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Properties

Label 12.12.1115343245...1856.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{24}\cdot 3^{8}\cdot 11^{20}\cdot 197^{4}$
Root discriminant $2634.01$
Ramified primes $2, 3, 11, 197$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $M_{12}$ (as 12T295)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![632319724608, 19295198208, -114286943232, -247374336, 4128059232, 0, -56221440, 0, 346060, 0, -968, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608)
 
gp: K = bnfinit(x^12 - 968*x^10 + 346060*x^8 - 56221440*x^6 + 4128059232*x^4 - 247374336*x^3 - 114286943232*x^2 + 19295198208*x + 632319724608, 1)
 

Normalized defining polynomial

\( x^{12} - 968 x^{10} + 346060 x^{8} - 56221440 x^{6} + 4128059232 x^{4} - 247374336 x^{3} - 114286943232 x^{2} + 19295198208 x + 632319724608 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(111534324540107369492059712937081339641856=2^{24}\cdot 3^{8}\cdot 11^{20}\cdot 197^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2634.01$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 11, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{22} a^{3}$, $\frac{1}{44} a^{4}$, $\frac{1}{484} a^{5}$, $\frac{1}{968} a^{6}$, $\frac{1}{21296} a^{7} - \frac{1}{968} a^{5} - \frac{1}{2} a$, $\frac{1}{276848} a^{8} - \frac{5}{276848} a^{7} - \frac{3}{12584} a^{6} + \frac{1}{1144} a^{5} - \frac{1}{572} a^{4} + \frac{5}{286} a^{3} + \frac{3}{26} a^{2} + \frac{1}{26} a$, $\frac{1}{6090656} a^{9} + \frac{1}{12584} a^{6} - \frac{5}{12584} a^{5} + \frac{1}{286} a^{4} - \frac{3}{572} a^{3} - \frac{2}{13} a^{2} - \frac{11}{26} a$, $\frac{1}{103541152} a^{10} + \frac{1}{12942644} a^{9} - \frac{1}{2353208} a^{8} - \frac{5}{276848} a^{7} + \frac{6}{26741} a^{6} - \frac{109}{213928} a^{5} - \frac{3}{286} a^{4} - \frac{46}{2431} a^{3} + \frac{21}{221} a^{2} - \frac{3}{26} a - \frac{5}{17}$, $\frac{1}{61739186989955904} a^{11} - \frac{6112003}{2806326681361632} a^{10} + \frac{72331165}{1403163340680816} a^{9} - \frac{33810641}{127560303698256} a^{8} - \frac{12724963}{3270777017904} a^{7} + \frac{178788053}{966365937108} a^{6} - \frac{153608237}{644243958072} a^{5} + \frac{6829369}{2662165116} a^{4} + \frac{90450011}{9761272092} a^{3} - \frac{499045985}{1996623837} a^{2} - \frac{114366082}{1996623837} a + \frac{22247857}{153586449}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 328442425589000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{12}$ (as 12T295):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 95040
The 15 conjugacy class representatives for $M_{12}$
Character table for $M_{12}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 12 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.24.155$x^{12} - 12 x^{11} + 16 x^{10} + 4 x^{9} + 16 x^{7} - 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 8$$4$$3$$24$12T60$[2, 2, 2, 3, 3]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.20.8$x^{11} - 11 x^{10} + 1221$$11$$1$$20$$C_{11}$$[2]$
$197$197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197.4.2.1$x^{4} + 985 x^{2} + 349281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$