Properties

Label 12.0.50551744653.1
Degree $12$
Signature $[0, 6]$
Discriminant $50551744653$
Root discriminant \(7.80\)
Ramified primes $3,37$
Class number $1$
Class group trivial
Galois group $C_6\wr C_2$ (as 12T42)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 11*x^10 - 21*x^9 + 32*x^8 - 40*x^7 + 45*x^6 - 46*x^5 + 40*x^4 - 26*x^3 + 12*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^12 - 4*y^11 + 11*y^10 - 21*y^9 + 32*y^8 - 40*y^7 + 45*y^6 - 46*y^5 + 40*y^4 - 26*y^3 + 12*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 11*x^10 - 21*x^9 + 32*x^8 - 40*x^7 + 45*x^6 - 46*x^5 + 40*x^4 - 26*x^3 + 12*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 + 11*x^10 - 21*x^9 + 32*x^8 - 40*x^7 + 45*x^6 - 46*x^5 + 40*x^4 - 26*x^3 + 12*x^2 - 4*x + 1)
 

\( x^{12} - 4 x^{11} + 11 x^{10} - 21 x^{9} + 32 x^{8} - 40 x^{7} + 45 x^{6} - 46 x^{5} + 40 x^{4} - 26 x^{3} + 12 x^{2} - 4 x + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(50551744653\) \(\medspace = 3^{6}\cdot 37^{5}\) 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:  \(7.80\)
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:  $3^{1/2}37^{5/6}\approx 35.10713565663159$
Ramified primes:   \(3\), \(37\) 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:  \(\Q(\sqrt{37}) \)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{37}a^{11}-\frac{16}{37}a^{10}+\frac{18}{37}a^{9}-\frac{15}{37}a^{8}-\frac{10}{37}a^{7}+\frac{6}{37}a^{6}+\frac{10}{37}a^{5}-\frac{18}{37}a^{4}-\frac{3}{37}a^{3}+\frac{10}{37}a^{2}+\frac{3}{37}a-\frac{3}{37}$ 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

Trivial group, which has order $1$

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:   \( -\frac{43}{37} a^{11} + \frac{133}{37} a^{10} - \frac{330}{37} a^{9} + \frac{534}{37} a^{8} - \frac{717}{37} a^{7} + \frac{778}{37} a^{6} - \frac{837}{37} a^{5} + \frac{774}{37} a^{4} - \frac{537}{37} a^{3} + \frac{162}{37} a^{2} - \frac{18}{37} a + \frac{18}{37} \)  (order $6$) 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:   $\frac{48}{37}a^{11}-\frac{176}{37}a^{10}+\frac{457}{37}a^{9}-\frac{831}{37}a^{8}+\frac{1185}{37}a^{7}-\frac{1414}{37}a^{6}+\frac{1516}{37}a^{5}-\frac{1493}{37}a^{4}+\frac{1188}{37}a^{3}-\frac{630}{37}a^{2}+\frac{218}{37}a-\frac{33}{37}$, $\frac{19}{37}a^{11}-\frac{82}{37}a^{10}+\frac{231}{37}a^{9}-\frac{433}{37}a^{8}+\frac{661}{37}a^{7}-\frac{774}{37}a^{6}+\frac{856}{37}a^{5}-\frac{823}{37}a^{4}+\frac{720}{37}a^{3}-\frac{402}{37}a^{2}+\frac{94}{37}a+\frac{17}{37}$, $\frac{106}{37}a^{11}-\frac{364}{37}a^{10}+\frac{946}{37}a^{9}-\frac{1664}{37}a^{8}+\frac{2381}{37}a^{7}-\frac{2805}{37}a^{6}+\frac{3058}{37}a^{5}-\frac{3018}{37}a^{4}+\frac{2383}{37}a^{3}-\frac{1308}{37}a^{2}+\frac{503}{37}a-\frac{133}{37}$, $a^{11}-3a^{10}+8a^{9}-13a^{8}+19a^{7}-21a^{6}+24a^{5}-22a^{4}+18a^{3}-8a^{2}+3a$, $\frac{4}{37}a^{11}-\frac{64}{37}a^{10}+\frac{146}{37}a^{9}-\frac{356}{37}a^{8}+\frac{478}{37}a^{7}-\frac{642}{37}a^{6}+\frac{632}{37}a^{5}-\frac{701}{37}a^{4}+\frac{580}{37}a^{3}-\frac{330}{37}a^{2}+\frac{86}{37}a-\frac{12}{37}$ 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:  \( 3.95685086216 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3.95685086216 \cdot 1}{6\cdot\sqrt{50551744653}}\cr\approx \mathstrut & 0.180471890197 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 11*x^10 - 21*x^9 + 32*x^8 - 40*x^7 + 45*x^6 - 46*x^5 + 40*x^4 - 26*x^3 + 12*x^2 - 4*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^12 - 4*x^11 + 11*x^10 - 21*x^9 + 32*x^8 - 40*x^7 + 45*x^6 - 46*x^5 + 40*x^4 - 26*x^3 + 12*x^2 - 4*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^12 - 4*x^11 + 11*x^10 - 21*x^9 + 32*x^8 - 40*x^7 + 45*x^6 - 46*x^5 + 40*x^4 - 26*x^3 + 12*x^2 - 4*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^12 - 4*x^11 + 11*x^10 - 21*x^9 + 32*x^8 - 40*x^7 + 45*x^6 - 46*x^5 + 40*x^4 - 26*x^3 + 12*x^2 - 4*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

$C_6\wr C_2$ (as 12T42):

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 72
The 27 conjugacy class representatives for $C_6\wr C_2$
Character table for $C_6\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 6.0.36963.1

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

Degree 12 sibling: data not computed
Degree 24 sibling: data not computed
Degree 36 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 ${\href{/padicField/2.12.0.1}{12} }$ R ${\href{/padicField/5.12.0.1}{12} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.4.0.1}{4} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ R ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }$

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 3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$[\ ]$
37.6.5.1$x^{6} + 37$$6$$1$$5$$C_6$$[\ ]_{6}$