Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 + x^13 + x^12 - 18*x^11 + 36*x^10 - 33*x^9 + 23*x^8 - 42*x^7 + 92*x^6 - 129*x^5 + 120*x^4 - 76*x^3 + 32*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + 3*y^14 + y^13 + y^12 - 18*y^11 + 36*y^10 - 33*y^9 + 23*y^8 - 42*y^7 + 92*y^6 - 129*y^5 + 120*y^4 - 76*y^3 + 32*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 3*x^14 + x^13 + x^12 - 18*x^11 + 36*x^10 - 33*x^9 + 23*x^8 - 42*x^7 + 92*x^6 - 129*x^5 + 120*x^4 - 76*x^3 + 32*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 3*x^14 + x^13 + x^12 - 18*x^11 + 36*x^10 - 33*x^9 + 23*x^8 - 42*x^7 + 92*x^6 - 129*x^5 + 120*x^4 - 76*x^3 + 32*x^2 - 8*x + 1)
\( x^{16} - 3 x^{15} + 3 x^{14} + x^{13} + x^{12} - 18 x^{11} + 36 x^{10} - 33 x^{9} + 23 x^{8} - 42 x^{7} + \cdots + 1 \)
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: |
\(8067379209183488\)
\(\medspace = 2^{8}\cdot 17^{3}\cdot 283^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.87\) | 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: | $2^{7/4}17^{3/4}283^{1/2}\approx 473.73165033466563$ | ||
| Ramified primes: |
\(2\), \(17\), \(283\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7951}a^{15}-\frac{297}{7951}a^{14}-\frac{140}{7951}a^{13}+\frac{1406}{7951}a^{12}+\frac{89}{7951}a^{11}-\frac{2331}{7951}a^{10}+\frac{1564}{7951}a^{9}+\frac{1309}{7951}a^{8}-\frac{3175}{7951}a^{7}+\frac{3141}{7951}a^{6}-\frac{1046}{7951}a^{5}-\frac{2694}{7951}a^{4}-\frac{2944}{7951}a^{3}-\frac{1199}{7951}a^{2}+\frac{2694}{7951}a+\frac{3056}{7951}$
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: |
\( -1 \)
(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: |
$\frac{2987}{7951}a^{15}+\frac{11324}{7951}a^{14}-\frac{28581}{7951}a^{13}+\frac{9545}{7951}a^{12}+\frac{51166}{7951}a^{11}+\frac{18281}{7951}a^{10}-\frac{202295}{7951}a^{9}+\frac{236621}{7951}a^{8}-\frac{45888}{7951}a^{7}+\frac{23840}{7951}a^{6}-\frac{349503}{7951}a^{5}+\frac{691171}{7951}a^{4}-\frac{683708}{7951}a^{3}+\frac{417940}{7951}a^{2}-\frac{142552}{7951}a+\frac{16426}{7951}$, $\frac{43531}{7951}a^{15}-\frac{103744}{7951}a^{14}+\frac{59734}{7951}a^{13}+\frac{93200}{7951}a^{12}+\frac{97534}{7951}a^{11}-\frac{739542}{7951}a^{10}+\frac{1087358}{7951}a^{9}-\frac{662671}{7951}a^{8}+\frac{470417}{7951}a^{7}-\frac{1489313}{7951}a^{6}+\frac{3031282}{7951}a^{5}-\frac{3541410}{7951}a^{4}+\frac{2662539}{7951}a^{3}-\frac{1283416}{7951}a^{2}+\frac{353059}{7951}a-\frac{37200}{7951}$, $\frac{11235}{7951}a^{15}-\frac{21228}{7951}a^{14}+\frac{9349}{7951}a^{13}+\frac{21626}{7951}a^{12}+\frac{37844}{7951}a^{11}-\frac{165162}{7951}a^{10}+\frac{214507}{7951}a^{9}-\frac{122000}{7951}a^{8}+\frac{132228}{7951}a^{7}-\frac{363149}{7951}a^{6}+\frac{643799}{7951}a^{5}-\frac{713223}{7951}a^{4}+\frac{556890}{7951}a^{3}-\frac{303909}{7951}a^{2}+\frac{116898}{7951}a-\frac{22111}{7951}$, $\frac{14485}{7951}a^{15}-\frac{24407}{7951}a^{14}-\frac{395}{7951}a^{13}+\frac{35203}{7951}a^{12}+\frac{56760}{7951}a^{11}-\frac{211315}{7951}a^{10}+\frac{200916}{7951}a^{9}-\frac{42025}{7951}a^{8}+\frac{86170}{7951}a^{7}-\frac{411689}{7951}a^{6}+\frac{687082}{7951}