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Properties

Label 12.0.36754698481...0000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 5^{9}\cdot 17^{9}$
Root discriminant $290.92$
Ramified primes $2, 3, 5, 17$
Class number $1822080$ (GRH)
Class group $[2, 2, 2, 2, 2, 56940]$ (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![4029274125, 0, 1343091375, 0, 125823375, 0, 4768500, 0, 78795, 0, 510, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 510*x^10 + 78795*x^8 + 4768500*x^6 + 125823375*x^4 + 1343091375*x^2 + 4029274125)
 
gp: K = bnfinit(x^12 + 510*x^10 + 78795*x^8 + 4768500*x^6 + 125823375*x^4 + 1343091375*x^2 + 4029274125, 1)
 

Normalized defining polynomial

\( x^{12} + 510 x^{10} + 78795 x^{8} + 4768500 x^{6} + 125823375 x^{4} + 1343091375 x^{2} + 4029274125 \)

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:  \(367546984815514776264000000000=2^{12}\cdot 3^{18}\cdot 5^{9}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $290.92$
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:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3060=2^{2}\cdot 3^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3060}(1,·)$, $\chi_{3060}(1189,·)$, $\chi_{3060}(2209,·)$, $\chi_{3060}(169,·)$, $\chi_{3060}(2027,·)$, $\chi_{3060}(1007,·)$, $\chi_{3060}(2903,·)$, $\chi_{3060}(3047,·)$, $\chi_{3060}(2041,·)$, $\chi_{3060}(1883,·)$, $\chi_{3060}(1021,·)$, $\chi_{3060}(863,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{85} a^{4}$, $\frac{1}{85} a^{5}$, $\frac{1}{1785} a^{6} + \frac{1}{595} a^{4} - \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{5355} a^{7} + \frac{8}{1785} a^{5} + \frac{2}{7} a^{3} + \frac{1}{21} a$, $\frac{1}{1365525} a^{8} - \frac{1}{5355} a^{6} + \frac{1}{255} a^{4} - \frac{2}{9} a^{2} - \frac{3}{7}$, $\frac{1}{4096575} a^{9} - \frac{1}{16065} a^{7} + \frac{1}{765} a^{5} - \frac{11}{27} a^{3} - \frac{10}{21} a$, $\frac{1}{146874503475} a^{10} - \frac{971}{2879892225} a^{8} - \frac{33548}{191992815} a^{6} + \frac{35849}{33881085} a^{4} - \frac{29382}{83657} a^{2} + \frac{865}{4921}$, $\frac{1}{440623510425} a^{11} - \frac{971}{8639676675} a^{9} - \frac{33548}{575978445} a^{7} - \frac{362752}{101643255} a^{5} - \frac{113039}{250971} a^{3} - \frac{1352}{4921} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{56940}$, which has order $1822080$ (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:  \( 6391.506052875455 \) (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

\(\Q(\sqrt{85}) \), \(\Q(\zeta_{9})^+\), 4.0.88434000.2, 6.6.4029274125.2

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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.12.0.1}{12} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$5$5.12.9.4$x^{12} + 30 x^{8} + 275 x^{4} + 1000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$17$17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$