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Properties

Label 12.0.29170965190...0000.4
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 7^{6}$
Root discriminant $61.48$
Ramified primes $2, 3, 5, 7$
Class number $2232$ (GRH)
Class group $[6, 372]$ (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![1615461, -623538, 739656, -245226, 156027, -42768, 18835, -4182, 1374, -230, 57, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^11 + 57*x^10 - 230*x^9 + 1374*x^8 - 4182*x^7 + 18835*x^6 - 42768*x^5 + 156027*x^4 - 245226*x^3 + 739656*x^2 - 623538*x + 1615461)
 
gp: K = bnfinit(x^12 - 6*x^11 + 57*x^10 - 230*x^9 + 1374*x^8 - 4182*x^7 + 18835*x^6 - 42768*x^5 + 156027*x^4 - 245226*x^3 + 739656*x^2 - 623538*x + 1615461, 1)
 

Normalized defining polynomial

\( x^{12} - 6 x^{11} + 57 x^{10} - 230 x^{9} + 1374 x^{8} - 4182 x^{7} + 18835 x^{6} - 42768 x^{5} + 156027 x^{4} - 245226 x^{3} + 739656 x^{2} - 623538 x + 1615461 \)

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:  \(2917096519063104000000=2^{12}\cdot 3^{18}\cdot 5^{6}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.48$
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:  \(1260=2^{2}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1260}(1,·)$, $\chi_{1260}(419,·)$, $\chi_{1260}(421,·)$, $\chi_{1260}(769,·)$, $\chi_{1260}(841,·)$, $\chi_{1260}(491,·)$, $\chi_{1260}(911,·)$, $\chi_{1260}(839,·)$, $\chi_{1260}(1259,·)$, $\chi_{1260}(71,·)$, $\chi_{1260}(349,·)$, $\chi_{1260}(1189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3426158341} a^{10} - \frac{5}{3426158341} a^{9} + \frac{1447501058}{3426158341} a^{8} + \frac{1062312480}{3426158341} a^{7} - \frac{866354134}{3426158341} a^{6} - \frac{1119031299}{3426158341} a^{5} + \frac{392120505}{3426158341} a^{4} - \frac{1105982616}{3426158341} a^{3} + \frac{1291202433}{3426158341} a^{2} - \frac{1101768423}{3426158341} a - \frac{840087337}{3426158341}$, $\frac{1}{265551254535887} a^{11} + \frac{38748}{265551254535887} a^{10} + \frac{5003638485153}{265551254535887} a^{9} + \frac{33579358196102}{265551254535887} a^{8} - \frac{69395200480764}{265551254535887} a^{7} + \frac{60147620631854}{265551254535887} a^{6} + \frac{24239528574970}{265551254535887} a^{5} - \frac{108722554929639}{265551254535887} a^{4} + \frac{58918606562331}{265551254535887} a^{3} + \frac{16772294784857}{265551254535887} a^{2} + \frac{2195439679976}{5010401028979} a - \frac{101007122065600}{265551254535887}$

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

Class group and class number

$C_{6}\times C_{372}$, which has order $2232$ (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:  \( 325.67540279491664 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-105}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-35})\), \(\Q(\zeta_{36})^+\), 6.0.281302875.3, 6.0.54010152000.17

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

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$$[2]^{6}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$