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Properties

Label 12.0.79912527430...336.16
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 23^{6}$
Root discriminant $118.91$
Ramified primes $2, 3, 7, 23$
Class number $176400$ (GRH)
Class group $[10, 17640]$ (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![12452899201, -529942832, 1429811640, -49564380, 70585208, -1949106, 1919630, -40216, 30354, -436, 265, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201)
 
gp: K = bnfinit(x^12 - 2*x^11 + 265*x^10 - 436*x^9 + 30354*x^8 - 40216*x^7 + 1919630*x^6 - 1949106*x^5 + 70585208*x^4 - 49564380*x^3 + 1429811640*x^2 - 529942832*x + 12452899201, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 265 x^{10} - 436 x^{9} + 30354 x^{8} - 40216 x^{7} + 1919630 x^{6} - 1949106 x^{5} + 70585208 x^{4} - 49564380 x^{3} + 1429811640 x^{2} - 529942832 x + 12452899201 \)

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:  \(7991252743075419172110336=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.91$
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:  $2, 3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3864=2^{3}\cdot 3\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{3864}(2945,·)$, $\chi_{3864}(2437,·)$, $\chi_{3864}(1885,·)$, $\chi_{3864}(1,·)$, $\chi_{3864}(2209,·)$, $\chi_{3864}(781,·)$, $\chi_{3864}(1517,·)$, $\chi_{3864}(3313,·)$, $\chi_{3864}(2393,·)$, $\chi_{3864}(185,·)$, $\chi_{3864}(2621,·)$, $\chi_{3864}(965,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{14105903584293276857807066621637637} a^{11} - \frac{6863976047878900126586925554663280}{14105903584293276857807066621637637} a^{10} - \frac{64818270185585467074783894794671}{169950645593894901901289959296839} a^{9} + \frac{4784162706128434298904953944419702}{14105903584293276857807066621637637} a^{8} + \frac{4655031972725876852489980333253515}{14105903584293276857807066621637637} a^{7} + \frac{4682092958829040479603703900389623}{14105903584293276857807066621637637} a^{6} - \frac{167582147654616986808273437535995}{1085069506484098219831312817049049} a^{5} - \frac{2044114408120355201243683048702610}{14105903584293276857807066621637637} a^{4} - \frac{2962710205750324356124071079284849}{14105903584293276857807066621637637} a^{3} - \frac{3033399614334926962539583335147323}{14105903584293276857807066621637637} a^{2} + \frac{1958379922906467169972920290418540}{14105903584293276857807066621637637} a + \frac{92210772632851886967315068060306}{328044269402169229251327130735759}$

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

Class group and class number

$C_{10}\times C_{17640}$, which has order $176400$ (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:  \( 140.7987960054707 \) (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

\(\Q(\sqrt{-46}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-966}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-46})\), 6.0.14957039104.1, \(\Q(\zeta_{21})^+\), 6.0.2826880390656.6

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$23$23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$