↑

Properties

Label 12.0.36705367363...0000.7
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 23^{6}$
Root discriminant $111.45$
Ramified primes $2, 3, 5, 23$
Class number $106400$ (GRH)
Class group $[20, 5320]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1914274729, -382535442, 416049516, -68396122, 37645887, -4215756, 1536671, -114270, 30606, -1370, 285, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 285*x^10 - 1370*x^9 + 30606*x^8 - 114270*x^7 + 1536671*x^6 - 4215756*x^5 + 37645887*x^4 - 68396122*x^3 + 416049516*x^2 - 382535442*x + 1914274729)
 
gp: K = bnfinit(x^12 - 6*x^11 + 285*x^10 - 1370*x^9 + 30606*x^8 - 114270*x^7 + 1536671*x^6 - 4215756*x^5 + 37645887*x^4 - 68396122*x^3 + 416049516*x^2 - 382535442*x + 1914274729, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 285 x^{10} - 1370 x^{9} + 30606 x^{8} - 114270 x^{7} + 1536671 x^{6} - 4215756 x^{5} + 37645887 x^{4} - 68396122 x^{3} + 416049516 x^{2} - 382535442 x + 1914274729 \)

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:  \(3670536736379502144000000=2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.45$
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:  $2, 3, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4140=2^{2}\cdot 3^{2}\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{4140}(2209,·)$, $\chi_{4140}(1379,·)$, $\chi_{4140}(3589,·)$, $\chi_{4140}(1,·)$, $\chi_{4140}(2761,·)$, $\chi_{4140}(551,·)$, $\chi_{4140}(1931,·)$, $\chi_{4140}(2759,·)$, $\chi_{4140}(3311,·)$, $\chi_{4140}(4139,·)$, $\chi_{4140}(829,·)$, $\chi_{4140}(1381,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{23} a^{4} - \frac{2}{23} a^{3} - \frac{1}{23} a^{2} + \frac{2}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{5} - \frac{5}{23} a^{3} + \frac{5}{23} a + \frac{2}{23}$, $\frac{1}{23} a^{6} - \frac{10}{23} a^{3} - \frac{11}{23} a + \frac{5}{23}$, $\frac{1}{23} a^{7} + \frac{3}{23} a^{3} + \frac{2}{23} a^{2} + \frac{2}{23} a + \frac{10}{23}$, $\frac{1}{529} a^{8} - \frac{4}{529} a^{7} + \frac{2}{529} a^{6} + \frac{8}{529} a^{5} - \frac{5}{529} a^{4} - \frac{8}{529} a^{3} + \frac{2}{529} a^{2} + \frac{4}{529} a + \frac{1}{529}$, $\frac{1}{529} a^{9} + \frac{9}{529} a^{7} - \frac{7}{529} a^{6} + \frac{4}{529} a^{5} - \frac{5}{529} a^{4} - \frac{191}{529} a^{3} + \frac{35}{529} a^{2} + \frac{247}{529} a + \frac{96}{529}$, $\frac{1}{18695294705221} a^{10} - \frac{5}{18695294705221} a^{9} + \frac{17530351989}{18695294705221} a^{8} - \frac{70121407926}{18695294705221} a^{7} - \frac{107579166639}{18695294705221} a^{6} - \frac{244676472590}{18695294705221} a^{5} + \frac{154564079307}{18695294705221} a^{4} + \frac{2726320653889}{18695294705221} a^{3} - \frac{1961042039230}{18695294705221} a^{2} - \frac{1327834899023}{18695294705221} a + \frac{1288607047433}{18695294705221}$, $\frac{1}{51054400756352167049} a^{11} + \frac{1365429}{51054400756352167049} a^{10} + \frac{20743136144991624}{51054400756352167049} a^{9} - \frac{21151988005825382}{51054400756352167049} a^{8} - \frac{842080697926463076}{51054400756352167049} a^{7} - \frac{1007817426406123029}{51054400756352167049} a^{6} + \frac{307708349307863258}{51054400756352167049} a^{5} - \frac{158643987538315559}{51054400756352167049} a^{4} - \frac{4731045379178028732}{51054400756352167049} a^{3} - \frac{10190689096632781168}{51054400756352167049} a^{2} + \frac{21694929205525572205}{51054400756352167049} a + \frac{9260692534944494477}{51054400756352167049}$

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

Class group and class number

$C_{20}\times C_{5320}$, which has order $106400$ (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:  \( 201.0008347866989 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{-345}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{5}, \sqrt{-69})\), 6.6.820125.1, 6.0.15326915904.6, 6.0.1915864488000.6

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$23$23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$