Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 2*x^8 - 9*x^6 - 8*x^5 - 17*x^4 - 14*x^3 + 4*x^2 + 4*x + 2)
gp: K = bnfinit(y^10 + 2*y^8 - 9*y^6 - 8*y^5 - 17*y^4 - 14*y^3 + 4*y^2 + 4*y + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 2*x^8 - 9*x^6 - 8*x^5 - 17*x^4 - 14*x^3 + 4*x^2 + 4*x + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 + 2*x^8 - 9*x^6 - 8*x^5 - 17*x^4 - 14*x^3 + 4*x^2 + 4*x + 2)
\( x^{10} + 2x^{8} - 9x^{6} - 8x^{5} - 17x^{4} - 14x^{3} + 4x^{2} + 4x + 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1680066179072\)
\(\medspace = -\,2^{12}\cdot 743^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.69\) | 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: | $2^{3/2}743^{1/2}\approx 77.09734106958553$ | ||
| Ramified primes: |
\(2\), \(743\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-743}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{691}a^{9}-\frac{51}{691}a^{8}-\frac{161}{691}a^{7}-\frac{81}{691}a^{6}-\frac{24}{691}a^{5}-\frac{166}{691}a^{4}+\frac{157}{691}a^{3}+\frac{271}{691}a^{2}+\frac{3}{691}a-\frac{149}{691}$
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: | $6$ | 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: |
$\frac{152}{691}a^{9}-\frac{151}{691}a^{8}+\frac{404}{691}a^{7}-\frac{565}{691}a^{6}-\frac{884}{691}a^{5}-\frac{356}{691}a^{4}-\frac{1703}{691}a^{3}+\frac{1114}{691}a^{2}+\frac{1147}{691}a+\frac{155}{691}$, $\frac{327}{691}a^{9}-\frac{93}{691}a^{8}+\frac{560}{691}a^{7}-\frac{229}{691}a^{6}-\frac{3011}{691}a^{5}-\frac{1766}{691}a^{4}-\frac{3941}{691}a^{3}-\frac{1904}{691}a^{2}+\frac{3745}{691}a+\frac{1029}{691}$, $\frac{423}{691}a^{9}-\frac{152}{691}a^{8}+\frac{997}{691}a^{7}-\frac{404}{691}a^{6}-\frac{3242}{691}a^{5}-\frac{2500}{691}a^{4}-\frac{6835}{691}a^{3}-\frac{4219}{691}a^{2}+\frac{578}{691}a+\frac{545}{691}$, $\frac{268}{691}a^{9}+\frac{152}{691}a^{8}+\frac{385}{691}a^{7}+\frac{404}{691}a^{6}-\frac{2977}{691}a^{5}-\frac{3028}{691}a^{4}-\frac{4912}{691}a^{3}-\frac{5455}{691}a^{2}+\frac{2186}{691}a+\frac{2219}{691}$, $\frac{487}{691}a^{9}-\frac{652}{691}a^{8}+\frac{1749}{691}a^{7}-\frac{2133}{691}a^{6}-\frac{2014}{691}a^{5}-\frac{686}{691}a^{4}-\frac{7152}{691}a^{3}+\frac{2069}{691}a^{2}+\frac{770}{691}a+\frac{683}{691}$, $\frac{541}{691}a^{9}-\frac{642}{691}a^{8}+\frac{2038}{691}a^{7}-\frac{2361}{691}a^{6}-\frac{1928}{691}a^{5}-\frac{2049}{691}a^{4}-\frac{8348}{691}a^{3}+\frac{119}{691}a^{2}+\frac{932}{691}a+\frac{929}{691}$
| 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: | \( 386.348534519 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 386.348534519 \cdot 1}{2\cdot\sqrt{1680066179072}}\cr\approx \mathstrut & 0.591487630246 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 + 2*x^8 - 9*x^6 - 8*x^5 - 17*x^4 - 14*x^3 + 4*x^2 + 4*x + 2)
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^10 + 2*x^8 - 9*x^6 - 8*x^5 - 17*x^4 - 14*x^3 + 4*x^2 + 4*x + 2, 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^10 + 2*x^8 - 9*x^6 - 8*x^5 - 17*x^4 - 14*x^3 + 4*x^2 + 4*x + 2);
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^10 + 2*x^8 - 9*x^6 - 8*x^5 - 17*x^4 - 14*x^3 + 4*x^2 + 4*x + 2);
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 720 |
| The 11 conjugacy class representatives for $S_{6}$ |
| Character table for $S_{6}$ |
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 6 siblings: | 6.4.26251034048.1, 6.0.47552.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 15 siblings: | deg 15, deg 15 |
| Degree 20 siblings: | 20.0.2822622366061589570781184.1, deg 20, deg 20 |
| Degree 30 siblings: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Degree 45 sibling: | data not computed |
| Minimal sibling: | 6.0.47552.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
| 2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
|
\(743\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |