↑

Properties

Label 12.0.151939915084881.2
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 7^{6}\cdot 11^{6}$
Root discriminant $15.20$
Ramified primes $3, 7, 11$
Class number $2$
Class group $[2]$
Galois group $D_6$ (as 12T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 98, 259, 0, 193, 46, 43, 22, 22, -12, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 11*x^10 - 12*x^9 + 22*x^8 + 22*x^7 + 43*x^6 + 46*x^5 + 193*x^4 + 259*x^2 + 98*x + 49)
 
gp: K = bnfinit(x^12 - 3*x^11 + 11*x^10 - 12*x^9 + 22*x^8 + 22*x^7 + 43*x^6 + 46*x^5 + 193*x^4 + 259*x^2 + 98*x + 49, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + 11 x^{10} - 12 x^{9} + 22 x^{8} + 22 x^{7} + 43 x^{6} + 46 x^{5} + 193 x^{4} + 259 x^{2} + 98 x + 49 \)

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:  \(151939915084881=3^{6}\cdot 7^{6}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.20$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{63} a^{8} + \frac{1}{9} a^{6} + \frac{23}{63} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{2}{21} a^{2} + \frac{1}{9} a + \frac{2}{9}$, $\frac{1}{189} a^{9} - \frac{1}{27} a^{7} - \frac{26}{189} a^{6} - \frac{8}{27} a^{5} - \frac{2}{9} a^{4} - \frac{13}{189} a^{3} + \frac{8}{27} a^{2} + \frac{7}{27} a - \frac{10}{27}$, $\frac{1}{567} a^{10} - \frac{1}{567} a^{9} - \frac{1}{567} a^{8} + \frac{23}{567} a^{7} - \frac{10}{189} a^{6} - \frac{79}{567} a^{5} + \frac{50}{567} a^{4} - \frac{4}{63} a^{3} - \frac{190}{567} a^{2} - \frac{26}{81} a - \frac{29}{81}$, $\frac{1}{141183} a^{11} + \frac{94}{141183} a^{10} + \frac{55}{47061} a^{9} + \frac{34}{15687} a^{8} + \frac{5053}{141183} a^{7} - \frac{14062}{141183} a^{6} + \frac{3026}{6723} a^{5} + \frac{43837}{141183} a^{4} - \frac{7759}{141183} a^{3} - \frac{62395}{141183} a^{2} - \frac{3266}{6723} a - \frac{9172}{20169}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{1466}{141183} a^{11} - \frac{4126}{141183} a^{10} + \frac{4934}{47061} a^{9} - \frac{1616}{15687} a^{8} + \frac{28085}{141183} a^{7} + \frac{29947}{141183} a^{6} + \frac{23677}{47061} a^{5} + \frac{52175}{141183} a^{4} + \frac{241903}{141183} a^{3} - \frac{44993}{141183} a^{2} + \frac{14507}{6723} a + \frac{16282}{20169} \) (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:  \( 312.443391661 \)
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

\(\Q(\sqrt{-231}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{-3}) \), 3.1.231.1 x3, \(\Q(\sqrt{-3}, \sqrt{77})\), 6.0.12326391.1, 6.2.4108797.2 x3, 6.0.160083.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$