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Properties

Label 12.0.19610762759...8237.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{11}\cdot 149^{6}$
Root discriminant $128.15$
Ramified primes $13, 149$
Class number $178850$ (GRH)
Class group $[35, 5110]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40013832041, -6659388724, 6659388724, -349088637, 349088637, -7991335, 7991335, -89467, 89467, -482, 482, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 482*x^10 - 482*x^9 + 89467*x^8 - 89467*x^7 + 7991335*x^6 - 7991335*x^5 + 349088637*x^4 - 349088637*x^3 + 6659388724*x^2 - 6659388724*x + 40013832041)
 
gp: K = bnfinit(x^12 - x^11 + 482*x^10 - 482*x^9 + 89467*x^8 - 89467*x^7 + 7991335*x^6 - 7991335*x^5 + 349088637*x^4 - 349088637*x^3 + 6659388724*x^2 - 6659388724*x + 40013832041, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 482 x^{10} - 482 x^{9} + 89467 x^{8} - 89467 x^{7} + 7991335 x^{6} - 7991335 x^{5} + 349088637 x^{4} - 349088637 x^{3} + 6659388724 x^{2} - 6659388724 x + 40013832041 \)

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:  \(19610762759203196764498237=13^{11}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.15$
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:  $13, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1937=13\cdot 149\)
Dirichlet character group:    $\lbrace$$\chi_{1937}(1,·)$, $\chi_{1937}(1191,·)$, $\chi_{1937}(297,·)$, $\chi_{1937}(1489,·)$, $\chi_{1937}(1042,·)$, $\chi_{1937}(1491,·)$, $\chi_{1937}(148,·)$, $\chi_{1937}(597,·)$, $\chi_{1937}(1193,·)$, $\chi_{1937}(1044,·)$, $\chi_{1937}(1787,·)$, $\chi_{1937}(1342,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{4157457641} a^{7} - \frac{1558069783}{4157457641} a^{6} + \frac{259}{4157457641} a^{5} - \frac{822507623}{4157457641} a^{4} + \frac{19166}{4157457641} a^{3} - \frac{1995867846}{4157457641} a^{2} + \frac{354571}{4157457641} a + \frac{219361608}{4157457641}$, $\frac{1}{4157457641} a^{8} + \frac{296}{4157457641} a^{6} - \frac{555825003}{4157457641} a^{5} + \frac{27380}{4157457641} a^{4} + \frac{1108815470}{4157457641} a^{3} + \frac{810448}{4157457641} a^{2} - \frac{548404020}{4157457641} a + \frac{3748322}{4157457641}$, $\frac{1}{4157457641} a^{9} - \frac{844967386}{4157457641} a^{6} - \frac{49284}{4157457641} a^{5} - \frac{13564697}{78442597} a^{4} - \frac{4862688}{4157457641} a^{3} - \frac{130506626}{4157457641} a^{2} - \frac{101204694}{4157457641} a + \frac{1588286288}{4157457641}$, $\frac{1}{4157457641} a^{10} - \frac{61605}{4157457641} a^{6} + \frac{1939826701}{4157457641} a^{5} - \frac{7597950}{4157457641} a^{4} + \frac{1216901755}{4157457641} a^{3} - \frac{253011735}{4157457641} a^{2} - \frac{508133330}{4157457641} a - \frac{1248191226}{4157457641}$, $\frac{1}{4157457641} a^{11} + \frac{275402753}{4157457641} a^{6} + \frac{8357745}{4157457641} a^{5} + \frac{1728515348}{4157457641} a^{4} + \frac{927709695}{4157457641} a^{3} + \frac{862946415}{4157457641} a^{2} - \frac{192132976}{4157457641} a + \frac{2034527590}{4157457641}$

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

Class group and class number

$C_{35}\times C_{5110}$, which has order $178850$ (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, 4.0.48775597.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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}$$[\ ]_{12}$
$149$149.12.6.2$x^{12} - 73439775749 x^{2} + 153195372212414$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$