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Properties

Label 12.0.74666441181...1616.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{22}\cdot 3^{18}\cdot 11^{16}$
Root discriminant $453.00$
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![214386723, 6368382, 1589841, -2029698, 1081080, -35442, 24013, -7854, 2112, -110, 33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723)
 
gp: K = bnfinit(x^12 - 6*x^11 + 33*x^10 - 110*x^9 + 2112*x^8 - 7854*x^7 + 24013*x^6 - 35442*x^5 + 1081080*x^4 - 2029698*x^3 + 1589841*x^2 + 6368382*x + 214386723, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 2112 x^{8} - 7854 x^{7} + 24013 x^{6} - 35442 x^{5} + 1081080 x^{4} - 2029698 x^{3} + 1589841 x^{2} + 6368382 x + 214386723 \)

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:  \(74666441181916948967589849071616=2^{22}\cdot 3^{18}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $453.00$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{36} a^{6} + \frac{1}{36} a^{5} + \frac{1}{36} a^{4} + \frac{1}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{4} a$, $\frac{1}{36} a^{7} - \frac{1}{36} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{432} a^{8} - \frac{1}{72} a^{6} - \frac{1}{108} a^{5} + \frac{1}{144} a^{4} + \frac{5}{36} a^{3} - \frac{7}{24} a^{2} - \frac{1}{12} a + \frac{7}{16}$, $\frac{1}{432} a^{9} - \frac{1}{72} a^{7} - \frac{1}{108} a^{6} + \frac{1}{144} a^{5} - \frac{1}{36} a^{4} + \frac{1}{24} a^{3} - \frac{1}{4} a^{2} + \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{2592} a^{10} - \frac{1}{2592} a^{9} - \frac{1}{864} a^{8} - \frac{11}{1296} a^{7} - \frac{11}{2592} a^{6} + \frac{5}{288} a^{5} - \frac{1}{288} a^{4} - \frac{1}{16} a^{3} + \frac{15}{32} a^{2} - \frac{35}{96} a - \frac{19}{96}$, $\frac{1}{15276831131851677216} a^{11} - \frac{888239361157447}{7638415565925838608} a^{10} - \frac{18719020370717}{3819207782962919304} a^{9} + \frac{10785111057448919}{15276831131851677216} a^{8} - \frac{3347511072410341}{412887327887883168} a^{7} + \frac{15214603753674265}{1909603891481459652} a^{6} + \frac{13097586263873603}{424356420329213256} a^{5} + \frac{805951915828951}{62867617826550112} a^{4} + \frac{91749789750730127}{565808560438951008} a^{3} - \frac{228450352024571}{70726070054868876} a^{2} - \frac{36908562549533587}{282904280219475504} a + \frac{134602296782855953}{565808560438951008}$

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:  \( 39976757877.1 \) (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 ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.22.79$x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{6} + 4 x^{4} + 4$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$3$3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$