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Properties

Label 12.0.33185084969...8496.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{24}\cdot 3^{16}\cdot 11^{16}$
Root discriminant $423.39$
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![74124864, 106043904, 52313184, 2860704, -2099592, -491040, 44000, 22176, 2046, -396, -66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 66*x^10 - 396*x^9 + 2046*x^8 + 22176*x^7 + 44000*x^6 - 491040*x^5 - 2099592*x^4 + 2860704*x^3 + 52313184*x^2 + 106043904*x + 74124864)
 
gp: K = bnfinit(x^12 - 66*x^10 - 396*x^9 + 2046*x^8 + 22176*x^7 + 44000*x^6 - 491040*x^5 - 2099592*x^4 + 2860704*x^3 + 52313184*x^2 + 106043904*x + 74124864, 1)
 

Normalized defining polynomial

\( x^{12} - 66 x^{10} - 396 x^{9} + 2046 x^{8} + 22176 x^{7} + 44000 x^{6} - 491040 x^{5} - 2099592 x^{4} + 2860704 x^{3} + 52313184 x^{2} + 106043904 x + 74124864 \)

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:  \(33185084969740866207817710698496=2^{24}\cdot 3^{16}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $423.39$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{6} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{12} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{7} + \frac{1}{6} a^{3}$, $\frac{1}{24} a^{8} + \frac{1}{12} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{48} a^{9} - \frac{1}{24} a^{5} - \frac{5}{12} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{288} a^{10} + \frac{1}{72} a^{8} - \frac{1}{144} a^{6} - \frac{1}{24} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{27848012191213954263374643552} a^{11} - \frac{811887365230382494630387}{13924006095606977131687321776} a^{10} + \frac{11807115685598607393812429}{13924006095606977131687321776} a^{9} + \frac{16974942174015930147724598}{870250380975436070730457611} a^{8} - \frac{496500176836257659557684261}{13924006095606977131687321776} a^{7} - \frac{174607926513880584303036713}{6962003047803488565843660888} a^{6} - \frac{2345485156434170928406561}{386777947100193809213536716} a^{5} - \frac{1836900988161381180663733}{32231495591682817434461393} a^{4} + \frac{7590781870073205177169859}{128925982366731269737845572} a^{3} + \frac{81464748370418164632270241}{193388973550096904606768358} a^{2} + \frac{25536627918660487157703940}{96694486775048452303384179} a - \frac{667814846299600618001330}{96694486775048452303384179}$

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:  \( 298140290899 \) (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 ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.1$x^{4} + 2 x^{2} + 4 x + 10$$4$$1$$8$$C_2^2$$[2, 3]$
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$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}$