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Properties

Label 12.0.1094032426497921.3
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 107^{6}$
Root discriminant $17.92$
Ramified primes $3, 107$
Class number $3$
Class group $[3]$
Galois group $D_6$ (as 12T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1089, -1683, 1545, -1368, 820, -194, -20, 77, -40, 8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 4*x^10 + 8*x^9 - 40*x^8 + 77*x^7 - 20*x^6 - 194*x^5 + 820*x^4 - 1368*x^3 + 1545*x^2 - 1683*x + 1089)
 
gp: K = bnfinit(x^12 - 2*x^11 + 4*x^10 + 8*x^9 - 40*x^8 + 77*x^7 - 20*x^6 - 194*x^5 + 820*x^4 - 1368*x^3 + 1545*x^2 - 1683*x + 1089, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 4 x^{10} + 8 x^{9} - 40 x^{8} + 77 x^{7} - 20 x^{6} - 194 x^{5} + 820 x^{4} - 1368 x^{3} + 1545 x^{2} - 1683 x + 1089 \)

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:  \(1094032426497921=3^{6}\cdot 107^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.92$
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:  $3, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{63} a^{8} - \frac{2}{63} a^{7} - \frac{2}{63} a^{6} + \frac{13}{63} a^{5} - \frac{8}{63} a^{4} - \frac{20}{63} a^{3} + \frac{10}{21} a^{2} - \frac{8}{21} a + \frac{2}{7}$, $\frac{1}{63} a^{9} - \frac{2}{21} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{10}{63} a^{3} - \frac{3}{7} a^{2} - \frac{10}{21} a - \frac{3}{7}$, $\frac{1}{63} a^{10} - \frac{1}{21} a^{7} + \frac{2}{21} a^{6} - \frac{1}{3} a^{5} + \frac{5}{63} a^{4} - \frac{1}{3} a^{3} + \frac{8}{21} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{19998148401} a^{11} + \frac{11900177}{6666049467} a^{10} - \frac{5713874}{2856878343} a^{9} + \frac{1851998}{240941547} a^{8} - \frac{2947879832}{19998148401} a^{7} - \frac{195016198}{1818013491} a^{6} - \frac{1446758074}{6666049467} a^{5} - \frac{386883872}{2856878343} a^{4} + \frac{1918285492}{6666049467} a^{3} + \frac{1039911758}{6666049467} a^{2} - \frac{98667710}{317430927} a - \frac{9734083}{202001499}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{3438437}{2856878343} a^{11} + \frac{1500350}{952292781} a^{10} - \frac{13560038}{2856878343} a^{9} - \frac{445048}{34420221} a^{8} + \frac{112006348}{2856878343} a^{7} - \frac{19751011}{259716213} a^{6} - \frac{8645917}{952292781} a^{5} + \frac{538217056}{2856878343} a^{4} - \frac{866366318}{952292781} a^{3} + \frac{1042965824}{952292781} a^{2} - \frac{548817209}{317430927} a + \frac{56490677}{28857357} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1921.32424147 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_6$ (as 12T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{321}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-107}) \), 3.3.321.1 x3, \(\Q(\sqrt{-3}, \sqrt{-107})\), 6.6.33076161.2, 6.0.309123.1 x3, 6.0.11025387.1 x3

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

Sibling fields

Degree 6 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$107$107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$