Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 - 5*x^13 + 4*x^12 + 5*x^11 - 4*x^10 - 9*x^8 + 12*x^7 + 2*x^6 - 9*x^5 + 8*x^4 - 7*x^3 + x^2 + x + 1)
gp: K = bnfinit(y^16 - y^15 + y^14 - 5*y^13 + 4*y^12 + 5*y^11 - 4*y^10 - 9*y^8 + 12*y^7 + 2*y^6 - 9*y^5 + 8*y^4 - 7*y^3 + y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + x^14 - 5*x^13 + 4*x^12 + 5*x^11 - 4*x^10 - 9*x^8 + 12*x^7 + 2*x^6 - 9*x^5 + 8*x^4 - 7*x^3 + x^2 + x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 + x^14 - 5*x^13 + 4*x^12 + 5*x^11 - 4*x^10 - 9*x^8 + 12*x^7 + 2*x^6 - 9*x^5 + 8*x^4 - 7*x^3 + x^2 + x + 1)
\( x^{16} - x^{15} + x^{14} - 5 x^{13} + 4 x^{12} + 5 x^{11} - 4 x^{10} - 9 x^{8} + 12 x^{7} + 2 x^{6} + \cdots + 1 \)
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: |
\(26987918400000000\)
\(\medspace = 2^{12}\cdot 3^{2}\cdot 5^{8}\cdot 37^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.64\) | 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: | $2^{4/3}3^{1/2}5^{1/2}37^{1/2}\approx 59.363543824214226$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{39}a^{14}+\frac{1}{13}a^{13}-\frac{2}{13}a^{12}-\frac{8}{39}a^{11}+\frac{8}{39}a^{10}-\frac{2}{13}a^{9}+\frac{5}{13}a^{8}+\frac{6}{13}a^{7}+\frac{4}{13}a^{6}-\frac{3}{13}a^{5}+\frac{11}{39}a^{4}+\frac{11}{39}a^{3}-\frac{1}{39}a^{2}+\frac{14}{39}a-\frac{2}{39}$, $\frac{1}{577707}a^{15}+\frac{843}{192569}a^{14}+\frac{4656}{192569}a^{13}+\frac{10030}{577707}a^{12}+\frac{134552}{577707}a^{11}-\frac{4963}{14813}a^{10}-\frac{13562}{192569}a^{9}+\frac{1898}{14813}a^{8}+\frac{77300}{192569}a^{7}-\frac{5099}{14813}a^{6}+\frac{286664}{577707}a^{5}-\frac{73447}{577707}a^{4}-\frac{125134}{577707}a^{3}+\frac{246230}{577707}a^{2}+\frac{223381}{577707}a-\frac{86093}{192569}$
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: |
$\frac{201106}{577707}a^{15}+\frac{7532}{577707}a^{14}+\frac{64245}{192569}a^{13}-\frac{748493}{577707}a^{12}-\frac{77726}{577707}a^{11}+\frac{1075390}{577707}a^{10}-\frac{15199}{192569}a^{9}+\frac{89956}{192569}a^{8}-\frac{497879}{192569}a^{7}+\frac{201489}{192569}a^{6}+\frac{1179878}{577707}a^{5}-\frac{24246}{14813}a^{4}+\frac{493821}{192569}a^{3}-\frac{65150}{192569}a^{2}-\frac{984868}{577707}a+\frac{16853}{577707}$, $\frac{36808}{192569}a^{15}+\frac{32166}{192569}a^{14}+\frac{34166}{192569}a^{13}-\frac{6849}{14813}a^{12}-\frac{121721}{192569}a^{11}+\frac{251432}{192569}a^{10}+\frac{112890}{192569}a^{9}+\frac{46686}{192569}a^{8}-\frac{260489}{192569}a^{7}-\frac{83559}{192569}a^{6}+\frac{495246}{192569}a^{5}-\frac{64676}{192569}a^{4}+\frac{21948}{192569}a^{3}+\frac{18294}{192569}a^{2}-\frac{147753}{192569}a+\frac{101838}{192569}$, $\frac{230270}{577707}a^{15}-\frac{183208}{577707}a^{14}+\frac{90684}{192569}a^{13}-\frac{1131022}{577707}a^{12}+\frac{262855}{192569}a^{11}+\frac{954139}{577707}a^{10}-\frac{193210}{192569}a^{9}+\frac{52139}{192569}a^{8}-\frac{718531}{192569}a^{7}+\frac{785319}{192569}a^{6}+\frac{295363}{577707}a^{5}-\frac{1394053}{577707}a^{4}+\frac{2026313}{577707}a^{3}-\frac{1197154}{577707}a^{2}+\frac{17397}{192569}a-\frac{288037}{577707}$, $\frac{154265}{577707}a^{15}+\frac{21017}{577707}a^{14}+\frac{392}{14813}a^{13}-\frac{576859}{577707}a^{12}-\frac{66700}{192569}a^{11}+\frac{1300681}{577707}a^{10}+\frac{61003}{192569}a^{9}-\frac{221252