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Properties

Label 12.0.33511670005...8329.2
Degree $12$
Signature $[0, 6]$
Discriminant $13^{10}\cdot 79^{6}$
Root discriminant $75.35$
Ramified primes $13, 79$
Class number $720$ (GRH)
Class group $[2, 6, 60]$ (GRH)
Galois group $C_2^2 \times A_4$ (as 12T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7821839, -2650793, -224616, 54721, 149825, -23332, -6138, -1302, 1066, -84, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 15*x^10 - 84*x^9 + 1066*x^8 - 1302*x^7 - 6138*x^6 - 23332*x^5 + 149825*x^4 + 54721*x^3 - 224616*x^2 - 2650793*x + 7821839)
 
gp: K = bnfinit(x^12 - 3*x^11 - 15*x^10 - 84*x^9 + 1066*x^8 - 1302*x^7 - 6138*x^6 - 23332*x^5 + 149825*x^4 + 54721*x^3 - 224616*x^2 - 2650793*x + 7821839, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} - 15 x^{10} - 84 x^{9} + 1066 x^{8} - 1302 x^{7} - 6138 x^{6} - 23332 x^{5} + 149825 x^{4} + 54721 x^{3} - 224616 x^{2} - 2650793 x + 7821839 \)

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:  \(33511670005535928548329=13^{10}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.35$
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:  $13, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{7} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} + \frac{1}{5}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{50} a^{9} - \frac{1}{50} a^{7} - \frac{2}{25} a^{6} - \frac{1}{5} a^{5} - \frac{21}{50} a^{4} - \frac{23}{50} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a + \frac{17}{50}$, $\frac{1}{52467500} a^{10} + \frac{5671}{26233750} a^{9} + \frac{426736}{13116875} a^{8} - \frac{828253}{26233750} a^{7} - \frac{4894009}{26233750} a^{6} - \frac{287144}{13116875} a^{5} + \frac{348239}{1049350} a^{4} + \frac{6109137}{26233750} a^{3} + \frac{1027531}{10493500} a^{2} + \frac{8857601}{26233750} a + \frac{22909769}{52467500}$, $\frac{1}{2982072277817494212500} a^{11} + \frac{2162576379481}{745518069454373553125} a^{10} - \frac{5046454319961282003}{745518069454373553125} a^{9} + \frac{21116296681853902451}{1491036138908747106250} a^{8} + \frac{4748097086771836699}{298207227781749421250} a^{7} - \frac{155739059630407765763}{745518069454373553125} a^{6} - \frac{118484482234549756641}{1491036138908747106250} a^{5} + \frac{677805231328471117587}{1491036138908747106250} a^{4} + \frac{857816700067790175623}{2982072277817494212500} a^{3} + \frac{195805815777311510978}{745518069454373553125} a^{2} - \frac{8554631012023317457}{96195879929596587500} a - \frac{292144087973160677471}{1491036138908747106250}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{60}$, which has order $720$ (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:  \( 12243.1450711 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times A_4$ (as 12T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 16 conjugacy class representatives for $C_2^2 \times A_4$
Character table for $C_2^2 \times A_4$

Intermediate fields

\(\Q(\sqrt{-79}) \), 3.3.169.1, 6.0.14081686879.2, 6.6.2317239613.1, 6.0.29332147.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 12.0.33511670005535928548329.1, 12.0.5369599424056389769.1, 12.0.5369599424056389769.2, 12.0.860374847629609.1
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
$79$79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.2.1.2$x^{2} + 158$$2$$1$$1$$C_2$$[\ ]_{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$