↑

Properties

Label 12.0.13179165217...000.10
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}$
Root discriminant $39.20$
Ramified primes $2, 3, 5, 7$
Class number $104$
Class group $[2, 52]$
Galois group $C_6\times C_2$ (as 12T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48721, -18702, 24313, -12124, 8459, -3284, 1805, -674, 257, -70, 25, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 257*x^8 - 674*x^7 + 1805*x^6 - 3284*x^5 + 8459*x^4 - 12124*x^3 + 24313*x^2 - 18702*x + 48721)
 
gp: K = bnfinit(x^12 - 6*x^11 + 25*x^10 - 70*x^9 + 257*x^8 - 674*x^7 + 1805*x^6 - 3284*x^5 + 8459*x^4 - 12124*x^3 + 24313*x^2 - 18702*x + 48721, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 25 x^{10} - 70 x^{9} + 257 x^{8} - 674 x^{7} + 1805 x^{6} - 3284 x^{5} + 8459 x^{4} - 12124 x^{3} + 24313 x^{2} - 18702 x + 48721 \)

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:  \(13179165217344000000=2^{12}\cdot 3^{6}\cdot 5^{6}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(419,·)$, $\chi_{420}(389,·)$, $\chi_{420}(391,·)$, $\chi_{420}(361,·)$, $\chi_{420}(299,·)$, $\chi_{420}(271,·)$, $\chi_{420}(149,·)$, $\chi_{420}(121,·)$, $\chi_{420}(59,·)$, $\chi_{420}(29,·)$, $\chi_{420}(31,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{4}{13} a^{7} - \frac{2}{13} a^{6} - \frac{6}{13} a^{5} - \frac{3}{13} a^{4} - \frac{6}{13} a^{3} - \frac{1}{13} a^{2} - \frac{5}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{9} - \frac{5}{13} a^{7} - \frac{1}{13} a^{6} - \frac{1}{13} a^{5} - \frac{5}{13} a^{4} + \frac{1}{13} a^{3} + \frac{4}{13} a^{2} - \frac{1}{13} a - \frac{2}{13}$, $\frac{1}{3827473} a^{10} - \frac{5}{3827473} a^{9} + \frac{2402}{93353} a^{8} - \frac{393898}{3827473} a^{7} - \frac{46017}{93353} a^{6} - \frac{14331}{89011} a^{5} + \frac{477402}{3827473} a^{4} - \frac{38677}{89011} a^{3} + \frac{1166106}{3827473} a^{2} - \frac{1009520}{3827473} a - \frac{980131}{3827473}$, $\frac{1}{51605818459} a^{11} + \frac{6736}{51605818459} a^{10} + \frac{308912406}{51605818459} a^{9} + \frac{1258498105}{51605818459} a^{8} + \frac{21489196358}{51605818459} a^{7} + \frac{6519510693}{51605818459} a^{6} + \frac{15445173456}{51605818459} a^{5} - \frac{13757750142}{51605818459} a^{4} + \frac{392561776}{1258678499} a^{3} - \frac{16493321989}{51605818459} a^{2} - \frac{1417072467}{51605818459} a - \frac{724816238}{3969678343}$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 246.505463083 \)
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

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\zeta_{28})^+\), 6.0.3630312000.2, 6.0.8103375.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$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}$