Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 18*x^6 + 68*x^5 + 56*x^4 - 230*x^3 - 46*x^2 + 173*x - 22)
gp: K = bnfinit(y^8 - 4*y^7 - 18*y^6 + 68*y^5 + 56*y^4 - 230*y^3 - 46*y^2 + 173*y - 22, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 4*x^7 - 18*x^6 + 68*x^5 + 56*x^4 - 230*x^3 - 46*x^2 + 173*x - 22);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^8 - 4*x^7 - 18*x^6 + 68*x^5 + 56*x^4 - 230*x^3 - 46*x^2 + 173*x - 22)
\( x^{8} - 4x^{7} - 18x^{6} + 68x^{5} + 56x^{4} - 230x^{3} - 46x^{2} + 173x - 22 \)
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: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(48501766991041\)
\(\medspace = 7^{4}\cdot 13^{4}\cdot 29^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(51.37\) | 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: | $7^{2/3}13^{1/2}29^{1/2}\approx 71.05086481765254$ | ||
| Ramified primes: |
\(7\), \(13\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{29}a^{6}-\frac{3}{29}a^{5}-\frac{14}{29}a^{4}+\frac{4}{29}a^{3}-\frac{9}{29}a^{2}-\frac{8}{29}a-\frac{1}{29}$, $\frac{1}{2523}a^{7}+\frac{40}{2523}a^{6}+\frac{872}{2523}a^{5}+\frac{226}{841}a^{4}-\frac{823}{2523}a^{3}+\frac{446}{2523}a^{2}-\frac{115}{841}a-\frac{478}{2523}$
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
$C_{4}$, which has order $4$
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: |
$\frac{47}{2523}a^{7}-\frac{121}{2523}a^{6}-\frac{950}{2523}a^{5}+\frac{617}{841}a^{4}+\frac{3775}{2523}a^{3}-\frac{6443}{2523}a^{2}-\frac{1751}{841}a+\frac{4765}{2523}$, $\frac{121}{2523}a^{7}-\frac{467}{2523}a^{6}-\frac{2194}{2523}a^{5}+\frac{2493}{841}a^{4}+\frac{7862}{2523}a^{3}-\frac{19375}{2523}a^{2}-\frac{5650}{841}a+\frac{2975}{2523}$, $\frac{61}{841}a^{7}-\frac{112}{841}a^{6}-\frac{1386}{841}a^{5}+\frac{1396}{841}a^{4}+\frac{6869}{841}a^{3}-\frac{2809}{841}a^{2}-\frac{5675}{841}a+\frac{2829}{841}$, $\frac{80}{841}a^{7}-\frac{193}{841}a^{6}-\frac{1638}{841}a^{5}+\frac{2504}{841}a^{4}+\frac{6370}{841}a^{3}-\frac{3586}{841}a^{2}-\frac{2979}{841}a+\frac{475}{841}$, $\frac{274}{2523}a^{7}-\frac{785}{2523}a^{6}-\frac{5890}{2523}a^{5}+\frac{4040}{841}a^{4}+\frac{30278}{2523}a^{3}-\frac{31960}{2523}a^{2}-\frac{17851}{841}a-\frac{3169}{2523}$, $\frac{9}{841}a^{7}-\frac{17}{841}a^{6}-\frac{272}{841}a^{5}+\frac{447}{841}a^{4}+\frac{2018}{841}a^{3}-\frac{3526}{841}a^{2}-\frac{1771}{841}a+\frac{2803}{841}$, $\frac{9}{841}a^{7}-\frac{46}{841}a^{6}-\frac{185}{841}a^{5}+\frac{853}{841}a^{4}+\frac{1061}{841}a^{3}-\frac{2424}{841}a^{2}-\frac{2380}{841}a+\frac{309}{841}$
| 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: | \( 18482.1278063 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 18482.1278063 \cdot 4}{2\cdot\sqrt{48501766991041}}\cr\approx \mathstrut & 1.35876123987 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 18*x^6 + 68*x^5 + 56*x^4 - 230*x^3 - 46*x^2 + 173*x - 22)
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 - 18*x^6 + 68*x^5 + 56*x^4 - 230*x^3 - 46*x^2 + 173*x - 22, 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 - 18*x^6 + 68*x^5 + 56*x^4 - 230*x^3 - 46*x^2 + 173*x - 22);
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 - 18*x^6 + 68*x^5 + 56*x^4 - 230*x^3 - 46*x^2 + 173*x - 22);
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
$C_2\times A_4$ (as 8T13):
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 24 |
| The 8 conjugacy class representatives for $A_4\times C_2$ |
| Character table for $A_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{377}) \), 4.4.6964321.1 |
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 24 |
| Degree 6 sibling: | 6.6.905177.1 |
| Degree 12 siblings: | 12.12.116452742545489441.3, 12.12.16551311845247868759889.4 |
| Minimal sibling: | 6.6.905177.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(13\)
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(29\)
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |