Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 - 3*y^13 + 11*y^12 - 19*y^11 + 10*y^10 + 21*y^9 - 44*y^8 + 21*y^7 + 40*y^6 - 89*y^5 + 92*y^4 - 60*y^3 + 26*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*x + 1)
\( x^{16} - 2 x^{15} + 2 x^{14} - 3 x^{13} + 11 x^{12} - 19 x^{11} + 10 x^{10} + 21 x^{9} - 44 x^{8} + \cdots + 1 \)
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: |
\(7058653305387264\)
\(\medspace = 2^{8}\cdot 3^{14}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.78\) | 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: | $2\cdot 3^{7/8}7^{1/2}\approx 13.837579007333664$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{40937}a^{15}-\frac{197}{3149}a^{14}+\frac{3681}{40937}a^{13}-\frac{4172}{40937}a^{12}-\frac{646}{3149}a^{11}-\frac{1462}{40937}a^{10}+\frac{16001}{40937}a^{9}-\frac{9538}{40937}a^{8}+\frac{138}{611}a^{7}+\frac{1093}{40937}a^{6}-\frac{13231}{40937}a^{5}+\frac{3141}{40937}a^{4}-\frac{14075}{40937}a^{3}-\frac{515}{3149}a^{2}-\frac{1544}{3149}a-\frac{11694}{40937}$
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: |
\( -\frac{324}{3149} a^{15} - \frac{4721}{3149} a^{14} + \frac{3976}{3149} a^{13} + \frac{807}{3149} a^{12} + \frac{6514}{3149} a^{11} - \frac{39599}{3149} a^{10} + \frac{36718}{3149} a^{9} + \frac{32633}{3149} a^{8} - \frac{1566}{47} a^{7} + \frac{67834}{3149} a^{6} + \frac{86078}{3149} a^{5} - \frac{180050}{3149} a^{4} + \frac{139104}{3149} a^{3} - \frac{54014}{3149} a^{2} + \frac{3792}{3149} a + \frac{3758}{3149} \)
(order $6$)
| 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: |
$\frac{82388}{40937}a^{15}-\frac{3639}{3149}a^{14}+\frac{8932}{40937}a^{13}-\frac{138495}{40937}a^{12}+\frac{48985}{3149}a^{11}-\frac{464909}{40937}a^{10}-\frac{617878}{40937}a^{9}+\frac{1770199}{40937}a^{8}-\frac{15219}{611}a^{7}-\frac{1607859}{40937}a^{6}+\frac{3187894}{40937}a^{5}-\frac{2438289}{40937}a^{4}+\frac{584417}{40937}a^{3}+\frac{43892}{3149}a^{2}-\frac{34707}{3149}a+\frac{129834}{40937}$, $\frac{101181}{40937}a^{15}+\frac{6811}{3149}a^{14}-\frac{202250}{40937}a^{13}-\frac{66662}{40937}a^{12}+\frac{38655}{3149}a^{11}+\frac{838436}{40937}a^{10}-\frac{2640200}{40937}a^{9}+\frac{1908499}{40937}a^{8}+\frac{38899}{611}a^{7}-\frac{6161491}{40937}a^{6}+\frac{2779679}{40937}a^{5}+\frac{4273038}{40937}a^{4}-\frac{7743312}{40937}a^{3}+\frac{461191}{3149}a^{2}-\frac{184216}{3149}a+\frac{482741}{40937}$, $\frac{191458}{40937}a^{15}-\frac{20547}{3149}a^{14}+\frac{108317}{40937}a^{13}-\frac{368465}{40937}a^{12}+\frac{139961}{3149}a^{11}-\frac{2317799}{40937}a^{10}-\frac{533118}{40937}a^{9}+\frac{4944669}{40937}a^{8}-\frac{76033}{611}a^{7}-\frac{1807578}{40937}a^{6}+\frac{9293461}{40937}a^{5}-\frac{10352013}{40937}a^{4}+\frac{6292847}{40937}a^{3}-\frac{153683}{3149}a^{2}+\frac{16968}{3149}a+\frac{57489}{40937}$, $\frac{5369}{3149}a^{15}-\frac{4624}{3149}a^{14}+\frac{165}{3149}a^{13}-\frac{10078}{3149}a^{12}+\frac{45755}{3149}a^{11}-\frac{39958}{3149}a^{10}-\frac{39437}{3149}a^{9}+\frac{119178}{3149}a^{8}-\frac{1114}{47}a^{7}-\frac{99038}{3149}a^{6}+\frac{208886}{3149}a^{5}-\frac{168912}{3149}a^{4}+\frac{73454}{3149}a^{3}-\frac{12216}{3149}a^{2}-\frac{1490}{3149}a-\frac{324}{3149}$, $\frac{77852}{40937}a^{15}-\frac{7512}{3149}a^{14}+\frac{96086}{40937}a^{13}-\frac{209071}{40937}a^{12}+\frac{56969}{3149}a^{11}-\frac{956315}{40937}a^{10}+\frac{242564}{40937}a^{9}+\frac{1395725}{40937}a^{8}-\frac{32020}{611}a^{7}+\frac{557331}{40937}a^{6}+\frac{2453202}{40937}a^{5}-\frac{4691324}{40937}a^{4}+\frac{4824345}{40937}a^{3}-\frac{236887}{3149}a^{2}+\frac{94610}{3149}a-\frac{248967}{40937}$, $\frac{122432}{40937}a^{15}-\frac{913}{3149}a^{14}-\frac{126052}{40937}a^{13}-\frac{138166}{40937}a^{12}+\frac{65341}{3149}a^{11}-\frac{19020}{40937}a^{10}-\frac{2093490}{40937}a^{9}+\frac{2755225}{40937}a^{8}+\frac{8187}{611}a^{7}-\frac{5040128}{40937}a^{6}+\frac{4605179}{40937}a^{5}+\frac{447041}{40937}a^{4}-\frac{4122022}{40937}a^{3}+\frac{302251}{3149}a^{2}-\frac{132796}{3149}a+\frac{379263}{40937}$, $\frac{367187}{40937}a^{15}-\frac{41097}{3149}a^{14}+\frac{366851}{40937}a^{13}-\frac{819427}{40937}a^{12}+\frac{272735}{3149}a^{11}-\frac{4892016}{40937}a^{10}+\frac{326569}{40937}a^{9}+\frac{8650325}{40937}a^{8}-\frac{167671}{611}a^{7}-\frac{133768}{40937}a^{6}+\frac{16322078}{40937}a^{5}-\frac{23154076}{40937}a^{4}+\frac{17987757}{40937}a^{3}-\frac{658847}{3149}a^{2}+\frac{191874}{3149}a-\frac{330344}{40937}$
| 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: | \( 37.0149239843 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 37.0149239843 \cdot 1}{6\cdot\sqrt{7058653305387264}}\cr\approx \mathstrut & 0.178362533511 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*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 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*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 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*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 + 2*x^14 - 3*x^13 + 11*x^12 - 19*x^11 + 10*x^10 + 21*x^9 - 44*x^8 + 21*x^7 + 40*x^6 - 89*x^5 + 92*x^4 - 60*x^3 + 26*x^2 - 7*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
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 32 |
| The 14 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 4.2.1323.1 x2, 4.0.189.1 x2, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.1750329.1 |
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
| Galois closure: | deg 32 |
| Degree 16 siblings: | 16.4.1807015246179139584.1, 16.0.36877862166921216.1 |
| Minimal sibling: | 16.0.36877862166921216.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.4 | $x^{8} + 8 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 224 x^{3} + 144 x^{2} - 224 x + 752$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(3\)
| 3.16.14.1 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
|
\(7\)
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |