Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 114*x^12 - 312*x^10 + 608*x^8 - 624*x^6 + 456*x^4 - 96*x^2 + 16)
gp: K = bnfinit(y^16 - 12*y^14 + 114*y^12 - 312*y^10 + 608*y^8 - 624*y^6 + 456*y^4 - 96*y^2 + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 + 114*x^12 - 312*x^10 + 608*x^8 - 624*x^6 + 456*x^4 - 96*x^2 + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 12*x^14 + 114*x^12 - 312*x^10 + 608*x^8 - 624*x^6 + 456*x^4 - 96*x^2 + 16)
\( x^{16} - 12x^{14} + 114x^{12} - 312x^{10} + 608x^{8} - 624x^{6} + 456x^{4} - 96x^{2} + 16 \)
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: |
\(45086848686489600000000\)
\(\medspace = 2^{44}\cdot 3^{8}\cdot 5^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.05\) | 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: | $2^{11/4}3^{1/2}5^{1/2}\approx 26.054222497305222$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| 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: | \(240=2^{4}\cdot 3\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{240}(19,·)$, $\chi_{240}(1,·)$, $\chi_{240}(131,·)$, $\chi_{240}(139,·)$, $\chi_{240}(11,·)$, $\chi_{240}(209,·)$, $\chi_{240}(211,·)$, $\chi_{240}(89,·)$, $\chi_{240}(91,·)$, $\chi_{240}(161,·)$, $\chi_{240}(41,·)$, $\chi_{240}(49,·)$, $\chi_{240}(179,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$, $\chi_{240}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{896}a^{12}+\frac{17}{112}a^{6}-\frac{55}{112}$, $\frac{1}{896}a^{13}+\frac{17}{112}a^{7}-\frac{55}{112}a$, $\frac{1}{27776}a^{14}+\frac{11}{27776}a^{12}-\frac{7}{62}a^{10}-\frac{375}{3472}a^{8}+\frac{131}{3472}a^{6}+\frac{6}{31}a^{4}+\frac{1625}{3472}a^{2}-\frac{829}{3472}$, $\frac{1}{27776}a^{15}+\frac{11}{27776}a^{13}-\frac{7}{62}a^{11}-\frac{375}{3472}a^{9}+\frac{131}{3472}a^{7}+\frac{6}{31}a^{5}+\frac{1625}{3472}a^{3}-\frac{829}{3472}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: |
\( -\frac{417}{27776} a^{14} + \frac{4899}{27776} a^{12} - \frac{207}{124} a^{10} + \frac{14891}{3472} a^{8} - \frac{28773}{3472} a^{6} + \frac{483}{62} a^{4} - \frac{21417}{3472} a^{2} + \frac{4507}{3472} \)
(order $6$)
| 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: |
$\frac{25}{27776}a^{14}+\frac{213}{27776}a^{12}-\frac{9}{124}a^{10}+\frac{4513}{3472}a^{8}-\frac{1251}{3472}a^{6}+\frac{21}{62}a^{4}+\frac{9377}{3472}a^{2}+\frac{45}{3472}$, $\frac{135}{3472}a^{14}-\frac{761}{1736}a^{12}+\frac{126}{31}a^{10}-\frac{3753}{434}a^{8}+\frac{5409}{434}a^{6}-\frac{216}{31}a^{4}+\frac{639}{434}a^{2}+\frac{44}{217}$, $\frac{439}{27776}a^{14}-\frac{1569}{13888}a^{12}+\frac{29}{31}a^{10}+\frac{10711}{3472}a^{8}-\frac{12925}{1736}a^{6}+\frac{835}{62}a^{4}-\frac{22689}{3472}a^{2}+\frac{2407}{1736}$, $\frac{75}{3968}a^{15}+\frac{417}{27776}a^{14}-\frac{849}{3968}a^{13}-\frac{4899}{27776}a^{12}+\frac{245}{124}a^{11}+\frac{207}{124}a^{10}-\frac{2085}{496}a^{9}-\frac{14891}{3472}a^{8}+\frac{2695}{496}a^{7}+\frac{28773}{3472}a^{6}-\frac{105}{31}a^{5}-\frac{483}{62}a^{4}+\frac{355}{496}a^{3}+\frac{21417}{3472}a^{2}-\frac{857}{496}a+\frac{2437}{3472}$, $\frac{15}{868}a^{15}-\frac{417}{27776}a^{14}-\frac{5353}{27776}a^{13}+\frac{4899}{27776}a^{12}+\frac{56}{31}a^{11}-\frac{207}{124}a^{10}-\frac{834}{217}a^{9}+\frac{14891}{3472}a^{8}+\frac{24471}{3472}a^{7}-\frac{28773}{3472}a^{6}-\frac{96}{31}a^{5}+\frac{483}{62}a^{4}+\frac{142}{217}a^{3}-\frac{21417}{3472}a^{2}+\frac{13295}{3472}a-\frac{2437}{3472}$, $\frac{15}{3472}a^{15}+\frac{13}{6944}a^{14}+\frac{71}{13888}a^{13}+\frac{2215}{27776}a^{12}-\frac{3}{62}a^{11}-\frac{27}{31}a^{10}+\frac{768}{217}a^{9}+\frac{8145}{868}a^{8}-\frac{417}{1736}a^{7}-\frac{52057}{3472}a^{6}+\frac{7}{31}a^{5}+\frac{653}{31}a^{4}+\frac{5713}{434}a^{3}-\frac{1443}{868}a^{2}+\frac{15}{1736}a+\frac{1935}{3472}$, $\frac{417}{27776}a^{15}-\frac{809}{27776}a^{14}-\frac{4899}{27776}a^{13}+\frac{10011}{27776}a^{12}+\frac{207}{124}a^{11}-\frac{423}{124}a^{10}-\frac{14891}{3472}a^{9}+\frac{34295}{3472}a^{8}+\frac{28773}{3472}a^{7}-\frac{58797}{3472}a^{6}-\frac{483}{62}a^{5}+\frac{987}{62}a^{4}+\frac{21417}{3472}a^{3}-\frac{29985}{3472}a^{2}+\frac{2437}{3472}a+\frac{2115}{3472}$
| 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: | \( 67549.6577055 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 67549.6577055 \cdot 2}{6\cdot\sqrt{45086848686489600000000}}\cr\approx \mathstrut & 0.257582233098 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 114*x^12 - 312*x^10 + 608*x^8 - 624*x^6 + 456*x^4 - 96*x^2 + 16)
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 - 12*x^14 + 114*x^12 - 312*x^10 + 608*x^8 - 624*x^6 + 456*x^4 - 96*x^2 + 16, 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 - 12*x^14 + 114*x^12 - 312*x^10 + 608*x^8 - 624*x^6 + 456*x^4 - 96*x^2 + 16);
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 - 12*x^14 + 114*x^12 - 312*x^10 + 608*x^8 - 624*x^6 + 456*x^4 - 96*x^2 + 16);
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 | R | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.22.7 | $x^{8} + 20 x^{7} + 154 x^{6} + 592 x^{5} + 1331 x^{4} + 2128 x^{3} + 2386 x^{2} + 764 x + 103$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ |
| 2.8.22.7 | $x^{8} + 20 x^{7} + 154 x^{6} + 592 x^{5} + 1331 x^{4} + 2128 x^{3} + 2386 x^{2} + 764 x + 103$ | $4$ | $2$ | $22$ | $C_4\times C_2$ | $[3, 4]^{2}$ | |
|
\(3\)
| 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |