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Properties

Label 12.12.8928969524...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{16}\cdot 5^{9}\cdot 17^{8}$
Root discriminant $55.71$
Ramified primes $2, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22405, 32370, -27645, -39580, 16066, 17254, -5373, -3088, 890, 198, -53, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 53*x^10 + 198*x^9 + 890*x^8 - 3088*x^7 - 5373*x^6 + 17254*x^5 + 16066*x^4 - 39580*x^3 - 27645*x^2 + 32370*x + 22405)
 
gp: K = bnfinit(x^12 - 4*x^11 - 53*x^10 + 198*x^9 + 890*x^8 - 3088*x^7 - 5373*x^6 + 17254*x^5 + 16066*x^4 - 39580*x^3 - 27645*x^2 + 32370*x + 22405, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 53 x^{10} + 198 x^{9} + 890 x^{8} - 3088 x^{7} - 5373 x^{6} + 17254 x^{5} + 16066 x^{4} - 39580 x^{3} - 27645 x^{2} + 32370 x + 22405 \)

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:  \(892896952448000000000=2^{16}\cdot 5^{9}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.71$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{3}{10} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a$, $\frac{1}{844810} a^{10} - \frac{25311}{844810} a^{9} - \frac{4783}{422405} a^{8} + \frac{90937}{844810} a^{7} + \frac{96931}{844810} a^{6} + \frac{37182}{84481} a^{5} + \frac{305827}{844810} a^{4} + \frac{22771}{168962} a^{3} + \frac{32337}{84481} a^{2} - \frac{40105}{168962} a - \frac{77473}{168962}$, $\frac{1}{17914196050} a^{11} - \frac{3393}{8957098025} a^{10} + \frac{33766109}{17914196050} a^{9} + \frac{24335405}{716567842} a^{8} - \frac{320809674}{1791419605} a^{7} + \frac{1003483826}{8957098025} a^{6} - \frac{1976967866}{8957098025} a^{5} - \frac{2689405531}{8957098025} a^{4} - \frac{113415809}{358283921} a^{3} + \frac{1664718789}{3582839210} a^{2} - \frac{1465250667}{3582839210} a - \frac{384189396}{1791419605}$

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:  \( 4137428.77768 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.5780.1 x3, \(\Q(\zeta_{20})^+\), 6.6.167042000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.12.8.1$x^{12} - 51 x^{9} + 867 x^{6} - 4913 x^{3} + 111166451$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$