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Properties

Label 12.0.34878787205...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 5^{9}\cdot 17^{8}$
Root discriminant $35.09$
Ramified primes $2, 5, 17$
Class number $16$
Class group $[2, 2, 2, 2]$
Galois group $C_3 : C_4$ (as 12T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -1375, 3150, -7080, 15931, 156, 1427, 78, 145, -18, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 12*x^10 - 18*x^9 + 145*x^8 + 78*x^7 + 1427*x^6 + 156*x^5 + 15931*x^4 - 7080*x^3 + 3150*x^2 - 1375*x + 625)
 
gp: K = bnfinit(x^12 - x^11 + 12*x^10 - 18*x^9 + 145*x^8 + 78*x^7 + 1427*x^6 + 156*x^5 + 15931*x^4 - 7080*x^3 + 3150*x^2 - 1375*x + 625, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 12 x^{10} - 18 x^{9} + 145 x^{8} + 78 x^{7} + 1427 x^{6} + 156 x^{5} + 15931 x^{4} - 7080 x^{3} + 3150 x^{2} - 1375 x + 625 \)

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:  \(3487878720500000000=2^{8}\cdot 5^{9}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.09$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{2}{5} a$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{3}$, $\frac{1}{10912775} a^{9} + \frac{557756}{10912775} a^{8} + \frac{405589}{10912775} a^{7} + \frac{19664}{436511} a^{6} + \frac{129501}{2182555} a^{5} + \frac{4485478}{10912775} a^{4} - \frac{1600147}{10912775} a^{3} + \frac{2729067}{10912775} a^{2} - \frac{977494}{2182555} a + \frac{168316}{436511}$, $\frac{1}{10912775} a^{10} - \frac{2}{25} a^{8} + \frac{1}{25} a^{7} - \frac{120007}{10912775} a^{5} - \frac{11}{25} a^{3} - \frac{7}{25} a^{2} - \frac{147659}{436511}$, $\frac{1}{54563875} a^{11} - \frac{1}{54563875} a^{10} + \frac{2}{54563875} a^{9} - \frac{5432153}{54563875} a^{8} + \frac{773351}{10912775} a^{7} + \frac{5228303}{54563875} a^{6} - \frac{5132648}{54563875} a^{5} - \frac{25949924}{54563875} a^{4} + \frac{16442701}{54563875} a^{3} - \frac{113423}{2182555} a^{2} - \frac{55350}{436511} a + \frac{9556}{436511}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$

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:  \( -\frac{29}{2182555} a^{11} - \frac{2377}{436511} a^{6} + \frac{4909201}{2182555} a \) (order $10$)
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:  \( 7502.80335755 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:C_4$ (as 12T5):

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 $C_3 : C_4$
Character table for $C_3 : C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.5780.1 x3, \(\Q(\zeta_{5})\), 6.6.167042000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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
$2$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.12.8.1$x^{12} - 51 x^{9} + 867 x^{6} - 4913 x^{3} + 111166451$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$