a^{5}-\frac{619260}{7951}a^{4}+\frac{315413}{7951}a^{3}-\frac{58188}{7951}a^{2}-\frac{32722}{7951}a+\frac{10894}{7951}$, $\frac{27794}{7951}a^{15}-\frac{57337}{7951}a^{14}+\frac{20732}{7951}a^{13}+\frac{62807}{7951}a^{12}+\frac{80415}{7951}a^{11}-\frac{440371}{7951}a^{10}+\frac{558269}{7951}a^{9}-\frac{271764}{7951}a^{8}+\frac{224827}{7951}a^{7}-\frac{859734}{7951}a^{6}+\frac{1658091}{7951}a^{5}-\frac{1775542}{7951}a^{4}+\frac{1190855}{7951}a^{3}-\frac{455572}{7951}a^{2}+\frac{58126}{7951}a+\frac{13833}{7951}$, $\frac{11536}{7951}a^{15}-\frac{7262}{7951}a^{14}-\frac{16889}{7951}a^{13}+\frac{23429}{7951}a^{12}+\frac{72584}{7951}a^{11}-\frac{111448}{7951}a^{10}+\frac{1485}{7951}a^{9}+\frac{89136}{7951}a^{8}+\frac{67065}{7951}a^{7}-\frac{244612}{7951}a^{6}+\frac{217639}{7951}a^{5}-\frac{5476}{7951}a^{4}-\frac{122528}{7951}a^{3}+\frac{130292}{7951}a^{2}-\frac{58132}{7951}a+\frac{7233}{7951}$, $\frac{39029}{7951}a^{15}-\frac{78565}{7951}a^{14}+\frac{30081}{7951}a^{13}+\frac{84433}{7951}a^{12}+\frac{118259}{7951}a^{11}-\frac{605533}{7951}a^{10}+\frac{772776}{7951}a^{9}-\frac{393764}{7951}a^{8}+\frac{357055}{7951}a^{7}-\frac{1222883}{7951}a^{6}+\frac{2301890}{7951}a^{5}-\frac{2488765}{7951}a^{4}+\frac{1747745}{7951}a^{3}-\frac{759481}{7951}a^{2}+\frac{175024}{7951}a-\frac{16229}{7951}$
| 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: | \( 13.0020519042 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 13.0020519042 \cdot 1}{2\cdot\sqrt{8067379209183488}}\cr\approx \mathstrut & 0.175814467991 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 + x^13 + x^12 - 18*x^11 + 36*x^10 - 33*x^9 + 23*x^8 - 42*x^7 + 92*x^6 - 129*x^5 + 120*x^4 - 76*x^3 + 32*x^2 - 8*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 + 3*x^14 + x^13 + x^12 - 18*x^11 + 36*x^10 - 33*x^9 + 23*x^8 - 42*x^7 + 92*x^6 - 129*x^5 + 120*x^4 - 76*x^3 + 32*x^2 - 8*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 + 3*x^14 + x^13 + x^12 - 18*x^11 + 36*x^10 - 33*x^9 + 23*x^8 - 42*x^7 + 92*x^6 - 129*x^5 + 120*x^4 - 76*x^3 + 32*x^2 - 8*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 + 3*x^14 + x^13 + x^12 - 18*x^11 + 36*x^10 - 33*x^9 + 23*x^8 - 42*x^7 + 92*x^6 - 129*x^5 + 120*x^4 - 76*x^3 + 32*x^2 - 8*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
$C_2^7.C_2\wr S_4$ (as 16T1851):
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 49152 |
| The 116 conjugacy class representatives for $C_2^7.C_2\wr S_4$ |
| Character table for $C_2^7.C_2\wr S_4$ |
Intermediate fields
| 4.2.283.1, 8.0.1361513.1 |
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 | R | $16$ | $16$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.6 | $x^{8} + 2 x^{7} + 24 x^{6} + 84 x^{5} + 264 x^{4} + 408 x^{3} + 384 x^{2} - 208 x + 80$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x^{2} + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(17\)
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.4.3.2 | $x^{4} + 34$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(283\)
| $\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{283}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |