Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 276*x^12 + 1776*x^10 + 6881*x^8 + 15072*x^6 + 16704*x^4 + 3072*x^2 + 256)
gp: K = bnfinit(y^16 + 24*y^14 + 276*y^12 + 1776*y^10 + 6881*y^8 + 15072*y^6 + 16704*y^4 + 3072*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 24*x^14 + 276*x^12 + 1776*x^10 + 6881*x^8 + 15072*x^6 + 16704*x^4 + 3072*x^2 + 256);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 24*x^14 + 276*x^12 + 1776*x^10 + 6881*x^8 + 15072*x^6 + 16704*x^4 + 3072*x^2 + 256)
\( x^{16} + 24x^{14} + 276x^{12} + 1776x^{10} + 6881x^{8} + 15072x^{6} + 16704x^{4} + 3072x^{2} + 256 \)
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: |
\(10646188457767892278050816\)
\(\medspace = 2^{48}\cdot 3^{8}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.66\) | 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: | $2^{3}3^{1/2}7^{1/2}\approx 36.66060555964672$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(336=2^{4}\cdot 3\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{336}(1,·)$, $\chi_{336}(323,·)$, $\chi_{336}(71,·)$, $\chi_{336}(265,·)$, $\chi_{336}(13,·)$, $\chi_{336}(335,·)$, $\chi_{336}(83,·)$, $\chi_{336}(85,·)$, $\chi_{336}(155,·)$, $\chi_{336}(97,·)$, $\chi_{336}(167,·)$, $\chi_{336}(169,·)$, $\chi_{336}(239,·)$, $\chi_{336}(181,·)$, $\chi_{336}(251,·)$, $\chi_{336}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{2}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{12}+\frac{1}{4}a^{8}+\frac{1}{16}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{13}+\frac{1}{8}a^{9}-\frac{1}{2}a^{7}+\frac{1}{32}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{131740406848}a^{14}-\frac{288317131}{16467550856}a^{12}+\frac{3477765633}{32935101712}a^{10}-\frac{2630820465}{8233775428}a^{8}-\frac{783465785}{5727843776}a^{6}+\frac{3721719239}{8233775428}a^{4}-\frac{3824511325}{8233775428}a^{2}-\frac{758504595}{2058443857}$, $\frac{1}{263480813696}a^{15}-\frac{288317131}{32935101712}a^{13}+\frac{3477765633}{65870203424}a^{11}-\frac{2630820465}{16467550856}a^{9}+\frac{4944377991}{11455687552}a^{7}+\frac{3721719239}{16467550856}a^{5}-\frac{3824511325}{16467550856}a^{3}-\frac{758504595}{4116887714}a$
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
$C_{4}\times C_{4}\times C_{8}$, which has order $128$ (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: | $7$ | 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: |
$\frac{10031865}{131740406848}a^{14}+\frac{9631167}{4116887714}a^{12}+\frac{971132253}{32935101712}a^{10}+\frac{1635605153}{8233775428}a^{8}+\frac{3957468879}{5727843776}a^{6}+\frac{18428854717}{16467550856}a^{4}+\frac{438338082}{2058443857}a^{2}+\frac{1103096384}{2058443857}$, $\frac{457506371}{263480813696}a^{15}+\frac{2784184685}{65870203424}a^{13}+\frac{32359760771}{65870203424}a^{11}+\frac{13163478911}{4116887714}a^{9}+\frac{142814966165}{11455687552}a^{7}+\frac{1792134708547}{65870203424}a^{5}+\frac{467018857015}{16467550856}a^{3}+\frac{4183940842}{2058443857}a$, $\frac{5002089}{4116887714}a^{15}+\frac{59306438}{2058443857}a^{13}+\frac{5394133739}{16467550856}a^{11}+\frac{4269983224}{2058443857}a^{9}+\frac{704830600}{89497559}a^{7}+\frac{34342691200}{2058443857}a^{5}+\frac{286600651155}{16467550856}a^{3}+\frac{2566542133}{2058443857}a$, $\frac{331527}{178995118}a^{15}-\frac{15705}{80673856}a^{14}+\frac{3925864}{89497559}a^{13}-\frac{12439}{2521058}a^{12}+\frac{356717709}{715980472}a^{11}-\frac{1149777}{20168464}a^{10}+\frac{563993821}{178995118}a^{9}-\frac{1794777}{5042116}a^{8}+\frac{1069044024}{89497559}a^{7}-\frac{98278713}{80673856}a^{6}+\frac{2258217632}{89497559}a^{5}-\frac{19612893}{10084232}a^{4}+\frac{18898398181}{715980472}a^{3}-\frac{1864017}{5042116}a^{2}+\frac{338415801}{178995118}a+\frac{482225}{1260529}$, $\frac{331527}{178995118}a^{15}+\frac{3925864}{89497559}a^{13}+\frac{356717709}{715980472}a^{11}+\frac{563993821}{178995118}a^{9}+\frac{1069044024}{89497559}a^{7}+\frac{2258217632}{89497559}a^{5}+\frac{18898398181}{715980472}a^{3}+\frac{338415801}{178995118}a-1$, $\frac{945514115}{263480813696}a^{15}-\frac{15705}{80673856}a^{14}+\frac{5673620589}{65870203424}a^{13}-\frac{12439}{2521058}a^{12}+\frac{65177789999}{65870203424}a^{11}-\frac{1149777}{20168464}a^{10}+\frac{13067668397}{2058443857}a^{9}-\frac{1794777}{5042116}a^{8}+\frac{279652601237}{11455687552}a^{7}-\frac{98278713}{80673856}a^{6}+\frac{3454182885699}{65870203424}a^{5}-\frac{19612893}{10084232}a^{4}+\frac{450841007589}{8233775428}a^{3}-\frac{1864017}{5042116}a^{2}+\frac{16151445107}{4116887714}a+\frac{1742754}{1260529}$, $\frac{10248153}{4116887714}a^{15}-\frac{315}{1099009}a^{14}+\frac{121283306}{2058443857}a^{13}-\frac{114047}{17584144}a^{12}+\frac{11014880875}{16467550856}a^{11}-\frac{154299}{2198018}a^{10}+\frac{8701874659}{2058443857}a^{9}-\frac{1815787}{4396036}a^{8}+\frac{1433257448}{89497559}a^{7}-\frac{66360}{47783}a^{6}+\frac{69535319872}{2058443857}a^{5}-\frac{38437391}{17584144}a^{4}+\frac{582725665171}{16467550856}a^{3}-\frac{456153}{1099009}a^{2}+\frac{5217021290}{2058443857}a+\frac{765505}{1099009}$
| 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: | \( 11964.3106427 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11964.3106427 \cdot 128}{2\cdot\sqrt{10646188457767892278050816}}\cr\approx \mathstrut & 0.570045506429 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 276*x^12 + 1776*x^10 + 6881*x^8 + 15072*x^6 + 16704*x^4 + 3072*x^2 + 256)
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 + 24*x^14 + 276*x^12 + 1776*x^10 + 6881*x^8 + 15072*x^6 + 16704*x^4 + 3072*x^2 + 256, 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 + 24*x^14 + 276*x^12 + 1776*x^10 + 6881*x^8 + 15072*x^6 + 16704*x^4 + 3072*x^2 + 256);
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 + 24*x^14 + 276*x^12 + 1776*x^10 + 6881*x^8 + 15072*x^6 + 16704*x^4 + 3072*x^2 + 256);
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
$C_2^2\times C_4$ (as 16T2):
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)
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
Intermediate fields
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]
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.24.9 | $x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ |
| 2.8.24.9 | $x^{8} + 8 x^{6} + 14 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |