Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 + 158*x^12 - 624*x^10 + 1287*x^8 - 1336*x^6 + 650*x^4 - 116*x^2 + 1)
gp: K = bnfinit(y^16 - 20*y^14 + 158*y^12 - 624*y^10 + 1287*y^8 - 1336*y^6 + 650*y^4 - 116*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 20*x^14 + 158*x^12 - 624*x^10 + 1287*x^8 - 1336*x^6 + 650*x^4 - 116*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 20*x^14 + 158*x^12 - 624*x^10 + 1287*x^8 - 1336*x^6 + 650*x^4 - 116*x^2 + 1)
\( x^{16} - 20x^{14} + 158x^{12} - 624x^{10} + 1287x^{8} - 1336x^{6} + 650x^{4} - 116x^{2} + 1 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(175415570277650636210176\)
\(\medspace = 2^{52}\cdot 79^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.36\) | 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: | $2^{27/8}79^{1/2}\approx 92.21249671842511$ | ||
| Ramified primes: |
\(2\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{4}$
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 $1$ (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: | $15$ | 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: |
$a^{14}-\frac{77}{4}a^{12}+\frac{573}{4}a^{10}-512a^{8}+\frac{3519}{4}a^{6}-\frac{1257}{2}a^{4}+\frac{583}{4}a^{2}-\frac{7}{4}$, $a^{14}-\frac{77}{4}a^{12}+\frac{573}{4}a^{10}-512a^{8}+\frac{3519}{4}a^{6}-\frac{1257}{2}a^{4}+\frac{583}{4}a^{2}-\frac{3}{4}$, $\frac{1}{2}a^{14}-\frac{19}{2}a^{12}+\frac{139}{2}a^{10}-\frac{485}{2}a^{8}+\frac{801}{2}a^{6}-\frac{525}{2}a^{4}+48a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{2}a^{14}-\frac{19}{4}a^{13}+\frac{19}{2}a^{12}+35a^{11}-\frac{139}{2}a^{10}-\frac{499}{4}a^{9}+\frac{485}{2}a^{8}+\frac{435}{2}a^{7}-\frac{801}{2}a^{6}-\frac{667}{4}a^{5}+\frac{525}{2}a^{4}+\frac{211}{4}a^{3}-48a^{2}-\frac{17}{2}a$, $\frac{1}{4}a^{15}-\frac{19}{4}a^{13}+35a^{11}-\frac{499}{4}a^{9}+\frac{435}{2}a^{7}-\frac{667}{4}a^{5}+\frac{211}{4}a^{3}-\frac{17}{2}a+1$, $\frac{1}{4}a^{15}-5a^{13}+\frac{79}{2}a^{11}-\frac{1}{4}a^{10}-156a^{9}+\frac{7}{2}a^{8}+\frac{643}{2}a^{7}-\frac{67}{4}a^{6}-\frac{663}{2}a^{5}+31a^{4}+\frac{623}{4}a^{3}-\frac{69}{4}a^{2}-26a+2$, $\frac{1}{4}a^{14}-5a^{12}+\frac{1}{2}a^{11}+\frac{77}{2}a^{10}-\frac{29}{4}a^{9}-\frac{283}{2}a^{8}+37a^{7}+\frac{495}{2}a^{6}-\frac{311}{4}a^{5}-176a^{4}+\frac{119}{2}a^{3}+\frac{147}{4}a^{2}-\frac{57}{4}a+\frac{5}{2}$, $\frac{1}{4}a^{15}-\frac{19}{4}a^{13}+35a^{11}-\frac{499}{4}a^{9}+\frac{435}{2}a^{7}-\frac{667}{4}a^{5}+\frac{207}{4}a^{3}-\frac{9}{2}a$, $\frac{1}{4}a^{15}-5a^{13}+39a^{11}-\frac{595}{4}a^{9}-\frac{1}{4}a^{8}+\frac{569}{2}a^{7}+\frac{5}{2}a^{6}-\frac{1015}{4}a^{5}-\frac{27}{4}a^{4}+\frac{385}{4}a^{3}+3a^{2}-\frac{47}{4}a+\frac{3}{4}$, $a^{15}+\frac{1}{2}a^{14}-\frac{39}{2}a^{13}-\frac{19}{2}a^{12}+\frac{297}{2}a^{11}+\frac{279}{4}a^{10}-\frac{2217}{4}a^{9}-\frac{983}{4}a^{8}+1041a^{7}+\frac{1659}{4}a^{6}-\frac{3675}{4}a^{5}-\frac{1143}{4}a^{4}+\frac{713}{2}a^{3}+\frac{229}{4}a^{2}-\frac{203}{4}a+\frac{5}{4}$, $\frac{1}{2}a^{15}-\frac{39}{4}a^{13}+\frac{149}{2}a^{11}-\frac{1}{4}a^{10}-\frac{1123}{4}a^{9}+\frac{7}{2}a^{8}+539a^{7}-\frac{67}{4}a^{6}-\frac{1993}{4}a^{5}+31a^{4}+\frac{415}{2}a^{3}-\frac{69}{4}a^{2}-\frac{61}{2}a+2$, $\frac{1}{4}a^{14}-5a^{12}-\frac{1}{2}a^{11}+\frac{77}{2}a^{10}+\frac{29}{4}a^{9}-\frac{283}{2}a^{8}-37a^{7}+\frac{495}{2}a^{6}+\frac{311}{4}a^{5}-176a^{4}-\frac{119}{2}a^{3}+\frac{147}{4}a^{2}+\frac{57}{4}a+\frac{5}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{19}{4}a^{13}+5a^{12}+\frac{69}{2}a^{11}-\frac{77}{2}a^{10}-\frac{235}{2}a^{9}+\frac{283}{2}a^{8}+\frac{361}{2}a^{7}-\frac{495}{2}a^{6}-89a^{5}+176a^{4}-\frac{27}{4}a^{3}-\frac{147}{4}a^{2}+\frac{23}{4}a-\frac{3}{2}$, $\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{19}{4}a^{13}-5a^{12}+\frac{69}{2}a^{11}+\frac{77}{2}a^{10}-\frac{235}{2}a^{9}-\frac{283}{2}a^{8}+\frac{361}{2}a^{7}+\frac{495}{2}a^{6}-89a^{5}-176a^{4}-\frac{27}{4}a^{3}+\frac{147}{4}a^{2}+\frac{23}{4}a+\frac{3}{2}$, $\frac{3}{4}a^{15}+\frac{1}{2}a^{14}-\frac{29}{2}a^{13}-\frac{39}{4}a^{12}+\frac{435}{4}a^{11}+\frac{147}{2}a^{10}-\frac{789}{2}a^{9}-\frac{1065}{4}a^{8}+\frac{2797}{4}a^{7}+465a^{6}-\frac{1079}{2}a^{5}-\frac{1371}{4}a^{4}+\frac{305}{2}a^{3}+\frac{177}{2}a^{2}-\frac{13}{2}a-2$
| 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: | \( 2484052.66943 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 2484052.66943 \cdot 1}{2\cdot\sqrt{175415570277650636210176}}\cr\approx \mathstrut & 0.194346473721 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 + 158*x^12 - 624*x^10 + 1287*x^8 - 1336*x^6 + 650*x^4 - 116*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 - 20*x^14 + 158*x^12 - 624*x^10 + 1287*x^8 - 1336*x^6 + 650*x^4 - 116*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(Rationals()); K<a> := NumberField(x^16 - 20*x^14 + 158*x^12 - 624*x^10 + 1287*x^8 - 1336*x^6 + 650*x^4 - 116*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 - 20*x^14 + 158*x^12 - 624*x^10 + 1287*x^8 - 1336*x^6 + 650*x^4 - 116*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
$C_4\times S_4$ (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
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 4.4.80896.1, 8.8.6544162816.1 |
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: | 12.12.1338314592572407808.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $8$ | $2$ | $52$ | |||
|
\(79\)
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |