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Properties

Label 12.4.434774896093890625.2
Degree $12$
Signature $[4, 4]$
Discriminant $5^{6}\cdot 7^{8}\cdot 13^{6}$
Root discriminant $29.50$
Ramified primes $5, 7, 13$
Class number $2$ (GRH)
Class group $[2]$ (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![63001, -36646, -77255, 8567, 11949, 67, -1799, -225, 161, -20, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + x^10 - 20*x^9 + 161*x^8 - 225*x^7 - 1799*x^6 + 67*x^5 + 11949*x^4 + 8567*x^3 - 77255*x^2 - 36646*x + 63001)
 
gp: K = bnfinit(x^12 - 3*x^11 + x^10 - 20*x^9 + 161*x^8 - 225*x^7 - 1799*x^6 + 67*x^5 + 11949*x^4 + 8567*x^3 - 77255*x^2 - 36646*x + 63001, 1)
 

Normalized defining polynomial

\( x^{12} - 3 x^{11} + x^{10} - 20 x^{9} + 161 x^{8} - 225 x^{7} - 1799 x^{6} + 67 x^{5} + 11949 x^{4} + 8567 x^{3} - 77255 x^{2} - 36646 x + 63001 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(434774896093890625=5^{6}\cdot 7^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.50$
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:  $5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8} - \frac{2}{41} a^{7} - \frac{16}{41} a^{6} - \frac{11}{41} a^{5} + \frac{6}{41} a^{4} - \frac{6}{41} a^{3} - \frac{5}{41} a^{2} - \frac{13}{41} a + \frac{1}{41}$, $\frac{1}{82} a^{9} + \frac{21}{82} a^{7} - \frac{1}{41} a^{6} + \frac{25}{82} a^{5} - \frac{35}{82} a^{4} + \frac{12}{41} a^{3} + \frac{9}{41} a^{2} + \frac{8}{41} a - \frac{39}{82}$, $\frac{1}{82} a^{10} - \frac{1}{82} a^{8} - \frac{20}{41} a^{7} - \frac{33}{82} a^{6} - \frac{39}{82} a^{5} - \frac{13}{41} a^{4} - \frac{7}{41} a^{3} - \frac{19}{41} a^{2} + \frac{1}{82} a - \frac{11}{41}$, $\frac{1}{60889744898344186571878} a^{11} + \frac{18551235584170418430}{30444872449172093285939} a^{10} + \frac{370763442041787281209}{60889744898344186571878} a^{9} + \frac{288030784920132865726}{30444872449172093285939} a^{8} - \frac{25357234611891739504761}{60889744898344186571878} a^{7} - \frac{10804859005587967166857}{60889744898344186571878} a^{6} - \frac{14341939335374388831569}{30444872449172093285939} a^{5} + \frac{8720503949562437424433}{30444872449172093285939} a^{4} + \frac{10313813819744237600732}{30444872449172093285939} a^{3} + \frac{2978023137266619898439}{60889744898344186571878} a^{2} - \frac{10613907706885060565203}{30444872449172093285939} a - \frac{47104597366504789095}{121294312546502363689}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
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:  \( 6172.51259567 \) (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

\(\Q(\sqrt{65}) \), \(\Q(\zeta_{7})^+\), 6.2.50721125.1, 6.2.31213.1, 6.6.659374625.1

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

Sibling fields

Degree 12 siblings: 12.4.15222677640625.1, 12.0.2572632521265625.1, 12.0.2572632521265625.2, 12.4.434774896093890625.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$