Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 10*x^13 + 29*x^12 + 70*x^11 + 25*x^10 - 105*x^9 - 179*x^8 - 105*x^7 + 25*x^6 + 70*x^5 + 29*x^4 - 10*x^3 - 10*x^2 + 1)
gp: K = bnfinit(y^16 - 10*y^14 - 10*y^13 + 29*y^12 + 70*y^11 + 25*y^10 - 105*y^9 - 179*y^8 - 105*y^7 + 25*y^6 + 70*y^5 + 29*y^4 - 10*y^3 - 10*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 10*x^14 - 10*x^13 + 29*x^12 + 70*x^11 + 25*x^10 - 105*x^9 - 179*x^8 - 105*x^7 + 25*x^6 + 70*x^5 + 29*x^4 - 10*x^3 - 10*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 10*x^14 - 10*x^13 + 29*x^12 + 70*x^11 + 25*x^10 - 105*x^9 - 179*x^8 - 105*x^7 + 25*x^6 + 70*x^5 + 29*x^4 - 10*x^3 - 10*x^2 + 1)
\( x^{16} - 10 x^{14} - 10 x^{13} + 29 x^{12} + 70 x^{11} + 25 x^{10} - 105 x^{9} - 179 x^{8} - 105 x^{7} + \cdots + 1 \)
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: | $[10, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-202794005063232421875\)
\(\medspace = -\,3^{8}\cdot 5^{12}\cdot 29^{4}\cdot 179\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.59\) | 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: | $3^{1/2}5^{3/4}29^{1/2}179^{1/2}\approx 417.26662108060015$ | ||
| Ramified primes: |
\(3\), \(5\), \(29\), \(179\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-179}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $12$ | 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: |
$33a^{15}+19a^{14}-324a^{13}-516a^{12}+708a^{11}+2762a^{10}+2279a^{9}-2474a^{8}-7453a^{7}-7298a^{6}-2570a^{5}+1365a^{4}+1747a^{3}+459a^{2}-178a-89$, $7a^{15}-69a^{13}-70a^{12}+193a^{11}+480a^{10}+204a^{9}-665a^{8}-1228a^{7}-840a^{6}-4a^{5}+385a^{4}+227a^{3}-38a-6$, $a^{15}-10a^{13}-10a^{12}+29a^{11}+70a^{10}+25a^{9}-105a^{8}-179a^{7}-105a^{6}+25a^{5}+70a^{4}+29a^{3}-10a^{2}-10a$, $52a^{15}+25a^{14}-510a^{13}-765a^{12}+1160a^{11}+4216a^{10}+3270a^{9}-4020a^{8}-11286a^{7}-10693a^{6}-3515a^{5}+2155a^{4}+2536a^{3}+611a^{2}-267a-122$, $26a^{15}+13a^{14}-254a^{13}-387a^{12}+565a^{11}+2108a^{10}+1694a^{9}-1918a^{8}-5640a^{7}-5514a^{6}-2010a^{5}+904a^{4}+1228a^{3}+331a^{2}-114a-58$, $17a^{15}+6a^{14}-167a^{13}-229a^{12}+403a^{11}+1325a^{10}+921a^{9}-1405a^{8}-3529a^{7}-3119a^{6}-808a^{5}+832a^{4}+799a^{3}+151a^{2}-99a-37$, $6a^{15}-59a^{13}-60a^{12}+164a^{11}+410a^{10}+179a^{9}-560a^{8}-1049a^{7}-735a^{6}-29a^{5}+315a^{4}+199a^{3}+9a^{2}-31a-7$, $2a^{15}-3a^{14}-19a^{13}+9a^{12}+78a^{11}+53a^{10}-121a^{9}-244a^{8}-88a^{7}+198a^{6}+291a^{5}+132a^{4}-28a^{3}-52a^{2}-7a+6$, $24a^{15}+11a^{14}-236a^{13}-347a^{12}+546a^{11}+1930a^{10}+1456a^{9}-1888a^{8}-5151a^{7}-4805a^{6}-1516a^{5}+1022a^{4}+1147a^{3}+262a^{2}-127a-53$, $8a^{15}+9a^{14}-79a^{13}-168a^{12}+132a^{11}+792a^{10}+838a^{9}-496a^{8}-2213a^{7}-2477a^{6}-1095a^{5}+315a^{4}+599a^{3}+204a^{2}-51a-36$, $33a^{15}+13a^{14}-323a^{13}-457a^{12}+758a^{11}+2588a^{10}+1898a^{9}-2583a^{8}-6868a^{7}-6354a^{6}-2014a^{5}+1289a^{4}+1456a^{3}+330a^{2}-155a-65$, $19a^{15}-a^{14}-184a^{13}-182a^{12}+503a^{11}+1260a^{10}+580a^{9}-1669a^{8}-3227a^{7}-2386a^{6}-274a^{5}+837a^{4}+575a^{3}+34a^{2}-82a-18$
| 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: | \( 19452.3804844 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{3}\cdot 19452.3804844 \cdot 1}{2\cdot\sqrt{202794005063232421875}}\cr\approx \mathstrut & 0.173482137914 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 10*x^13 + 29*x^12 + 70*x^11 + 25*x^10 - 105*x^9 - 179*x^8 - 105*x^7 + 25*x^6 + 70*x^5 + 29*x^4 - 10*x^3 - 10*x^2 + 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 - 10*x^14 - 10*x^13 + 29*x^12 + 70*x^11 + 25*x^10 - 105*x^9 - 179*x^8 - 105*x^7 + 25*x^6 + 70*x^5 + 29*x^4 - 10*x^3 - 10*x^2 + 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 - 10*x^14 - 10*x^13 + 29*x^12 + 70*x^11 + 25*x^10 - 105*x^9 - 179*x^8 - 105*x^7 + 25*x^6 + 70*x^5 + 29*x^4 - 10*x^3 - 10*x^2 + 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 - 10*x^14 - 10*x^13 + 29*x^12 + 70*x^11 + 25*x^10 - 105*x^9 - 179*x^8 - 105*x^7 + 25*x^6 + 70*x^5 + 29*x^4 - 10*x^3 - 10*x^2 + 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
$C_2^2\wr C_2^2.C_4$ (as 16T1616):
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 4096 |
| The 94 conjugacy class representatives for $C_2^2\wr C_2^2.C_4$ |
| Character table for $C_2^2\wr C_2^2.C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.725.1, 4.4.32625.1, 8.8.1064390625.1 |
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: | 16.8.17925988595712890625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(5\)
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(179\)
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |