↑

Properties

Label 12.12.2173896069...0000.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{16}\cdot 5^{9}\cdot 19^{8}$
Root discriminant $59.99$
Ramified primes $2, 5, 19$
Class number $3$ (GRH)
Class group $[3]$ (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![4496, 15424, -36000, -47040, 25256, 27160, -6416, -4720, 836, 300, -50, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 - 50*x^10 + 300*x^9 + 836*x^8 - 4720*x^7 - 6416*x^6 + 27160*x^5 + 25256*x^4 - 47040*x^3 - 36000*x^2 + 15424*x + 4496)
 
gp: K = bnfinit(x^12 - 6*x^11 - 50*x^10 + 300*x^9 + 836*x^8 - 4720*x^7 - 6416*x^6 + 27160*x^5 + 25256*x^4 - 47040*x^3 - 36000*x^2 + 15424*x + 4496, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} - 50 x^{10} + 300 x^{9} + 836 x^{8} - 4720 x^{7} - 6416 x^{6} + 27160 x^{5} + 25256 x^{4} - 47040 x^{3} - 36000 x^{2} + 15424 x + 4496 \)

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:  \(2173896069248000000000=2^{16}\cdot 5^{9}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.99$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{9}$, $\frac{1}{328} a^{10} + \frac{2}{41} a^{9} - \frac{1}{328} a^{8} + \frac{11}{164} a^{7} + \frac{3}{41} a^{6} + \frac{6}{41} a^{5} - \frac{1}{82} a^{4} + \frac{8}{41} a^{3} - \frac{1}{82} a^{2} + \frac{8}{41} a - \frac{11}{41}$, $\frac{1}{496571837747504} a^{11} + \frac{55083118427}{124142959436876} a^{10} - \frac{7513404575611}{124142959436876} a^{9} - \frac{766326679117}{124142959436876} a^{8} - \frac{14159517365587}{124142959436876} a^{7} - \frac{14764399925039}{124142959436876} a^{6} + \frac{30501544697685}{124142959436876} a^{5} + \frac{2035827607197}{31035739859219} a^{4} + \frac{2019236401577}{31035739859219} a^{3} - \frac{2064640694024}{31035739859219} a^{2} + \frac{8989428538283}{31035739859219} a + \frac{2648858591085}{31035739859219}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 4041408.2696 \) (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

\(\Q(\sqrt{5}) \), 3.3.7220.1 x3, \(\Q(\zeta_{20})^+\), 6.6.260642000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$