Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 8*x^6 + 28*x^5 + 50*x^4 - 92*x^3 - 32*x^2 - 124*x - 11)
gp: K = bnfinit(y^8 - 4*y^7 - 8*y^6 + 28*y^5 + 50*y^4 - 92*y^3 - 32*y^2 - 124*y - 11, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 4*x^7 - 8*x^6 + 28*x^5 + 50*x^4 - 92*x^3 - 32*x^2 - 124*x - 11);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 4*x^7 - 8*x^6 + 28*x^5 + 50*x^4 - 92*x^3 - 32*x^2 - 124*x - 11)
\( x^{8} - 4x^{7} - 8x^{6} + 28x^{5} + 50x^{4} - 92x^{3} - 32x^{2} - 124x - 11 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-3542940000000\)
\(\medspace = -\,2^{8}\cdot 3^{11}\cdot 5^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.04\) | 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: | $2^{31/28}3^{25/12}5^{71/60}\approx 142.692231128093$ | ||
| 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: | \(\Q(\sqrt{-15}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{2}+\frac{1}{8}$, $\frac{1}{8}a^{5}-\frac{1}{4}a^{2}-\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{4}-\frac{1}{16}a^{2}+\frac{1}{16}$, $\frac{1}{32}a^{7}-\frac{1}{32}a^{6}+\frac{1}{32}a^{5}-\frac{1}{32}a^{4}+\frac{3}{32}a^{3}-\frac{3}{32}a^{2}-\frac{5}{32}a+\frac{5}{32}$
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 $1$ (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: | $4$ | 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: |
$\frac{1}{32}a^{7}-\frac{3}{32}a^{6}-\frac{7}{32}a^{5}+\frac{5}{32}a^{4}+\frac{51}{32}a^{3}-\frac{41}{32}a^{2}+\frac{19}{32}a-\frac{57}{32}$, $\frac{1}{16}a^{6}-\frac{1}{8}a^{5}-\frac{5}{16}a^{4}+\frac{1}{2}a^{3}+\frac{3}{16}a^{2}+\frac{5}{8}a+\frac{1}{16}$, $\frac{3}{8}a^{7}-\frac{3}{2}a^{6}-\frac{11}{4}a^{5}+\frac{21}{2}a^{4}+\frac{129}{8}a^{3}-\frac{155}{4}a^{2}-\frac{43}{4}a-\frac{189}{4}$, $\frac{5}{32}a^{7}+\frac{3}{32}a^{6}-\frac{91}{32}a^{5}-\frac{137}{32}a^{4}+\frac{399}{32}a^{3}+\frac{897}{32}a^{2}+\frac{135}{32}a+\frac{5}{32}$
| 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: | \( 6646.76218873 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 6646.76218873 \cdot 1}{2\cdot\sqrt{3542940000000}}\cr\approx \mathstrut & 1.75185373164 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 8*x^6 + 28*x^5 + 50*x^4 - 92*x^3 - 32*x^2 - 124*x - 11)
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^8 - 4*x^7 - 8*x^6 + 28*x^5 + 50*x^4 - 92*x^3 - 32*x^2 - 124*x - 11, 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^8 - 4*x^7 - 8*x^6 + 28*x^5 + 50*x^4 - 92*x^3 - 32*x^2 - 124*x - 11);
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^8 - 4*x^7 - 8*x^6 + 28*x^5 + 50*x^4 - 92*x^3 - 32*x^2 - 124*x - 11);
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
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 non-solvable group of order 40320 |
| The 22 conjugacy class representatives for $S_8$ |
| Character table for $S_8$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 sibling: | deg 16 |
| Degree 28 sibling: | deg 28 |
| Degree 30 sibling: | deg 30 |
| Degree 35 sibling: | deg 35 |
| 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 | R | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.13 | $x^{8} + 2 x + 2$ | $8$ | $1$ | $8$ | $C_2^3:(C_7: C_3)$ | $[8/7, 8/7, 8/7]_{7}^{3}$ |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.6.10.11 | $x^{6} + 3 x^{5} + 9 x^{2} + 9 x + 24$ | $6$ | $1$ | $10$ | $C_3^2:D_4$ | $[9/4, 9/4]_{4}^{2}$ | |
|
\(5\)
| 5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |