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Properties

Label 12.12.1203064005...0625.1
Degree $12$
Signature $[12, 0]$
Discriminant $3^{28}\cdot 5^{18}\cdot 13^{10}$
Root discriminant $1230.34$
Ramified primes $3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{12}$ (as 12T300)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59049, 3542940, 11868849, 15943230, 10235160, 2519424, -561330, -527796, -138510, -16920, -864, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 864*x^10 - 16920*x^9 - 138510*x^8 - 527796*x^7 - 561330*x^6 + 2519424*x^5 + 10235160*x^4 + 15943230*x^3 + 11868849*x^2 + 3542940*x + 59049)
 
gp: K = bnfinit(x^12 - 864*x^10 - 16920*x^9 - 138510*x^8 - 527796*x^7 - 561330*x^6 + 2519424*x^5 + 10235160*x^4 + 15943230*x^3 + 11868849*x^2 + 3542940*x + 59049, 1)
 

Normalized defining polynomial

\( x^{12} - 864 x^{10} - 16920 x^{9} - 138510 x^{8} - 527796 x^{7} - 561330 x^{6} + 2519424 x^{5} + 10235160 x^{4} + 15943230 x^{3} + 11868849 x^{2} + 3542940 x + 59049 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12030640053495428917361789703369140625=3^{28}\cdot 5^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1230.34$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{27} a^{5}$, $\frac{1}{162} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{486} a^{7} - \frac{1}{27} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{1458} a^{8} - \frac{1}{81} a^{5} - \frac{1}{18} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{4374} a^{9} + \frac{1}{486} a^{6} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{13122} a^{10} + \frac{1}{1458} a^{7} - \frac{1}{54} a^{5} - \frac{1}{18} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{196830} a^{11} + \frac{1}{65610} a^{10} - \frac{1}{4374} a^{8} - \frac{1}{1458} a^{7} - \frac{1}{405} a^{6} + \frac{2}{405} a^{5} - \frac{1}{54} a^{4} - \frac{1}{18} a^{3} - \frac{1}{5} a - \frac{1}{10}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 1657889266260000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{12}$ (as 12T300):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 239500800
The 43 conjugacy class representatives for $A_{12}$
Character table for $A_{12}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$3.3.5.1$x^{3} + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.9.23.33$x^{9} + 6 x^{6} + 3$$9$$1$$23$$(C_3^2:C_3):C_2$$[2, 5/2, 17/6, 19/6]_{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.10.18.23$x^{10} - 20 x^{9} + 105$$10$$1$$18$$(C_5^2 : C_8):C_2$$[17/8, 17/8]_{8}^{2}$
$13$13.5.4.1$x^{5} - 13$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$