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Properties

Label 12.0.27354472316015625.1
Degree $12$
Signature $[0, 6]$
Discriminant $3^{14}\cdot 5^{8}\cdot 11^{4}$
Root discriminant $23.43$
Ramified primes $3, 5, 11$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_3:S_3.C_2$ (as 12T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -6144, 3840, -1088, -240, 600, -423, 150, -15, -17, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 15*x^10 - 17*x^9 - 15*x^8 + 150*x^7 - 423*x^6 + 600*x^5 - 240*x^4 - 1088*x^3 + 3840*x^2 - 6144*x + 4096)
 
gp: K = bnfinit(x^12 - 6*x^11 + 15*x^10 - 17*x^9 - 15*x^8 + 150*x^7 - 423*x^6 + 600*x^5 - 240*x^4 - 1088*x^3 + 3840*x^2 - 6144*x + 4096, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 15 x^{10} - 17 x^{9} - 15 x^{8} + 150 x^{7} - 423 x^{6} + 600 x^{5} - 240 x^{4} - 1088 x^{3} + 3840 x^{2} - 6144 x + 4096 \)

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:  \(27354472316015625=3^{14}\cdot 5^{8}\cdot 11^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.43$
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:  $3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{1}{4} a^{5} - \frac{1}{12} a^{4} + \frac{5}{12} a^{3} - \frac{1}{2} a^{2} + \frac{1}{12} a + \frac{1}{3}$, $\frac{1}{48} a^{8} - \frac{1}{24} a^{7} + \frac{7}{48} a^{6} + \frac{11}{48} a^{5} - \frac{19}{48} a^{4} - \frac{11}{24} a^{3} + \frac{1}{48} a^{2} - \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{192} a^{9} - \frac{1}{96} a^{8} + \frac{7}{192} a^{7} + \frac{11}{192} a^{6} + \frac{29}{192} a^{5} + \frac{37}{96} a^{4} + \frac{1}{192} a^{3} + \frac{7}{48} a^{2} + \frac{1}{3} a$, $\frac{1}{5376} a^{10} - \frac{1}{896} a^{9} - \frac{17}{5376} a^{8} + \frac{25}{768} a^{7} + \frac{529}{5376} a^{6} + \frac{859}{2688} a^{5} + \frac{569}{5376} a^{4} - \frac{11}{96} a^{3} + \frac{79}{336} a^{2} + \frac{11}{84} a + \frac{4}{21}$, $\frac{1}{236544} a^{11} + \frac{1}{16896} a^{10} - \frac{195}{78848} a^{9} - \frac{503}{78848} a^{8} - \frac{163}{7168} a^{7} - \frac{4969}{39424} a^{6} - \frac{901}{78848} a^{5} - \frac{173}{1792} a^{4} - \frac{445}{4928} a^{3} + \frac{1}{528} a^{2} + \frac{17}{924} a + \frac{30}{77}$

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:  \( \frac{5}{8448} a^{11} - \frac{151}{29568} a^{10} + \frac{601}{59136} a^{9} - \frac{181}{19712} a^{8} - \frac{3}{256} a^{7} + \frac{963}{9856} a^{6} - \frac{6147}{19712} a^{5} + \frac{149}{448} a^{4} - \frac{23}{704} a^{3} - \frac{2683}{3696} a^{2} + \frac{1223}{462} a - \frac{809}{231} \) (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:  \( 3486.97473965 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3.C_2$ (as 12T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$
Character table for $C_2\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.6.55130625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$3.12.14.4$x^{12} + 6 x^{11} - 6 x^{10} + 6 x^{9} - 3 x^{8} + 9 x^{7} - 6 x^{6} + 9 x^{5} + 9 x^{4} + 9 x^{3} + 9 x^{2} + 9$$6$$2$$14$12T40$[3/2, 3/2]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$