Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81)
gp: K = bnfinit(y^16 + 3*y^14 - 9*y^12 + 62*y^10 + 327*y^8 - 126*y^6 - 146*y^4 - 36*y^2 + 81, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81)
\( x^{16} + 3x^{14} - 9x^{12} + 62x^{10} + 327x^{8} - 126x^{6} - 146x^{4} - 36x^{2} + 81 \)
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: |
\(10017750154516748107776\)
\(\medspace = 2^{16}\cdot 3^{8}\cdot 13^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.72\) | 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: | $2\cdot 3^{1/2}13^{3/4}\approx 23.71636563583009$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| 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: | \(156=2^{2}\cdot 3\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{156}(1,·)$, $\chi_{156}(131,·)$, $\chi_{156}(5,·)$, $\chi_{156}(73,·)$, $\chi_{156}(77,·)$, $\chi_{156}(79,·)$, $\chi_{156}(83,·)$, $\chi_{156}(151,·)$, $\chi_{156}(25,·)$, $\chi_{156}(155,·)$, $\chi_{156}(31,·)$, $\chi_{156}(103,·)$, $\chi_{156}(109,·)$, $\chi_{156}(47,·)$, $\chi_{156}(53,·)$, $\chi_{156}(125,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{2}$, $\frac{1}{9}a^{9}+\frac{1}{9}a^{3}$, $\frac{1}{9}a^{10}+\frac{1}{9}a^{4}$, $\frac{1}{27}a^{11}+\frac{1}{27}a^{9}+\frac{1}{9}a^{7}-\frac{2}{27}a^{5}+\frac{4}{27}a^{3}$, $\frac{1}{81}a^{12}-\frac{2}{81}a^{10}+\frac{1}{27}a^{8}-\frac{11}{81}a^{6}+\frac{10}{81}a^{4}-\frac{4}{9}a^{2}$, $\frac{1}{81}a^{13}+\frac{1}{81}a^{11}-\frac{1}{27}a^{9}-\frac{2}{81}a^{7}+\frac{4}{81}a^{5}-\frac{11}{27}a^{3}$, $\frac{1}{1042551}a^{14}-\frac{3557}{1042551}a^{12}-\frac{17879}{347517}a^{10}-\frac{6374}{1042551}a^{8}+\frac{64549}{1042551}a^{6}-\frac{9376}{38613}a^{4}-\frac{10727}{115839}a^{2}-\frac{4306}{12871}$, $\frac{1}{1042551}a^{15}-\frac{3557}{1042551}a^{13}-\frac{5008}{347517}a^{11}+\frac{32239}{1042551}a^{9}-\frac{167129}{1042551}a^{7}+\frac{5713}{347517}a^{5}-\frac{96536}{347517}a^{3}-\frac{4306}{12871}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{4}$, which has order $4$ (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: |
\( \frac{2756}{347517} a^{15} + \frac{26701}{1042551} a^{13} - \frac{64766}{1042551} a^{11} + \frac{56498}{115839} a^{9} + \frac{2775253}{1042551} a^{7} - \frac{184652}{1042551} a^{5} - \frac{9646}{115839} a^{3} - \frac{53950}{38613} a \)
(order $12$)
| 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: |
$\frac{4181}{1042551}a^{15}+\frac{2353}{347517}a^{13}-\frac{56348}{1042551}a^{11}+\frac{302180}{1042551}a^{9}+\frac{343307}{347517}a^{7}-\frac{2465930}{1042551}a^{5}-\frac{162650}{347517}a^{3}-\frac{3442}{38613}a+1$, $\frac{1651}{1042551}a^{14}+\frac{9440}{1042551}a^{12}-\frac{2207}{1042551}a^{10}+\frac{56488}{1042551}a^{8}+\frac{860876}{1042551}a^{6}+\frac{1125808}{1042551}a^{4}-\frac{128491}{115839}a^{2}+\frac{8457}{12871}$, $\frac{5630}{1042551}a^{14}+\frac{1004}{38613}a^{12}-\frac{9779}{1042551}a^{10}+\frac{294761}{1042551}a^{8}+\frac{810268}{347517}a^{6}+\frac{3177820}{1042551}a^{4}+\frac{177916}{115839}a^{2}+\frac{6184}{12871}$, $\frac{383}{115839}a^{15}-\frac{1910}{347517}a^{14}+\frac{665}{38613}a^{13}-\frac{18925}{1042551}a^{12}-\frac{95}{12871}a^{11}+\frac{44885}{1042551}a^{10}+\frac{17119}{115839}a^{9}-\frac{39155}{115839}a^{8}+\frac{59090}{38613}a^{7}-\frac{1935364}{1042551}a^{6}+\frac{25745}{12871}a^{5}+\frac{127970}{1042551}a^{4}-\frac{74812}{115839}a^{3}+\frac{6685}{115839}a^{2}-\frac{15390}{12871}a+\frac{12544}{12871}$, $\frac{3602}{347517}a^{15}+\frac{121}{17091}a^{14}+\frac{34477}{1042551}a^{13}+\frac{155}{5697}a^{12}-\frac{84647}{1042551}a^{11}-\frac{772}{17091}a^{10}+\frac{73841}{115839}a^{9}+\frac{6703}{17091}a^{8}+\frac{3615142}{1042551}a^{7}+\frac{15280}{5697}a^{6}-\frac{241334}{1042551}a^{5}+\frac{16358}{17091}a^{4}-\frac{12607}{115839}a^{3}-\frac{2216}{1899}a^{2}-\frac{31655}{38613}a-\frac{67}{211}$, $\frac{2756}{347517}a^{15}+\frac{6239}{1042551}a^{14}+\frac{26701}{1042551}a^{13}+\frac{7741}{347517}a^{12}-\frac{64766}{1042551}a^{11}-\frac{46727}{1042551}a^{10}+\frac{56498}{115839}a^{9}+\frac{351521}{1042551}a^{8}+\frac{2775253}{1042551}a^{7}+\frac{772424}{347517}a^{6}-\frac{184652}{1042551}a^{5}+\frac{371212}{1042551}a^{4}-\frac{9646}{115839}a^{3}-\frac{12488}{12871}a^{2}-\frac{15337}{38613}a-\frac{3357}{12871}$, $\frac{2291}{347517}a^{15}-\frac{4975}{1042551}a^{14}+\frac{20510}{1042551}a^{13}-\frac{4807}{347517}a^{12}-\frac{60274}{1042551}a^{11}+\frac{41116}{1042551}a^{10}+\frac{147332}{347517}a^{9}-\frac{338140}{1042551}a^{8}+\frac{2247203}{1042551}a^{7}-\frac{173024}{115839}a^{6}-\frac{874273}{1042551}a^{5}+\frac{621658}{1042551}a^{4}-\frac{69104}{347517}a^{3}-\frac{33103}{38613}a^{2}+\frac{11615}{38613}a+\frac{5006}{12871}$
| 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: | \( 46563.6853863 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 46563.6853863 \cdot 4}{12\cdot\sqrt{10017750154516748107776}}\cr\approx \mathstrut & 0.376686399766 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81)
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 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81, 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 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81);
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 + 3*x^14 - 9*x^12 + 62*x^10 + 327*x^8 - 126*x^6 - 146*x^4 - 36*x^2 + 81);
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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |