Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + x^12 + 2*x^10 - 5*x^9 + 7*x^8 - 5*x^7 - 3*x^6 + 10*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - 3*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 3*y^14 - 2*y^13 + y^12 + 2*y^10 - 5*y^9 + 7*y^8 - 5*y^7 - 3*y^6 + 10*y^5 - 9*y^4 + 2*y^3 + 3*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + x^12 + 2*x^10 - 5*x^9 + 7*x^8 - 5*x^7 - 3*x^6 + 10*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + x^12 + 2*x^10 - 5*x^9 + 7*x^8 - 5*x^7 - 3*x^6 + 10*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - 3*x + 1)
\( x^{16} - 2 x^{15} + 3 x^{14} - 2 x^{13} + x^{12} + 2 x^{10} - 5 x^{9} + 7 x^{8} - 5 x^{7} - 3 x^{6} + \cdots + 1 \)
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: |
\(117135062500000000\)
\(\medspace = 2^{8}\cdot 5^{12}\cdot 37^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.66\) | 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: | $2^{2/3}5^{3/4}37^{1/2}\approx 32.28605847922135$ | ||
| Ramified primes: |
\(2\), \(5\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{41}a^{15}+\frac{3}{41}a^{14}+\frac{18}{41}a^{13}+\frac{6}{41}a^{12}-\frac{10}{41}a^{11}-\frac{9}{41}a^{10}-\frac{2}{41}a^{9}-\frac{15}{41}a^{8}+\frac{14}{41}a^{7}-\frac{17}{41}a^{6}-\frac{6}{41}a^{5}-\frac{20}{41}a^{4}+\frac{14}{41}a^{3}-\frac{10}{41}a^{2}-\frac{6}{41}a+\frac{8}{41}$
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: |
\( \frac{54}{41} a^{15} - \frac{125}{41} a^{14} + \frac{193}{41} a^{13} - \frac{168}{41} a^{12} + \frac{116}{41} a^{11} - \frac{76}{41} a^{10} + \frac{179}{41} a^{9} - \frac{359}{41} a^{8} + \frac{510}{41} a^{7} - \frac{426}{41} a^{6} + \frac{45}{41} a^{5} + \frac{437}{41} a^{4} - \frac{433}{41} a^{3} + \frac{157}{41} a^{2} + \frac{86}{41} a - \frac{60}{41} \)
(order $10$)
| 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: |
$\frac{60}{41}a^{15}-\frac{66}{41}a^{14}+\frac{55}{41}a^{13}+\frac{73}{41}a^{12}-\frac{108}{41}a^{11}+\frac{116}{41}a^{10}+\frac{44}{41}a^{9}-\frac{121}{41}a^{8}+\frac{61}{41}a^{7}+\frac{210}{41}a^{6}-\frac{606}{41}a^{5}+\frac{645}{41}a^{4}-\frac{103}{41}a^{3}-\frac{313}{41}a^{2}+\frac{337}{41}a-\frac{94}{41}$, $\frac{54}{41}a^{15}-\frac{125}{41}a^{14}+\frac{193}{41}a^{13}-\frac{168}{41}a^{12}+\frac{116}{41}a^{11}-\frac{76}{41}a^{10}+\frac{179}{41}a^{9}-\frac{359}{41}a^{8}+\frac{510}{41}a^{7}-\frac{426}{41}a^{6}+\frac{45}{41}a^{5}+\frac{437}{41}a^{4}-\frac{433}{41}a^{3}+\frac{157}{41}a^{2}+\frac{86}{41}a-\frac{101}{41}$, $\frac{57}{41}a^{15}-\frac{75}{41}a^{14}+\frac{83}{41}a^{13}+\frac{14}{41}a^{12}-\frac{37}{41}a^{11}+\frac{61}{41}a^{10}+\frac{91}{41}a^{9}-\frac{158}{41}a^{8}+\frac{183}{41}a^{7}+\frac{56}{41}a^{6}-\frac{383}{41}a^{5}+\frac{541}{41}a^{4}-\frac{145}{41}a^{3}-\frac{201}{41}a^{2}+\frac{232}{41}a-\frac{77}{41}$, $\frac{66}{41}a^{15}-\frac{48}{41}a^{14}+\frac{40}{41}a^{13}+\frac{109}{41}a^{12}-\frac{127}{41}a^{11}+\frac{144}{41}a^{10}+\frac{32}{41}a^{9}-\frac{88}{41}a^{8}-\frac{19}{41}a^{7}+\frac{272}{41}a^{6}-\frac{724}{41}a^{5}+\frac{607}{41}a^{4}-\frac{60}{41}a^{3}-\frac{373}{41}a^{2}+\frac{342}{41}a-\frac{128}{41}$, $\frac{61}{41}a^{15}-\frac{145}{41}a^{14}+\frac{155}{41}a^{13}-\frac{85}{41}a^{12}-\frac{36}{41}a^{11}+\frac{25}{41}a^{10}+\frac{83}{41}a^{9}-\frac{341}{41}a^{8}+\frac{362}{41}a^{7}-\frac{176}{41}a^{6}-\frac{407}{41}a^{5}+\frac{830}{41}a^{4}-\frac{499}{41}a^{3}-\frac{159}{41}a^{2}+\frac{331}{41}a-\frac{127}{41}$, $\frac{53}{41}a^{15}-\frac{87}{41}a^{14}+\frac{134}{41}a^{13}-\frac{51}{41}a^{12}+\frac{44}{41}a^{11}+\frac{15}{41}a^{10}+\frac{140}{41}a^{9}-\frac{221}{41}a^{8}+\frac{332}{41}a^{7}-\frac{163}{41}a^{6}-\frac{195}{41}a^{5}+\frac{416}{41}a^{4}-\frac{283}{41}a^{3}-\frac{79}{41}a^{2}+\frac{174}{41}a-\frac{109}{41}$, $\frac{3}{41}a^{15}-\frac{73}{41}a^{14}+\frac{54}{41}a^{13}-\frac{105}{41}a^{12}-\frac{30}{41}a^{11}-\frac{27}{41}a^{10}-\frac{47}{41}a^{9}-\frac{168}{41}a^{8}+\frac{165}{41}a^{7}-\frac{174}{41}a^{6}-\frac{18}{41}a^{5}+\frac{391}{41}a^{4}-\frac{327}{41}a^{3}+\frac{93}{41}a^{2}+\frac{146}{41}a-\frac{140}{41}$
| 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: | \( 333.853446138 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 333.853446138 \cdot 1}{10\cdot\sqrt{117135062500000000}}\cr\approx \mathstrut & 0.236947075908 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + x^12 + 2*x^10 - 5*x^9 + 7*x^8 - 5*x^7 - 3*x^6 + 10*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - 3*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 - 2*x^15 + 3*x^14 - 2*x^13 + x^12 + 2*x^10 - 5*x^9 + 7*x^8 - 5*x^7 - 3*x^6 + 10*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - 3*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 - 2*x^15 + 3*x^14 - 2*x^13 + x^12 + 2*x^10 - 5*x^9 + 7*x^8 - 5*x^7 - 3*x^6 + 10*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - 3*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 - 2*x^15 + 3*x^14 - 2*x^13 + x^12 + 2*x^10 - 5*x^9 + 7*x^8 - 5*x^7 - 3*x^6 + 10*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - 3*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
$C_4\times S_4$ (as 16T181):
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 96 |
| The 20 conjugacy class representatives for $C_4\times S_4$ |
| Character table for $C_4\times S_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.3700.1, 8.0.13690000.1 |
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 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Minimal sibling: | 12.8.937080500000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(37\)
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.8.4.1 | $x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |