Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + 6*x^11 - 4*x^10 - 4*x^9 + 15*x^8 - 4*x^7 - 4*x^6 + 6*x^5 - 2*x^3 + 3*x^2 - 2*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 3*y^14 - 2*y^13 + 6*y^11 - 4*y^10 - 4*y^9 + 15*y^8 - 4*y^7 - 4*y^6 + 6*y^5 - 2*y^3 + 3*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + 6*x^11 - 4*x^10 - 4*x^9 + 15*x^8 - 4*x^7 - 4*x^6 + 6*x^5 - 2*x^3 + 3*x^2 - 2*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + 6*x^11 - 4*x^10 - 4*x^9 + 15*x^8 - 4*x^7 - 4*x^6 + 6*x^5 - 2*x^3 + 3*x^2 - 2*x + 1)
\( x^{16} - 2 x^{15} + 3 x^{14} - 2 x^{13} + 6 x^{11} - 4 x^{10} - 4 x^{9} + 15 x^{8} - 4 x^{7} - 4 x^{6} + \cdots + 1 \)
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: |
\(152812774400000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 11\cdot 13^{4}\cdot 19\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.86\) | 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: | $2^{3/2}5^{1/2}11^{1/2}13^{1/2}19^{1/2}\approx 329.66649814623264$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\), \(13\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{209}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{178}a^{14}+\frac{21}{89}a^{13}-\frac{19}{178}a^{12}+\frac{5}{89}a^{11}-\frac{75}{178}a^{10}+\frac{39}{89}a^{9}+\frac{16}{89}a^{8}+\frac{40}{89}a^{7}+\frac{16}{89}a^{6}+\frac{39}{89}a^{5}-\frac{75}{178}a^{4}+\frac{5}{89}a^{3}-\frac{19}{178}a^{2}+\frac{21}{89}a+\frac{1}{178}$, $\frac{1}{178}a^{15}-\frac{3}{178}a^{13}+\frac{7}{178}a^{12}+\frac{39}{178}a^{11}+\frac{12}{89}a^{10}-\frac{20}{89}a^{9}+\frac{71}{178}a^{8}+\frac{27}{89}a^{7}+\frac{69}{178}a^{6}+\frac{31}{178}a^{5}+\frac{45}{178}a^{4}-\frac{83}{178}a^{3}-\frac{25}{89}a^{2}+\frac{17}{178}a+\frac{47}{178}$
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
Trivial group, which has order $1$
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: |
$\frac{121}{178}a^{15}-\frac{79}{89}a^{14}+\frac{105}{89}a^{13}+\frac{11}{89}a^{12}-\frac{243}{178}a^{11}+\frac{435}{89}a^{10}-\frac{165}{178}a^{9}-\frac{324}{89}a^{8}+\frac{1637}{178}a^{7}+2a^{6}-\frac{326}{89}a^{5}+\frac{415}{89}a^{4}+\frac{125}{178}a^{3}-\frac{189}{89}a^{2}+\frac{69}{89}a-\frac{39}{89}$, $a^{15}-2a^{14}+3a^{13}-2a^{12}+6a^{10}-4a^{9}-4a^{8}+15a^{7}-4a^{6}-4a^{5}+6a^{4}-2a^{2}+3a-2$, $\frac{21}{178}a^{15}+\frac{33}{178}a^{14}-\frac{101}{178}a^{13}+\frac{143}{178}a^{12}-\frac{97}{178}a^{11}-\frac{13}{178}a^{10}+\frac{244}{89}a^{9}-\frac{284}{89}a^{8}-\frac{71}{89}a^{7}+\frac{496}{89}a^{6}-\frac{335}{178}a^{5}-\frac{729}{178}a^{4}+\frac{189}{178}a^{3}-\frac{75}{178}a^{2}-\frac{37}{178}a-\frac{137}{178}$, $\frac{118}{89}a^{15}-\frac{311}{178}a^{14}+\frac{381}{178}a^{13}-\frac{2}{89}a^{12}-\frac{157}{89}a^{11}+\frac{1399}{178}a^{10}+\frac{33}{178}a^{9}-\frac{1651}{178}a^{8}+\frac{2905}{178}a^{7}+\frac{1259}{178}a^{6}-\frac{1367}{178}a^{5}+\frac{285}{89}a^{4}+\frac{310}{89}a^{3}-\frac{373}{178}a^{2}+\frac{473}{178}a-\frac{172}{89}$, $\frac{21}{178}a^{15}+\frac{33}{178}a^{14}-\frac{101}{178}a^{13}+\frac{143}{178}a^{12}-\frac{97}{178}a^{11}-\frac{13}{178}a^{10}+\frac{244}{89}a^{9}-\frac{284}{89}a^{8}-\frac{71}{89}a^{7}+\frac{496}{89}a^{6}-\frac{335}{178}a^{5}-\frac{729}{178}a^{4}+\frac{189}{178}a^{3}-\frac{75}{178}a^{2}-\frac{37}{178}a+\frac{41}{178}$, $\frac{126}{89}a^{15}-\frac{138}{89}a^{14}+\frac{234}{89}a^{13}-\frac{56}{89}a^{12}-\frac{26}{89}a^{11}+\frac{647}{89}a^{10}+\frac{127}{89}a^{9}-\frac{454}{89}a^{8}+\frac{1193}{89}a^{7}+\frac{807}{89}a^{6}+\frac{84}{89}a^{5}+3a^{4}+\frac{266}{89}a^{3}-\frac{29}{89}a^{2}+\frac{173}{89}a-\frac{90}{89}$, $\frac{20}{89}a^{15}-\frac{93}{89}a^{14}+\frac{128}{89}a^{13}-\frac{140}{89}a^{12}+\frac{28}{89}a^{11}+\frac{157}{89}a^{10}-\frac{400}{89}a^{9}-\frac{43}{89}a^{8}+\frac{493}{89}a^{7}-\frac{617}{89}a^{6}-\frac{315}{89}a^{5}+\frac{132}{89}a^{4}-\frac{98}{89}a^{3}-\frac{34}{89}a^{2}-\frac{6}{89}a-\frac{132}{89}$
| 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: | \( 101.255505806 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 101.255505806 \cdot 1}{2\cdot\sqrt{152812774400000000}}\cr\approx \mathstrut & 0.314592068024 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 2*x^13 + 6*x^11 - 4*x^10 - 4*x^9 + 15*x^8 - 4*x^7 - 4*x^6 + 6*x^5 - 2*x^3 + 3*x^2 - 2*x + 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 - 2*x^15 + 3*x^14 - 2*x^13 + 6*x^11 - 4*x^10 - 4*x^9 + 15*x^8 - 4*x^7 - 4*x^6 + 6*x^5 - 2*x^3 + 3*x^2 - 2*x + 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 - 2*x^15 + 3*x^14 - 2*x^13 + 6*x^11 - 4*x^10 - 4*x^9 + 15*x^8 - 4*x^7 - 4*x^6 + 6*x^5 - 2*x^3 + 3*x^2 - 2*x + 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 - 2*x^15 + 3*x^14 - 2*x^13 + 6*x^11 - 4*x^10 - 4*x^9 + 15*x^8 - 4*x^7 - 4*x^6 + 6*x^5 - 2*x^3 + 3*x^2 - 2*x + 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
$C_2^7.C_2\wr D_4$ (as 16T1778):
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 16384 |
| The 148 conjugacy class representatives for $C_2^7.C_2\wr D_4$ |
| Character table for $C_2^7.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.400.1, 8.0.27040000.1 |
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
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.4.188981478400000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
| 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(11\)
| 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(13\)
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.8.0.1 | $x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |