Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 + 46*x^12 + 15*x^10 - 79*x^8 + 15*x^6 + 46*x^4 - 20*x^2 + 1)
gp: K = bnfinit(y^16 - 20*y^14 + 46*y^12 + 15*y^10 - 79*y^8 + 15*y^6 + 46*y^4 - 20*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 20*x^14 + 46*x^12 + 15*x^10 - 79*x^8 + 15*x^6 + 46*x^4 - 20*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 20*x^14 + 46*x^12 + 15*x^10 - 79*x^8 + 15*x^6 + 46*x^4 - 20*x^2 + 1)
\( x^{16} - 20x^{14} + 46x^{12} + 15x^{10} - 79x^{8} + 15x^{6} + 46x^{4} - 20x^{2} + 1 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(105747725158992197265625\)
\(\medspace = 5^{10}\cdot 101^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.48\) | 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: | $5^{3/4}101^{1/2}\approx 33.60378443921746$ | ||
| Ramified primes: |
\(5\), \(101\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{550}a^{12}+\frac{49}{275}a^{10}+\frac{59}{550}a^{8}-\frac{1}{2}a^{7}-\frac{271}{550}a^{6}-\frac{1}{2}a^{5}-\frac{108}{275}a^{4}+\frac{49}{275}a^{2}-\frac{1}{2}a-\frac{137}{275}$, $\frac{1}{550}a^{13}+\frac{49}{275}a^{11}+\frac{59}{550}a^{9}-\frac{271}{550}a^{7}-\frac{1}{2}a^{6}+\frac{59}{550}a^{5}-\frac{1}{2}a^{4}-\frac{177}{550}a^{3}-\frac{1}{2}a^{2}-\frac{137}{275}a-\frac{1}{2}$, $\frac{1}{550}a^{14}+\frac{8}{55}a^{10}-\frac{3}{550}a^{8}-\frac{29}{275}a^{6}+\frac{91}{550}a^{4}-\frac{1}{2}a^{3}-\frac{23}{50}a^{2}-\frac{1}{2}a-\frac{49}{275}$, $\frac{1}{550}a^{15}+\frac{8}{55}a^{11}-\frac{3}{550}a^{9}-\frac{29}{275}a^{7}+\frac{91}{550}a^{5}-\frac{1}{2}a^{4}-\frac{23}{50}a^{3}-\frac{1}{2}a^{2}-\frac{49}{275}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
Trivial group, which has order $1$ (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: | $11$ | 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: |
$\frac{32}{55}a^{15}-\frac{632}{55}a^{13}+\frac{2633}{110}a^{11}+\frac{1517}{110}a^{9}-\frac{4423}{110}a^{7}-\frac{56}{55}a^{5}+\frac{1303}{55}a^{3}-\frac{661}{110}a+\frac{1}{2}$, $a$, $\frac{108}{275}a^{15}-\frac{39}{5}a^{13}+\frac{936}{55}a^{11}+\frac{2096}{275}a^{9}-\frac{8244}{275}a^{7}+\frac{1248}{275}a^{5}+\frac{456}{25}a^{3}-\frac{2004}{275}a$, $\frac{9}{110}a^{15}+\frac{49}{275}a^{14}-\frac{82}{55}a^{13}-\frac{1917}{550}a^{12}+\frac{103}{110}a^{11}+\frac{167}{25}a^{10}+\frac{346}{55}a^{9}+\frac{1464}{275}a^{8}-\frac{39}{55}a^{7}-\frac{5927}{550}a^{6}-\frac{937}{110}a^{5}-\frac{106}{55}a^{4}+\frac{38}{55}a^{3}+\frac{533}{110}a^{2}+\frac{247}{55}a+\frac{152}{275}$, $\frac{12}{275}a^{15}-\frac{151}{550}a^{14}-\frac{419}{550}a^{13}+\frac{2979}{550}a^{12}-\frac{46}{275}a^{11}-\frac{279}{25}a^{10}+\frac{2707}{550}a^{9}-\frac{3911}{550}a^{8}-\frac{159}{275}a^{7}+\frac{5387}{275}a^{6}-\frac{2737}{550}a^{5}+\frac{119}{110}a^{4}-\frac{109}{550}a^{3}-\frac{673}{55}a^{2}+\frac{402}{275}a+\frac{226}{275}$, $\frac{31}{275}a^{15}-\frac{7}{550}a^{14}-\frac{1193}{550}a^{13}+\frac{29}{110}a^{12}+\frac{948}{275}a^{11}-\frac{15}{22}a^{10}+\frac{1701}{275}a^{9}-\frac{387}{275}a^{8}-\frac{4793}{550}a^{7}+\frac{1043}{275}a^{6}-\frac{232}{55}a^{5}-\frac{57}{550}a^{4}+\frac{76}{11}a^{3}-\frac{1069}{550}a^{2}-\frac{119}{550}a-\frac{269}{550}$, $\frac{157}{275}a^{15}-\frac{21}{275}a^{14}-\frac{6167}{550}a^{13}+\frac{741}{550}a^{12}+\frac{12279}{550}a^{11}-\frac{21}{275}a^{10}+\frac{1621}{110}a^{9}-\frac{691}{110}a^{8}-\frac{4121}{110}a^{7}-\frac{37}{22}a^{6}+\frac{221}{550}a^{5}+\frac{151}{25}a^{4}+\frac{12217}{550}a^{3}+\frac{1019}{550}a^{2}-\frac{4489}{550}a-\frac{459}{275}$, $\frac{9}{110}a^{15}-\frac{49}{275}a^{14}-\frac{82}{55}a^{13}+\frac{1917}{550}a^{12}+\frac{103}{110}a^{11}-\frac{167}{25}a^{10}+\frac{346}{55}a^{9}-\frac{1464}{275}a^{8}-\frac{39}{55}a^{7}+\frac{5927}{550}a^{6}-\frac{937}{110}a^{5}+\frac{106}{55}a^{4}+\frac{38}{55}a^{3}-\frac{533}{110}a^{2}+\frac{247}{55}a-\frac{152}{275}$, $\frac{223}{550}a^{15}-\frac{189}{550}a^{14}-\frac{2166}{275}a^{13}+\frac{67}{10}a^{12}+\frac{7729}{550}a^{11}-\frac{1363}{110}a^{10}+\frac{4284}{275}a^{9}-\frac{6693}{550}a^{8}-\frac{13487}{550}a^{7}+\frac{5426}{275}a^{6}-\frac{622}{55}a^{5}+\frac{5791}{550}a^{4}+\frac{1379}{110}a^{3}-\frac{423}{50}a^{2}-\frac{611}{550}a-\frac{34}{275}$, $\frac{83}{275}a^{15}+\frac{147}{550}a^{14}-\frac{3283}{550}a^{13}-\frac{2903}{550}a^{12}+\frac{6971}{550}a^{11}+\frac{278}{25}a^{10}+\frac{541}{110}a^{9}+\frac{1041}{275}a^{8}-\frac{1857}{110}a^{7}-\frac{7763}{550}a^{6}-\frac{391}{550}a^{5}+\frac{75}{22}a^{4}+\frac{3209}{275}a^{3}+\frac{254}{55}a^{2}-\frac{838}{275}a-\frac{267}{275}$, $\frac{9}{55}a^{15}+\frac{9}{275}a^{14}-\frac{157}{50}a^{13}-\frac{287}{550}a^{12}+\frac{2679}{550}a^{11}-\frac{51}{50}a^{10}+\frac{2406}{275}a^{9}+\frac{2263}{550}a^{8}-\frac{3864}{275}a^{7}+\frac{1933}{550}a^{6}-\frac{753}{550}a^{5}-\frac{749}{110}a^{4}+\frac{519}{50}a^{3}+\frac{87}{55}a^{2}-\frac{1011}{275}a+\frac{212}{275}$
| 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: | \( 872718.117676 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 872718.117676 \cdot 1}{2\cdot\sqrt{105747725158992197265625}}\cr\approx \mathstrut & 0.535387119148 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 + 46*x^12 + 15*x^10 - 79*x^8 + 15*x^6 + 46*x^4 - 20*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 - 20*x^14 + 46*x^12 + 15*x^10 - 79*x^8 + 15*x^6 + 46*x^4 - 20*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 - 20*x^14 + 46*x^12 + 15*x^10 - 79*x^8 + 15*x^6 + 46*x^4 - 20*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 - 20*x^14 + 46*x^12 + 15*x^10 - 79*x^8 + 15*x^6 + 46*x^4 - 20*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
$C_4\wr C_2$ (as 16T42):
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 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), 4.4.51005.1 x2, 4.4.2525.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.4.325188753125.1, 8.4.13007550125.1, 8.8.65037750625.1 |
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
| Galois closure: | deg 32 |
| Degree 8 siblings: | 8.4.325188753125.1, 8.4.13007550125.1 |
| Degree 16 sibling: | 16.0.259160193017822265625.1 |
| Minimal sibling: | 8.4.13007550125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(101\)
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |