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Properties

Label 12.0.37176212245...3808.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{16}\cdot 7^{6}\cdot 13^{11}$
Root discriminant $240.36$
Ramified primes $2, 3, 7, 13$
Class number $1263600$ (GRH)
Class group $[3, 6, 6, 30, 390]$ (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![3926087841, 1474874154, 456229527, -6234410, 3543930, -589134, 943592, -3354, 20787, -52, 195, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 195*x^10 - 52*x^9 + 20787*x^8 - 3354*x^7 + 943592*x^6 - 589134*x^5 + 3543930*x^4 - 6234410*x^3 + 456229527*x^2 + 1474874154*x + 3926087841)
 
gp: K = bnfinit(x^12 + 195*x^10 - 52*x^9 + 20787*x^8 - 3354*x^7 + 943592*x^6 - 589134*x^5 + 3543930*x^4 - 6234410*x^3 + 456229527*x^2 + 1474874154*x + 3926087841, 1)
 

Normalized defining polynomial

\( x^{12} + 195 x^{10} - 52 x^{9} + 20787 x^{8} - 3354 x^{7} + 943592 x^{6} - 589134 x^{5} + 3543930 x^{4} - 6234410 x^{3} + 456229527 x^{2} + 1474874154 x + 3926087841 \)

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:  \(37176212245675331887703543808=2^{12}\cdot 3^{16}\cdot 7^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.36$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3276=2^{2}\cdot 3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3276}(1,·)$, $\chi_{3276}(811,·)$, $\chi_{3276}(2407,·)$, $\chi_{3276}(2857,·)$, $\chi_{3276}(1735,·)$, $\chi_{3276}(1933,·)$, $\chi_{3276}(1681,·)$, $\chi_{3276}(307,·)$, $\chi_{3276}(1849,·)$, $\chi_{3276}(2521,·)$, $\chi_{3276}(475,·)$, $\chi_{3276}(895,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{39} a^{8} - \frac{1}{39} a^{7} + \frac{1}{39} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{897} a^{9} - \frac{8}{897} a^{8} + \frac{14}{897} a^{7} - \frac{20}{897} a^{6} - \frac{19}{69} a^{5} + \frac{19}{69} a^{4} + \frac{16}{69} a^{3} - \frac{1}{23} a^{2} + \frac{8}{23} a + \frac{11}{23}$, $\frac{1}{897} a^{10} - \frac{4}{897} a^{8} - \frac{1}{39} a^{7} - \frac{16}{897} a^{6} + \frac{28}{69} a^{5} + \frac{7}{69} a^{4} + \frac{10}{69} a^{3} + \frac{6}{23} a - \frac{4}{23}$, $\frac{1}{103038253536858848146249454907731464263} a^{11} + \frac{675515072133754046839222303593892}{7926019502835296011249958069825497251} a^{10} + \frac{23439142483588401635259076326294334}{103038253536858848146249454907731464263} a^{9} + \frac{128326341140839339661194108842988525}{11448694837428760905138828323081273807} a^{8} + \frac{447799047217520771878681198365662246}{34346084512286282715416484969243821421} a^{7} - \frac{88495398616595005128780928297052371}{11448694837428760905138828323081273807} a^{6} + \frac{1369826104945501904863845439389052142}{7926019502835296011249958069825497251} a^{5} - \frac{2750353876358598810389158872879137290}{7926019502835296011249958069825497251} a^{4} + \frac{2640095527789625236041973774403556611}{7926019502835296011249958069825497251} a^{3} - \frac{32745372687498012879806247565115734}{880668833648366223472217563313944139} a^{2} - \frac{1202848082653566067296070157888756677}{2642006500945098670416652689941832417} a - \frac{532789015926427185090503281383053119}{2642006500945098670416652689941832417}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}\times C_{30}\times C_{390}$, which has order $1263600$ (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:  \( 9116.238746847283 \) (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

\(\Q(\sqrt{13}) \), 3.3.13689.2, 4.0.1722448.1, 6.6.2436053373.2

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$2$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$13$13.12.11.5$x^{12} - 3328$$12$$1$$11$$C_{12}$$[\ ]_{12}$