Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 763*x^10 - 966*x^9 + 1007*x^8 - 850*x^7 + 543*x^6 - 218*x^5 + 5*x^4 + 60*x^3 - 28*x^2 + 1)
gp: K = bnfinit(y^16 - 8*y^15 + 36*y^14 - 112*y^13 + 265*y^12 - 498*y^11 + 763*y^10 - 966*y^9 + 1007*y^8 - 850*y^7 + 543*y^6 - 218*y^5 + 5*y^4 + 60*y^3 - 28*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 763*x^10 - 966*x^9 + 1007*x^8 - 850*x^7 + 543*x^6 - 218*x^5 + 5*x^4 + 60*x^3 - 28*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 763*x^10 - 966*x^9 + 1007*x^8 - 850*x^7 + 543*x^6 - 218*x^5 + 5*x^4 + 60*x^3 - 28*x^2 + 1)
\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 265 x^{12} - 498 x^{11} + 763 x^{10} - 966 x^{9} + 1007 x^{8} + \cdots + 1 \)
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: | $[6, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-17307213524714210420160\)
\(\medspace = -\,2^{6}\cdot 3^{8}\cdot 5\cdot 43\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.54\) | 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: | $2^{3/2}3^{1/2}5^{1/2}43^{1/2}61^{2/3}\approx 1113.1270607407046$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(43\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-215}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{26}a^{15}-\frac{1}{26}a^{14}+\frac{3}{26}a^{13}+\frac{5}{26}a^{11}+\frac{5}{26}a^{10}+\frac{5}{26}a^{9}+\frac{5}{26}a^{8}-\frac{11}{26}a^{7}+\frac{9}{26}a^{6}+\frac{4}{13}a^{5}+\frac{7}{26}a^{4}-\frac{11}{26}a^{3}+\frac{9}{26}a^{2}-\frac{2}{13}a-\frac{1}{13}$
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 $1$ (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: | $10$ | 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: |
$a-1$, $a$, $a^{3}-a^{2}+2a$, $\frac{7}{13}a^{15}-\frac{46}{13}a^{14}+\frac{367}{26}a^{13}-39a^{12}+\frac{2163}{26}a^{11}-\frac{3661}{26}a^{10}+\frac{5075}{26}a^{9}-\frac{5793}{26}a^{8}+\frac{5371}{26}a^{7}-\frac{1991}{13}a^{6}+\frac{2023}{26}a^{5}-\frac{276}{13}a^{4}-\frac{90}{13}a^{3}+\frac{89}{13}a^{2}+\frac{24}{13}a-\frac{1}{13}$, $\frac{47}{26}a^{15}-\frac{173}{13}a^{14}+\frac{740}{13}a^{13}-\frac{337}{2}a^{12}+\frac{9907}{26}a^{11}-\frac{8885}{13}a^{10}+\frac{25975}{26}a^{9}-\frac{31277}{26}a^{8}+\frac{30761}{26}a^{7}-\frac{12106}{13}a^{6}+\frac{13727}{26}a^{5}-\frac{4247}{26}a^{4}-\frac{486}{13}a^{3}+\frac{1697}{26}a^{2}-\frac{120}{13}a-\frac{55}{26}$, $\frac{1}{2}a^{14}-\frac{7}{2}a^{13}+\frac{29}{2}a^{12}-\frac{83}{2}a^{11}+91a^{10}-158a^{9}+224a^{8}-\frac{523}{2}a^{7}+249a^{6}-189a^{5}+101a^{4}-\frac{57}{2}a^{3}-\frac{17}{2}a^{2}+\frac{21}{2}a-\frac{5}{2}$, $\frac{1}{2}a^{14}-\frac{7}{2}a^{13}+15a^{12}-\frac{89}{2}a^{11}+\frac{203}{2}a^{10}-183a^{9}+269a^{8}-\frac{649}{2}a^{7}+\frac{639}{2}a^{6}-252a^{5}+\frac{281}{2}a^{4}-\frac{85}{2}a^{3}-14a^{2}+18a-3$, $\frac{2}{13}a^{15}-\frac{43}{26}a^{14}+\frac{110}{13}a^{13}-29a^{12}+\frac{1905}{26}a^{11}-\frac{1888}{13}a^{10}+\frac{3013}{13}a^{9}-\frac{3955}{13}a^{8}+\frac{8549}{26}a^{7}-\frac{3752}{13}a^{6}+\frac{2551}{13}a^{5}-\frac{2299}{26}a^{4}+\frac{186}{13}a^{3}+\frac{174}{13}a^{2}-\frac{73}{13}a-\frac{17}{13}$, $\frac{2}{13}a^{15}-\frac{17}{26}a^{14}+\frac{19}{13}a^{13}-\frac{1}{2}a^{12}-\frac{175}{26}a^{11}+\frac{348}{13}a^{10}-\frac{1605}{26}a^{9}+\frac{1375}{13}a^{8}-\frac{3749}{26}a^{7}+\frac{2046}{13}a^{6}-\frac{3621}{26}a^{5}+\frac{2381}{26}a^{4}-\frac{516}{13}a^{3}+\frac{127}{26}a^{2}+\frac{109}{13}a-\frac{30}{13}$, $\frac{1}{2}a^{12}-3a^{11}+\frac{21}{2}a^{10}-25a^{9}+45a^{8}-63a^{7}+\frac{141}{2}a^{6}-63a^{5}+\frac{81}{2}a^{4}-16a^{3}-\frac{5}{2}a^{2}+\frac{11}{2}a-\frac{1}{2}$
| 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: | \( 218886.822294 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{5}\cdot 218886.822294 \cdot 1}{2\cdot\sqrt{17307213524714210420160}}\cr\approx \mathstrut & 0.521381192011 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 763*x^10 - 966*x^9 + 1007*x^8 - 850*x^7 + 543*x^6 - 218*x^5 + 5*x^4 + 60*x^3 - 28*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 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 763*x^10 - 966*x^9 + 1007*x^8 - 850*x^7 + 543*x^6 - 218*x^5 + 5*x^4 + 60*x^3 - 28*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 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 763*x^10 - 966*x^9 + 1007*x^8 - 850*x^7 + 543*x^6 - 218*x^5 + 5*x^4 + 60*x^3 - 28*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 - 8*x^15 + 36*x^14 - 112*x^13 + 265*x^12 - 498*x^11 + 763*x^10 - 966*x^9 + 1007*x^8 - 850*x^7 + 543*x^6 - 218*x^5 + 5*x^4 + 60*x^3 - 28*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
$C_4^4.C_2\wr A_4$ (as 16T1845):
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 140 conjugacy class representatives for $C_4^4.C_2\wr A_4$ |
| Character table for $C_4^4.C_2\wr A_4$ |
Intermediate fields
| 4.4.33489.1, 8.6.8972104968.1 |
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 | R | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.2 | $x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.12.0.1 | $x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(43\)
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.12.8.1 | $x^{12} + 9 x^{10} + 364 x^{9} + 33 x^{8} + 720 x^{7} - 16731 x^{6} - 1002 x^{5} - 64041 x^{4} - 1125646 x^{3} + 428523 x^{2} - 4549998 x + 62837938$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |