Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*x^2 + 1)
gp: K = bnfinit(y^16 - 13*y^14 + 129*y^12 - 494*y^10 + 1430*y^8 - 494*y^6 + 129*y^4 - 13*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*x^2 + 1)
\( x^{16} - 13x^{14} + 129x^{12} - 494x^{10} + 1430x^{8} - 494x^{6} + 129x^{4} - 13x^{2} + 1 \)
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: |
\(250516897691366976454656\)
\(\medspace = 2^{16}\cdot 3^{8}\cdot 17^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.00\) | 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: | $2\cdot 3^{1/2}17^{3/4}\approx 29.001957651613843$ | ||
| Ramified primes: |
\(2\), \(3\), \(17\)
| 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: | \(204=2^{2}\cdot 3\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{204}(1,·)$, $\chi_{204}(67,·)$, $\chi_{204}(137,·)$, $\chi_{204}(203,·)$, $\chi_{204}(13,·)$, $\chi_{204}(149,·)$, $\chi_{204}(89,·)$, $\chi_{204}(157,·)$, $\chi_{204}(35,·)$, $\chi_{204}(101,·)$, $\chi_{204}(103,·)$, $\chi_{204}(169,·)$, $\chi_{204}(47,·)$, $\chi_{204}(115,·)$, $\chi_{204}(55,·)$, $\chi_{204}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8}a^{6}-\frac{3}{8}$, $\frac{1}{8}a^{7}-\frac{3}{8}a$, $\frac{1}{8}a^{8}-\frac{3}{8}a^{2}$, $\frac{1}{8}a^{9}-\frac{3}{8}a^{3}$, $\frac{1}{8}a^{10}-\frac{3}{8}a^{4}$, $\frac{1}{8}a^{11}-\frac{3}{8}a^{5}$, $\frac{1}{1664}a^{12}-\frac{3}{64}a^{6}+\frac{545}{1664}$, $\frac{1}{1664}a^{13}-\frac{3}{64}a^{7}+\frac{545}{1664}a$, $\frac{1}{1664}a^{14}-\frac{3}{64}a^{8}+\frac{545}{1664}a^{2}$, $\frac{1}{1664}a^{15}-\frac{3}{64}a^{9}+\frac{545}{1664}a^{3}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $13$ |
Class group and class number
$C_{20}$, which has order $20$ (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: |
\( -\frac{733}{1664} a^{15} + \frac{4817}{832} a^{13} - \frac{461}{8} a^{11} + \frac{14431}{64} a^{9} - \frac{21071}{32} a^{7} + \frac{2383}{8} a^{5} - \frac{98925}{1664} a^{3} + \frac{4985}{832} a \)
(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: |
$\frac{625}{1664}a^{15}+\frac{87}{832}a^{14}-\frac{3961}{832}a^{13}-\frac{2247}{1664}a^{12}+\frac{375}{8}a^{11}+\frac{107}{8}a^{10}-\frac{10875}{64}a^{9}-\frac{1617}{32}a^{8}+\frac{15295}{32}a^{7}+\frac{9309}{64}a^{6}-\frac{125}{8}a^{5}-\frac{321}{8}a^{4}+\frac{2625}{1664}a^{3}+\frac{10911}{832}a^{2}+\frac{2895}{832}a-\frac{535}{1664}$, $\frac{331}{1664}a^{14}-\frac{1071}{416}a^{12}+\frac{51}{2}a^{10}-\frac{6185}{64}a^{8}+\frac{4437}{16}a^{6}-\frac{153}{2}a^{4}+\frac{7963}{1664}a^{2}-\frac{255}{416}$, $\frac{105}{1664}a^{14}-\frac{1331}{1664}a^{12}+\frac{63}{8}a^{10}-\frac{1827}{64}a^{8}+\frac{5137}{64}a^{6}-\frac{21}{8}a^{4}+\frac{441}{1664}a^{2}+\frac{733}{1664}$, $\frac{127}{104}a^{15}-\frac{6573}{416}a^{13}+\frac{313}{2}a^{11}-\frac{4743}{8}a^{9}+\frac{27231}{16}a^{7}-\frac{939}{2}a^{5}+\frac{2309}{52}a^{3}-\frac{1565}{416}a$, $\frac{853}{1664}a^{15}-\frac{15}{1664}a^{14}-\frac{11025}{1664}a^{13}+\frac{95}{832}a^{12}+\frac{525}{8}a^{11}-\frac{9}{8}a^{10}-\frac{15887}{64}a^{9}+\frac{261}{64}a^{8}+\frac{45675}{64}a^{7}-\frac{369}{32}a^{6}-\frac{1575}{8}a^{5}+\frac{3}{8}a^{4}+\frac{73429}{1664}a^{3}-\frac{3391}{1664}a^{2}-\frac{2625}{1664}a+\frac{87}{832}$, $\frac{2703}{1664}a^{15}+\frac{1589}{1664}a^{14}-\frac{2663}{128}a^{13}-\frac{10245}{832}a^{12}+\frac{411}{2}a^{11}+\frac{975}{8}a^{10}-\frac{48789}{64}a^{9}-\frac{29359}{64}a^{8}+\frac{138905}{64}a^{7}+\frac{42095}{32}a^{6}-\frac{725}{2}a^{5}-\frac{2621}{8}a^{4}+\frac{128415}{1664}a^{3}+\frac{116789}{1664}a^{2}+\frac{1225}{128}a-\frac{805}{832}$, $\frac{285}{832}a^{15}+\frac{479}{1664}a^{14}-\frac{3563}{832}a^{13}-\frac{6113}{1664}a^{12}+42a^{11}+\frac{145}{4}a^{10}-\frac{4723}{32}a^{9}-\frac{8541}{64}a^{8}+\frac{13101}{32}a^{7}+\frac{24235}{64}a^{6}+60a^{5}-\frac{191}{4}a^{4}+\frac{1925}{832}a^{3}+\frac{21439}{1664}a^{2}+\frac{2445}{832}a-\frac{881}{1664}$
| 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: | \( 165082.730127 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 165082.730127 \cdot 20}{12\cdot\sqrt{250516897691366976454656}}\cr\approx \mathstrut & 1.33527533427 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*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 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*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 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*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 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*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
$C_2^2\times C_4$ (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 | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.1.0.1}{1} }^{16}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(17\)
| 17.8.6.1 | $x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74840 x^{4} + 51968 x^{3} + 39432 x^{2} + 270464 x + 1062564$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 17.8.6.1 | $x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74840 x^{4} + 51968 x^{3} + 39432 x^{2} + 270464 x + 1062564$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |