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Properties

Label 12.0.86063856338...3936.4
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{10}\cdot 11^{18}$
Root discriminant $257.77$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![136284024, -102293600, 49941848, -19653392, 5076852, -897336, 145860, -14652, 1848, -528, 88, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 88*x^10 - 528*x^9 + 1848*x^8 - 14652*x^7 + 145860*x^6 - 897336*x^5 + 5076852*x^4 - 19653392*x^3 + 49941848*x^2 - 102293600*x + 136284024)
 
gp: K = bnfinit(x^12 - 4*x^11 + 88*x^10 - 528*x^9 + 1848*x^8 - 14652*x^7 + 145860*x^6 - 897336*x^5 + 5076852*x^4 - 19653392*x^3 + 49941848*x^2 - 102293600*x + 136284024, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 88 x^{10} - 528 x^{9} + 1848 x^{8} - 14652 x^{7} + 145860 x^{6} - 897336 x^{5} + 5076852 x^{4} - 19653392 x^{3} + 49941848 x^{2} - 102293600 x + 136284024 \)

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

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(86063856338705521500898983936=2^{18}\cdot 3^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $257.77$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{42} a^{7} - \frac{1}{7} a^{6} - \frac{3}{14} a^{5} - \frac{5}{21} a^{4} - \frac{3}{7} a^{3} + \frac{5}{21} a - \frac{2}{7}$, $\frac{1}{84} a^{8} + \frac{3}{14} a^{6} + \frac{5}{21} a^{5} + \frac{1}{14} a^{4} - \frac{2}{7} a^{3} - \frac{8}{21} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{84} a^{9} + \frac{1}{42} a^{6} - \frac{1}{7} a^{4} + \frac{10}{21} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7}$, $\frac{1}{84} a^{10} + \frac{1}{7} a^{6} + \frac{1}{14} a^{5} + \frac{3}{14} a^{4} + \frac{1}{3} a + \frac{2}{7}$, $\frac{1}{21753445813456454143547058067524} a^{11} - \frac{61486830408669675750391995037}{10876722906728227071773529033762} a^{10} + \frac{3301695170618477480434581751}{1553817558104032438824789861966} a^{9} + \frac{63207956768607941882121580885}{10876722906728227071773529033762} a^{8} + \frac{71804085227713261720092572963}{10876722906728227071773529033762} a^{7} + \frac{1208410691192502143529774105160}{5438361453364113535886764516881} a^{6} + \frac{870057041197709710263787710631}{5438361453364113535886764516881} a^{5} + \frac{164728168508437990347835581745}{1553817558104032438824789861966} a^{4} - \frac{2425061610944913974951600292196}{5438361453364113535886764516881} a^{3} - \frac{719136253636448239646985884246}{1812787151121371178628921505627} a^{2} + \frac{621939433410051863154398538113}{5438361453364113535886764516881} a + \frac{679676627982849938046953569842}{1812787151121371178628921505627}$

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:  $5$
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:  \( 7036301978.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,11)$ (as 12T179):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 660
The 8 conjugacy class representatives for $\PSL(2,11)$
Character table for $\PSL(2,11)$

Intermediate fields

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

Sibling fields

Degree 11 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.12.18.59$x^{12} - 2 x^{11} + 6 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} - 4 x^{6} - 8 x^{3} + 16 x^{2} - 8$$4$$3$$18$$A_4$$[2, 2]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.18.1$x^{11} + 66 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$