↑

Properties

Label 12.0.10819730946...864.16
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 17^{6}$
Root discriminant $216.86$
Ramified primes $2, 3, 7, 17$
Class number $1740960$ (GRH)
Class group $[2, 6, 6, 24180]$ (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![6075013344, -458315280, 476053380, -35929344, 19106076, -1393704, 504889, -35142, 9519, -500, 111, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 111*x^10 - 500*x^9 + 9519*x^8 - 35142*x^7 + 504889*x^6 - 1393704*x^5 + 19106076*x^4 - 35929344*x^3 + 476053380*x^2 - 458315280*x + 6075013344)
 
gp: K = bnfinit(x^12 - 6*x^11 + 111*x^10 - 500*x^9 + 9519*x^8 - 35142*x^7 + 504889*x^6 - 1393704*x^5 + 19106076*x^4 - 35929344*x^3 + 476053380*x^2 - 458315280*x + 6075013344, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 111 x^{10} - 500 x^{9} + 9519 x^{8} - 35142 x^{7} + 504889 x^{6} - 1393704 x^{5} + 19106076 x^{4} - 35929344 x^{3} + 476053380 x^{2} - 458315280 x + 6075013344 \)

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:  \(10819730946497133891723300864=2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.86$
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:  $2, 3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4284=2^{2}\cdot 3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{4284}(1,·)$, $\chi_{4284}(2651,·)$, $\chi_{4284}(2243,·)$, $\chi_{4284}(1189,·)$, $\chi_{4284}(1633,·)$, $\chi_{4284}(2279,·)$, $\chi_{4284}(985,·)$, $\chi_{4284}(3299,·)$, $\chi_{4284}(2005,·)$, $\chi_{4284}(3095,·)$, $\chi_{4284}(2041,·)$, $\chi_{4284}(4283,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{4} - \frac{5}{14} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{14} a^{5} - \frac{1}{14} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{196} a^{6} - \frac{3}{196} a^{5} - \frac{5}{196} a^{4} - \frac{13}{196} a^{3} - \frac{5}{98} a^{2} - \frac{13}{98} a - \frac{5}{49}$, $\frac{1}{196} a^{7} - \frac{1}{28} a^{3} - \frac{2}{7} a^{2} + \frac{3}{14} a + \frac{20}{49}$, $\frac{1}{784} a^{8} + \frac{1}{56} a^{5} + \frac{3}{112} a^{4} + \frac{1}{56} a^{3} - \frac{1}{8} a^{2} + \frac{12}{49} a + \frac{1}{7}$, $\frac{1}{5488} a^{9} - \frac{1}{5488} a^{8} - \frac{3}{1372} a^{7} - \frac{1}{392} a^{6} + \frac{3}{112} a^{5} + \frac{19}{784} a^{4} + \frac{1}{98} a^{3} + \frac{61}{2744} a^{2} - \frac{131}{343} a + \frac{115}{343}$, $\frac{1}{1320779162416} a^{10} - \frac{5}{1320779162416} a^{9} - \frac{184889245}{330194790604} a^{8} + \frac{1479113975}{660389581208} a^{7} + \frac{68256437}{188682737488} a^{6} - \frac{1683883289}{188682737488} a^{5} - \frac{41061065}{11792671093} a^{4} + \frac{16147124775}{660389581208} a^{3} - \frac{5245925297}{82548697651} a^{2} + \frac{8166263935}{165097395302} a - \frac{18775071586}{82548697651}$, $\frac{1}{133196616192166352} a^{11} + \frac{25209}{66598308096083176} a^{10} - \frac{298945037277}{19028088027452336} a^{9} - \frac{995510137189}{4757022006863084} a^{8} + \frac{274409074061793}{133196616192166352} a^{7} - \frac{8458754600777}{9514044013726168} a^{6} + \frac{45874099696597}{19028088027452336} a^{5} - \frac{195788903423611}{33299154048041588} a^{4} + \frac{3905161957328855}{66598308096083176} a^{3} + \frac{808793109099979}{4757022006863084} a^{2} - \frac{662840162454947}{2378511003431542} a - \frac{558106946725697}{8324788512010397}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{24180}$, which has order $1740960$ (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:  \( 38984.4787659612 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-357}) \), \(\Q(\sqrt{-119}) \), 3.3.3969.1, \(\Q(\sqrt{3}, \sqrt{-119})\), 6.6.3024568512.1, 6.0.104017935696192.3, 6.0.541760081751.3

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
$3$3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$