Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*x^2 - 6*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + 7*y^14 - 12*y^13 + 17*y^12 - 18*y^11 + 13*y^10 + 3*y^9 - 29*y^8 + 45*y^7 - 34*y^6 + 3*y^5 + 23*y^4 - 27*y^3 + 17*y^2 - 6*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*x^2 - 6*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*x^2 - 6*x + 1)
\( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 17 x^{12} - 18 x^{11} + 13 x^{10} + 3 x^{9} - 29 x^{8} + \cdots + 1 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{258287}a^{15}+\frac{30004}{258287}a^{14}-\frac{58447}{258287}a^{13}-\frac{50411}{258287}a^{12}+\frac{104099}{258287}a^{11}-\frac{24303}{258287}a^{10}-\frac{115907}{258287}a^{9}+\frac{71396}{258287}a^{8}-\frac{110922}{258287}a^{7}+\frac{108160}{258287}a^{6}-\frac{77356}{258287}a^{5}+\frac{3780}{258287}a^{4}+\frac{38490}{258287}a^{3}-\frac{90061}{258287}a^{2}-\frac{3529}{258287}a+\frac{2961}{258287}$
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: |
\( -\frac{2168}{1427} a^{15} + \frac{6831}{1427} a^{14} - \frac{15920}{1427} a^{13} + \frac{27085}{1427} a^{12} - \frac{37976}{1427} a^{11} + \frac{39739}{1427} a^{10} - \frac{26875}{1427} a^{9} - \frac{9827}{1427} a^{8} + \frac{69352}{1427} a^{7} - \frac{106130}{1427} a^{6} + \frac{76691}{1427} a^{5} + \frac{4502}{1427} a^{4} - \frac{63856}{1427} a^{3} + \frac{62907}{1427} a^{2} - \frac{32096}{1427} a + \frac{7760}{1427} \)
(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: |
$\frac{742714}{258287}a^{15}-\frac{2161226}{258287}a^{14}+\frac{4765237}{258287}a^{13}-\frac{7937118}{258287}a^{12}+\frac{10743873}{258287}a^{11}-\frac{10639401}{258287}a^{10}+\frac{6471242}{258287}a^{9}+\frac{4620236}{258287}a^{8}-\frac{21796896}{258287}a^{7}+\frac{29226505}{258287}a^{6}-\frac{16812559}{258287}a^{5}-\frac{5028223}{258287}a^{4}+\frac{17936790}{258287}a^{3}-\frac{15445236}{258287}a^{2}+\frac{7032519}{258287}a-\frac{1687373}{258287}$, $\frac{511103}{258287}a^{15}-\frac{914500}{258287}a^{14}+\frac{2070527}{258287}a^{13}-\frac{2893092}{258287}a^{12}+\frac{3613224}{258287}a^{11}-\frac{2638962}{258287}a^{10}+\frac{807473}{258287}a^{9}+\frac{4313307}{258287}a^{8}-\frac{9935094}{258287}a^{7}+\frac{7482480}{258287}a^{6}-\frac{634291}{258287}a^{5}-\frac{4408299}{258287}a^{4}+\frac{5349142}{258287}a^{3}-\frac{2929022}{258287}a^{2}+\frac{1485356}{258287}a-\frac{185837}{258287}$, $\frac{51764}{258287}a^{15}+\frac{47325}{258287}a^{14}-\frac{134877}{258287}a^{13}+\frac{515131}{258287}a^{12}-\frac{835906}{258287}a^{11}+\frac{1386920}{258287}a^{10}-\frac{1352660}{258287}a^{9}+\frac{1205296}{258287}a^{8}+\frac{211889}{258287}a^{7}-\frac{2934216}{258287}a^{6}+\frac{4358256}{258287}a^{5}-\frac{2438209}{258287}a^{4}-\frac{1320993}{258287}a^{3}+\frac{3003903}{258287}a^{2}-\frac{1874256}{258287}a+\frac{625587}{258287}$, $\frac{198149}{258287}a^{15}-\frac{762431}{258287}a^{14}+\frac{1665912}{258287}a^{13}-\frac{2997245}{258287}a^{12}+\frac{4187236}{258287}a^{