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Properties

Label 12.0.21517004436...0000.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 7^{8}$
Root discriminant $59.94$
Ramified primes $2, 3, 5, 7$
Class number $1924$ (GRH)
Class group $[2, 962]$ (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![12267361, -2965388, 4423460, -876724, 705880, -116656, 64062, -7972, 3259, -276, 88, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361)
 
gp: K = bnfinit(x^12 - 4*x^11 + 88*x^10 - 276*x^9 + 3259*x^8 - 7972*x^7 + 64062*x^6 - 116656*x^5 + 705880*x^4 - 876724*x^3 + 4423460*x^2 - 2965388*x + 12267361, 1)
 

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 88 x^{10} - 276 x^{9} + 3259 x^{8} - 7972 x^{7} + 64062 x^{6} - 116656 x^{5} + 705880 x^{4} - 876724 x^{3} + 4423460 x^{2} - 2965388 x + 12267361 \)

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:  \(2151700443648000000000=2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.94$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(323,·)$, $\chi_{840}(289,·)$, $\chi_{840}(169,·)$, $\chi_{840}(683,·)$, $\chi_{840}(827,·)$, $\chi_{840}(529,·)$, $\chi_{840}(107,·)$, $\chi_{840}(361,·)$, $\chi_{840}(121,·)$, $\chi_{840}(347,·)$, $\chi_{840}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{523090086} a^{10} - \frac{7784673}{174363362} a^{9} + \frac{4010603}{87181681} a^{8} - \frac{6887283}{174363362} a^{7} - \frac{25129711}{523090086} a^{6} + \frac{19144715}{261545043} a^{5} + \frac{112177363}{523090086} a^{4} - \frac{54857551}{523090086} a^{3} - \frac{37048355}{523090086} a^{2} + \frac{33284309}{261545043} a - \frac{115934053}{261545043}$, $\frac{1}{21421337278617929526} a^{11} + \frac{8513813668}{10710668639308964763} a^{10} - \frac{231075758704265237}{21421337278617929526} a^{9} - \frac{368855657260611278}{10710668639308964763} a^{8} + \frac{86804426530010599}{10710668639308964763} a^{7} + \frac{538468455804252221}{7140445759539309842} a^{6} - \frac{9917832233446320899}{21421337278617929526} a^{5} - \frac{4516793539104128107}{21421337278617929526} a^{4} - \frac{465417604888436825}{3570222879769654921} a^{3} - \frac{1064866043047275032}{3570222879769654921} a^{2} + \frac{917999379264084012}{3570222879769654921} a + \frac{5546327631166434523}{21421337278617929526}$

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

Class group and class number

$C_{2}\times C_{962}$, which has order $1924$ (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:  \( 104.882003477 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 4.0.72000.2, 6.6.300125.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
$3$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$