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Properties

Label 12.0.20327905131...3424.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 71^{6}$
Root discriminant $87.57$
Ramified primes $2, 3, 71$
Class number $33600$ (GRH)
Class group $[2, 2, 2, 2, 10, 210]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![62195361, -15389676, 17104653, -3539604, 2048187, -346086, 135691, -18006, 5235, -500, 111, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 111*x^10 - 500*x^9 + 5235*x^8 - 18006*x^7 + 135691*x^6 - 346086*x^5 + 2048187*x^4 - 3539604*x^3 + 17104653*x^2 - 15389676*x + 62195361)
 
gp: K = bnfinit(x^12 - 6*x^11 + 111*x^10 - 500*x^9 + 5235*x^8 - 18006*x^7 + 135691*x^6 - 346086*x^5 + 2048187*x^4 - 3539604*x^3 + 17104653*x^2 - 15389676*x + 62195361, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 111 x^{10} - 500 x^{9} + 5235 x^{8} - 18006 x^{7} + 135691 x^{6} - 346086 x^{5} + 2048187 x^{4} - 3539604 x^{3} + 17104653 x^{2} - 15389676 x + 62195361 \)

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:  \(203279051316071044583424=2^{12}\cdot 3^{18}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.57$
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:  $2, 3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2556=2^{2}\cdot 3^{2}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{2556}(1,·)$, $\chi_{2556}(995,·)$, $\chi_{2556}(709,·)$, $\chi_{2556}(1703,·)$, $\chi_{2556}(1705,·)$, $\chi_{2556}(2413,·)$, $\chi_{2556}(143,·)$, $\chi_{2556}(851,·)$, $\chi_{2556}(853,·)$, $\chi_{2556}(1847,·)$, $\chi_{2556}(1561,·)$, $\chi_{2556}(2555,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17} a^{9} + \frac{4}{17} a^{8} + \frac{7}{17} a^{7} + \frac{5}{17} a^{6} + \frac{8}{17} a^{5} + \frac{8}{17} a^{4} + \frac{2}{17} a^{3} + \frac{3}{17} a^{2} + \frac{3}{17} a + \frac{5}{17}$, $\frac{1}{176680763173} a^{10} - \frac{5}{176680763173} a^{9} - \frac{8912828114}{176680763173} a^{8} + \frac{35651312486}{176680763173} a^{7} + \frac{4693984016}{176680763173} a^{6} + \frac{37819217403}{176680763173} a^{5} - \frac{87760013327}{176680763173} a^{4} - \frac{81493155338}{176680763173} a^{3} - \frac{49486592143}{176680763173} a^{2} - \frac{27192688152}{176680763173} a - \frac{52927603389}{176680763173}$, $\frac{1}{5085402406408459} a^{11} + \frac{14386}{5085402406408459} a^{10} + \frac{3223305767390}{5085402406408459} a^{9} + \frac{2399927361124576}{5085402406408459} a^{8} + \frac{1442413939246091}{5085402406408459} a^{7} - \frac{574739167816817}{5085402406408459} a^{6} + \frac{1132224535389404}{5085402406408459} a^{5} + \frac{110366085115948}{5085402406408459} a^{4} - \frac{1409112416326085}{5085402406408459} a^{3} - \frac{771220901089985}{5085402406408459} a^{2} - \frac{1606437297111680}{5085402406408459} a - \frac{1125030326329408}{5085402406408459}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{210}$, which has order $33600$ (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:  \( 325.67540279491664 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-71}) \), \(\Q(\sqrt{-213}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-71})\), \(\Q(\zeta_{36})^+\), 6.0.2348254071.3, 6.0.450864781632.8

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
$71$71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$