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Properties

Label 12.0.98875186240...5625.3
Degree $12$
Signature $[0, 6]$
Discriminant $5^{9}\cdot 7^{10}\cdot 13^{11}$
Root discriminant $177.66$
Ramified primes $5, 7, 13$
Class number $1089120$ (GRH)
Class group $[2, 4, 136140]$ (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![28891136, -6304000, -3977856, -639536, 428416, 82432, 37220, -8113, -1767, -222, 66, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 66*x^10 - 222*x^9 - 1767*x^8 - 8113*x^7 + 37220*x^6 + 82432*x^5 + 428416*x^4 - 639536*x^3 - 3977856*x^2 - 6304000*x + 28891136)
 
gp: K = bnfinit(x^12 - x^11 + 66*x^10 - 222*x^9 - 1767*x^8 - 8113*x^7 + 37220*x^6 + 82432*x^5 + 428416*x^4 - 639536*x^3 - 3977856*x^2 - 6304000*x + 28891136, 1)
 

Normalized defining polynomial

\( x^{12} - x^{11} + 66 x^{10} - 222 x^{9} - 1767 x^{8} - 8113 x^{7} + 37220 x^{6} + 82432 x^{5} + 428416 x^{4} - 639536 x^{3} - 3977856 x^{2} - 6304000 x + 28891136 \)

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:  \(988751862409257207447265625=5^{9}\cdot 7^{10}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.66$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(227,·)$, $\chi_{455}(4,·)$, $\chi_{455}(453,·)$, $\chi_{455}(423,·)$, $\chi_{455}(64,·)$, $\chi_{455}(327,·)$, $\chi_{455}(398,·)$, $\chi_{455}(16,·)$, $\chi_{455}(114,·)$, $\chi_{455}(447,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} - \frac{1}{64} a^{6} + \frac{3}{64} a^{5} - \frac{3}{64} a^{4} - \frac{3}{16} a^{3} + \frac{3}{16} a^{2}$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} + \frac{1}{32} a^{5} + \frac{5}{128} a^{4} - \frac{5}{32} a^{3} + \frac{7}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{3}{128} a^{6} - \frac{13}{256} a^{5} + \frac{7}{128} a^{4} + \frac{15}{64} a^{3} - \frac{3}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{21248} a^{10} + \frac{1}{5312} a^{9} + \frac{7}{10624} a^{8} + \frac{29}{10624} a^{7} - \frac{161}{21248} a^{6} + \frac{425}{10624} a^{5} - \frac{195}{10624} a^{4} - \frac{20}{83} a^{3} - \frac{171}{2656} a^{2} + \frac{15}{664} a - \frac{28}{83}$, $\frac{1}{35318526507468113354752} a^{11} + \frac{430181553951147077}{35318526507468113354752} a^{10} + \frac{707669981221057061}{1103703953358378542336} a^{9} - \frac{61990659799606150703}{17659263253734056677376} a^{8} + \frac{80815716536069650021}{35318526507468113354752} a^{7} + \frac{504267302686089387245}{35318526507468113354752} a^{6} + \frac{742918352703190248217}{17659263253734056677376} a^{5} + \frac{462469168652345802043}{8829631626867028338688} a^{4} - \frac{66248105152554577423}{4414815813433514169344} a^{3} + \frac{9925860120965190787}{68981497084898658896} a^{2} + \frac{75737818113169256087}{275925988339594635584} a - \frac{1986790662258181801}{34490748542449329448}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{136140}$, which has order $1089120$ (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:  \( 767947.3557001817 \) (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

\(\Q(\sqrt{65}) \), 3.3.8281.1, 4.0.13456625.2, 6.6.111434311625.2

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/3.12.0.1}{12} }$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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
$5$5.12.9.4$x^{12} + 30 x^{8} + 275 x^{4} + 1000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.12.11.8$x^{12} + 104$$12$$1$$11$$C_{12}$$[\ ]_{12}$