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Properties

Label 12.0.81018273851...5776.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{16}\cdot 11^{16}$
Root discriminant $211.70$
Ramified primes $2, 3, 11$
Class number $12$ (GRH)
Class group $[12]$ (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![63954000, -46951920, 1638648, 2610564, 1261161, -4092, 28952, -1716, 726, -132, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 132*x^9 + 726*x^8 - 1716*x^7 + 28952*x^6 - 4092*x^5 + 1261161*x^4 + 2610564*x^3 + 1638648*x^2 - 46951920*x + 63954000)
 
gp: K = bnfinit(x^12 - 132*x^9 + 726*x^8 - 1716*x^7 + 28952*x^6 - 4092*x^5 + 1261161*x^4 + 2610564*x^3 + 1638648*x^2 - 46951920*x + 63954000, 1)
 

Normalized defining polynomial

\( x^{12} - 132 x^{9} + 726 x^{8} - 1716 x^{7} + 28952 x^{6} - 4092 x^{5} + 1261161 x^{4} + 2610564 x^{3} + 1638648 x^{2} - 46951920 x + 63954000 \)

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:  \(8101827385190641164017995776=2^{12}\cdot 3^{16}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $211.70$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{88} a^{7} - \frac{1}{44} a^{6} - \frac{1}{44} a^{5} + \frac{9}{44} a^{4} + \frac{1}{88} a^{3} - \frac{7}{22} a^{2} + \frac{1}{22} a$, $\frac{1}{528} a^{8} - \frac{1}{528} a^{7} - \frac{13}{264} a^{6} - \frac{7}{132} a^{5} - \frac{113}{528} a^{4} + \frac{17}{528} a^{3} + \frac{7}{88} a^{2} + \frac{15}{44} a - \frac{1}{2}$, $\frac{1}{3168} a^{9} + \frac{5}{1056} a^{7} + \frac{7}{176} a^{6} + \frac{13}{1056} a^{5} - \frac{5}{24} a^{4} + \frac{365}{3168} a^{3} + \frac{193}{528} a^{2} - \frac{27}{88} a - \frac{5}{12}$, $\frac{1}{19008} a^{10} - \frac{1}{6336} a^{9} - \frac{1}{2112} a^{8} - \frac{1}{6336} a^{7} + \frac{35}{6336} a^{6} - \frac{227}{6336} a^{5} + \frac{355}{1728} a^{4} + \frac{641}{6336} a^{3} + \frac{65}{352} a^{2} + \frac{265}{1584} a - \frac{3}{8}$, $\frac{1}{6370665135521681434682880} a^{11} - \frac{15995122426747871395}{1274133027104336286936576} a^{10} + \frac{55597357834832403439}{424711009034778762312192} a^{9} + \frac{2489602613541323471}{2123555045173893811560960} a^{8} + \frac{3310458389099803864657}{2123555045173893811560960} a^{7} - \frac{2766157079835071355419}{64350152884057388229120} a^{6} - \frac{119969859245117943370633}{6370665135521681434682880} a^{5} + \frac{458717347237749940870283}{6370665135521681434682880} a^{4} - \frac{47800348626846107850757}{530888761293473452890240} a^{3} + \frac{47071742294883812147111}{265444380646736726445120} a^{2} + \frac{8124554860203782816363}{66361095161684181611280} a + \frac{105954835395923791959}{268125637016905784288}$

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

Class group and class number

$C_{12}$, 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:  \( 44249324260.0 \) (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 ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[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}$