↑

Properties

Label 12.0.15685137817...2336.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{16}\cdot 3^{16}\cdot 11^{18}$
Root discriminant $397.76$
Ramified primes $2, 3, 11$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $\PSL(2,11)$ (as 12T179)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19395024, 777312, 9291216, -9843328, 4900368, -535920, -106656, 27984, 9768, -792, -132, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 132*x^10 - 792*x^9 + 9768*x^8 + 27984*x^7 - 106656*x^6 - 535920*x^5 + 4900368*x^4 - 9843328*x^3 + 9291216*x^2 + 777312*x + 19395024)
 
gp: K = bnfinit(x^12 - 132*x^10 - 792*x^9 + 9768*x^8 + 27984*x^7 - 106656*x^6 - 535920*x^5 + 4900368*x^4 - 9843328*x^3 + 9291216*x^2 + 777312*x + 19395024, 1)
 

Normalized defining polynomial

\( x^{12} - 132 x^{10} - 792 x^{9} + 9768 x^{8} + 27984 x^{7} - 106656 x^{6} - 535920 x^{5} + 4900368 x^{4} - 9843328 x^{3} + 9291216 x^{2} + 777312 x + 19395024 \)

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:  \(15685137817729081293538839822336=2^{16}\cdot 3^{16}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $397.76$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{12} a^{6} + \frac{1}{6} a^{3}$, $\frac{1}{72} a^{7} + \frac{1}{36} a^{6} - \frac{1}{4} a^{5} - \frac{1}{18} a^{4} + \frac{1}{18} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{216} a^{8} + \frac{1}{216} a^{7} - \frac{1}{108} a^{6} + \frac{25}{108} a^{5} + \frac{1}{27} a^{4} - \frac{2}{27} a^{3} - \frac{1}{2} a^{2} - \frac{1}{18} a + \frac{1}{9}$, $\frac{1}{1944} a^{9} + \frac{1}{648} a^{8} + \frac{1}{216} a^{7} + \frac{8}{243} a^{6} + \frac{1}{36} a^{5} + \frac{11}{54} a^{4} + \frac{41}{243} a^{3} - \frac{1}{162} a^{2} - \frac{5}{18} a + \frac{38}{81}$, $\frac{1}{104976} a^{10} + \frac{1}{52488} a^{9} + \frac{19}{17496} a^{8} - \frac{5}{6561} a^{7} + \frac{551}{26244} a^{6} - \frac{305}{1458} a^{5} + \frac{1657}{26244} a^{4} + \frac{1511}{6561} a^{3} + \frac{556}{2187} a^{2} - \frac{422}{2187} a - \frac{775}{2187}$, $\frac{1}{3108900519915085003871376} a^{11} + \frac{106745877256920073}{777225129978771250967844} a^{10} + \frac{48381577304868046463}{388612564989385625483922} a^{9} - \frac{690103903153147071587}{777225129978771250967844} a^{8} + \frac{5597241757989075801431}{1554450259957542501935688} a^{7} + \frac{8028064642010038866025}{777225129978771250967844} a^{6} + \frac{577687861223841626663}{388612564989385625483922} a^{5} + \frac{26165736569666532720592}{194306282494692812741961} a^{4} + \frac{40213074915982694512102}{194306282494692812741961} a^{3} + \frac{10529688161591430195805}{21589586943854756971329} a^{2} - \frac{21269952597022848713821}{129537521663128541827974} a - \frac{22570719317164286754362}{64768760831564270913987}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 61492581932.6 \) (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 ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$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$Trivial$[\ ]$
11.11.18.2$x^{11} + 77 x^{8} + 11$$11$$1$$18$$C_{11}:C_5$$[9/5]_{5}$