Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4900*x^7 - 3*x^6 - 630*x^5 + 9800*x^4 - 24*x^3 + 630*x^2 - 4900*x - 1)
gp: K = bnfinit(y^9 - 4900*y^7 - 3*y^6 - 630*y^5 + 9800*y^4 - 24*y^3 + 630*y^2 - 4900*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 4900*x^7 - 3*x^6 - 630*x^5 + 9800*x^4 - 24*x^3 + 630*x^2 - 4900*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 4900*x^7 - 3*x^6 - 630*x^5 + 9800*x^4 - 24*x^3 + 630*x^2 - 4900*x - 1)
\( x^{9} - 4900x^{7} - 3x^{6} - 630x^{5} + 9800x^{4} - 24x^{3} + 630x^{2} - 4900x - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-43987937064626389889214456471715203\)
\(\medspace = -\,3^{3}\cdot 37^{6}\cdot 73^{6}\cdot 127^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(7067.39\) | 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: | $3^{1/2}37^{2/3}73^{2/3}127^{2/3}\approx 8487.494335737842$ | ||
| Ramified primes: |
\(3\), \(37\), \(73\), \(127\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}-\frac{4}{9}$, $\frac{1}{6174243}a^{8}+\frac{25411}{686027}a^{7}-\frac{156806}{6174243}a^{6}+\frac{1}{6174243}a^{5}-\frac{2516039}{6174243}a^{4}+\frac{176407}{2058081}a^{3}+\frac{2058061}{6174243}a^{2}+\frac{228839}{6174243}a-\frac{1754269}{6174243}$
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
$C_{6}\times C_{6}\times C_{18}\times C_{18}\times C_{2232}$, which has order $26034048$ (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: | $5$ | 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: |
$\frac{20476}{6174243}a^{8}-\frac{20422}{6174243}a^{7}+\frac{33437746}{2058081}a^{6}+\frac{33379822}{2058081}a^{5}+\frac{35621719}{2058081}a^{4}-\frac{237898073}{6174243}a^{3}-\frac{237641849}{6174243}a^{2}-\frac{250542809}{6174243}a+\frac{38324}{6174243}$, $\frac{234661}{2058081}a^{8}+\frac{38378}{6174243}a^{7}+\frac{3450183511}{6174243}a^{6}-\frac{186617300}{6174243}a^{5}-\frac{2823998933}{6174243}a^{4}-\frac{3606976238}{6174243}a^{3}+\frac{68275793}{6174243}a^{2}+\frac{1035075616}{2058081}a+\frac{6898702}{6174243}$, $\frac{235501}{2058081}a^{8}-\frac{235483}{2058081}a^{7}+\frac{384573279}{686027}a^{6}+\frac{1154347940}{2058081}a^{5}+\frac{1300655507}{2058081}a^{4}-\frac{1055287103}{2058081}a^{3}-\frac{349904493}{686027}a^{2}-\frac{400045850}{686027}a+\frac{614060}{686027}$, $\frac{17956}{6174243}a^{8}+\frac{71961}{686027}a^{7}-\frac{88651211}{6174243}a^{6}-\frac{3173542946}{6174243}a^{5}+\frac{3254823889}{6174243}a^{4}-\frac{62251024}{2058081}a^{3}+\frac{3311093209}{6174243}a^{2}-\frac{3123063550}{6174243}a-\frac{724459}{6174243}$, $60\!\cdots\!75a^{8}-63\!\cdots\!62a^{7}+29\!\cdots\!00a^{6}+31\!\cdots\!25a^{5}+39\!\cdots\!74a^{4}-19\!\cdots\!40a^{3}-31\!\cdots\!25a^{2}-21\!\cdots\!38a-43\!\cdots\!28$
| 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: | \( 1675675.4950873558 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1675675.4950873558 \cdot 26034048}{2\cdot\sqrt{43987937064626389889214456471715203}}\cr\approx \mathstrut & 0.206378484398527 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 4900*x^7 - 3*x^6 - 630*x^5 + 9800*x^4 - 24*x^3 + 630*x^2 - 4900*x - 1)
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^9 - 4900*x^7 - 3*x^6 - 630*x^5 + 9800*x^4 - 24*x^3 + 630*x^2 - 4900*x - 1, 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^9 - 4900*x^7 - 3*x^6 - 630*x^5 + 9800*x^4 - 24*x^3 + 630*x^2 - 4900*x - 1);
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^9 - 4900*x^7 - 3*x^6 - 630*x^5 + 9800*x^4 - 24*x^3 + 630*x^2 - 4900*x - 1);
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
$C_3\times S_3$ (as 9T4):
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 18 |
| The 9 conjugacy class representatives for $S_3\times C_3$ |
| Character table for $S_3\times C_3$ |
Intermediate fields
| 3.3.22080601.1, 3.1.353002568187.1 |
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: | data not computed |
| Degree 6 sibling: | data not computed |
| Minimal sibling: | 6.0.16930356172816707.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(37\)
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(73\)
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(127\)
| 127.3.2.2 | $x^{3} + 1143$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 127.3.2.2 | $x^{3} + 1143$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 127.3.2.2 | $x^{3} + 1143$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |