↑

Properties

Label 12.0.10331448031...1637.1
Degree $12$
Signature $[0, 6]$
Discriminant $7^{8}\cdot 13^{11}$
Root discriminant $38.42$
Ramified primes $7, 13$
Class number $3$
Class group $[3]$
Galois group $C_{12}$ (as 12T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1847, -5409, 1600, 3886, 768, -1353, 53, 90, 27, -27, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + x^10 - 27*x^9 + 27*x^8 + 90*x^7 + 53*x^6 - 1353*x^5 + 768*x^4 + 3886*x^3 + 1600*x^2 - 5409*x + 1847)
 
gp: K = bnfinit(x^12 - x^11 + x^10 - 27*x^9 + 27*x^8 + 90*x^7 + 53*x^6 - 1353*x^5 + 768*x^4 + 3886*x^3 + 1600*x^2 - 5409*x + 1847, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + x^{10} - 27 x^{9} + 27 x^{8} + 90 x^{7} + 53 x^{6} - 1353 x^{5} + 768 x^{4} + 3886 x^{3} + 1600 x^{2} - 5409 x + 1847 \)

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:  \(10331448031704891637=7^{8}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.42$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(2,·)$, $\chi_{91}(4,·)$, $\chi_{91}(37,·)$, $\chi_{91}(32,·)$, $\chi_{91}(8,·)$, $\chi_{91}(74,·)$, $\chi_{91}(46,·)$, $\chi_{91}(16,·)$, $\chi_{91}(23,·)$, $\chi_{91}(57,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{214} a^{10} + \frac{1}{214} a^{9} + \frac{27}{214} a^{8} - \frac{28}{107} a^{7} - \frac{79}{214} a^{6} + \frac{43}{107} a^{5} - \frac{33}{107} a^{4} + \frac{22}{107} a^{3} + \frac{21}{214} a^{2} + \frac{31}{107} a + \frac{44}{107}$, $\frac{1}{27344698816253902} a^{11} - \frac{2859051731071}{27344698816253902} a^{10} + \frac{3101394661288989}{13672349408126951} a^{9} + \frac{3085608482553841}{13672349408126951} a^{8} + \frac{10821088590919255}{27344698816253902} a^{7} - \frac{3764859107484957}{27344698816253902} a^{6} - \frac{9403887914549579}{27344698816253902} a^{5} + \frac{12637473191337649}{27344698816253902} a^{4} + \frac{1626984196298641}{27344698816253902} a^{3} - \frac{5582427606695312}{13672349408126951} a^{2} - \frac{12393527311060433}{27344698816253902} a + \frac{7322987159715161}{27344698816253902}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6176.96689838 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.8281.1, 4.0.2197.1, 6.6.891474493.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$7$7.12.8.2$x^{12} + 49 x^{6} - 1029 x^{3} + 12005$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$