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Properties

Label 12.0.40153952813...8016.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{16}\cdot 11^{18}$
Root discriminant $631.41$
Ramified primes $2, 3, 11$
Class number $12$ (GRH)
Class group $[2, 6]$ (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![806177988, -358726104, 227670696, -49073244, 15987609, -4461996, 625328, -123420, 23958, -660, 264, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 264*x^10 - 660*x^9 + 23958*x^8 - 123420*x^7 + 625328*x^6 - 4461996*x^5 + 15987609*x^4 - 49073244*x^3 + 227670696*x^2 - 358726104*x + 806177988)
 
gp: K = bnfinit(x^12 + 264*x^10 - 660*x^9 + 23958*x^8 - 123420*x^7 + 625328*x^6 - 4461996*x^5 + 15987609*x^4 - 49073244*x^3 + 227670696*x^2 - 358726104*x + 806177988, 1)
 

Normalized defining polynomial

\( x^{12} + 264 x^{10} - 660 x^{9} + 23958 x^{8} - 123420 x^{7} + 625328 x^{6} - 4461996 x^{5} + 15987609 x^{4} - 49073244 x^{3} + 227670696 x^{2} - 358726104 x + 806177988 \)

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:  \(4015395281338644811145942994518016=2^{24}\cdot 3^{16}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $631.41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} - \frac{1}{3} a^{5} - \frac{5}{12} a^{4} - \frac{1}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{24} a^{6} - \frac{1}{3} a^{5} + \frac{7}{24} a^{4} - \frac{1}{12} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{650752362344662619596097292434640787128} a^{11} + \frac{59397489534850877234814163257704267}{4929942138974716815121949185110915054} a^{10} + \frac{51629359918302927895062569991297029}{6573256185299622420162598913481220072} a^{9} - \frac{666111363045535108869662990026147}{3286628092649811210081299456740610036} a^{8} + \frac{474651497878081477145428408902339301}{19719768555898867260487796740443660216} a^{7} - \frac{7229961234818320896817233958385163}{821657023162452802520324864185152509} a^{6} - \frac{21427514742913759696755847963324204559}{59159305667696601781463390221330980648} a^{5} + \frac{1542606530260919965649216737937589203}{9859884277949433630243898370221830108} a^{4} - \frac{282954836548001322645451182395170939}{1643314046324905605040649728370305018} a^{3} - \frac{588641119543118578391459332757286427}{4929942138974716815121949185110915054} a^{2} - \frac{310619990956629840803556740424895781}{1643314046324905605040649728370305018} a + \frac{123763066775147050004303339162884757}{821657023162452802520324864185152509}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 3119906243710 \) (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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.18.2$x^{11} + 77 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$