Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*x^2 - 8*x + 1)
gp: K = bnfinit(y^16 - 5*y^15 + 14*y^14 - 26*y^13 + 39*y^12 - 53*y^11 + 74*y^10 - 107*y^9 + 153*y^8 - 196*y^7 + 213*y^6 - 191*y^5 + 138*y^4 - 77*y^3 + 31*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*x^2 - 8*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*x^2 - 8*x + 1)
\( x^{16} - 5 x^{15} + 14 x^{14} - 26 x^{13} + 39 x^{12} - 53 x^{11} + 74 x^{10} - 107 x^{9} + 153 x^{8} + \cdots + 1 \)
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: |
\(2779733938787757\)
\(\medspace = 3^{8}\cdot 7^{2}\cdot 37^{2}\cdot 151^{2}\cdot 277\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.23\) | 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: | $3^{1/2}7^{1/2}37^{1/2}151^{1/2}277^{1/2}\approx 5700.840201233499$ | ||
| Ramified primes: |
\(3\), \(7\), \(37\), \(151\), \(277\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{277}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7}a^{15}-\frac{3}{7}a^{14}+\frac{1}{7}a^{13}-\frac{3}{7}a^{12}-\frac{2}{7}a^{11}-\frac{1}{7}a^{10}+\frac{2}{7}a^{9}+\frac{2}{7}a^{8}+\frac{3}{7}a^{7}-\frac{1}{7}a^{6}+\frac{1}{7}a^{5}-\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{2}{7}a+\frac{3}{7}$
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: |
\( -19 a^{15} + 73 a^{14} - 172 a^{13} + 259 a^{12} - 358 a^{11} + 470 a^{10} - 694 a^{9} + 1009 a^{8} - 1411 a^{7} + 1614 a^{6} - 1515 a^{5} + 1126 a^{4} - 629 a^{3} + 235 a^{2} - 49 a + 4 \)
(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: |
$\frac{61}{7}a^{15}-\frac{190}{7}a^{14}+\frac{362}{7}a^{13}-\frac{365}{7}a^{12}+\frac{403}{7}a^{11}-\frac{488}{7}a^{10}+\frac{878}{7}a^{9}-\frac{1299}{7}a^{8}+\frac{1667}{7}a^{7}-\frac{1160}{7}a^{6}+\frac{96}{7}a^{5}+135a^{4}-\frac{1473}{7}a^{3}+\frac{1240}{7}a^{2}-\frac{599}{7}a+\frac{134}{7}$, $\frac{36}{7}a^{15}-\frac{115}{7}a^{14}+\frac{218}{7}a^{13}-\frac{213}{7}a^{12}+\frac{215}{7}a^{11}-\frac{253}{7}a^{10}+\frac{478}{7}a^{9}-\frac{712}{7}a^{8}+\frac{906}{7}a^{7}-\frac{554}{7}a^{6}-\frac{160}{7}a^{5}+122a^{4}-\frac{1136}{7}a^{3}+\frac{948}{7}a^{2}-\frac{467}{7}a+\frac{108}{7}$, $\frac{26}{7}a^{15}-\frac{99}{7}a^{14}+\frac{201}{7}a^{13}-\frac{239}{7}a^{12}+\frac{242}{7}a^{11}-\frac{306}{7}a^{10}+\frac{493}{7}a^{9}-\frac{781}{7}a^{8}+\frac{1016}{7}a^{7}-\frac{887}{7}a^{6}+\frac{271}{7}a^{5}+53a^{4}-\frac{773}{7}a^{3}+\frac{750}{7}a^{2}-\frac{410}{7}a+\frac{106}{7}$, $3a^{15}+4a^{14}-31a^{13}+94a^{12}-142a^{11}+199a^{10}-248a^{9}+373a^{8}-550a^{7}+823a^{6}-975a^{5}+942a^{4}-712a^{3}+398a^{2}-142a+25$, $15a^{15}-72a^{14}+189a^{13}-327a^{12}+461a^{11}-616a^{10}+868a^{9}-1275a^{8}+1808a^{7}-2241a^{6}+2277a^{5}-1877a^{4}+1203a^{3}-556a^{2}+160a-21$, $\frac{68}{7}a^{15}-\frac{225}{7}a^{14}+\frac{453}{7}a^{13}-\frac{519}{7}a^{12}+\frac{613}{7}a^{11}-\frac{768}{7}a^{10}+\frac{1277}{7}a^{9}-\frac{1887}{7}a^{8}+\frac{2500}{7}a^{7}-\frac{2182}{7}a^{6}+\frac{1104}{7}a^{5}+21a^{4}-\frac{976}{7}a^{3}+\frac{1030}{7}a^{2}-\frac{550}{7}a+\frac{120}{7}$, $\frac{124}{7}a^{15}-\frac{568}{7}a^{14}+\frac{1482}{7}a^{13}-\frac{2528}{7}a^{12}+\frac{3588}{7}a^{11}-\frac{4758}{7}a^{10}+\frac{6765}{7}a^{9}-\frac{9881}{7}a^{8}+\frac{14043}{7}a^{7}-\frac{17246}{7}a^{6}+\frac{17533}{7}a^{5}-2050a^{4}+\frac{9195}{7}a^{3}-\frac{4241}{7}a^{2}+\frac{1235}{7}a-\frac{167}{7}$
| 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: | \( 20.5837782538 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 20.5837782538 \cdot 1}{6\cdot\sqrt{2779733938787757}}\cr\approx \mathstrut & 0.158056068962 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*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 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*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 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*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 - 5*x^15 + 14*x^14 - 26*x^13 + 39*x^12 - 53*x^11 + 74*x^10 - 107*x^9 + 153*x^8 - 196*x^7 + 213*x^6 - 191*x^5 + 138*x^4 - 77*x^3 + 31*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
$C_2^6.S_4^2:D_4$ (as 16T1905):
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 294912 |
| The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$ |
| Character table for $C_2^6.S_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 8.0.3167829.1 |
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 | $16$ | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.3.0.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(151\)
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{151}$ | $x + 145$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 151.3.0.1 | $x^{3} + x + 145$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 151.3.0.1 | $x^{3} + x + 145$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 151.4.0.1 | $x^{4} + 13 x^{2} + 89 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 151.4.2.1 | $x^{4} + 298 x^{3} + 22515 x^{2} + 46786 x + 3373376$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(277\)
| $\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |