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Properties

Label 12.0.84346657391...0000.7
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}$
Root discriminant $55.44$
Ramified primes $2, 3, 5, 7$
Class number $1800$ (GRH)
Class group $[30, 60]$ (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![3101281, -565856, 1131480, -153516, 188576, -20370, 19118, -1624, 1230, -76, 49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 49*x^10 - 76*x^9 + 1230*x^8 - 1624*x^7 + 19118*x^6 - 20370*x^5 + 188576*x^4 - 153516*x^3 + 1131480*x^2 - 565856*x + 3101281)
 
gp: K = bnfinit(x^12 - 2*x^11 + 49*x^10 - 76*x^9 + 1230*x^8 - 1624*x^7 + 19118*x^6 - 20370*x^5 + 188576*x^4 - 153516*x^3 + 1131480*x^2 - 565856*x + 3101281, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 49 x^{10} - 76 x^{9} + 1230 x^{8} - 1624 x^{7} + 19118 x^{6} - 20370 x^{5} + 188576 x^{4} - 153516 x^{3} + 1131480 x^{2} - 565856 x + 3101281 \)

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:  \(843466573910016000000=2^{18}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.44$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(739,·)$, $\chi_{840}(521,·)$, $\chi_{840}(379,·)$, $\chi_{840}(361,·)$, $\chi_{840}(299,·)$, $\chi_{840}(419,·)$, $\chi_{840}(499,·)$, $\chi_{840}(41,·)$, $\chi_{840}(761,·)$, $\chi_{840}(121,·)$, $\chi_{840}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{6045375996279051848780629} a^{11} - \frac{2666423034525762009516689}{6045375996279051848780629} a^{10} - \frac{459100650616937839541665}{6045375996279051848780629} a^{9} - \frac{531738216540945336771851}{6045375996279051848780629} a^{8} - \frac{2178166944102515602842844}{6045375996279051848780629} a^{7} + \frac{1514876637542337628856836}{6045375996279051848780629} a^{6} + \frac{1558601925143931512101234}{6045375996279051848780629} a^{5} - \frac{1312684671400023040539684}{6045375996279051848780629} a^{4} - \frac{2529332881770812072990762}{6045375996279051848780629} a^{3} - \frac{1114187531458523558147620}{6045375996279051848780629} a^{2} + \frac{2082156371623264373047007}{6045375996279051848780629} a + \frac{2543473179089188307971735}{6045375996279051848780629}$

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

Class group and class number

$C_{30}\times C_{60}$, which has order $1800$ (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:  \( 140.7987960054707 \) (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

\(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-10}, \sqrt{21})\), 6.0.29042496000.9, 6.0.153664000.1, \(\Q(\zeta_{21})^+\)

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$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}$