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Properties

Label 12.0.95632159177...464.16
Degree $12$
Signature $[0, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 11^{6}$
Root discriminant $82.23$
Ramified primes $2, 3, 7, 11$
Class number $12152$ (GRH)
Class group $[2, 6076]$ (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![196996801, -17138192, 41712696, -2880444, 3900776, -214002, 207374, -8728, 6618, -196, 121, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 121*x^10 - 196*x^9 + 6618*x^8 - 8728*x^7 + 207374*x^6 - 214002*x^5 + 3900776*x^4 - 2880444*x^3 + 41712696*x^2 - 17138192*x + 196996801)
 
gp: K = bnfinit(x^12 - 2*x^11 + 121*x^10 - 196*x^9 + 6618*x^8 - 8728*x^7 + 207374*x^6 - 214002*x^5 + 3900776*x^4 - 2880444*x^3 + 41712696*x^2 - 17138192*x + 196996801, 1)
 

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 121 x^{10} - 196 x^{9} + 6618 x^{8} - 8728 x^{7} + 207374 x^{6} - 214002 x^{5} + 3900776 x^{4} - 2880444 x^{3} + 41712696 x^{2} - 17138192 x + 196996801 \)

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:  \(95632159177126518718464=2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.23$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1848=2^{3}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1848}(1,·)$, $\chi_{1848}(373,·)$, $\chi_{1848}(353,·)$, $\chi_{1848}(461,·)$, $\chi_{1848}(109,·)$, $\chi_{1848}(529,·)$, $\chi_{1848}(1781,·)$, $\chi_{1848}(793,·)$, $\chi_{1848}(89,·)$, $\chi_{1848}(1517,·)$, $\chi_{1848}(881,·)$, $\chi_{1848}(1429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{337237438151933739827174521573} a^{11} - \frac{113772380308794756182402037738}{337237438151933739827174521573} a^{10} + \frac{47105866733165559017013862505}{337237438151933739827174521573} a^{9} + \frac{84188685607424662676852503234}{337237438151933739827174521573} a^{8} + \frac{19163483017920506183801779560}{337237438151933739827174521573} a^{7} - \frac{117825196109725502206051520583}{337237438151933739827174521573} a^{6} + \frac{41505998465247814607977825167}{337237438151933739827174521573} a^{5} + \frac{70050535989583981248259270508}{337237438151933739827174521573} a^{4} - \frac{29875105293834916235365501170}{337237438151933739827174521573} a^{3} - \frac{69365584422841851887358176184}{337237438151933739827174521573} a^{2} + \frac{132466496353849182161147428008}{337237438151933739827174521573} a - \frac{130404926922038127655676522633}{337237438151933739827174521573}$

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

Class group and class number

$C_{2}\times C_{6076}$, which has order $12152$ (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:  \( 140.7987960054707 \) (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

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-462}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-22})\), \(\Q(\zeta_{21})^+\), 6.0.1636214272.1, 6.0.309244497408.7

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$