}{192569}a^{8}-\frac{358207}{192569}a^{7}+\frac{153487}{192569}a^{6}+\frac{1950718}{577707}a^{5}-\frac{1526437}{577707}a^{4}-\frac{354064}{577707}a^{3}+\frac{598643}{577707}a^{2}-\frac{23094}{14813}a+\frac{212327}{577707}$, $\frac{400469}{577707}a^{15}-\frac{304595}{577707}a^{14}+\frac{145419}{192569}a^{13}-\frac{1914700}{577707}a^{12}+\frac{1247063}{577707}a^{11}+\frac{1917548}{577707}a^{10}-\frac{359266}{192569}a^{9}+\frac{145643}{192569}a^{8}-\frac{1242431}{192569}a^{7}+\frac{1347851}{192569}a^{6}+\frac{865594}{577707}a^{5}-\frac{71554}{14813}a^{4}+\frac{1062357}{192569}a^{3}-\frac{896252}{192569}a^{2}+\frac{406966}{577707}a+\frac{91372}{577707}$, $\frac{140507}{577707}a^{15}-\frac{56599}{192569}a^{14}+\frac{14073}{192569}a^{13}-\frac{719821}{577707}a^{12}+\frac{658435}{577707}a^{11}+\frac{377436}{192569}a^{10}-\frac{218996}{192569}a^{9}-\frac{290688}{192569}a^{8}-\frac{455963}{192569}a^{7}+\frac{656517}{192569}a^{6}+\frac{1410949}{577707}a^{5}-\frac{2103926}{577707}a^{4}+\frac{176290}{577707}a^{3}-\frac{456011}{577707}a^{2}+\frac{128369}{577707}a+\frac{6604}{14813}$, $\frac{43434}{192569}a^{15}+\frac{211142}{577707}a^{14}+\frac{64136}{192569}a^{13}-\frac{81375}{192569}a^{12}-\frac{775738}{577707}a^{11}+\frac{430984}{577707}a^{10}+\frac{301810}{192569}a^{9}+\frac{176532}{192569}a^{8}-\frac{154701}{192569}a^{7}-\frac{342390}{192569}a^{6}+\frac{311790}{192569}a^{5}+\frac{832696}{577707}a^{4}+\frac{243721}{577707}a^{3}+\frac{755134}{577707}a^{2}-\frac{613190}{577707}a-\frac{35956}{44439}$
| 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: | \( 31.5416353238 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 31.5416353238 \cdot 1}{2\cdot\sqrt{26987918400000000}}\cr\approx \mathstrut & 0.233189216512 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 - 5*x^13 + 4*x^12 + 5*x^11 - 4*x^10 - 9*x^8 + 12*x^7 + 2*x^6 - 9*x^5 + 8*x^4 - 7*x^3 + x^2 + 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 - x^15 + x^14 - 5*x^13 + 4*x^12 + 5*x^11 - 4*x^10 - 9*x^8 + 12*x^7 + 2*x^6 - 9*x^5 + 8*x^4 - 7*x^3 + x^2 + 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 - x^15 + x^14 - 5*x^13 + 4*x^12 + 5*x^11 - 4*x^10 - 9*x^8 + 12*x^7 + 2*x^6 - 9*x^5 + 8*x^4 - 7*x^3 + x^2 + 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 - x^15 + x^14 - 5*x^13 + 4*x^12 + 5*x^11 - 4*x^10 - 9*x^8 + 12*x^7 + 2*x^6 - 9*x^5 + 8*x^4 - 7*x^3 + x^2 + 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
$(C_2^3\times C_4):S_4$ (as 16T1046):
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 768 |
| The 40 conjugacy class representatives for $(C_2^3\times C_4):S_4$ |
| Character table for $(C_2^3\times C_4):S_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.3700.1, 8.0.13690000.1 |
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 | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ |
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(3\)
| 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.12.0.1 | $x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(37\)
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.8.4.1 | $x^{8} + 3700 x^{7} + 5133910 x^{6} + 3166256548 x^{5} + 732510094073 x^{4} + 136269235536 x^{3} + 4476368972260 x^{2} + 17928293629116 x + 2173698901413$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |