Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 9*x^12 - 16*x^10 + 24*x^8 - 34*x^6 + 29*x^4 - 6*x^2 + 1)
gp: K = bnfinit(y^16 - 4*y^14 + 9*y^12 - 16*y^10 + 24*y^8 - 34*y^6 + 29*y^4 - 6*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 9*x^12 - 16*x^10 + 24*x^8 - 34*x^6 + 29*x^4 - 6*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 + 9*x^12 - 16*x^10 + 24*x^8 - 34*x^6 + 29*x^4 - 6*x^2 + 1)
\( x^{16} - 4x^{14} + 9x^{12} - 16x^{10} + 24x^{8} - 34x^{6} + 29x^{4} - 6x^{2} + 1 \)
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: |
\(11007531417600000000\)
\(\medspace = 2^{32}\cdot 3^{8}\cdot 5^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.49\) | 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: | $2^{9/4}3^{1/2}5^{1/2}\approx 18.42311740638763$ | ||
| 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10}a^{12}+\frac{2}{5}a^{6}-\frac{1}{10}$, $\frac{1}{10}a^{13}+\frac{2}{5}a^{7}-\frac{1}{10}a$, $\frac{1}{170}a^{14}-\frac{7}{170}a^{12}+\frac{3}{17}a^{10}+\frac{32}{85}a^{8}+\frac{1}{85}a^{6}-\frac{4}{17}a^{4}-\frac{21}{170}a^{2}+\frac{57}{170}$, $\frac{1}{170}a^{15}-\frac{7}{170}a^{13}+\frac{3}{17}a^{11}+\frac{32}{85}a^{9}+\frac{1}{85}a^{7}-\frac{4}{17}a^{5}-\frac{21}{170}a^{3}+\frac{57}{170}a$
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
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{53}{170} a^{14} + \frac{92}{85} a^{12} - \frac{40}{17} a^{10} + \frac{344}{85} a^{8} - \frac{512}{85} a^{6} + \frac{144}{17} a^{4} - \frac{1097}{170} a^{2} + \frac{113}{85} \)
(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: |
$\frac{5}{34}a^{14}-\frac{107}{170}a^{12}+\frac{24}{17}a^{10}-\frac{44}{17}a^{8}+\frac{331}{85}a^{6}-\frac{100}{17}a^{4}+\frac{167}{34}a^{2}-\frac{173}{170}$, $\frac{7}{170}a^{15}-\frac{16}{85}a^{13}+\frac{4}{17}a^{11}-\frac{31}{85}a^{9}+\frac{41}{85}a^{7}-\frac{11}{17}a^{5}+\frac{23}{170}a^{3}+\frac{106}{85}a$, $\frac{23}{85}a^{15}-\frac{169}{170}a^{13}+\frac{36}{17}a^{11}-\frac{313}{85}a^{9}+\frac{437}{85}a^{7}-\frac{116}{17}a^{5}+\frac{367}{85}a^{3}+\frac{259}{170}a$, $\frac{99}{170}a^{15}+\frac{53}{170}a^{14}-\frac{421}{170}a^{13}-\frac{92}{85}a^{12}+\frac{93}{17}a^{11}+\frac{40}{17}a^{10}-\frac{827}{85}a^{9}-\frac{344}{85}a^{8}+\frac{1238}{85}a^{7}+\frac{512}{85}a^{6}-\frac{345}{17}a^{5}-\frac{144}{17}a^{4}+\frac{3021}{170}a^{3}+\frac{927}{170}a^{2}-\frac{409}{170}a-\frac{113}{85}$, $\frac{23}{85}a^{15}+\frac{2}{85}a^{14}-\frac{76}{85}a^{13}+\frac{23}{170}a^{12}+\frac{36}{17}a^{11}-\frac{5}{17}a^{10}-\frac{313}{85}a^{9}+\frac{43}{85}a^{8}+\frac{471}{85}a^{7}-\frac{64}{85}a^{6}-\frac{133}{17}a^{5}+\frac{18}{17}a^{4}+\frac{537}{85}a^{3}-\frac{127}{85}a^{2}-\frac{219}{85}a-\frac{163}{170}$, $\frac{43}{170}a^{15}-\frac{1}{170}a^{14}-\frac{199}{170}a^{13}+\frac{12}{85}a^{12}+\frac{44}{17}a^{11}-\frac{3}{17}a^{10}-\frac{409}{85}a^{9}+\frac{53}{85}a^{8}+\frac{587}{85}a^{7}-\frac{52}{85}a^{6}-\frac{172}{17}a^{5}+\frac{21}{17}a^{4}+\frac{1477}{170}a^{3}-\frac{149}{170}a^{2}-\frac{201}{170}a-\frac{37}{85}$, $\frac{23}{85}a^{15}-\frac{1}{170}a^{14}-\frac{22}{17}a^{13}+\frac{7}{170}a^{12}+\frac{53}{17}a^{11}-\frac{3}{17}a^{10}-\frac{483}{85}a^{9}+\frac{53}{85}a^{8}+\frac{152}{17}a^{7}-\frac{86}{85}a^{6}-\frac{218}{17}a^{5}+\frac{21}{17}a^{4}+\frac{1047}{85}a^{3}-\frac{319}{170}a^{2}-\frac{71}{17}a+\frac{283}{170}$
| 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: | \( 3309.02321892 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3309.02321892 \cdot 1}{6\cdot\sqrt{11007531417600000000}}\cr\approx \mathstrut & 0.403777899249 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 9*x^12 - 16*x^10 + 24*x^8 - 34*x^6 + 29*x^4 - 6*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 - 4*x^14 + 9*x^12 - 16*x^10 + 24*x^8 - 34*x^6 + 29*x^4 - 6*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 - 4*x^14 + 9*x^12 - 16*x^10 + 24*x^8 - 34*x^6 + 29*x^4 - 6*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 - 4*x^14 + 9*x^12 - 16*x^10 + 24*x^8 - 34*x^6 + 29*x^4 - 6*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
$C_2^2\times D_4$ (as 16T25):
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 20 conjugacy class representatives for $C_2^2 \times D_4$ |
| Character table for $C_2^2 \times D_4$ |
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]
Sibling fields
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 | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.11 | $x^{8} + 2 x^{6} - 4 x^{5} + 20 x^{4} + 8 x^{3} + 44 x^{2} - 8 x + 76$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ |
| 2.8.16.11 | $x^{8} + 2 x^{6} - 4 x^{5} + 20 x^{4} + 8 x^{3} + 44 x^{2} - 8 x + 76$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
|
\(3\)
| 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 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |