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Properties

Label 12.0.31599259021...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{16}\cdot 5^{6}\cdot 13^{11}$
Root discriminant $287.28$
Ramified primes $2, 3, 5, 13$
Class number $2034288$ (GRH)
Class group $[2, 6, 6, 28254]$ (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![17237397684, -2628689544, 1641488550, 26701324, 38233377, 794976, 2843672, 35256, 45123, -52, 312, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 312*x^10 - 52*x^9 + 45123*x^8 + 35256*x^7 + 2843672*x^6 + 794976*x^5 + 38233377*x^4 + 26701324*x^3 + 1641488550*x^2 - 2628689544*x + 17237397684)
 
gp: K = bnfinit(x^12 + 312*x^10 - 52*x^9 + 45123*x^8 + 35256*x^7 + 2843672*x^6 + 794976*x^5 + 38233377*x^4 + 26701324*x^3 + 1641488550*x^2 - 2628689544*x + 17237397684, 1)
 

Normalized defining polynomial

\( x^{12} + 312 x^{10} - 52 x^{9} + 45123 x^{8} + 35256 x^{7} + 2843672 x^{6} + 794976 x^{5} + 38233377 x^{4} + 26701324 x^{3} + 1641488550 x^{2} - 2628689544 x + 17237397684 \)

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:  \(315992590210501847764992000000=2^{18}\cdot 3^{16}\cdot 5^{6}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $287.28$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1,·)$, $\chi_{4680}(2521,·)$, $\chi_{4680}(2749,·)$, $\chi_{4680}(841,·)$, $\chi_{4680}(109,·)$, $\chi_{4680}(4669,·)$, $\chi_{4680}(121,·)$, $\chi_{4680}(3349,·)$, $\chi_{4680}(601,·)$, $\chi_{4680}(3481,·)$, $\chi_{4680}(349,·)$, $\chi_{4680}(3829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{26} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{26} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{156} a^{8} + \frac{1}{78} a^{7} + \frac{1}{156} a^{6} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{8268} a^{9} - \frac{3}{2756} a^{8} + \frac{19}{2756} a^{7} + \frac{29}{2756} a^{6} - \frac{91}{212} a^{5} - \frac{39}{212} a^{4} + \frac{145}{318} a^{3} - \frac{25}{53} a^{2} - \frac{5}{53} a + \frac{13}{53}$, $\frac{1}{8268} a^{10} - \frac{2}{689} a^{8} - \frac{3}{689} a^{7} + \frac{8}{689} a^{6} - \frac{5}{106} a^{5} + \frac{191}{636} a^{4} - \frac{39}{106} a^{3} + \frac{17}{106} a^{2} + \frac{21}{53} a + \frac{11}{53}$, $\frac{1}{1195909492017567713344650674049834756756} a^{11} + \frac{2400878947725003724337313180961963}{298977373004391928336162668512458689189} a^{10} - \frac{68451733988165020900448014220817365}{1195909492017567713344650674049834756756} a^{9} + \frac{391685994336774705501242032416619071}{132878832446396412593850074894426084084} a^{8} - \frac{2105686117663753575925674695778746785}{398636497339189237781550224683278252252} a^{7} - \frac{255371257669050432748620221565123811}{132878832446396412593850074894426084084} a^{6} - \frac{8624589487587835816703195906883248117}{45996518923752604359409641309609029106} a^{5} - \frac{38035941437585139447160695452070180703}{91993037847505208718819282619218058212} a^{4} + \frac{187667883945346206597105725458617667}{45996518923752604359409641309609029106} a^{3} + \frac{339086007547893896410709855184580226}{2555362162430700242189424517200501617} a^{2} - \frac{3481782106309055157901504374640282795}{7666086487292100726568273551601504851} a - \frac{2522325162971943330907398078833167666}{7666086487292100726568273551601504851}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{28254}$, which has order $2034288$ (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:  \( 16557.868316895623 \) (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

\(\Q(\sqrt{13}) \), 3.3.13689.1, 4.0.3515200.1, 6.6.2436053373.1

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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$2$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$3$3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
3.3.4.3$x^{3} - 3 x^{2} + 12$$3$$1$$4$$C_3$$[2]$
$5$5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$13$13.12.11.6$x^{12} - 13312$$12$$1$$11$$C_{12}$$[\ ]_{12}$