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Properties

Label 12.0.9845005879332009.3
Degree $12$
Signature $[0, 6]$
Discriminant $9.845\times 10^{15}$
Root discriminant \(21.52\)
Ramified primes $3,71$
Class number $4$
Class group [4]
Galois group $A_4 \times C_2$ (as 12T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249)
 
gp: K = bnfinit(y^12 + 18*y^10 - 32*y^9 + 267*y^8 - 324*y^7 + 1168*y^6 - 384*y^5 + 2799*y^4 - 228*y^3 + 4545*y^2 + 2052*y + 3249, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249)
 

\( x^{12} + 18 x^{10} - 32 x^{9} + 267 x^{8} - 324 x^{7} + 1168 x^{6} - 384 x^{5} + 2799 x^{4} + \cdots + 3249 \) 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:   \(9845005879332009\) \(\medspace = 3^{18}\cdot 71^{4}\) 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:  \(21.52\)
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:  $3^{3/2}71^{1/2}\approx 43.78355855797927$
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:  \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})\)$^{3}$, 8.0.1500534351369.1$^{4}$, 12.0.9845005879332009.3$^{12}$, deg 24$^{12}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{7}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{18}a^{8}+\frac{1}{18}a^{7}+\frac{1}{18}a^{6}-\frac{7}{18}a^{5}+\frac{5}{18}a^{4}-\frac{2}{9}a^{3}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{18}a^{9}+\frac{1}{18}a^{6}-\frac{1}{3}a^{5}+\frac{2}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}$, $\frac{1}{18}a^{10}+\frac{1}{18}a^{7}+\frac{2}{9}a^{4}+\frac{1}{6}a$, $\frac{1}{7543105483554}a^{11}+\frac{54890755}{22055863987}a^{10}-\frac{29683722487}{3771552741777}a^{9}+\frac{48708538721}{2514368494518}a^{8}+\frac{46381138840}{3771552741777}a^{7}-\frac{103968134609}{2514368494518}a^{6}+\frac{1591507266322}{3771552741777}a^{5}+\frac{3439199298349}{7543105483554}a^{4}-\frac{886023392771}{3771552741777}a^{3}-\frac{5741902536}{22055863987}a^{2}+\frac{1000086484321}{2514368494518}a+\frac{54324281147}{132335183922}$ 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

$C_{4}$, which has order $4$

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:   \( \frac{4102188952}{3771552741777} a^{11} - \frac{66531530}{66167591961} a^{10} + \frac{16329264793}{1257184247259} a^{9} - \frac{223429805149}{3771552741777} a^{8} + \frac{1680842399177}{7543105483554} a^{7} - \frac{3342446322127}{7543105483554} a^{6} + \frac{388918882651}{838122831506} a^{5} - \frac{4364133541997}{7543105483554} a^{4} + \frac{4443930623839}{7543105483554} a^{3} - \frac{86127830828}{66167591961} a^{2} - \frac{191409735199}{1257184247259} a + \frac{14420559703}{132335183922} \)  (order $18$) 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:   $\frac{1819518389}{2514368494518}a^{11}-\frac{38584684}{22055863987}a^{10}+\frac{9754595642}{1257184247259}a^{9}-\frac{469776076877}{7543105483554}a^{8}+\frac{632018934989}{3771552741777}a^{7}-\frac{4465225268849}{7543105483554}a^{6}+\frac{2286482945881}{3771552741777}a^{5}-\frac{13725514460287}{7543105483554}a^{4}+\frac{3925262807548}{3771552741777}a^{3}-\frac{83980801139}{22055863987}a^{2}+\frac{1334014430465}{2514368494518}a-\frac{259208115403}{132335183922}$, $\frac{2499617569}{7543105483554}a^{11}+\frac{241221076}{198502775883}a^{10}+\frac{3455435263}{838122831506}a^{9}-\frac{6630582839}{2514368494518}a^{8}-\frac{11611063453}{3771552741777}a^{7}+\frac{428530507685}{7543105483554}a^{6}-\frac{804627898328}{3771552741777}a^{5}-\frac{143449239417}{419061415753}a^{4}-\frac{1250734126228}{3771552741777}a^{3}-\frac{129118937531}{132335183922}a^{2}-\frac{604361239162}{419061415753}a-\frac{331376723369}{132335183922}$, $\frac{9946082005}{2514368494518}a^{11}+\frac{3276934771}{397005551766}a^{10}+\frac{172564567003}{2514368494518}a^{9}-\frac{210536457349}{7543105483554}a^{8}+\frac{1587872290927}{2514368494518}a^{7}+\frac{1046547032615}{7543105483554}a^{6}+\frac{5960877532019}{3771552741777}a^{5}+\frac{5669318742571}{3771552741777}a^{4}+\frac{10445528214005}{3771552741777}a^{3}+\frac{146384766059}{44111727974}a^{2}+\frac{6798134882669}{2514368494518}a+\frac{214311504697}{132335183922}$, $\frac{2570915044}{3771552741777}a^{11}-\frac{1037951705}{397005551766}a^{10}+\frac{968637773}{838122831506}a^{9}-\frac{27067780119}{419061415753}a^{8}+\frac{465238161211}{3771552741777}a^{7}-\frac{2819841734363}{7543105483554}a^{6}-\frac{3379247823839}{7543105483554}a^{5}+\frac{992028453730}{1257184247259}a^{4}-\frac{14957613581857}{7543105483554}a^{3}+\frac{21378803435}{132335183922}a^{2}-\frac{334816641307}{419061415753}a+\frac{260035349909}{132335183922}$, $\frac{182285582}{66167591961}a^{11}+\frac{261490154}{22055863987}a^{10}+\frac{8410909279}{132335183922}a^{9}+\frac{6167744393}{66167591961}a^{8}+\frac{9885617766}{22055863987}a^{7}+\frac{46111520417}{44111727974}a^{6}+\frac{112658611430}{66167591961}a^{5}+\frac{75871618207}{22055863987}a^{4}+\frac{310744368103}{66167591961}a^{3}+\frac{173229374057}{22055863987}a^{2}+\frac{122775524376}{22055863987}a+\frac{171372272401}{44111727974}$ 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:  \( 3039.60843665 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 3039.60843665 \cdot 4}{18\cdot\sqrt{9845005879332009}}\cr\approx \mathstrut & 0.418867194287 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249)
 
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 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249, 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 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249);
 
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 + 18*x^10 - 32*x^9 + 267*x^8 - 324*x^7 + 1168*x^6 - 384*x^5 + 2799*x^4 - 228*x^3 + 4545*x^2 + 2052*x + 3249);
 
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 12T7):

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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 6.0.99222003.2, 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: 6.0.99222003.2
Degree 8 sibling: 8.0.1500534351369.1
Degree 12 sibling: 12.0.49628674637712657369.3
Minimal sibling: 6.0.99222003.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ R ${\href{/padicField/5.6.0.1}{6} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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 3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
\(71\) Copy content Toggle raw display 71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$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$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$