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Properties

Label 12.0.49589822592...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{9}$
Root discriminant $43.78$
Ramified primes $2, 3, 5$
Class number $64$
Class group $[2, 2, 4, 4]$
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![19036, -9504, 24156, -4456, 5112, -36, 936, -108, 234, -28, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 18*x^10 - 28*x^9 + 234*x^8 - 108*x^7 + 936*x^6 - 36*x^5 + 5112*x^4 - 4456*x^3 + 24156*x^2 - 9504*x + 19036)
 
gp: K = bnfinit(x^12 + 18*x^10 - 28*x^9 + 234*x^8 - 108*x^7 + 936*x^6 - 36*x^5 + 5112*x^4 - 4456*x^3 + 24156*x^2 - 9504*x + 19036, 1)
 

Normalized defining polynomial

\( x^{12} + 18 x^{10} - 28 x^{9} + 234 x^{8} - 108 x^{7} + 936 x^{6} - 36 x^{5} + 5112 x^{4} - 4456 x^{3} + 24156 x^{2} - 9504 x + 19036 \)

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:  \(49589822592000000000=2^{16}\cdot 3^{18}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.78$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{30} a^{8} + \frac{1}{30} a^{7} - \frac{1}{15} a^{6} + \frac{1}{3} a^{5} - \frac{7}{15} a^{4} + \frac{1}{3} a^{3} - \frac{4}{15} a^{2} - \frac{4}{15} a - \frac{7}{15}$, $\frac{1}{330} a^{9} - \frac{13}{330} a^{7} + \frac{7}{330} a^{6} - \frac{24}{55} a^{5} - \frac{38}{165} a^{4} + \frac{56}{165} a^{3} + \frac{1}{11} a^{2} + \frac{7}{165} a - \frac{6}{55}$, $\frac{1}{3300} a^{10} - \frac{1}{1650} a^{9} - \frac{2}{275} a^{8} - \frac{2}{75} a^{7} - \frac{34}{825} a^{6} - \frac{74}{275} a^{5} - \frac{31}{150} a^{4} - \frac{406}{825} a^{3} + \frac{86}{275} a^{2} - \frac{104}{825} a + \frac{304}{825}$, $\frac{1}{129577123826700} a^{11} - \frac{2289232069}{129577123826700} a^{10} - \frac{49552199}{6478856191335} a^{9} + \frac{5254017917}{863847492178} a^{8} - \frac{294396097679}{12957712382670} a^{7} - \frac{1261612474879}{32394280956675} a^{6} + \frac{853023821639}{1963289754950} a^{5} - \frac{5362965258971}{12957712382670} a^{4} + \frac{2541015414823}{6478856191335} a^{3} + \frac{234911172245}{1295771238267} a^{2} + \frac{23701921868}{410054189325} a + \frac{1828384122304}{10798093652225}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}$, which has order $64$

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:  \( 1592.03444914 \)
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

\(\Q(\sqrt{5}) \), 3.3.1620.1 x3, 4.0.18000.1, 6.6.13122000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.12.18.68$x^{12} + 21 x^{11} - 21 x^{10} + 21 x^{9} - 27 x^{7} + 15 x^{6} + 18 x^{5} - 27 x^{4} + 27 x^{3} + 27 x^{2} + 27 x - 36$$6$$2$$18$$C_3 : C_4$$[2]_{2}^{2}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$