11}-\frac{4503198}{258287}a^{10}+\frac{3123341}{258287}a^{9}+\frac{667014}{258287}a^{8}-\frac{7383149}{258287}a^{7}+\frac{12054930}{258287}a^{6}-\frac{8495500}{258287}a^{5}-\frac{29080}{258287}a^{4}+\frac{6771936}{258287}a^{3}-\frac{7163721}{258287}a^{2}+\frac{3531106}{258287}a-\frac{883736}{258287}$, $\frac{151638}{258287}a^{15}-\frac{237240}{258287}a^{14}+\frac{590506}{258287}a^{13}-\frac{736027}{258287}a^{12}+\frac{929018}{258287}a^{11}-\frac{535972}{258287}a^{10}+\frac{8110}{258287}a^{9}+\frac{1538478}{258287}a^{8}-\frac{2923666}{258287}a^{7}+\frac{2007876}{258287}a^{6}-\frac{5023}{258287}a^{5}-\frac{1755222}{258287}a^{4}+\frac{2101577}{258287}a^{3}-\frac{777941}{258287}a^{2}+\frac{40162}{258287}a+\frac{355599}{258287}$, $\frac{624254}{258287}a^{15}-\frac{1631085}{258287}a^{14}+\frac{3722387}{258287}a^{13}-\frac{6037489}{258287}a^{12}+\frac{8247991}{258287}a^{11}-\frac{7990053}{258287}a^{10}+\frac{4986107}{258287}a^{9}+\frac{3883030}{258287}a^{8}-\frac{16645587}{258287}a^{7}+\frac{21687504}{258287}a^{6}-\frac{12852680}{258287}a^{5}-\frac{2871069}{258287}a^{4}+\frac{13044348}{258287}a^{3}-\frac{12005980}{258287}a^{2}+\frac{6136345}{258287}a-\frac{1435400}{258287}$, $\frac{744774}{258287}a^{15}-\frac{1825292}{258287}a^{14}+\frac{3951298}{258287}a^{13}-\frac{6144395}{258287}a^{12}+\frac{7968446}{258287}a^{11}-\frac{6979885}{258287}a^{10}+\frac{3260566}{258287}a^{9}+\frac{6280415}{258287}a^{8}-\frac{19129351}{258287}a^{7}+\frac{20869240}{258287}a^{6}-\frac{7504508}{258287}a^{5}-\frac{8347764}{258287}a^{4}+\frac{14316063}{258287}a^{3}-\frac{9063655}{258287}a^{2}+\frac{3120510}{258287}a-\frac{495166}{258287}$
| 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: | \( 33.0234170129 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 33.0234170129 \cdot 1}{6\cdot\sqrt{6158959248447369}}\cr\approx \mathstrut & 0.170355453107 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*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 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*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 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*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 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 - 18*x^11 + 13*x^10 + 3*x^9 - 29*x^8 + 45*x^7 - 34*x^6 + 3*x^5 + 23*x^4 - 27*x^3 + 17*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
$C_4^2:D_4$ (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
| \(\Q(\sqrt{-3}) \), 4.0.117.1, 4.0.189.1, 4.0.2457.1, 8.0.1601613.1, 8.0.8719893.1, 8.0.6036849.2 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 36 x^{12} + 120 x^{11} + 312 x^{10} + 352 x^{9} - 522 x^{8} - 2664 x^{7} - 3672 x^{6} - 1440 x^{5} + 4292 x^{4} + 7720 x^{3} + 6408 x^{2} + 2592 x + 433$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $[\ ]_{4}^{4}$ |
|
\(7\)
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.8.6.2 | $x^{8} + 130 x^{4} - 1521$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |