↑

Properties

Label 12.0.17374411772...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{22}\cdot 5^{8}\cdot 13^{9}$
Root discriminant $71.34$
Ramified primes $2, 5, 13$
Class number $512$ (GRH)
Class group $[2, 16, 16]$ (GRH)
Galois group $C_3 : C_4$ (as 12T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1284583, -672574, 740767, -119238, 60484, 7090, 18759, -2582, 2036, -262, 79, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 79*x^10 - 262*x^9 + 2036*x^8 - 2582*x^7 + 18759*x^6 + 7090*x^5 + 60484*x^4 - 119238*x^3 + 740767*x^2 - 672574*x + 1284583)
 
gp: K = bnfinit(x^12 - 6*x^11 + 79*x^10 - 262*x^9 + 2036*x^8 - 2582*x^7 + 18759*x^6 + 7090*x^5 + 60484*x^4 - 119238*x^3 + 740767*x^2 - 672574*x + 1284583, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 79 x^{10} - 262 x^{9} + 2036 x^{8} - 2582 x^{7} + 18759 x^{6} + 7090 x^{5} + 60484 x^{4} - 119238 x^{3} + 740767 x^{2} - 672574 x + 1284583 \)

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:  \(17374411772723200000000=2^{22}\cdot 5^{8}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.34$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{43899434} a^{10} + \frac{3669610}{21949717} a^{9} + \frac{441642}{21949717} a^{8} + \frac{1629595}{43899434} a^{7} - \frac{3730279}{21949717} a^{6} - \frac{3369736}{21949717} a^{5} + \frac{7660305}{43899434} a^{4} + \frac{7779079}{21949717} a^{3} - \frac{8886171}{21949717} a^{2} - \frac{20968683}{43899434} a - \frac{629762}{21949717}$, $\frac{1}{594018915348568913749474} a^{11} + \frac{2720514033474857}{297009457674284456874737} a^{10} + \frac{41860378428143139875187}{594018915348568913749474} a^{9} - \frac{57207645376304696749802}{297009457674284456874737} a^{8} + \frac{40544321302396544924639}{297009457674284456874737} a^{7} - \frac{11253548992672152436080}{297009457674284456874737} a^{6} - \frac{35347578032984301494684}{297009457674284456874737} a^{5} + \frac{75145273255417001261663}{594018915348568913749474} a^{4} - \frac{4995692132401816169742}{297009457674284456874737} a^{3} - \frac{14922889795232199575507}{594018915348568913749474} a^{2} + \frac{11346259023691007424263}{594018915348568913749474} a - \frac{125605221264804322326743}{594018915348568913749474}$

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

Class group and class number

$C_{2}\times C_{16}\times C_{16}$, which has order $512$ (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:  \( 1640.26926328 \) (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

\(\Q(\sqrt{13}) \), 3.3.1300.1 x3, 4.0.140608.2, 6.6.21970000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$2$2.12.22.63$x^{12} - 60 x^{6} + 52$$6$$2$$22$$C_3 : C_4$$[3]_{3}^{2}$
$5$5.12.8.1$x^{12} - 30 x^{9} + 175 x^{6} + 500 x^{3} + 5000$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$