Properties

Label 12.0.1093889542148001.2
Degree $12$
Signature $[0, 6]$
Discriminant $1.094\times 10^{15}$
Root discriminant \(17.92\)
Ramified primes $3,71$
Class number $1$
Class group trivial
Galois group $A_4\times C_2$ (as 12T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 15*x^10 - 18*x^9 - 3*x^8 + 24*x^7 + 3*x^6 + 6*x^5 + 33*x^4 + 152*x^3 + 273*x^2 + 198*x + 51)
 
gp: K = bnfinit(y^12 - 6*y^11 + 15*y^10 - 18*y^9 - 3*y^8 + 24*y^7 + 3*y^6 + 6*y^5 + 33*y^4 + 152*y^3 + 273*y^2 + 198*y + 51, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 6*x^11 + 15*x^10 - 18*x^9 - 3*x^8 + 24*x^7 + 3*x^6 + 6*x^5 + 33*x^4 + 152*x^3 + 273*x^2 + 198*x + 51);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 6*x^11 + 15*x^10 - 18*x^9 - 3*x^8 + 24*x^7 + 3*x^6 + 6*x^5 + 33*x^4 + 152*x^3 + 273*x^2 + 198*x + 51)
 

\( x^{12} - 6 x^{11} + 15 x^{10} - 18 x^{9} - 3 x^{8} + 24 x^{7} + 3 x^{6} + 6 x^{5} + 33 x^{4} + 152 x^{3} + 273 x^{2} + 198 x + 51 \) 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:   \(1093889542148001\) \(\medspace = 3^{16}\cdot 71^{4}\) 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:  \(17.92\)
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:  $3^{4/3}71^{1/2}\approx 36.45783266912841$
Ramified primes:   \(3\), \(71\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  8.0.166726039041.1$^{8}$, 12.0.1093889542148001.2$^{24}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{3}a^{4}-\frac{1}{2}$, $\frac{1}{18}a^{8}+\frac{1}{18}a^{7}+\frac{1}{18}a^{6}-\frac{7}{18}a^{5}+\frac{5}{18}a^{4}+\frac{5}{18}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{54}a^{9}-\frac{1}{18}a^{7}+\frac{2}{27}a^{6}-\frac{1}{9}a^{5}+\frac{2}{9}a^{4}-\frac{4}{27}a^{3}+\frac{5}{18}a^{2}+\frac{1}{3}a+\frac{7}{18}$, $\frac{1}{162}a^{10}-\frac{1}{162}a^{9}-\frac{1}{54}a^{8}+\frac{7}{162}a^{7}-\frac{5}{81}a^{6}-\frac{2}{9}a^{5}-\frac{10}{81}a^{4}-\frac{31}{162}a^{3}+\frac{19}{54}a^{2}+\frac{19}{54}a-\frac{7}{54}$, $\frac{1}{486}a^{11}+\frac{1}{486}a^{10}+\frac{2}{243}a^{9}+\frac{1}{486}a^{8}-\frac{23}{486}a^{7}-\frac{10}{243}a^{6}-\frac{73}{243}a^{5}-\frac{125}{486}a^{4}-\frac{239}{486}a^{3}-\frac{10}{27}a^{2}-\frac{77}{162}a-\frac{5}{162}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

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:   \( -1 \)  (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:   $\frac{1}{9}a^{11}-\frac{115}{162}a^{10}+\frac{313}{162}a^{9}-\frac{143}{54}a^{8}+\frac{25}{81}a^{7}+\frac{529}{162}a^{6}-\frac{29}{18}a^{5}+\frac{257}{162}a^{4}+\frac{212}{81}a^{3}+\frac{451}{27}a^{2}+\frac{667}{27}a+\frac{607}{54}$, $\frac{29}{486}a^{11}-\frac{95}{243}a^{10}+\frac{533}{486}a^{9}-\frac{386}{243}a^{8}+\frac{203}{486}a^{7}+\frac{863}{486}a^{6}-\frac{697}{486}a^{5}+\frac{310}{243}a^{4}+\frac{785}{486}a^{3}+\frac{194}{27}a^{2}+\frac{1841}{162}a+\frac{415}{81}$, $\frac{113}{486}a^{11}-\frac{392}{243}a^{10}+\frac{2411}{486}a^{9}-\frac{2081}{243}a^{8}+\frac{1582}{243}a^{7}+\frac{214}{243}a^{6}-\frac{311}{243}a^{5}+\frac{1331}{486}a^{4}+\frac{1174}{243}a^{3}+\frac{553}{18}a^{2}+\frac{2707}{81}a+\frac{823}{81}$, $\frac{23}{486}a^{11}-\frac{80}{243}a^{10}+\frac{473}{486}a^{9}-\frac{362}{243}a^{8}+\frac{175}{243}a^{7}+\frac{190}{243}a^{6}-\frac{32}{243}a^{5}-\frac{79}{486}a^{4}+\frac{187}{243}a^{3}+\frac{277}{54}a^{2}+\frac{562}{81}a+\frac{223}{81}$, $\frac{205}{486}a^{11}-\frac{658}{243}a^{10}+\frac{3601}{486}a^{9}-\frac{2503}{243}a^{8}+\frac{811}{486}a^{7}+\frac{5971}{486}a^{6}-\frac{3497}{486}a^{5}+\frac{1763}{243}a^{4}+\frac{5221}{486}a^{3}+\frac{1595}{27}a^{2}+\frac{14365}{162}a+\frac{3416}{81}$ 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:  \( 1519.80421832 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1519.80421832 \cdot 1}{2\cdot\sqrt{1093889542148001}}\cr\approx \mathstrut & 1.41367678071 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 15*x^10 - 18*x^9 - 3*x^8 + 24*x^7 + 3*x^6 + 6*x^5 + 33*x^4 + 152*x^3 + 273*x^2 + 198*x + 51)
 
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 - 6*x^11 + 15*x^10 - 18*x^9 - 3*x^8 + 24*x^7 + 3*x^6 + 6*x^5 + 33*x^4 + 152*x^3 + 273*x^2 + 198*x + 51, 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 - 6*x^11 + 15*x^10 - 18*x^9 - 3*x^8 + 24*x^7 + 3*x^6 + 6*x^5 + 33*x^4 + 152*x^3 + 273*x^2 + 198*x + 51);
 
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 - 6*x^11 + 15*x^10 - 18*x^9 - 3*x^8 + 24*x^7 + 3*x^6 + 6*x^5 + 33*x^4 + 152*x^3 + 273*x^2 + 198*x + 51);
 
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

$C_2\times A_4$ (as 12T6):

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 24
The 8 conjugacy class representatives for $A_4\times C_2$
Character table for $A_4\times C_2$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.0.465831.1 x2, 6.6.33074001.1

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: deg 24
Degree 6 sibling: 6.0.465831.1
Degree 8 sibling: 8.0.166726039041.1
Degree 12 sibling: 12.0.5514297181968073041.1
Minimal sibling: 6.0.465831.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{4}$ R ${\href{/padicField/5.3.0.1}{3} }^{4}$ ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 3.3.4.2$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} + 6 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
\(71\) Copy content Toggle raw display 71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$