Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 312*x^7 + 1565*x^6 + 22118*x^5 - 146182*x^4 - 420319*x^3 + 3575354*x^2 + 518192*x - 17159528)
gp: K = bnfinit(y^9 - y^8 - 312*y^7 + 1565*y^6 + 22118*y^5 - 146182*y^4 - 420319*y^3 + 3575354*y^2 + 518192*y - 17159528, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 312*x^7 + 1565*x^6 + 22118*x^5 - 146182*x^4 - 420319*x^3 + 3575354*x^2 + 518192*x - 17159528);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - x^8 - 312*x^7 + 1565*x^6 + 22118*x^5 - 146182*x^4 - 420319*x^3 + 3575354*x^2 + 518192*x - 17159528)
\( x^{9} - x^{8} - 312 x^{7} + 1565 x^{6} + 22118 x^{5} - 146182 x^{4} - 420319 x^{3} + 3575354 x^{2} + \cdots - 17159528 \)
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: | $[9, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(59654416235884558133761\)
\(\medspace = 19^{8}\cdot 37^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(339.33\) | 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: | $19^{8/9}37^{8/9}\approx 339.33367110228545$ | ||
| Ramified primes: |
\(19\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $9$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(703=19\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{703}(256,·)$, $\chi_{703}(1,·)$, $\chi_{703}(581,·)$, $\chi_{703}(44,·)$, $\chi_{703}(16,·)$, $\chi_{703}(530,·)$, $\chi_{703}(403,·)$, $\chi_{703}(121,·)$, $\chi_{703}(157,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{44}a^{7}-\frac{1}{22}a^{6}+\frac{21}{44}a^{4}+\frac{9}{44}a^{3}+\frac{19}{44}a^{2}+\frac{3}{11}a-\frac{4}{11}$, $\frac{1}{85404695519396}a^{8}+\frac{189176772536}{21351173879849}a^{7}-\frac{1562162517835}{42702347759698}a^{6}-\frac{16481788250445}{85404695519396}a^{5}-\frac{5603987017891}{85404695519396}a^{4}+\frac{28472923054713}{85404695519396}a^{3}-\frac{4909856482767}{21351173879849}a^{2}+\frac{8475395790637}{21351173879849}a+\frac{285194116217}{1255951404697}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{9}$, which has order $9$ (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: | $8$ | 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: |
$\frac{32267778127}{42702347759698}a^{8}+\frac{332900943161}{85404695519396}a^{7}-\frac{4443110419733}{21351173879849}a^{6}-\frac{1168732379112}{21351173879849}a^{5}+\frac{121079457069007}{7764063229036}a^{4}-\frac{17\!\cdots\!17}{85404695519396}a^{3}-\frac{33\!\cdots\!23}{85404695519396}a^{2}+\frac{11\!\cdots\!91}{21351173879849}a+\frac{33\!\cdots\!48}{1255951404697}$, $\frac{10026678699}{42702347759698}a^{8}-\frac{119323361061}{85404695519396}a^{7}-\frac{1415743509489}{21351173879849}a^{6}+\frac{14620658089387}{21351173879849}a^{5}+\frac{149190561315811}{85404695519396}a^{4}-\frac{34\!\cdots\!07}{85404695519396}a^{3}+\frac{83\!\cdots\!91}{85404695519396}a^{2}+\frac{42\!\cdots\!67}{21351173879849}a-\frac{657263336632127}{1255951404697}$, $\frac{18979834276}{21351173879849}a^{8}-\frac{262468764289}{85404695519396}a^{7}-\frac{10459682054861}{42702347759698}a^{6}+\frac{85639200795893}{42702347759698}a^{5}+\frac{677356591206937}{85404695519396}a^{4}-\frac{10\!\cdots\!91}{85404695519396}a^{3}+\frac{22\!\cdots\!13}{85404695519396}a^{2}+\frac{22\!\cdots\!85}{42702347759698}a-\frac{20\!\cdots\!68}{1255951404697}$, $\frac{245376163379}{21351173879849}a^{8}+\frac{6387322961507}{85404695519396}a^{7}-\frac{64618297088239}{21351173879849}a^{6}-\frac{202665315485799}{42702347759698}a^{5}+\frac{18\!\cdots\!29}{85404695519396}a^{4}-\frac{270344515289175}{7764063229036}a^{3}-\frac{39\!\cdots\!39}{7764063229036}a^{2}+\frac{56\!\cdots\!38}{21351173879849}a+\frac{33\!\cdots\!22}{1255951404697}$, $\frac{989668930271}{85404695519396}a^{8}+\frac{6515243263939}{85404695519396}a^{7}-\frac{130082316643423}{42702347759698}a^{6}-\frac{424841966687555}{85404695519396}a^{5}+\frac{94\!\cdots\!63}{42702347759698}a^{4}-\frac{586065908802756}{21351173879849}a^{3}-\frac{44\!\cdots\!73}{85404695519396}a^{2}+\frac{11\!\cdots\!53}{42702347759698}a+\frac{33\!\cdots\!85}{1255951404697}$, $\frac{36988903805}{42702347759698}a^{8}+\frac{766590596743}{85404695519396}a^{7}-\frac{4592667542545}{21351173879849}a^{6}-\frac{54650633161175}{42702347759698}a^{5}+\frac{14\!\cdots\!17}{85404695519396}a^{4}+\frac{51\!\cdots\!31}{85404695519396}a^{3}-\frac{46\!\cdots\!81}{85404695519396}a^{2}-\frac{41\!\cdots\!19}{42702347759698}a+\frac{68\!\cdots\!16}{1255951404697}$, $\frac{112638791637}{85404695519396}a^{8}+\frac{876692149077}{85404695519396}a^{7}-\frac{6955191281986}{21351173879849}a^{6}-\frac{71843715233035}{85404695519396}a^{5}+\frac{44188065088899}{1941015807259}a^{4}+\frac{256537351308881}{21351173879849}a^{3}-\frac{43\!\cdots\!21}{85404695519396}a^{2}+\frac{22\!\cdots\!11}{42702347759698}a+\frac{29\!\cdots\!88}{1255951404697}$, $\frac{10411654353}{5023805618788}a^{8}+\frac{27102218639}{2511902809394}a^{7}-\frac{1412128591405}{2511902809394}a^{6}-\frac{770335532363}{5023805618788}a^{5}+\frac{202245021038373}{5023805618788}a^{4}-\frac{24645162961001}{456709601708}a^{3}-\frac{203780585403233}{228354800854}a^{2}+\frac{34\!\cdots\!51}{2511902809394}a+\frac{47\!\cdots\!28}{1255951404697}$
| 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: | \( 84671913.5934 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 84671913.5934 \cdot 9}{2\cdot\sqrt{59654416235884558133761}}\cr\approx \mathstrut & 0.798731017000 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 312*x^7 + 1565*x^6 + 22118*x^5 - 146182*x^4 - 420319*x^3 + 3575354*x^2 + 518192*x - 17159528)
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 - x^8 - 312*x^7 + 1565*x^6 + 22118*x^5 - 146182*x^4 - 420319*x^3 + 3575354*x^2 + 518192*x - 17159528, 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 - x^8 - 312*x^7 + 1565*x^6 + 22118*x^5 - 146182*x^4 - 420319*x^3 + 3575354*x^2 + 518192*x - 17159528);
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 - x^8 - 312*x^7 + 1565*x^6 + 22118*x^5 - 146182*x^4 - 420319*x^3 + 3575354*x^2 + 518192*x - 17159528);
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 cyclic group of order 9 |
| The 9 conjugacy class representatives for $C_9$ |
| Character table for $C_9$ |
Intermediate fields
| 3.3.494209.2 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{3}$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.1.0.1}{1} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.1.0.1}{1} }^{9}$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.9.8.2 | $x^{9} + 57$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
|
\(37\)
| 37.9.8.1 | $x^{9} + 37$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |