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Properties

Label 12.8.19839538883...7856.1
Degree $12$
Signature $[8, 2]$
Discriminant $2^{14}\cdot 3^{10}\cdot 7^{8}\cdot 596427121^{2}$
Root discriminant $595.38$
Ramified primes $2, 3, 7, 596427121$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{12}$ (as 12T300)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5751, 5400, 83862, 7200, -121905, -36720, 33012, 14880, -345, -1000, -138, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 138*x^10 - 1000*x^9 - 345*x^8 + 14880*x^7 + 33012*x^6 - 36720*x^5 - 121905*x^4 + 7200*x^3 + 83862*x^2 + 5400*x - 5751)
 
gp: K = bnfinit(x^12 - 138*x^10 - 1000*x^9 - 345*x^8 + 14880*x^7 + 33012*x^6 - 36720*x^5 - 121905*x^4 + 7200*x^3 + 83862*x^2 + 5400*x - 5751, 1)
 

Normalized defining polynomial

\( x^{12} - 138 x^{10} - 1000 x^{9} - 345 x^{8} + 14880 x^{7} + 33012 x^{6} - 36720 x^{5} - 121905 x^{4} + 7200 x^{3} + 83862 x^{2} + 5400 x - 5751 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1983953888340397738015626127097856=2^{14}\cdot 3^{10}\cdot 7^{8}\cdot 596427121^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $595.38$
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:  $2, 3, 7, 596427121$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{336} a^{6} - \frac{1}{28} a^{5} + \frac{3}{112} a^{4} - \frac{1}{21} a^{3} + \frac{3}{16} a^{2} - \frac{11}{28} a + \frac{29}{112}$, $\frac{1}{672} a^{7} - \frac{1}{672} a^{6} - \frac{13}{224} a^{5} - \frac{1}{672} a^{4} + \frac{55}{672} a^{3} + \frac{19}{224} a^{2} - \frac{13}{32} a + \frac{67}{224}$, $\frac{1}{1344} a^{8} - \frac{1}{84} a^{5} - \frac{15}{224} a^{4} + \frac{3}{28} a^{3} + \frac{3}{14} a^{2} + \frac{1}{7} a - \frac{173}{448}$, $\frac{1}{8064} a^{9} - \frac{1}{2688} a^{8} + \frac{1}{1008} a^{6} + \frac{19}{448} a^{5} - \frac{3}{64} a^{4} + \frac{1}{56} a^{3} + \frac{3}{16} a^{2} + \frac{39}{128} a - \frac{155}{896}$, $\frac{1}{16128} a^{10} + \frac{1}{5376} a^{8} + \frac{1}{2016} a^{7} - \frac{1}{896} a^{6} - \frac{41}{672} a^{5} - \frac{83}{896} a^{4} + \frac{73}{672} a^{3} + \frac{265}{1792} a^{2} - \frac{15}{32} a + \frac{59}{1792}$, $\frac{1}{677376} a^{11} - \frac{1}{75264} a^{10} + \frac{1}{25088} a^{9} - \frac{235}{677376} a^{8} + \frac{1}{112896} a^{7} - \frac{5}{5376} a^{6} + \frac{109}{1792} a^{5} + \frac{1775}{37632} a^{4} + \frac{1489}{75264} a^{3} + \frac{10919}{75264} a^{2} + \frac{1513}{25088} a + \frac{9991}{25088}$

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:  $9$
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:  \( 35339558705200 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{12}$ (as 12T300):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 239500800
The 43 conjugacy class representatives for $A_{12}$
Character table for $A_{12}$ is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/11.11.0.1}{11} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
7.5.4.1$x^{5} - 7$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
596427121Data not computed