Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 5*x^13 - x^12 - 10*x^11 + 3*x^10 - 4*x^9 + 9*x^8 - 4*x^7 + 8*x^6 - x^5 + 7*x^4 + 4*x^2 - x + 1)
gp: K = bnfinit(y^16 + 2*y^14 - 5*y^13 - y^12 - 10*y^11 + 3*y^10 - 4*y^9 + 9*y^8 - 4*y^7 + 8*y^6 - y^5 + 7*y^4 + 4*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 2*x^14 - 5*x^13 - x^12 - 10*x^11 + 3*x^10 - 4*x^9 + 9*x^8 - 4*x^7 + 8*x^6 - x^5 + 7*x^4 + 4*x^2 - x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 2*x^14 - 5*x^13 - x^12 - 10*x^11 + 3*x^10 - 4*x^9 + 9*x^8 - 4*x^7 + 8*x^6 - x^5 + 7*x^4 + 4*x^2 - x + 1)
\( x^{16} + 2 x^{14} - 5 x^{13} - x^{12} - 10 x^{11} + 3 x^{10} - 4 x^{9} + 9 x^{8} - 4 x^{7} + 8 x^{6} + \cdots + 1 \)
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: | $[2, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-115558273191796875\)
\(\medspace = -\,3^{4}\cdot 5^{8}\cdot 11^{4}\cdot 19^{2}\cdot 691\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.65\) | 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: | $3^{1/2}5^{1/2}11^{1/2}19^{1/2}691^{1/2}\approx 1471.8304929576639$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\), \(19\), \(691\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-691}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7}a^{14}-\frac{2}{7}a^{12}+\frac{2}{7}a^{11}+\frac{3}{7}a^{9}+\frac{3}{7}a^{8}-\frac{2}{7}a^{7}-\frac{3}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}-\frac{3}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{11109}a^{15}-\frac{788}{11109}a^{14}+\frac{1201}{3703}a^{13}+\frac{4735}{11109}a^{12}+\frac{1010}{3703}a^{11}-\frac{802}{11109}a^{10}-\frac{1234}{11109}a^{9}-\frac{290}{1587}a^{8}+\frac{3127}{11109}a^{7}+\frac{1235}{3703}a^{6}-\frac{1039}{11109}a^{5}+\frac{200}{483}a^{4}+\frac{3089}{11109}a^{3}-\frac{4435}{11109}a^{2}+\frac{1128}{3703}a+\frac{1154}{11109}$
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: | $8$ | 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: |
$\frac{910}{1587}a^{15}+\frac{6469}{11109}a^{14}+\frac{525}{529}a^{13}-\frac{19616}{11109}a^{12}-\frac{13752}{3703}a^{11}-\frac{9322}{1587}a^{10}-\frac{36670}{11109}a^{9}+\frac{2950}{11109}a^{8}+\frac{46576}{11109}a^{7}+\frac{8126}{3703}a^{6}+\frac{32687}{11109}a^{5}+\frac{1088}{483}a^{4}+\frac{33044}{11109}a^{3}+\frac{18281}{11109}a^{2}+\frac{9475}{3703}a+\frac{6344}{11109}$, $a$, $\frac{6469}{11109}a^{15}-\frac{245}{1587}a^{14}+\frac{4078}{3703}a^{13}-\frac{34886}{11109}a^{12}-\frac{74}{529}a^{11}-\frac{55780}{11109}a^{10}+\frac{28430}{11109}a^{9}-\frac{10754}{11109}a^{8}+\frac{49858}{11109}a^{7}-\frac{6091}{3703}a^{6}+\frac{31394}{11109}a^{5}-\frac{502}{483}a^{4}+\frac{18281}{11109}a^{3}+\frac{2945}{11109}a^{2}+\frac{4238}{3703}a+\frac{677}{1587}$, $\frac{2594}{11109}a^{15}+\frac{3158}{11109}a^{14}+\frac{1171}{3703}a^{13}-\frac{10312}{11109}a^{12}-\frac{7074}{3703}a^{11}-\frac{25223}{11109}a^{10}-\frac{3191}{11109}a^{9}+\frac{20477}{11109}a^{8}+\frac{4121}{1587}a^{7}+\frac{1024}{3703}a^{6}-\frac{743}{1587}a^{5}-\frac{80}{483}a^{4}+\frac{27082}{11109}a^{3}+\frac{6121}{11109}a^{2}+\frac{2249}{3703}a+\frac{394}{11109}$, $\frac{5921}{11109}a^{15}+\frac{7967}{11109}a^{14}+\frac{5064}{3703}a^{13}-\frac{19051}{11109}a^{12}-\frac{13360}{3703}a^{11}-\frac{93971}{11109}a^{10}-\frac{5663}{1587}a^{9}-\frac{20323}{11109}a^{8}+\frac{80375}{11109}a^{7}+\frac{1370}{529}a^{6}+\frac{45316}{11109}a^{5}+\frac{505}{483}a^{4}+\frac{8359}{1587}a^{3}+\frac{33781}{11109}a^{2}+\frac{11901}{3703}a+\frac{5560}{11109}$, $\frac{214}{1587}a^{15}+\frac{5065}{11109}a^{14}-\frac{80}{529}a^{13}+\frac{727}{11109}a^{12}-\frac{11069}{3703}a^{11}-\frac{232}{1587}a^{10}-\frac{25069}{11109}a^{9}+\frac{37840}{11109}a^{8}+\frac{2596}{11109}a^{7}+\frac{5407}{3703}a^{6}-\frac{13858}{11109}a^{5}+\frac{1244}{483}a^{4}+\frac{15500}{11109}a^{3}+\frac{9074}{11109}a^{2}-\frac{411}{3703}a+\frac{449}{11109}$, $\frac{3131}{11109}a^{15}+\frac{5318}{11109}a^{14}+\frac{1786}{3703}a^{13}-\frac{6817}{11109}a^{12}-\frac{10632}{3703}a^{11}-\frac{33755}{11109}a^{10}-\frac{4889}{1587}a^{9}+\frac{17462}{11109}a^{8}+\frac{13130}{11109}a^{7}+\frac{1331}{529}a^{6}-\frac{6107}{11109}a^{5}+\frac{1405}{483}a^{4}+\frac{4603}{1587}a^{3}+\frac{36766}{11109}a^{2}+\frac{2280}{3703}a+\frac{4336}{11109}$, $\frac{997}{11109}a^{15}+\frac{6277}{11109}a^{14}+\frac{1328}{3703}a^{13}+\frac{4231}{11109}a^{12}-\frac{9239}{3703}a^{11}-\frac{33073}{11109}a^{10}-\frac{43222}{11109}a^{9}-\frac{3659}{11109}a^{8}+\frac{22969}{11109}a^{7}+\frac{9834}{3703}a^{6}+\frac{9950}{11109}a^{5}+\frac{38}{69}a^{4}+\frac{26345}{11109}a^{3}+\frac{1769}{1587}a^{2}+\frac{7897}{3703}a+\frac{1550}{11109}$
| 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: | \( 84.8251848921 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 84.8251848921 \cdot 1}{2\cdot\sqrt{115558273191796875}}\cr\approx \mathstrut & 0.192936043092 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 5*x^13 - x^12 - 10*x^11 + 3*x^10 - 4*x^9 + 9*x^8 - 4*x^7 + 8*x^6 - x^5 + 7*x^4 + 4*x^2 - 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^14 - 5*x^13 - x^12 - 10*x^11 + 3*x^10 - 4*x^9 + 9*x^8 - 4*x^7 + 8*x^6 - x^5 + 7*x^4 + 4*x^2 - 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^14 - 5*x^13 - x^12 - 10*x^11 + 3*x^10 - 4*x^9 + 9*x^8 - 4*x^7 + 8*x^6 - x^5 + 7*x^4 + 4*x^2 - 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^14 - 5*x^13 - x^12 - 10*x^11 + 3*x^10 - 4*x^9 + 9*x^8 - 4*x^7 + 8*x^6 - x^5 + 7*x^4 + 4*x^2 - 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
$C_4^4.C_2\wr D_4$ (as 16T1823):
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 32768 |
| The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$ |
| Character table for $C_4^4.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.275.1, 8.2.12931875.1 |
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 | $16$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | R | $16$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(19\)
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(691\)
| $\Q_{691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{691}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |