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Properties

Label 12.0.24616829072...5625.2
Degree $12$
Signature $[0, 6]$
Discriminant $3^{6}\cdot 5^{6}\cdot 43^{10}$
Root discriminant $88.98$
Ramified primes $3, 5, 43$
Class number $16226$ (GRH)
Class group $[16226]$ (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![235264, 378688, 194304, 61904, 79168, 12956, -3136, -2377, 577, -226, 62, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264)
 
gp: K = bnfinit(x^12 - x^11 + 62*x^10 - 226*x^9 + 577*x^8 - 2377*x^7 - 3136*x^6 + 12956*x^5 + 79168*x^4 + 61904*x^3 + 194304*x^2 + 378688*x + 235264, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 62 x^{10} - 226 x^{9} + 577 x^{8} - 2377 x^{7} - 3136 x^{6} + 12956 x^{5} + 79168 x^{4} + 61904 x^{3} + 194304 x^{2} + 378688 x + 235264 \)

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:  \(246168290724753398765625=3^{6}\cdot 5^{6}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.98$
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:  $3, 5, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(645=3\cdot 5\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{645}(1,·)$, $\chi_{645}(386,·)$, $\chi_{645}(214,·)$, $\chi_{645}(424,·)$, $\chi_{645}(394,·)$, $\chi_{645}(44,·)$, $\chi_{645}(466,·)$, $\chi_{645}(436,·)$, $\chi_{645}(566,·)$, $\chi_{645}(596,·)$, $\chi_{645}(509,·)$, $\chi_{645}(479,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{5} - \frac{1}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{32} a^{4} + \frac{5}{32} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{64} a^{7} - \frac{1}{32} a^{5} - \frac{1}{16} a^{4} + \frac{5}{64} a^{3} - \frac{3}{16} a^{2} - \frac{1}{16} a + \frac{1}{4}$, $\frac{1}{768} a^{8} - \frac{1}{384} a^{7} + \frac{5}{384} a^{6} - \frac{1}{64} a^{5} - \frac{31}{768} a^{4} - \frac{61}{384} a^{3} + \frac{7}{64} a^{2} + \frac{15}{32} a + \frac{5}{24}$, $\frac{1}{3072} a^{9} + \frac{1}{512} a^{7} - \frac{5}{384} a^{6} - \frac{55}{3072} a^{5} - \frac{17}{384} a^{4} - \frac{23}{96} a^{3} + \frac{15}{64} a^{2} + \frac{7}{192} a - \frac{19}{48}$, $\frac{1}{73728} a^{10} - \frac{1}{8192} a^{9} + \frac{5}{36864} a^{8} - \frac{27}{4096} a^{7} + \frac{115}{24576} a^{6} + \frac{119}{73728} a^{5} - \frac{773}{18432} a^{4} - \frac{2047}{9216} a^{3} + \frac{1135}{4608} a^{2} + \frac{239}{4608} a + \frac{421}{1152}$, $\frac{1}{438019951951872} a^{11} - \frac{907094945}{146006650650624} a^{10} + \frac{9067956457}{109504987987968} a^{9} - \frac{24836684615}{73003325325312} a^{8} - \frac{506672895289}{146006650650624} a^{7} - \frac{6329006042707}{438019951951872} a^{6} + \frac{2216535372803}{219009975975936} a^{5} + \frac{42901020757}{27376246996992} a^{4} - \frac{2319334628251}{13688123498496} a^{3} - \frac{1515511855927}{27376246996992} a^{2} + \frac{1805368794239}{13688123498496} a + \frac{85245910287}{380225652736}$

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

Class group and class number

$C_{16226}$, which has order $16226$ (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:  \( 589381.2229376945 \) (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

\(\Q(\sqrt{-215}) \), \(\Q(\sqrt{129}) \), \(\Q(\sqrt{-15}) \), 3.3.1849.1, \(\Q(\sqrt{-15}, \sqrt{129})\), 6.0.18376055375.1, 6.6.3969227961.1, 6.0.11538453375.2

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$43$43.12.10.1$x^{12} - 430 x^{6} + 1347921$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$