Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 18*x^13 + 29*x^12 - 43*x^11 + 58*x^10 - 73*x^9 + 85*x^8 - 88*x^7 + 81*x^6 - 65*x^5 + 45*x^4 - 26*x^3 + 12*x^2 - 4*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 + 10*y^14 - 18*y^13 + 29*y^12 - 43*y^11 + 58*y^10 - 73*y^9 + 85*y^8 - 88*y^7 + 81*y^6 - 65*y^5 + 45*y^4 - 26*y^3 + 12*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 10*x^14 - 18*x^13 + 29*x^12 - 43*x^11 + 58*x^10 - 73*x^9 + 85*x^8 - 88*x^7 + 81*x^6 - 65*x^5 + 45*x^4 - 26*x^3 + 12*x^2 - 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 10*x^14 - 18*x^13 + 29*x^12 - 43*x^11 + 58*x^10 - 73*x^9 + 85*x^8 - 88*x^7 + 81*x^6 - 65*x^5 + 45*x^4 - 26*x^3 + 12*x^2 - 4*x + 1)
\( x^{16} - 4 x^{15} + 10 x^{14} - 18 x^{13} + 29 x^{12} - 43 x^{11} + 58 x^{10} - 73 x^{9} + 85 x^{8} + \cdots + 1 \)
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: |
\(3750900906670801\)
\(\medspace = 13^{2}\cdot 43^{2}\cdot 331^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.41\) | 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: | $13^{1/2}43^{1/2}331^{1/2}\approx 430.1499738463319$ | ||
| Ramified primes: |
\(13\), \(43\), \(331\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{113}a^{15}-\frac{31}{113}a^{14}+\frac{56}{113}a^{13}+\frac{52}{113}a^{12}-\frac{19}{113}a^{11}+\frac{18}{113}a^{10}+\frac{24}{113}a^{9}-\frac{43}{113}a^{8}+\frac{3}{113}a^{7}-\frac{56}{113}a^{6}+\frac{11}{113}a^{5}-\frac{23}{113}a^{4}-\frac{12}{113}a^{3}-\frac{41}{113}a^{2}-\frac{11}{113}a-\frac{46}{113}$
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
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: |
$\frac{56}{113}a^{15}-\frac{154}{113}a^{14}+\frac{198}{113}a^{13}-\frac{26}{113}a^{12}-\frac{160}{113}a^{11}+\frac{330}{113}a^{10}-\frac{690}{113}a^{9}+\frac{1321}{113}a^{8}-\frac{1866}{113}a^{7}+\frac{2627}{113}a^{6}-\frac{3226}{113}a^{5}+\frac{3006}{113}a^{4}-\frac{2367}{113}a^{3}+\frac{1546}{113}a^{2}-\frac{729}{113}a+\frac{249}{113}$, $\frac{287}{113}a^{15}-\frac{1100}{113}a^{14}+\frac{2625}{113}a^{13}-\frac{4399}{113}a^{12}+\frac{6751}{113}a^{11}-\frac{9750}{113}a^{10}+\frac{12877}{113}a^{9}-\frac{15731}{113}a^{8}+\frac{17585}{113}a^{7}-\frac{17315}{113}a^{6}+\frac{14683}{113}a^{5}-\frac{10669}{113}a^{4}+\frac{6613}{113}a^{3}-\frac{3179}{113}a^{2}+\frac{1137}{113}a-\frac{207}{113}$, $\frac{116}{113}a^{15}-\frac{545}{113}a^{14}+\frac{1298}{113}a^{13}-\frac{2330}{113}a^{12}+\frac{3559}{113}a^{11}-\frac{5370}{113}a^{10}+\frac{7078}{113}a^{9}-\frac{8717}{113}a^{8}+\frac{10066}{113}a^{7}-\frac{10112}{113}a^{6}+\frac{8847}{113}a^{5}-\frac{6849}{113}a^{4}+\frac{4258}{113}a^{3}-\frac{2157}{113}a^{2}+\frac{758}{113}a-\frac{138}{113}$, $\frac{91}{113}a^{15}-\frac{561}{113}a^{14}+\frac{1593}{113}a^{13}-\frac{3178}{113}a^{12}+\frac{5051}{113}a^{11}-\frac{7515}{113}a^{10}+\frac{10320}{113}a^{9}-\frac{13066}{113}a^{8}+\frac{15415}{113}a^{7}