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Properties

Label 12.0.27998342980...0656.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 101^{9}$
Root discriminant $50.57$
Ramified primes $2, 101$
Class number $80$ (GRH)
Class group $[4, 20]$ (GRH)
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![6859, 18411, 24757, 18097, 14153, 4230, 1649, 250, 107, -41, 42, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 42*x^10 - 41*x^9 + 107*x^8 + 250*x^7 + 1649*x^6 + 4230*x^5 + 14153*x^4 + 18097*x^3 + 24757*x^2 + 18411*x + 6859)
 
gp: K = bnfinit(x^12 - 2*x^11 + 42*x^10 - 41*x^9 + 107*x^8 + 250*x^7 + 1649*x^6 + 4230*x^5 + 14153*x^4 + 18097*x^3 + 24757*x^2 + 18411*x + 6859, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 42 x^{10} - 41 x^{9} + 107 x^{8} + 250 x^{7} + 1649 x^{6} + 4230 x^{5} + 14153 x^{4} + 18097 x^{3} + 24757 x^{2} + 18411 x + 6859 \)

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:  \(279983429807196390656=2^{8}\cdot 101^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.57$
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:  $2, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{511233} a^{10} - \frac{1864}{511233} a^{9} + \frac{1401}{170411} a^{8} - \frac{216793}{511233} a^{7} - \frac{14569}{170411} a^{6} + \frac{67090}{170411} a^{5} + \frac{52331}{170411} a^{4} - \frac{70801}{511233} a^{3} + \frac{58437}{170411} a^{2} - \frac{216382}{511233} a + \frac{1758}{8969}$, $\frac{1}{1689676984160439051} a^{11} + \frac{964746148043}{1689676984160439051} a^{10} - \frac{242199117932741858}{1689676984160439051} a^{9} + \frac{247369250135967554}{1689676984160439051} a^{8} + \frac{228129794440474691}{563225661386813017} a^{7} - \frac{417225530038841116}{1689676984160439051} a^{6} - \frac{244487360443680001}{1689676984160439051} a^{5} + \frac{527842593502193878}{1689676984160439051} a^{4} - \frac{223147503997066240}{1689676984160439051} a^{3} - \frac{1434394531893947}{563225661386813017} a^{2} + \frac{10662973277538163}{88930367587391529} a - \frac{143264195631254}{1560181887498097}$

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

Class group and class number

$C_{4}\times C_{20}$, which has order $80$ (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:  $5$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1356.3184458 \) (assuming GRH)
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

\(\Q(\sqrt{101}) \), 3.3.404.1 x3, 4.0.1030301.1, 6.6.16484816.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$$[\ ]_{3}^{4}$
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$