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Properties

Label 12.0.27121520661...8693.1
Degree $12$
Signature $[0, 6]$
Discriminant $13^{11}\cdot 73^{6}$
Root discriminant $89.70$
Ramified primes $13, 73$
Class number $23944$ (GRH)
Class group $[2, 2, 5986]$ (GRH)
Galois group $C_{12}$ (as 12T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![634146319, -191987407, 191987407, -20036719, 20036719, -931087, 931087, -21295, 21295, -235, 235, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 235*x^10 - 235*x^9 + 21295*x^8 - 21295*x^7 + 931087*x^6 - 931087*x^5 + 20036719*x^4 - 20036719*x^3 + 191987407*x^2 - 191987407*x + 634146319)
 
gp: K = bnfinit(x^12 - x^11 + 235*x^10 - 235*x^9 + 21295*x^8 - 21295*x^7 + 931087*x^6 - 931087*x^5 + 20036719*x^4 - 20036719*x^3 + 191987407*x^2 - 191987407*x + 634146319, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 235 x^{10} - 235 x^{9} + 21295 x^{8} - 21295 x^{7} + 931087 x^{6} - 931087 x^{5} + 20036719 x^{4} - 20036719 x^{3} + 191987407 x^{2} - 191987407 x + 634146319 \)

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:  \(271215206617378764238693=13^{11}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.70$
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:  $13, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(949=13\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{949}(1,·)$, $\chi_{949}(291,·)$, $\chi_{949}(583,·)$, $\chi_{949}(72,·)$, $\chi_{949}(74,·)$, $\chi_{949}(656,·)$, $\chi_{949}(145,·)$, $\chi_{949}(147,·)$, $\chi_{949}(437,·)$, $\chi_{949}(439,·)$, $\chi_{949}(731,·)$, $\chi_{949}(220,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{81546139} a^{7} + \frac{19352569}{81546139} a^{6} + \frac{126}{81546139} a^{5} - \frac{30122162}{81546139} a^{4} + \frac{4536}{81546139} a^{3} + \frac{2163016}{81546139} a^{2} + \frac{40824}{81546139} a + \frac{8652064}{81546139}$, $\frac{1}{81546139} a^{8} + \frac{144}{81546139} a^{6} - \frac{22161686}{81546139} a^{5} + \frac{6480}{81546139} a^{4} - \frac{37444404}{81546139} a^{3} + \frac{93312}{81546139} a^{2} - \frac{21630160}{81546139} a + \frac{209952}{81546139}$, $\frac{1}{81546139} a^{9} - \frac{36362896}{81546139} a^{6} - \frac{11664}{81546139} a^{5} - \frac{21798443}{81546139} a^{4} - \frac{559872}{81546139} a^{3} - \frac{6919908}{81546139} a^{2} - \frac{5668704}{81546139} a - \frac{22705131}{81546139}$, $\frac{1}{81546139} a^{10} - \frac{14580}{81546139} a^{6} - \frac{6657331}{81546139} a^{5} - \frac{874800}{81546139} a^{4} - \frac{32662849}{81546139} a^{3} - \frac{14171760}{81546139} a^{2} - \frac{9753183}{81546139} a - \frac{34012224}{81546139}$, $\frac{1}{81546139} a^{11} + \frac{4157749}{81546139} a^{6} + \frac{962280}{81546139} a^{5} - \frac{6280155}{81546139} a^{4} - \frac{29583019}{81546139} a^{3} - \frac{31335696}{81546139} a^{2} - \frac{9621277}{81546139} a - \frac{4783913}{81546139}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{5986}$, which has order $23944$ (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:  \( 120.784031363 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{12}$ (as 12T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 4.0.11707813.1, \(\Q(\zeta_{13})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$73$73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73.4.2.2$x^{4} - 73 x^{2} + 58619$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$