Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 5*x^8 + 15*x^6 - 12*x^5 + 45*x^4 + 180*x^3 + 486)
gp: K = bnfinit(y^10 + 5*y^8 + 15*y^6 - 12*y^5 + 45*y^4 + 180*y^3 + 486, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 5*x^8 + 15*x^6 - 12*x^5 + 45*x^4 + 180*x^3 + 486);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 + 5*x^8 + 15*x^6 - 12*x^5 + 45*x^4 + 180*x^3 + 486)
\( x^{10} + 5x^{8} + 15x^{6} - 12x^{5} + 45x^{4} + 180x^{3} + 486 \)
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: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-17496000000000000\)
\(\medspace = -\,2^{15}\cdot 3^{7}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.10\) | 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: | $2^{11/4}3^{11/6}5^{271/200}\approx 446.33315037681194$ | ||
| 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: | \(\Q(\sqrt{-6}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{3}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{4}$, $\frac{1}{18}a^{7}-\frac{1}{18}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{108}a^{8}-\frac{1}{27}a^{6}+\frac{17}{36}a^{4}-\frac{1}{9}a^{3}+\frac{1}{6}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{237168}a^{9}-\frac{283}{79056}a^{8}-\frac{1937}{118584}a^{7}+\frac{1367}{39528}a^{6}+\frac{2207}{79056}a^{5}+\frac{10421}{79056}a^{4}+\frac{127}{1464}a^{3}+\frac{235}{13176}a^{2}-\frac{625}{4392}a-\frac{269}{1464}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$ (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: |
$\frac{1465}{4392}a^{9}-\frac{4255}{13176}a^{8}+\frac{4163}{2196}a^{7}-\frac{12829}{6588}a^{6}+\frac{22237}{4392}a^{5}-\frac{44621}{4392}a^{4}+\frac{28343}{2196}a^{3}+\frac{33157}{732}a^{2}-\frac{60037}{732}a+\frac{24907}{244}$, $\frac{9467}{237168}a^{9}-\frac{12485}{79056}a^{8}-\frac{200815}{118584}a^{7}+\frac{31837}{39528}a^{6}+\frac{79981}{79056}a^{5}+\frac{465859}{79056}a^{4}-\frac{153113}{4392}a^{3}-\frac{17371}{13176}a^{2}+\frac{164581}{4392}a-\frac{142003}{1464}$, $\frac{7813}{26352}a^{9}+\frac{19667}{26352}a^{8}+\frac{41311}{13176}a^{7}+\frac{80285}{13176}a^{6}+\frac{11691}{976}a^{5}+\frac{21923}{2928}a^{4}+\frac{5921}{4392}a^{3}+\frac{31919}{1464}a^{2}+\frac{14537}{1464}a-\frac{19163}{488}$, $\frac{979}{13176}a^{9}-\frac{2303}{13176}a^{8}-\frac{1907}{6588}a^{7}-\frac{3557}{6588}a^{6}+\frac{7597}{4392}a^{5}-\frac{2165}{4392}a^{4}-\frac{15089}{2196}a^{3}+\frac{723}{244}a^{2}+\frac{3403}{732}a-\frac{3885}{244}$
| 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: | \( 183259.277733 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 183259.277733 \cdot 2}{2\cdot\sqrt{17496000000000000}}\cr\approx \mathstrut & 13.5673780728 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^10 + 5*x^8 + 15*x^6 - 12*x^5 + 45*x^4 + 180*x^3 + 486)
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 + 5*x^8 + 15*x^6 - 12*x^5 + 45*x^4 + 180*x^3 + 486, 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 + 5*x^8 + 15*x^6 - 12*x^5 + 45*x^4 + 180*x^3 + 486);
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 + 5*x^8 + 15*x^6 - 12*x^5 + 45*x^4 + 180*x^3 + 486);
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
$S_5\wr C_2$ (as 10T43):
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 28800 |
| The 35 conjugacy class representatives for $S_5^2 \wr C_2$ |
| Character table for $S_5^2 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \) |
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 12 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 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 | R | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.4.9.5 | $x^{4} + 10 x^{2} + 8 x + 6$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.3.5.2 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
|
\(5\)
| 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |