Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*x^2 - 3*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + 2*y^14 + 6*y^13 - 10*y^12 - 3*y^11 + 23*y^10 - 24*y^9 + 7*y^8 + 6*y^7 + y^6 - 15*y^5 + 20*y^4 - 15*y^3 + 7*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*x^2 - 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*x^2 - 3*x + 1)
\( x^{16} - 3 x^{15} + 2 x^{14} + 6 x^{13} - 10 x^{12} - 3 x^{11} + 23 x^{10} - 24 x^{9} + 7 x^{8} + \cdots + 1 \)
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: |
\(6158959248447369\)
\(\medspace = 3^{12}\cdot 7^{4}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.70\) | 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: | $3^{3/4}7^{1/2}13^{3/4}\approx 41.29024647011401$ | ||
| Ramified primes: |
\(3\), \(7\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{238663}a^{15}+\frac{107485}{238663}a^{14}-\frac{89485}{238663}a^{13}+\frac{32552}{238663}a^{12}-\frac{88877}{238663}a^{11}-\frac{495}{14039}a^{10}+\frac{21273}{238663}a^{9}-\frac{38003}{238663}a^{8}+\frac{89451}{238663}a^{7}-\frac{107187}{238663}a^{6}-\frac{98593}{238663}a^{5}+\frac{27453}{238663}a^{4}+\frac{38752}{238663}a^{3}-\frac{10378}{238663}a^{2}+\frac{405}{238663}a+\frac{95971}{238663}$
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: |
\( \frac{8884}{1717} a^{15} - \frac{19678}{1717} a^{14} + \frac{1713}{1717} a^{13} + \frac{56036}{1717} a^{12} - \frac{44856}{1717} a^{11} - \frac{3878}{101} a^{10} + \frac{155389}{1717} a^{9} - \frac{85641}{1717} a^{8} - \frac{15313}{1717} a^{7} + \frac{45251}{1717} a^{6} + \frac{46225}{1717} a^{5} - \frac{98399}{1717} a^{4} + \frac{96684}{1717} a^{3} - \frac{50196}{1717} a^{2} + \frac{16358}{1717} a - \frac{9477}{1717} \)
(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: |
$\frac{1083357}{238663}a^{15}-\frac{2391837}{238663}a^{14}+\frac{232129}{238663}a^{13}+\frac{6797422}{238663}a^{12}-\frac{5524607}{238663}a^{11}-\frac{463280}{14039}a^{10}+\frac{19092569}{238663}a^{9}-\frac{10756428}{238663}a^{8}-\frac{1982806}{238663}a^{7}+\frac{5901129}{238663}a^{6}+\frac{5788894}{238663}a^{5}-\frac{12239163}{238663}a^{4}+\frac{11691273}{238663}a^{3}-\frac{6108917}{238663}a^{2}+\frac{2006295}{238663}a-\frac{1487988}{238663}$, $\frac{573866}{238663}a^{15}-\frac{1181329}{238663}a^{14}+\frac{2711}{238663}a^{13}+\frac{3435641}{238663}a^{12}-\frac{2398697}{238663}a^{11}-\frac{237207}{14039}a^{10}+\frac{9069499}{238663}a^{9}-\frac{5093907}{238663}a^{8}-\frac{421115}{238663}a^{7}+\frac{2742667}{238663}a^{6}+\frac{2814839}{238663}a^{5}-\frac{5767907}{238663}a^{4}+\frac{5803467}{238663}a^{3}-\frac{3087465}{238663}a^{2}+\frac{1151283}{238663}a-\frac{811972}{238663}$, $\frac{676016}{238663}a^{15}-\frac{1518557}{238663}a^{14}+\frac{141524}{238663}a^{13}+\frac{4285514}{238663}a^{12}-\frac{3398379}{238663}a^{11}-\frac{303174}{14039}a^{10}+\frac{11705127}{238663}a^{9}-\frac{6201314}{238663}a^{8}-\frac{930409}{238663}a^{7}+\frac{2494405}{238663}a^{6}+\frac{3834478}{238663}a^{5}-\frac{7166185}{238663}a^{4}+\frac{7049064}{238663}a^{3}-\frac{4252434}{238663}a^{2}+\frac{1710660}{238663}a-\frac{934373}{238663}$, $\frac{699650}{238663}a^{15}-\frac{1547039}{238663}a^{14}+\frac{284540}{238663}a^{13}+\frac{4169970}{238663}a^{12}-\frac{3682997}{238663}a^{11}-\frac{251361}{14039}a^{10}+\frac{12085594}{238663}a^{9}-\frac{8184651}{238663}a^{8}+\frac{509649}{238663}a^{7}+\frac{3599244}{238663}a^{6}+\frac{3038396}{238663}a^{5}-\frac{7505343}{238663}a^{4}+\frac{8595879}{238663}a^{3}-\frac{4896511}{238663}a^{2}+\frac{1974573}{238663}a-\frac{1008911}{238663}$, $\frac{590026}{238663}a^{15}-\frac{1451706}{238663}a^{14}+\frac{461554}{238663}a^{13}+\frac{3701372}{238663}a^{12}-\frac{4047724}{238663}a^{11}-\frac{206099}{14039}a^{10}+\frac{11791752}{238663}a^{9}-\frac{8722433}{238663}a^{8}+\frac{2580}{238663}a^{7}+\frac{4248779}{238663}a^{6}+\frac{2150158}{238663}a^{5}-\frac{7711248}{238663}a^{4}+\frac{8409368}{238663}a^{3}-\frac{4686497}{238663}a^{2}+\frac{1490845}{238663}a-\frac{752786}{238663}$, $\frac{1837616}{238663}a^{15}-\frac{4206996}{238663}a^{14}+\frac{693892}{238663}a^{13}+\frac{11514862}{238663}a^{12}-\frac{10195581}{238663}a^{11}-\frac{749099}{14039}a^{10}+\frac{33211903}{238663}a^{9}-\frac{20504099}{238663}a^{8}-\frac{1597782}{238663}a^{7}+\frac{9812891}{238663}a^{6}+\frac{8760770}{238663}a^{5}-\frac{21276573}{238663}a^{4}+\frac{21702277}{238663}a^{3}-\frac{12106320}{238663}a^{2}+\frac{4379180}{238663}a-\frac{2656040}{238663}$, $\frac{2403860}{238663}a^{15}-\frac{5308916}{238663}a^{14}+\frac{652419}{238663}a^{13}+\frac{14810016}{238663}a^{12}-\frac{12260715}{238663}a^{11}-\frac{975868}{14039}a^{10}+\frac{41713447}{238663}a^{9}-\frac{24960033}{238663}a^{8}-\frac{1715565}{238663}a^{7}+\frac{12114097}{238663}a^{6}+\frac{11740331}{238663}a^{5}-\frac{26029143}{238663}a^{4}+\frac{27602794}{238663}a^{3}-\frac{15090122}{238663}a^{2}+\frac{5546172}{238663}a-\frac{3218216}{238663}$
| 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: | \( 33.7844971841 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 33.7844971841 \cdot 1}{6\cdot\sqrt{6158959248447369}}\cr\approx \mathstrut & 0.174281580963 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*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 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*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 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*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 - 3*x^15 + 2*x^14 + 6*x^13 - 10*x^12 - 3*x^11 + 23*x^10 - 24*x^9 + 7*x^8 + 6*x^7 + x^6 - 15*x^5 + 20*x^4 - 15*x^3 + 7*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
$C_4^2:D_4$ (as 16T211):
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 128 |
| The 44 conjugacy class representatives for $C_4^2:D_4$ |
| Character table for $C_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.117.1, 4.0.2457.2, 8.0.8719893.2, 8.0.1601613.1, 8.0.6036849.1 |
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: | 16.0.6158959248447369.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 36 x^{12} + 120 x^{11} + 312 x^{10} + 352 x^{9} - 522 x^{8} - 2664 x^{7} - 3672 x^{6} - 1440 x^{5} + 4292 x^{4} + 7720 x^{3} + 6408 x^{2} + 2592 x + 433$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $[\ ]_{4}^{4}$ |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.3.4 | $x^{4} + 91$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.4.3.4 | $x^{4} + 91$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |