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Properties

Label 12.0.42644551872...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{8}\cdot 5^{9}\cdot 31^{8}$
Root discriminant $52.38$
Ramified primes $2, 5, 31$
Class number $108$ (GRH)
Class group $[3, 6, 6]$ (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![9025, -7125, 47535, -119595, 145651, 19812, 8967, 310, 581, -65, 28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 28*x^10 - 65*x^9 + 581*x^8 + 310*x^7 + 8967*x^6 + 19812*x^5 + 145651*x^4 - 119595*x^3 + 47535*x^2 - 7125*x + 9025)
 
gp: K = bnfinit(x^12 - 4*x^11 + 28*x^10 - 65*x^9 + 581*x^8 + 310*x^7 + 8967*x^6 + 19812*x^5 + 145651*x^4 - 119595*x^3 + 47535*x^2 - 7125*x + 9025, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 28 x^{10} - 65 x^{9} + 581 x^{8} + 310 x^{7} + 8967 x^{6} + 19812 x^{5} + 145651 x^{4} - 119595 x^{3} + 47535 x^{2} - 7125 x + 9025 \)

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:  \(426445518720500000000=2^{8}\cdot 5^{9}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.38$
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:  $2, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{2}{15} a^{8} + \frac{1}{3} a^{7} - \frac{4}{15} a^{6} + \frac{2}{15} a^{4} + \frac{2}{15} a^{3} + \frac{1}{15} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{320120744927743599734225223735} a^{11} + \frac{3360379986472964195663738573}{106706914975914533244741741245} a^{10} + \frac{43850533807507998395776191461}{320120744927743599734225223735} a^{9} - \frac{48222814027642343488729025761}{320120744927743599734225223735} a^{8} + \frac{10094465523805264813634930562}{106706914975914533244741741245} a^{7} - \frac{26303247789574172619435975247}{320120744927743599734225223735} a^{6} - \frac{127849610610859340186896333088}{320120744927743599734225223735} a^{5} + \frac{97415485310465276318882208593}{320120744927743599734225223735} a^{4} + \frac{86084709109081857899896504487}{320120744927743599734225223735} a^{3} + \frac{149858461118616945536477275078}{320120744927743599734225223735} a^{2} + \frac{22581392133310973685555821129}{64024148985548719946845044747} a + \frac{273498671332440046971854486}{1123230683956995086786755171}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (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:  \( \frac{531937474867407624150086}{106706914975914533244741741245} a^{11} - \frac{2798653472520114149677332}{106706914975914533244741741245} a^{10} + \frac{3713963809355736886198665}{21341382995182906648948348249} a^{9} - \frac{56617826671168496088543219}{106706914975914533244741741245} a^{8} + \frac{379080550873169071801322106}{106706914975914533244741741245} a^{7} - \frac{283146328248990069941575503}{106706914975914533244741741245} a^{6} + \frac{5123297590539664311268713132}{106706914975914533244741741245} a^{5} + \frac{5265261205754200624907303826}{106706914975914533244741741245} a^{4} + \frac{14678034980672696868811961751}{21341382995182906648948348249} a^{3} - \frac{133570384400722236702636712563}{106706914975914533244741741245} a^{2} + \frac{51693015021337717923097464907}{21341382995182906648948348249} a - \frac{225099083529739137597994563}{1123230683956995086786755171} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10042.6435644 \) (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

\(\Q(\sqrt{5}) \), 3.3.19220.1 x3, \(\Q(\zeta_{5})\), 6.6.1847042000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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}$
$31$31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$