}\cdot 3^{4}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.56\) | 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: | $2^{9/4}3^{1/2}7^{1/2}\approx 21.798526485920096$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{12}a^{10}+\frac{1}{12}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{6}$, $\frac{1}{12}a^{11}+\frac{1}{12}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}+\frac{1}{6}a^{3}+\frac{1}{6}a$, $\frac{1}{84}a^{12}+\frac{1}{42}a^{10}-\frac{1}{6}a^{8}-\frac{1}{2}a^{5}+\frac{17}{84}a^{4}-\frac{1}{2}a^{3}-\frac{8}{21}a^{2}+\frac{1}{42}$, $\frac{1}{84}a^{13}+\frac{1}{42}a^{11}-\frac{1}{6}a^{9}-\frac{1}{2}a^{6}+\frac{17}{84}a^{5}-\frac{1}{2}a^{4}-\frac{8}{21}a^{3}+\frac{1}{42}a$, $\frac{1}{18396}a^{14}-\frac{29}{6132}a^{12}+\frac{223}{6132}a^{10}-\frac{173}{2628}a^{8}+\frac{1195}{9198}a^{6}+\frac{221}{1022}a^{4}-\frac{25}{1533}a^{2}+\frac{533}{4599}$, $\frac{1}{18396}a^{15}-\frac{29}{6132}a^{13}+\frac{223}{6132}a^{11}-\frac{173}{2628}a^{9}+\frac{1195}{9198}a^{7}+\frac{221}{1022}a^{5}-\frac{25}{1533}a^{3}+\frac{533}{4599}a$
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
$C_{2}$, which has order $2$
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: |
\( \frac{2461}{18396} a^{15} - \frac{815}{1533} a^{13} + \frac{4863}{2044} a^{11} + \frac{1735}{2628} a^{9} - \frac{8131}{4599} a^{7} - \frac{2055}{2044} a^{5} + \frac{3971}{511} a^{3} + \frac{7033}{9198} a \)
(order $8$)
| 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: |
$\frac{313}{18396}a^{14}-\frac{65}{1022}a^{12}+\frac{153}{511}a^{10}+\frac{191}{1314}a^{8}-\frac{1567}{18396}a^{6}+\frac{793}{1533}a^{4}+\frac{1767}{1022}a^{2}+\frac{1265}{4599}$, $\frac{167}{4599}a^{15}-\frac{43}{292}a^{13}+\frac{1993}{3066}a^{11}+\frac{253}{1314}a^{9}-\frac{6565}{9198}a^{7}+\frac{935}{6132}a^{5}+\frac{5200}{1533}a^{3}+\frac{995}{9198}a$, $\frac{347}{9198}a^{15}-\frac{15}{2044}a^{14}-\frac{191}{1022}a^{13}+\frac{17}{876}a^{12}+\frac{5185}{6132}a^{11}-\frac{197}{3066}a^{10}-\frac{1583}{2628}a^{9}-\frac{53}{146}a^{8}-\frac{1579}{18396}a^{7}+\frac{1453}{2044}a^{6}-\frac{359}{6132}a^{5}-\frac{2861}{6132}a^{4}+\frac{431}{219}a^{3}-\frac{2521}{3066}a^{2}-\frac{5015}{9198}a+\frac{289}{1022}$, $\frac{347}{9198}a^{15}+\frac{15}{2044}a^{14}-\frac{191}{1022}a^{13}-\frac{17}{876}a^{12}+\frac{5185}{6132}a^{11}+\frac{197}{3066}a^{10}-\frac{1583}{2628}a^{9}+\frac{53}{146}a^{8}-\frac{1579}{18396}a^{7}-\frac{1453}{2044}a^{6}-\frac{359}{6132}a^{5}+\frac{2861}{6132}a^{4}+\frac{431}{219}a^{3}+\frac{2521}{3066}a^{2}-\frac{5015}{9198}a-\frac{289}{1022}$, $\frac{1223}{18396}a^{15}+\frac{913}{9198}a^{14}-\frac{344}{1533}a^{13}-\frac{1219}{3066}a^{12}+\frac{3103}{3066}a^{11}+\frac{11027}{6132}a^{10}+\frac{1411}{1314}a^{9}+\frac{995}{2628}a^{8}-\frac{15791}{18396}a^{7}-\frac{18707}{18396}a^{6}-\frac{1952}{1533}a^{5}-\frac{2105}{6132}a^{4}+\frac{763}{219}a^{3}+\frac{2819}{438}a^{2}+\frac{12379}{4599}a+\frac{8585}{9198}$, $\frac{149}{6132}a^{15}+\frac{107}{2628}a^{14}-\frac{79}{876}a^{13}-\frac{229}{1533}a^{12}+\frac{593}{1533}a^{11}+\frac{4237}{6132}a^{10}+\frac{71}{219}a^{9}+\frac{947}{2628}a^{8}-\frac{4145}{6132}a^{7}-\frac{125}{657}a^{6}-\frac{1133}{6132}a^{5}-\frac{727}{2044}a^{4}+\frac{2476}{1533}a^{3}+\frac{3905}{1533}a^{2}+\frac{3125}{3066}a-\frac{697}{9198}$, $\frac{1049}{18396}a^{15}+\frac{115}{6132}a^{14}-\frac{79}{292}a^{13}-\frac{221}{3066}a^{12}+\frac{2445}{2044}a^{11}+\frac{97}{292}a^{10}-\frac{1459}{2628}a^{9}+\frac{107}{876}a^{8}-\frac{6571}{9198}a^{7}-\frac{545}{3066}a^{6}-\frac{649}{1533}a^{5}+\frac{709}{6132}a^{4}+\frac{3711}{1022}a^{3}+\frac{410}{511}a^{2}-\frac{7874}{4599}a+\frac{191}{438}$
| 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: | \( 10362.7919914 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10362.7919914 \cdot 2}{8\cdot\sqrt{32088482764780732416}}\cr\approx \mathstrut & 1.11091580340 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 18*x^12 + 4*x^10 - 11*x^8 - 8*x^6 + 60*x^4 + 8*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 - 4*x^14 + 18*x^12 + 4*x^10 - 11*x^8 - 8*x^6 + 60*x^4 + 8*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 - 4*x^14 + 18*x^12 + 4*x^10 - 11*x^8 - 8*x^6 + 60*x^4 + 8*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 - 4*x^14 + 18*x^12 + 4*x^10 - 11*x^8 - 8*x^6 + 60*x^4 + 8*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
$C_2^2\times D_4$ (as 16T25):
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 $C_2^2 \times D_4$ |
| Character table for $C_2^2 \times D_4$ |
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 | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.18.53 | $x^{8} + 2 x^{6} + 4 x^{3} + 2$ | $8$ | $1$ | $18$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ |
| 2.8.18.53 | $x^{8} + 2 x^{6} + 4 x^{3} + 2$ | $8$ | $1$ | $18$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
|
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |