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Properties

Label 12.0.33096636485...224.33
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}$
Root discriminant $288.39$
Ramified primes $2, 3, 7, 19$
Class number $1499904$ (GRH)
Class group $[4, 4, 12, 7812]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51646472991, -7788574854, 3740971227, -392352750, 102108541, -7745318, 1515592, -55720, 10316, 344, 49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 49*x^10 + 344*x^9 + 10316*x^8 - 55720*x^7 + 1515592*x^6 - 7745318*x^5 + 102108541*x^4 - 392352750*x^3 + 3740971227*x^2 - 7788574854*x + 51646472991)
 
gp: K = bnfinit(x^12 - 2*x^11 + 49*x^10 + 344*x^9 + 10316*x^8 - 55720*x^7 + 1515592*x^6 - 7745318*x^5 + 102108541*x^4 - 392352750*x^3 + 3740971227*x^2 - 7788574854*x + 51646472991, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 49 x^{10} + 344 x^{9} + 10316 x^{8} - 55720 x^{7} + 1515592 x^{6} - 7745318 x^{5} + 102108541 x^{4} - 392352750 x^{3} + 3740971227 x^{2} - 7788574854 x + 51646472991 \)

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:  \(330966364856493592168450228224=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $288.39$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3192=2^{3}\cdot 3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3192}(1,·)$, $\chi_{3192}(1285,·)$, $\chi_{3192}(961,·)$, $\chi_{3192}(1033,·)$, $\chi_{3192}(1217,·)$, $\chi_{3192}(2957,·)$, $\chi_{3192}(1265,·)$, $\chi_{3192}(2773,·)$, $\chi_{3192}(2705,·)$, $\chi_{3192}(2725,·)$, $\chi_{3192}(797,·)$, $\chi_{3192}(3029,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{57} a^{6} - \frac{1}{57} a^{5} + \frac{8}{19} a^{4} + \frac{25}{57} a^{3} - \frac{7}{57} a^{2} - \frac{8}{19} a + \frac{4}{19}$, $\frac{1}{57} a^{7} + \frac{4}{57} a^{5} - \frac{9}{19} a^{4} - \frac{20}{57} a^{3} - \frac{4}{19} a^{2} - \frac{4}{19} a + \frac{4}{19}$, $\frac{1}{2907} a^{8} - \frac{2}{2907} a^{7} + \frac{16}{2907} a^{6} - \frac{389}{2907} a^{5} - \frac{1160}{2907} a^{4} - \frac{128}{2907} a^{3} + \frac{144}{323} a^{2} - \frac{388}{969} a + \frac{6}{19}$, $\frac{1}{8721} a^{9} - \frac{1}{8721} a^{8} - \frac{37}{8721} a^{7} + \frac{35}{8721} a^{6} - \frac{223}{8721} a^{5} + \frac{191}{8721} a^{4} - \frac{215}{459} a^{3} - \frac{150}{323} a^{2} + \frac{734}{2907} a - \frac{4}{57}$, $\frac{1}{36479943} a^{10} - \frac{488}{12159981} a^{9} - \frac{254}{36479943} a^{8} - \frac{187733}{36479943} a^{7} + \frac{262576}{36479943} a^{6} - \frac{1795460}{36479943} a^{5} - \frac{1902035}{12159981} a^{4} - \frac{16484501}{36479943} a^{3} + \frac{5074259}{12159981} a^{2} - \frac{6865}{639999} a + \frac{54629}{238431}$, $\frac{1}{448999350837879107584789425819} a^{11} - \frac{402298606983238187017}{49888816759764345287198825091} a^{10} + \frac{13030423763663206786189657}{448999350837879107584789425819} a^{9} + \frac{54496616894290411142329351}{448999350837879107584789425819} a^{8} + \frac{770849374676860587987809734}{448999350837879107584789425819} a^{7} + \frac{3234120907513160891405556889}{448999350837879107584789425819} a^{6} - \frac{21834955220188193001833628229}{149666450279293035861596475273} a^{5} + \frac{104843082552305281806216824005}{448999350837879107584789425819} a^{4} - \frac{32733572714766014551366059584}{149666450279293035861596475273} a^{3} + \frac{53000535940452018660156349895}{149666450279293035861596475273} a^{2} - \frac{1409230137322553968621994480}{2934636279986137958070519123} a - \frac{303176463002960260334984}{975286234624838138275347}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{12}\times C_{7812}$, which has order $1499904$ (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:  \( 75910.8816311755 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-798}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{21}) \), 3.3.17689.1, \(\Q(\sqrt{21}, \sqrt{-38})\), 6.0.575296762424832.1, 6.0.3043898213888.1, 6.6.59138236269.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.12.10.2$x^{12} + 57 x^{6} + 1444$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$