-\frac{16283}{113}a^{6}+\frac{14900}{113}a^{5}-\frac{11698}{113}a^{4}+\frac{7722}{113}a^{3}-\frac{4183}{113}a^{2}+\frac{1598}{113}a-\frac{457}{113}$, $\frac{112}{113}a^{15}-\frac{308}{113}a^{14}+\frac{735}{113}a^{13}-\frac{1182}{113}a^{12}+\frac{1940}{113}a^{11}-\frac{2617}{113}a^{10}+\frac{3366}{113}a^{9}-\frac{4251}{113}a^{8}+\frac{4630}{113}a^{7}-\frac{4351}{113}a^{6}+\frac{3831}{113}a^{5}-\frac{2576}{113}a^{4}+\frac{1481}{113}a^{3}-\frac{637}{113}a^{2}+\frac{237}{113}a+\frac{46}{113}$, $\frac{124}{113}a^{15}-\frac{454}{113}a^{14}+\frac{1068}{113}a^{13}-\frac{1801}{113}a^{12}+\frac{2842}{113}a^{11}-\frac{4209}{113}a^{10}+\frac{5575}{113}a^{9}-\frac{6914}{113}a^{8}+\frac{7830}{113}a^{7}-\frac{7961}{113}a^{6}+\frac{7127}{113}a^{5}-\frac{5564}{113}a^{4}+\frac{3823}{113}a^{3}-\frac{2033}{113}a^{2}+\frac{896}{113}a-\frac{280}{113}$, $\frac{363}{113}a^{15}-\frac{1309}{113}a^{14}+\frac{3152}{113}a^{13}-\frac{5306}{113}a^{12}+\frac{8358}{113}a^{11}-\frac{11998}{113}a^{10}+\frac{15831}{113}a^{9}-\frac{19451}{113}a^{8}+\frac{21881}{113}a^{7}-\frac{21684}{113}a^{6}+\frac{18683}{113}a^{5}-\frac{13660}{113}a^{4}+\frac{8526}{113}a^{3}-\frac{4035}{113}a^{2}+\frac{1318}{113}a-\frac{200}{113}$
| 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: | \( 8.20527213707 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8.20527213707 \cdot 1}{2\cdot\sqrt{3750900906670801}}\cr\approx \mathstrut & 0.162717455844 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 18*x^13 + 29*x^12 - 43*x^11 + 58*x^10 - 73*x^9 + 85*x^8 - 88*x^7 + 81*x^6 - 65*x^5 + 45*x^4 - 26*x^3 + 12*x^2 - 4*x + 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 - 4*x^15 + 10*x^14 - 18*x^13 + 29*x^12 - 43*x^11 + 58*x^10 - 73*x^9 + 85*x^8 - 88*x^7 + 81*x^6 - 65*x^5 + 45*x^4 - 26*x^3 + 12*x^2 - 4*x + 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 - 4*x^15 + 10*x^14 - 18*x^13 + 29*x^12 - 43*x^11 + 58*x^10 - 73*x^9 + 85*x^8 - 88*x^7 + 81*x^6 - 65*x^5 + 45*x^4 - 26*x^3 + 12*x^2 - 4*x + 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 - 4*x^15 + 10*x^14 - 18*x^13 + 29*x^12 - 43*x^11 + 58*x^10 - 73*x^9 + 85*x^8 - 88*x^7 + 81*x^6 - 65*x^5 + 45*x^4 - 26*x^3 + 12*x^2 - 4*x + 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
$C_2^8.S_4$ (as 16T1664):
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 6144 |
| The 105 conjugacy class representatives for $C_2^8.S_4$ |
| Character table for $C_2^8.S_4$ |
Intermediate fields
| 4.2.331.1, 8.0.1424293.1, 8.2.61244599.1, 8.2.4711123.1 |
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
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| 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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(43\)
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(331\)
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{331}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |