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Properties

Label 12.0.28605102732...0000.5
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{6}\cdot 19^{10}$
Root discriminant $90.10$
Ramified primes $2, 3, 5, 19$
Class number $10976$ (GRH)
Class group $[2, 14, 392]$ (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![1184689, -527814, 715986, -407446, 285141, -108066, 53621, -15738, 4758, -560, 123, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 123*x^10 - 560*x^9 + 4758*x^8 - 15738*x^7 + 53621*x^6 - 108066*x^5 + 285141*x^4 - 407446*x^3 + 715986*x^2 - 527814*x + 1184689)
 
gp: K = bnfinit(x^12 - 6*x^11 + 123*x^10 - 560*x^9 + 4758*x^8 - 15738*x^7 + 53621*x^6 - 108066*x^5 + 285141*x^4 - 407446*x^3 + 715986*x^2 - 527814*x + 1184689, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 123 x^{10} - 560 x^{9} + 4758 x^{8} - 15738 x^{7} + 53621 x^{6} - 108066 x^{5} + 285141 x^{4} - 407446 x^{3} + 715986 x^{2} - 527814 x + 1184689 \)

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:  \(286051027323963456000000=2^{12}\cdot 3^{6}\cdot 5^{6}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.10$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1140=2^{2}\cdot 3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1140}(1,·)$, $\chi_{1140}(1091,·)$, $\chi_{1140}(229,·)$, $\chi_{1140}(961,·)$, $\chi_{1140}(911,·)$, $\chi_{1140}(49,·)$, $\chi_{1140}(179,·)$, $\chi_{1140}(1139,·)$, $\chi_{1140}(791,·)$, $\chi_{1140}(121,·)$, $\chi_{1140}(1019,·)$, $\chi_{1140}(349,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{21} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{21} a - \frac{8}{21}$, $\frac{1}{63} a^{8} - \frac{1}{63} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{13}{63} a^{2} + \frac{19}{63} a + \frac{22}{63}$, $\frac{1}{63} a^{9} + \frac{1}{63} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} + \frac{5}{21} a^{3} - \frac{5}{21} a^{2} + \frac{29}{63} a + \frac{13}{63}$, $\frac{1}{45586531683} a^{10} - \frac{5}{45586531683} a^{9} + \frac{208100357}{45586531683} a^{8} - \frac{832401398}{45586531683} a^{7} + \frac{32713700}{723595741} a^{6} - \frac{467069204}{6512361669} a^{5} + \frac{771217857}{5065170187} a^{4} - \frac{9403915667}{45586531683} a^{3} - \frac{3846968495}{45586531683} a^{2} + \frac{8142745822}{45586531683} a - \frac{359219771}{4144230153}$, $\frac{1}{429926580302373} a^{11} + \frac{1570}{143308860100791} a^{10} - \frac{1929621764465}{429926580302373} a^{9} - \frac{571752082588}{429926580302373} a^{8} + \frac{134489691317}{429926580302373} a^{7} + \frac{3805995756718}{61418082900339} a^{6} - \frac{17627222659213}{429926580302373} a^{5} - \frac{55447126428794}{429926580302373} a^{4} + \frac{18708738111671}{47769620033597} a^{3} + \frac{19257305895857}{47769620033597} a^{2} + \frac{137025324971923}{429926580302373} a - \frac{11929969698175}{39084234572943}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{392}$, which has order $10976$ (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:  \( 1234.5516326116856 \) (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

\(\Q(\sqrt{-285}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-57}) \), 3.3.361.1, \(\Q(\sqrt{5}, \sqrt{-57})\), 6.0.534837384000.9, 6.6.16290125.1, 6.0.4278699072.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$