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Properties

Label 12.0.33096636485...224.22
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}$
Root discriminant $288.39$
Ramified primes $2, 3, 7, 19$
Class number $3279360$ (GRH)
Class group $[2, 2, 2, 2, 2, 102480]$ (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![2117917600, -360419200, 232061456, -53662112, 20369536, -3407360, 810648, -75536, 11938, -840, 121, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 121*x^10 - 840*x^9 + 11938*x^8 - 75536*x^7 + 810648*x^6 - 3407360*x^5 + 20369536*x^4 - 53662112*x^3 + 232061456*x^2 - 360419200*x + 2117917600)
 
gp: K = bnfinit(x^12 - 2*x^11 + 121*x^10 - 840*x^9 + 11938*x^8 - 75536*x^7 + 810648*x^6 - 3407360*x^5 + 20369536*x^4 - 53662112*x^3 + 232061456*x^2 - 360419200*x + 2117917600, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 121 x^{10} - 840 x^{9} + 11938 x^{8} - 75536 x^{7} + 810648 x^{6} - 3407360 x^{5} + 20369536 x^{4} - 53662112 x^{3} + 232061456 x^{2} - 360419200 x + 2117917600 \)

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:  \(330966364856493592168450228224=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $288.39$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3192=2^{3}\cdot 3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3192}(1,·)$, $\chi_{3192}(2659,·)$, $\chi_{3192}(2273,·)$, $\chi_{3192}(521,·)$, $\chi_{3192}(2539,·)$, $\chi_{3192}(11,·)$, $\chi_{3192}(1873,·)$, $\chi_{3192}(1331,·)$, $\chi_{3192}(2393,·)$, $\chi_{3192}(787,·)$, $\chi_{3192}(121,·)$, $\chi_{3192}(1451,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{20} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a$, $\frac{1}{480} a^{6} - \frac{1}{48} a^{5} - \frac{3}{32} a^{4} + \frac{1}{8} a^{3} + \frac{41}{120} a^{2} + \frac{1}{3} a + \frac{5}{24}$, $\frac{1}{960} a^{7} - \frac{1}{960} a^{5} + \frac{3}{32} a^{4} + \frac{11}{240} a^{3} - \frac{1}{8} a^{2} - \frac{31}{240} a + \frac{1}{24}$, $\frac{1}{1920} a^{8} + \frac{1}{1920} a^{6} - \frac{13}{960} a^{5} - \frac{23}{960} a^{4} + \frac{1}{4} a^{3} - \frac{63}{160} a^{2} - \frac{1}{80} a + \frac{5}{48}$, $\frac{1}{19200} a^{9} + \frac{1}{4800} a^{8} + \frac{1}{19200} a^{7} + \frac{7}{9600} a^{6} + \frac{59}{3200} a^{5} - \frac{331}{4800} a^{4} + \frac{637}{1600} a^{3} + \frac{159}{800} a^{2} + \frac{161}{480} a - \frac{13}{48}$, $\frac{1}{148070400} a^{10} - \frac{21}{987136} a^{9} - \frac{5273}{29614080} a^{8} - \frac{893}{3701760} a^{7} - \frac{5033}{6169600} a^{6} + \frac{2899}{231360} a^{5} - \frac{199399}{3701760} a^{4} + \frac{62839}{308480} a^{3} + \frac{1107101}{9254400} a^{2} + \frac{43433}{308480} a + \frac{61067}{370176}$, $\frac{1}{16195791221589918720000} a^{11} + \frac{9738306816121}{4048947805397479680000} a^{10} + \frac{56156253726163099}{3239158244317983744000} a^{9} + \frac{268909067942842843}{1619579122158991872000} a^{8} + \frac{190238338625860457}{674824634232913280000} a^{7} + \frac{37144263703842919}{337412317116456640000} a^{6} - \frac{4763282714914624823}{404894780539747968000} a^{5} - \frac{4644237171311045221}{202447390269873984000} a^{4} + \frac{39194636492119627547}{337412317116456640000} a^{3} + \frac{9798380919976743743}{126529618918671240000} a^{2} + \frac{18684067649511467719}{40489478053974796800} a + \frac{1876621894440037523}{20244739026987398400}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{102480}$, which has order $3279360$ (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:  \( 928596.3639515013 \) (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

\(\Q(\sqrt{-266}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-399}) \), 3.3.17689.2, \(\Q(\sqrt{6}, \sqrt{-266})\), 6.0.21307287497216.2, 6.6.4325539567104.1, 6.0.1123626489111.2

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{12}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$