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Properties

Label 12.0.14455510594...024.10
Degree $12$
Signature $[0, 6]$
Discriminant $2^{20}\cdot 13^{10}$
Root discriminant $26.92$
Ramified primes $2, 13$
Class number $20$
Class group $[20]$
Galois group $C_2\times C_4^2:C_3:C_2$ (as 12T95)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 416, 584, 524, 500, 264, 184, 36, 28, -12, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 2*x^10 - 12*x^9 + 28*x^8 + 36*x^7 + 184*x^6 + 264*x^5 + 500*x^4 + 524*x^3 + 584*x^2 + 416*x + 256)
 
gp: K = bnfinit(x^12 - x^11 + 2*x^10 - 12*x^9 + 28*x^8 + 36*x^7 + 184*x^6 + 264*x^5 + 500*x^4 + 524*x^3 + 584*x^2 + 416*x + 256, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 2 x^{10} - 12 x^{9} + 28 x^{8} + 36 x^{7} + 184 x^{6} + 264 x^{5} + 500 x^{4} + 524 x^{3} + 584 x^{2} + 416 x + 256 \)

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:  \(144555105949057024=2^{20}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.92$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{335560352} a^{11} - \frac{16033825}{335560352} a^{10} + \frac{2249169}{167780176} a^{9} - \frac{1000053}{41945044} a^{8} + \frac{1019313}{41945044} a^{7} + \frac{5347549}{83890088} a^{6} - \frac{1549070}{10486261} a^{5} + \frac{3489313}{20972522} a^{4} - \frac{8162165}{83890088} a^{3} - \frac{1353197}{83890088} a^{2} - \frac{16159725}{41945044} a + \frac{3938111}{10486261}$

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

Class group and class number

$C_{20}$, which has order $20$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4078.83943299 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2:C_3:C_2$ (as 12T95):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 192
The 20 conjugacy class representatives for $C_2\times C_4^2:C_3:C_2$
Character table for $C_2\times C_4^2:C_3:C_2$

Intermediate fields

3.1.104.1, 6.2.7311616.1

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 24 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.20.54$x^{8} + 4 x^{5} + 6 x^{4} + 14$$8$$1$$20$$Q_8:C_2$$[2, 3, 3]^{2}$
$13$13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.7.4$x^{8} + 104$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$