↑

Properties

Label 12.0.11849506903...8533.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{11}\cdot 137^{6}$
Root discriminant $122.88$
Ramified primes $13, 137$
Class number $100402$ (GRH)
Class group $[100402]$ (GRH)
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![24466501151, -4384043743, 4384043743, -249420159, 249420159, -6207007, 6207007, -75583, 75583, -443, 443, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 443*x^10 - 443*x^9 + 75583*x^8 - 75583*x^7 + 6207007*x^6 - 6207007*x^5 + 249420159*x^4 - 249420159*x^3 + 4384043743*x^2 - 4384043743*x + 24466501151)
 
gp: K = bnfinit(x^12 - x^11 + 443*x^10 - 443*x^9 + 75583*x^8 - 75583*x^7 + 6207007*x^6 - 6207007*x^5 + 249420159*x^4 - 249420159*x^3 + 4384043743*x^2 - 4384043743*x + 24466501151, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 443 x^{10} - 443 x^{9} + 75583 x^{8} - 75583 x^{7} + 6207007 x^{6} - 6207007 x^{5} + 249420159 x^{4} - 249420159 x^{3} + 4384043743 x^{2} - 4384043743 x + 24466501151 \)

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:  \(11849506903407426861218533=13^{11}\cdot 137^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.88$
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:  $13, 137$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1781=13\cdot 137\)
Dirichlet character group:    $\lbrace$$\chi_{1781}(1,·)$, $\chi_{1781}(1506,·)$, $\chi_{1781}(549,·)$, $\chi_{1781}(136,·)$, $\chi_{1781}(1643,·)$, $\chi_{1781}(684,·)$, $\chi_{1781}(686,·)$, $\chi_{1781}(1234,·)$, $\chi_{1781}(821,·)$, $\chi_{1781}(823,·)$, $\chi_{1781}(410,·)$, $\chi_{1781}(412,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2595845771} a^{7} - \frac{1057780162}{2595845771} a^{6} + \frac{238}{2595845771} a^{5} - \frac{331954055}{2595845771} a^{4} + \frac{16184}{2595845771} a^{3} + \frac{1241263592}{2595845771} a^{2} + \frac{275128}{2595845771} a + \frac{148762176}{2595845771}$, $\frac{1}{2595845771} a^{8} + \frac{272}{2595845771} a^{6} - \frac{377315286}{2595845771} a^{5} + \frac{23120}{2595845771} a^{4} + \frac{752545655}{2595845771} a^{3} + \frac{628864}{2595845771} a^{2} - \frac{371905440}{2595845771} a + \frac{2672672}{2595845771}$, $\frac{1}{2595845771} a^{9} - \frac{799991803}{2595845771} a^{6} - \frac{41616}{2595845771} a^{5} + \frac{189446630}{2595845771} a^{4} - \frac{3773184}{2595845771} a^{3} - \frac{535652234}{2595845771} a^{2} - \frac{72162144}{2595845771} a + \frac{1070220464}{2595845771}$, $\frac{1}{2595845771} a^{10} - \frac{52020}{2595845771} a^{6} + \frac{1090754461}{2595845771} a^{5} - \frac{5895600}{2595845771} a^{4} + \frac{1048827541}{2595845771} a^{3} - \frac{180405360}{2595845771} a^{2} - \frac{547926842}{2595845771} a - \frac{817837632}{2595845771}$, $\frac{1}{2595845771} a^{11} - \frac{490464892}{2595845771} a^{6} + \frac{6485160}{2595845771} a^{5} + \frac{364955133}{2595845771} a^{4} + \frac{661486320}{2595845771} a^{3} + \frac{916421144}{2595845771} a^{2} + \frac{515092073}{2595845771} a + \frac{392152169}{2595845771}$

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

Class group and class number

$C_{100402}$, which has order $100402$ (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:  \( 120.784031363 \) (assuming GRH)
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

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.41235493.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$137$137.12.6.2$x^{12} - 48261724457 x^{2} + 191743831267661$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$