Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 4*x^8 - 2*x^7 + 5*x^6 + 4*x^5 + 5*x^4 + 10*x^3 - 22*x^2 - 12*x + 9)
gp: K = bnfinit(y^10 + 4*y^8 - 2*y^7 + 5*y^6 + 4*y^5 + 5*y^4 + 10*y^3 - 22*y^2 - 12*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 4*x^8 - 2*x^7 + 5*x^6 + 4*x^5 + 5*x^4 + 10*x^3 - 22*x^2 - 12*x + 9);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 + 4*x^8 - 2*x^7 + 5*x^6 + 4*x^5 + 5*x^4 + 10*x^3 - 22*x^2 - 12*x + 9)
\( x^{10} + 4x^{8} - 2x^{7} + 5x^{6} + 4x^{5} + 5x^{4} + 10x^{3} - 22x^{2} - 12x + 9 \)
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: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2339717120000\)
\(\medspace = 2^{17}\cdot 5^{4}\cdot 13^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.26\) | 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: | $2^{11/6}5^{1/2}13^{2/3}\approx 44.055712473468965$ | ||
| Ramified primes: |
\(2\), \(5\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{6855}a^{9}-\frac{182}{2285}a^{8}-\frac{2129}{6855}a^{7}-\frac{1547}{6855}a^{6}-\frac{248}{1371}a^{5}-\frac{2972}{6855}a^{4}+\frac{439}{1371}a^{3}-\frac{206}{6855}a^{2}+\frac{32}{6855}a+\frac{1027}{2285}$
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: | $5$ | 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: |
$\frac{133}{1371}a^{9}+\frac{15}{457}a^{8}+\frac{640}{1371}a^{7}-\frac{101}{1371}a^{6}+\frac{971}{1371}a^{5}+\frac{943}{1371}a^{4}+\frac{1283}{1371}a^{3}+\frac{2764}{1371}a^{2}-\frac{2599}{1371}a-\frac{52}{457}$, $\frac{968}{6855}a^{9}+\frac{226}{2285}a^{8}+\frac{3854}{6855}a^{7}+\frac{1007}{6855}a^{6}+\frac{4789}{6855}a^{5}+\frac{6317}{6855}a^{4}+\frac{12049}{6855}a^{3}+\frac{2071}{1371}a^{2}-\frac{10154}{6855}a-\frac{1156}{457}$, $\frac{1037}{2285}a^{9}-\frac{893}{2285}a^{8}+\frac{5021}{2285}a^{7}-\frac{6567}{2285}a^{6}+\frac{11086}{2285}a^{5}-\frac{5897}{2285}a^{4}+\frac{10866}{2285}a^{3}+\frac{325}{457}a^{2}-\frac{23941}{2285}a+\frac{1850}{457}$, $\frac{1513}{6855}a^{9}+\frac{662}{2285}a^{8}+\frac{6157}{6855}a^{7}+\frac{6541}{6855}a^{6}+\frac{3521}{6855}a^{5}+\frac{23551}{6855}a^{4}+\frac{11441}{6855}a^{3}+\frac{40669}{6855}a^{2}-\frac{13279}{6855}a-\frac{14573}{2285}$, $\frac{368}{6855}a^{9}+\frac{203}{2285}a^{8}+\frac{148}{1371}a^{7}+\frac{208}{1371}a^{6}+\frac{223}{6855}a^{5}+\frac{895}{1371}a^{4}+\frac{2983}{6855}a^{3}+\frac{968}{6855}a^{2}-\frac{1934}{6855}a-\frac{1831}{2285}$
| 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: | \( 344.74214707 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 344.74214707 \cdot 1}{2\cdot\sqrt{2339717120000}}\cr\approx \mathstrut & 0.70252563273 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 + 4*x^8 - 2*x^7 + 5*x^6 + 4*x^5 + 5*x^4 + 10*x^3 - 22*x^2 - 12*x + 9)
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 + 4*x^8 - 2*x^7 + 5*x^6 + 4*x^5 + 5*x^4 + 10*x^3 - 22*x^2 - 12*x + 9, 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 + 4*x^8 - 2*x^7 + 5*x^6 + 4*x^5 + 5*x^4 + 10*x^3 - 22*x^2 - 12*x + 9);
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 + 4*x^8 - 2*x^7 + 5*x^6 + 4*x^5 + 5*x^4 + 10*x^3 - 22*x^2 - 12*x + 9);
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 120 |
| The 7 conjugacy class representatives for $S_5$ |
| Character table for $S_5$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 5.1.135200.1 |
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 5 sibling: | 5.1.135200.1 |
| Degree 6 sibling: | 6.2.1462323200.1 |
| Degree 10 sibling: | data not computed |
| Degree 12 sibling: | data not computed |
| Degree 15 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | 5.1.135200.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.6.11.1 | $x^{6} + 4 x^{3} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ | |
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.6.4.2 | $x^{6} - 156 x^{3} + 338$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |