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Properties

Label 9.3.765...136.1
Degree $9$
Signature $[3, 3]$
Discriminant $-7.658\times 10^{32}$
Root discriminant \(4505.97\)
Ramified primes $2,43,71,103$
Class number $93830688$ (GRH)
Class group [2, 2, 42, 558516] (GRH)
Galois group $S_3\times C_3$ (as 9T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 1965*x^7 + 73862*x^6 - 1377684*x^5 + 15584739*x^4 - 111818439*x^3 + 498777897*x^2 - 1264170258*x + 1392458831)
 
gp: K = bnfinit(y^9 - y^8 - 1965*y^7 + 73862*y^6 - 1377684*y^5 + 15584739*y^4 - 111818439*y^3 + 498777897*y^2 - 1264170258*y + 1392458831, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 1965*x^7 + 73862*x^6 - 1377684*x^5 + 15584739*x^4 - 111818439*x^3 + 498777897*x^2 - 1264170258*x + 1392458831);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 - 1965*x^7 + 73862*x^6 - 1377684*x^5 + 15584739*x^4 - 111818439*x^3 + 498777897*x^2 - 1264170258*x + 1392458831)
 

\( x^{9} - x^{8} - 1965 x^{7} + 73862 x^{6} - 1377684 x^{5} + 15584739 x^{4} - 111818439 x^{3} + \cdots + 1392458831 \) Copy content Toggle raw display

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:   \(-765763711957563428907417238099136\) \(\medspace = -\,2^{6}\cdot 43^{7}\cdot 71^{3}\cdot 103^{7}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(4505.97\)
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:  $2^{2/3}43^{5/6}71^{1/2}103^{5/6}\approx 14618.536789485457$
Ramified primes:   \(2\), \(43\), \(71\), \(103\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-314459}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{71}a^{6}+\frac{23}{71}a^{5}+\frac{31}{71}a^{4}+\frac{32}{71}a^{3}-\frac{21}{71}a^{2}-\frac{29}{71}a-\frac{7}{71}$, $\frac{1}{71}a^{7}-\frac{1}{71}a^{5}+\frac{29}{71}a^{4}+\frac{24}{71}a^{3}+\frac{28}{71}a^{2}+\frac{21}{71}a+\frac{19}{71}$, $\frac{1}{71}a^{8}-\frac{19}{71}a^{5}-\frac{16}{71}a^{4}-\frac{11}{71}a^{3}-\frac{10}{71}a-\frac{7}{71}$ Copy content Toggle raw display

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

$C_{2}\times C_{2}\times C_{42}\times C_{558516}$, which has order $93830688$ (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 $2$) Copy content Toggle raw display
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:   $a-7$, $a^{8}+7a^{7}-1909a^{6}+58590a^{5}-908964a^{4}+8313027a^{3}-45314223a^{2}+136264113a-174057354$, $\frac{21}{71}a^{8}+\frac{69}{71}a^{7}-\frac{41186}{71}a^{6}+\frac{1371540}{71}a^{5}-\frac{22661224}{71}a^{4}+\frac{220237269}{71}a^{3}-\frac{1277530904}{71}a^{2}+\frac{4094526541}{71}a-\frac{5578327291}{71}$, $\frac{5}{71}a^{8}+\frac{33}{71}a^{7}-\frac{9574}{71}a^{6}+\frac{296550}{71}a^{5}-\frac{4635011}{71}a^{4}+\frac{42707447}{71}a^{3}-\frac{234646284}{71}a^{2}+\frac{711548834}{71}a-\frac{916912158}{71}$, $\frac{238}{71}a^{8}+\frac{1598}{71}a^{7}-\frac{455362}{71}a^{6}+\frac{14066106}{71}a^{5}-\frac{219343350}{71}a^{4}+\frac{2016174058}{71}a^{3}-\frac{11047516878}{71}a^{2}+\frac{33398547684}{71}a-\frac{42890027189}{71}$ Copy content Toggle raw display (assuming GRH)
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:  \( 74376.1833618562 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 74376.1833618562 \cdot 93830688}{2\cdot\sqrt{765763711957563428907417238099136}}\cr\approx \mathstrut & 0.250225170996319 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 1965*x^7 + 73862*x^6 - 1377684*x^5 + 15584739*x^4 - 111818439*x^3 + 498777897*x^2 - 1264170258*x + 1392458831)
 
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 - 1965*x^7 + 73862*x^6 - 1377684*x^5 + 15584739*x^4 - 111818439*x^3 + 498777897*x^2 - 1264170258*x + 1392458831, 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 - 1965*x^7 + 73862*x^6 - 1377684*x^5 + 15584739*x^4 - 111818439*x^3 + 498777897*x^2 - 1264170258*x + 1392458831);
 
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 - 1965*x^7 + 73862*x^6 - 1377684*x^5 + 15584739*x^4 - 111818439*x^3 + 498777897*x^2 - 1264170258*x + 1392458831);
 
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

3.3.19616041.2, 3.1.1257836.1

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: 6.0.9759407007223876480827824.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{3}$ ${\href{/padicField/5.3.0.1}{3} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{3}$ ${\href{/padicField/11.3.0.1}{3} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.3.0.1}{3} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ R ${\href{/padicField/47.3.0.1}{3} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.9.6.1$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
\(43\) Copy content Toggle raw display 43.3.2.2$x^{3} + 301$$3$$1$$2$$C_3$$[\ ]_{3}$
43.6.5.3$x^{6} + 301$$6$$1$$5$$C_6$$[\ ]_{6}$
\(71\) Copy content Toggle raw display $\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
\(103\) Copy content Toggle raw display 103.3.2.2$x^{3} + 206$$3$$1$$2$$C_3$$[\ ]_{3}$
103.6.5.4$x^{6} + 206$$6$$1$$5$$C_6$$[\ ]_{6}$