Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
gp: K = bnfinit(y^16 - 6*y^15 + 18*y^14 - 36*y^13 + 47*y^12 - 30*y^11 - 18*y^10 + 72*y^9 - 95*y^8 + 72*y^7 - 18*y^6 - 30*y^5 + 47*y^4 - 36*y^3 + 18*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 47 x^{12} - 30 x^{11} - 18 x^{10} + 72 x^{9} - 95 x^{8} + \cdots + 1 \)
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: |
\(4096000000000000\)
\(\medspace = 2^{24}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.46\) | 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: | $2^{3/2}5^{3/4}\approx 9.457416090031758$ | ||
| Ramified primes: |
\(2\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{101}a^{14}+\frac{30}{101}a^{13}-\frac{14}{101}a^{12}+\frac{36}{101}a^{11}+\frac{44}{101}a^{10}+\frac{3}{101}a^{9}+\frac{46}{101}a^{8}+\frac{8}{101}a^{7}+\frac{46}{101}a^{6}+\frac{3}{101}a^{5}+\frac{44}{101}a^{4}+\frac{36}{101}a^{3}-\frac{14}{101}a^{2}+\frac{30}{101}a+\frac{1}{101}$, $\frac{1}{101}a^{15}-\frac{5}{101}a^{13}-\frac{49}{101}a^{12}-\frac{26}{101}a^{11}-\frac{4}{101}a^{10}-\frac{44}{101}a^{9}+\frac{42}{101}a^{8}+\frac{8}{101}a^{7}+\frac{37}{101}a^{6}-\frac{46}{101}a^{5}+\frac{29}{101}a^{4}+\frac{17}{101}a^{3}+\frac{46}{101}a^{2}+\frac{10}{101}a-\frac{30}{101}$
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: |
\( -\frac{174}{101} a^{15} + \frac{807}{101} a^{14} - \frac{1887}{101} a^{13} + \frac{2985}{101} a^{12} - \frac{2380}{101} a^{11} - \frac{863}{101} a^{10} + \frac{4421}{101} a^{9} - \frac{5940}{101} a^{8} + \frac{3953}{101} a^{7} - \frac{424}{101} a^{6} - \frac{2705}{101} a^{5} + \frac{2889}{101} a^{4} - \frac{1580}{101} a^{3} + \frac{292}{101} a^{2} - \frac{53}{101} a - \frac{33}{101} \)
(order $20$)
| 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: |
$\frac{314}{101}a^{15}-\frac{1707}{101}a^{14}+\frac{4689}{101}a^{13}-\frac{8658}{101}a^{12}+\frac{9871}{101}a^{11}-\frac{3846}{101}a^{10}-\frac{7827}{101}a^{9}+\frac{18193}{101}a^{8}-\frac{19527}{101}a^{7}+\frac{11472}{101}a^{6}+\frac{1039}{101}a^{5}-\frac{9139}{101}a^{4}+\frac{9738}{101}a^{3}-\frac{5694}{101}a^{2}+\frac{2228}{101}a-\frac{421}{101}$, $\frac{29}{101}a^{15}-\frac{105}{101}a^{14}+\frac{240}{101}a^{13}-\frac{456}{101}a^{12}+\frac{516}{101}a^{11}-\frac{393}{101}a^{10}+\frac{25}{101}a^{9}+\frac{832}{101}a^{8}-\frac{1416}{101}a^{7}+\frac{1495}{101}a^{6}-\frac{639}{101}a^{5}-\frac{244}{101}a^{4}+\frac{1056}{101}a^{3}-\frac{832}{101}a^{2}+\frac{473}{101}a-\frac{66}{101}$, $\frac{204}{101}a^{15}-\frac{1022}{101}a^{14}+\frac{2660}{101}a^{13}-\frac{4778}{101}a^{12}+\frac{5172}{101}a^{11}-\frac{1748}{101}a^{10}-\frac{4164}{101}a^{9}+\frac{9430}{101}a^{8}-\frac{9978}{101}a^{7}+\frac{5986}{101}a^{6}+\frac{377}{101}a^{5}-\frac{4308}{101}a^{4}+\frac{4854}{101}a^{3}-\frac{2972}{101}a^{2}+\frac{1276}{101}a-\frac{274}{101}$, $\frac{422}{101}a^{15}-\frac{2377}{101}a^{14}+\frac{6673}{101}a^{13}-\frac{12448}{101}a^{12}+\frac{14455}{101}a^{11}-\frac{5882}{101}a^{10}-\frac{11458}{101}a^{9}+\frac{26855}{101}a^{8}-\frac{29073}{101}a^{7}+170a^{6}+\frac{1434}{101}a^{5}-\frac{13671}{101}a^{4}+\frac{14724}{101}a^{3}-\frac{8718}{101}a^{2}+\frac{3408}{101}a-\frac{695}{101}$, $\frac{258}{101}a^{15}-\frac{1291}{101}a^{14}+\frac{3309}{101}a^{13}-\frac{5779}{101}a^{12}+\frac{5901}{101}a^{11}-\frac{1175}{101}a^{10}-\frac{6236}{101}a^{9}+\frac{11949}{101}a^{8}-\frac{11496}{101}a^{7}+\frac{6013}{101}a^{6}+\frac{1631}{101}a^{5}-\frac{5690}{101}a^{4}+\frac{5582}{101}a^{3}-\frac{3287}{101}a^{2}+\frac{1523}{101}a-\frac{446}{101}$, $\frac{398}{101}a^{15}-\frac{2017}{101}a^{14}+\frac{5170}{101}a^{13}-\frac{8939}{101}a^{12}+\frac{8950}{101}a^{11}-\frac{1258}{101}a^{10}-\frac{10332}{101}a^{9}+\frac{18369}{101}a^{8}-\frac{16487}{101}a^{7}+\frac{7188}{101}a^{6}+\frac{4022}{101}a^{5}-\frac{8930}{101}a^{4}+\frac{7581}{101}a^{3}-\frac{3550}{101}a^{2}+\frac{1343}{101}a-\frac{322}{101}$, $\frac{74}{101}a^{15}-\frac{342}{101}a^{14}+\frac{783}{101}a^{13}-\frac{1161}{101}a^{12}+\frac{712}{101}a^{11}+\frac{917}{101}a^{10}-\frac{2565}{101}a^{9}+\frac{2930}{101}a^{8}-\frac{1538}{101}a^{7}-\frac{369}{101}a^{6}+\frac{1832}{101}a^{5}-\frac{1691}{101}a^{4}+\frac{965}{101}a^{3}-\frac{90}{101}a^{2}-\frac{26}{101}a+\frac{64}{101}$
| 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: | \( 87.2940978199 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 87.2940978199 \cdot 1}{20\cdot\sqrt{4096000000000000}}\cr\approx \mathstrut & 0.165658550937 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*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 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*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 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*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 - 6*x^15 + 18*x^14 - 36*x^13 + 47*x^12 - 30*x^11 - 18*x^10 + 72*x^9 - 95*x^8 + 72*x^7 - 18*x^6 - 30*x^5 + 47*x^4 - 36*x^3 + 18*x^2 - 6*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
$C_2^2:C_4$ (as 16T10):
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 16 |
| The 10 conjugacy class representatives for $C_2^2 : C_4$ |
| Character table for $C_2^2 : C_4$ |
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
| Degree 8 siblings: | 8.4.64000000.1, 8.0.4000000.2 |
| Minimal sibling: | 8.0.4000000.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $24$ | |||
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |