Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 1)
gp: K = bnfinit(y^16 - 4*y^13 + 3*y^12 + 8*y^10 - 8*y^9 + y^8 - 8*y^7 + 8*y^6 + 3*y^4 - 4*y^3 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 1)
\( x^{16} - 4x^{13} + 3x^{12} + 8x^{10} - 8x^{9} + x^{8} - 8x^{7} + 8x^{6} + 3x^{4} - 4x^{3} + 1 \)
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: |
\(5455162516701184\)
\(\medspace = 2^{32}\cdot 7^{4}\cdot 23^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.63\) | 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: | $2^{2}7^{1/2}23^{1/2}\approx 50.75431016179808$ | ||
| Ramified primes: |
\(2\), \(7\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7}a^{14}-\frac{2}{7}a^{13}+\frac{3}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}-\frac{3}{7}a^{9}-\frac{2}{7}a^{8}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}-\frac{1}{7}a^{3}+\frac{3}{7}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{49}a^{15}-\frac{1}{49}a^{14}-\frac{13}{49}a^{13}-\frac{12}{49}a^{12}+\frac{22}{49}a^{11}-\frac{8}{49}a^{10}-\frac{12}{49}a^{9}-\frac{3}{49}a^{8}-\frac{17}{49}a^{7}+\frac{23}{49}a^{6}+\frac{13}{49}a^{5}-\frac{20}{49}a^{4}-\frac{5}{49}a^{3}+\frac{15}{49}a^{2}-\frac{8}{49}a-\frac{13}{49}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
\( -\frac{31}{49} a^{15} + \frac{59}{49} a^{14} + \frac{4}{49} a^{13} + \frac{113}{49} a^{12} - \frac{318}{49} a^{11} + \frac{157}{49} a^{10} - \frac{202}{49} a^{9} + \frac{625}{49} a^{8} - \frac{432}{49} a^{7} + \frac{211}{49} a^{6} - \frac{536}{49} a^{5} + \frac{284}{49} a^{4} - \frac{20}{49} a^{3} + \frac{158}{49} a^{2} - \frac{102}{49} a - \frac{10}{49} \)
(order $8$)
| 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: |
$\frac{2}{49}a^{15}+\frac{40}{49}a^{14}-\frac{12}{49}a^{13}+\frac{4}{49}a^{12}-\frac{145}{49}a^{11}+\frac{166}{49}a^{10}-\frac{52}{49}a^{9}+\frac{302}{49}a^{8}-\frac{370}{49}a^{7}+\frac{158}{49}a^{6}-\frac{296}{49}a^{5}+\frac{338}{49}a^{4}-\frac{52}{49}a^{3}+\frac{58}{49}a^{2}-\frac{100}{49}a+\frac{16}{49}$, $\frac{2}{49}a^{15}+\frac{40}{49}a^{14}-\frac{12}{49}a^{13}+\frac{4}{49}a^{12}-\frac{145}{49}a^{11}+\frac{166}{49}a^{10}-\frac{52}{49}a^{9}+\frac{302}{49}a^{8}-\frac{370}{49}a^{7}+\frac{158}{49}a^{6}-\frac{296}{49}a^{5}+\frac{289}{49}a^{4}-\frac{52}{49}a^{3}+\frac{58}{49}a^{2}-\frac{51}{49}a-\frac{33}{49}$, $\frac{13}{49}a^{15}+\frac{29}{49}a^{14}-\frac{8}{49}a^{13}-\frac{30}{49}a^{12}-\frac{50}{49}a^{11}+\frac{127}{49}a^{10}+\frac{12}{49}a^{9}+\frac{122}{49}a^{8}-\frac{214}{49}a^{7}+\frac{117}{49}a^{6}-\frac{153}{49}a^{5}+\frac{216}{49}a^{4}-\frac{58}{49}a^{3}+\frac{27}{49}a^{2}-\frac{90}{49}a+\frac{20}{49}$, $\frac{94}{49}a^{15}+\frac{4}{49}a^{14}+\frac{3}{49}a^{13}-\frac{344}{49}a^{12}+\frac{255}{49}a^{11}-\frac{17}{49}a^{10}+\frac{636}{49}a^{9}-\frac{576}{49}a^{8}+\frac{117}{49}a^{7}-\frac{582}{49}a^{6}+\frac{389}{49}a^{5}+\frac{31}{49}a^{4}+\frac{167}{49}a^{3}-\frac{109}{49}a^{2}-\frac{17}{49}a+\frac{3}{49}$, $\frac{67}{49}a^{15}+\frac{17}{49}a^{14}-\frac{10}{49}a^{13}-\frac{258}{49}a^{12}+\frac{116}{49}a^{11}+\frac{73}{49}a^{10}+\frac{463}{49}a^{9}-\frac{320}{49}a^{8}-\frac{145}{49}a^{7}-\frac{391}{49}a^{6}+\frac{227}{49}a^{5}+\frac{200}{49}a^{4}+\frac{71}{49}a^{3}-\frac{66}{49}a^{2}-\frac{67}{49}a+\frac{46}{49}$, $\frac{3}{7}a^{15}-\frac{4}{7}a^{14}-\frac{2}{7}a^{13}-\frac{11}{7}a^{12}+\frac{25}{7}a^{11}-\frac{5}{7}a^{10}+\frac{16}{7}a^{9}-7a^{8}+\frac{20}{7}a^{7}-\frac{13}{7}a^{6}+6a^{5}-\frac{6}{7}a^{4}-a^{3}-2a^{2}-\frac{1}{7}a+\frac{9}{7}$, $\frac{37}{49}a^{15}+\frac{47}{49}a^{14}-\frac{12}{49}a^{13}-\frac{143}{49}a^{12}-\frac{54}{49}a^{11}+\frac{166}{49}a^{10}+\frac{235}{49}a^{9}+\frac{15}{49}a^{8}-\frac{272}{49}a^{7}-\frac{150}{49}a^{6}-\frac{16}{49}a^{5}+\frac{163}{49}a^{4}+\frac{123}{49}a^{3}-\frac{26}{49}a^{2}-\frac{23}{49}a-\frac{5}{49}$
| 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: | \( 40.8921494432 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 40.8921494432 \cdot 1}{8\cdot\sqrt{5455162516701184}}\cr\approx \mathstrut & 0.168106711465 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 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^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 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^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 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^13 + 3*x^12 + 8*x^10 - 8*x^9 + x^8 - 8*x^7 + 8*x^6 + 3*x^4 - 4*x^3 + 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
$D_4^2:C_2^2$ (as 16T509):
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 256 |
| The 40 conjugacy class representatives for $D_4^2:C_2^2$ |
| Character table for $D_4^2:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\), 8.2.4616192.1, 8.2.73859072.3, 8.0.3211264.1 |
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: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $32$ | |||
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |