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Properties

Label 6.0.24069811311.2
Degree $6$
Signature $[0, 3]$
Discriminant $-24069811311$
Root discriminant \(53.73\)
Ramified primes $3,7,13$
Class number $48$
Class group $[2, 2, 12]$
Galois group $C_6$ (as 6T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099)
 
gp: K = bnfinit(y^6 - y^5 + 34*y^4 - 9*y^3 + 786*y^2 + 738*y + 9099, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099)
 

\( x^{6} - x^{5} + 34x^{4} - 9x^{3} + 786x^{2} + 738x + 9099 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-24069811311\) \(\medspace = -\,3^{3}\cdot 7^{4}\cdot 13^{5}\) 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:  \(53.73\)
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))
 
Ramified primes:   \(3\), \(7\), \(13\) 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:  \(\Q(\sqrt{-39}) \)
$\card{ \Gal(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(1,·)$, $\chi_{273}(179,·)$, $\chi_{273}(100,·)$, $\chi_{273}(212,·)$, $\chi_{273}(155,·)$, $\chi_{273}(172,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-39}) \), 6.0.24069811311.2$^{3}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{66}a^{4}+\frac{23}{66}a^{3}+\frac{13}{66}a^{2}-\frac{1}{11}a-\frac{3}{22}$, $\frac{1}{39204}a^{5}-\frac{2}{891}a^{4}+\frac{3845}{19602}a^{3}-\frac{857}{13068}a^{2}-\frac{3587}{13068}a-\frac{439}{4356}$ 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

$C_{2}\times C_{2}\times C_{12}$, which has order $48$

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:  $2$
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:   $\frac{5}{1452}a^{5}-\frac{1}{66}a^{4}+\frac{37}{363}a^{3}-\frac{161}{1452}a^{2}+\frac{589}{484}a+\frac{389}{484}$, $\frac{1}{484}a^{5}+\frac{17}{242}a^{3}+\frac{25}{484}a^{2}+\frac{327}{484}a+\frac{581}{484}$ Copy content Toggle raw display
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:  \( 31.7983118742 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 31.7983118742 \cdot 48}{2\cdot\sqrt{24069811311}}\cr\approx \mathstrut & 1.22016501460 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099)
 
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^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099, 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^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099);
 
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^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099);
 
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

$C_6$ (as 6T1):

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 6
The 6 conjugacy class representatives for $C_6$
Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.3.8281.2

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]
 

Sibling algebras

Twin sextic algebra: 3.3.8281.2 $\times$ \(\Q(\sqrt{-39}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{2}$ R ${\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.1.0.1}{1} }^{6}$ R ${\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.6.0.1}{6} }$ ${\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
\(7\) Copy content Toggle raw display 7.6.4.1$x^{6} + 14 x^{3} - 245$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(13\) Copy content Toggle raw display 13.6.5.2$x^{6} + 13$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.39.2t1.a.a$1$ $ 3 \cdot 13 $ \(\Q(\sqrt{-39}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.91.3t1.a.a$1$ $ 7 \cdot 13 $ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
* 1.273.6t1.h.a$1$ $ 3 \cdot 7 \cdot 13 $ 6.0.24069811311.2 $C_6$ (as 6T1) $0$ $-1$
* 1.91.3t1.a.b$1$ $ 7 \cdot 13 $ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
* 1.273.6t1.h.b$1$ $ 3 \cdot 7 \cdot 13 $ 6.0.24069811311.2 $C_6$ (as 6T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.